UNIVERSITY OF CALIFORNIA 
 AT LOS ANGELES
 
 ELEMENTS OF GEODESY. 
 
 BY 
 
 J. HOWARD GORE, B.S., PH.D., 
 
 Professor of Mathematics in The Columbian University; sometime Astronomer 
 
 and Topographer U. S. Geological Survey; Acting Assistant U, S. 
 
 Coast and Geodetic Survey; Associate des Preussischen 
 
 Geoadtischen Institutes. 
 
 SECOND EDITION. REVISED AND CORRECTED. 
 
 NEW YORK: 
 
 JOHN WILEY & SONS, 
 
 15 ASTOR PLACE. 
 
 1889.
 
 COPYRIGHT, 1886, 
 BY IOHN WILEY & SONS
 
 3>o 
 
 PREFACE. 
 
 THE chief reason for making the following pages public is 
 the desire to put into better shape the principles of Geodesy, 
 and have accessible in a single book what heretofore has been 
 scattered through many. The advanced student and prac- 
 tised observer will find nothing new in this work, and may, 
 when accident throws it into their hands, lay it aside with feel- 
 ings of disappointment. But it is hoped that the beginner 
 , will be enabled to get a clear insight into the subject, and feel 
 * grateful that the discoveries and writings of many have been 
 ^ so condensed or elaborated as to make the study of Geodesy 
 pleasant. The plan pursued in the discussions that follow is 
 Pto take up each division in its logical order, develop each for- 
 *J mula step by step, and leave the* results or conclusion in the 
 shape that the majority of writers have considered the best. 
 In the text only occasional acknowledgments have been in- 
 serted, though at the end of each chapter a list of books will 
 be found to which reference has been frequently made. These 
 lists are by no means complete, so far as the literature of the 
 subject is concerned, but contain the titles of those books 
 which were found the most helpful while engaged in self- 
 instruction. The compilation of a complete Bibliography is
 
 j v PREFACE. 
 
 now in hand, forming a part of a History of Geodesy, which 
 will be finished in the course of a few years. 
 
 It is a pleasure to record the interest of Mr. Henry Gan- 
 nett, Chief Geographer of the U. S. Geological Survey, which 
 prompted him to read the manuscript and suggest important 
 improvements. 
 
 I desire to acknowledge my obligations to my associate, Pro- 
 fessor H. L. Hodgkins, A.M., for the interest he has shown in 
 the work, and for his careful revision of the proof-sheets as 
 they came from the press. 
 
 I also wish to express my indebtedness to my friend Miss 
 Lizzie P. Brown for her suggestions, and for the elimination of 
 errors that otherwise would have seriously blemished the work. 
 It is hoped that errors do not remain in sufficient number or 
 of such size as to impair the clearness or accuracy of the dis- 
 cussions that follow. 
 
 When page 102 was written, it was thought that a satisfac- 
 tory formula could be procured for the computation referred 
 to, but the increasing doubts regarding the coefficient of re- 
 fraction have induced me to omit further consideration of the 
 subject. 
 
 WASHINGTON, July, 1886.
 
 GEODETIC OPERATIONS. 
 
 CHAPTER I. 
 
 AN HISTORIC SKETCH OF GEODETIC OPERATIONS. 
 
 ONE of the first problems that suggested itself for solution 
 in the intellectual infancy of mankind was: "What is the 
 earth, its size and shape?" The possibility of examining the 
 constituency of the superficial strata answered with sufficient 
 exactness, for the time being, the first part of the question. 
 The natural conclusion deducible from daily experience and 
 observation is : were the earth deprived of the irregularities 
 produced by the valleys and mountains, its surface would be 
 a plane. The exact date of the abandonment of this theory 
 is unknown. Froriep refers to a Sanskrit manuscript contain- 
 ing the following sentence: "According to the Chaldeans, 
 4000 steps of a camel make a mile, 66f miles a degree, from 
 which the circumference of the earth is 24,000 miles." Of the 
 authenticated announcements of hypotheses, Pythagoras was 
 the first to declare that the earth is spherical. This honor 
 is sometimes assigned to Thales and Anaximander. Archi- 
 medes gave as an approximate value for the circumference 
 300,000 stadia. To Eratosthenes (B.C. 276) belongs the credit 
 of making the initial step towards a determination of the cir- 
 cumference. He observed that at Syene, in Southern Egypt, 
 an object on the day of summer solstice cast no shadow, while 
 1
 
 2 GEODETIC OPERATIONS. 
 
 at Alexandria the sun made an angle with the vertical equal to 
 one fiftieth of a circumference. Considering that Alexandria 
 was north of Syene, he reasoned that the entire circumference 
 of the earth was 50 times the distance between those places, or 
 250,000 stadia; this he afterwards increased to 252,000 stadia. 
 The neglect of the sun's diameter in the determination of dec- 
 lination, and the false supposition that Alexandria and Syene 
 were on the same meridian, introduced considerable inaccura- 
 cies in his results, the exact amount of which, however, \ve can- 
 not estimate owing to our ignorance as to the length of the 
 stadium. 
 
 About two hundred years later Posidonius determined the 
 amplitude of the arc between Rhodes and Alexandria from 
 observations on the star Canopus at both places. At Rhodes 
 he saw this star, when on the meridian, just visible above the 
 horizon, and at Alexandria its altitude at the same time was 
 fa of a great circle. From this he concluded that the circum- 
 ference was 48 times the distance these places were apart, or 
 48 X 500O stadia = 240,000 stadia. If we know the latitude 
 of two points on the same meridian, the difference will be the 
 amplitude of the arc passing through them, and the circumfer- 
 ence will bear the same ratio to the length of the arc that its 
 amplitude bears to four right angles. 
 
 Letronne has shown that the amplitude of the arc Posi- 
 donius used is only 5 = ^ of a great circle, and Strabo gives 
 40CO stadia as the length of the arc, making the circumference 
 288,000 stadia. 
 
 Ptolemy in the second century gave 180,000 stadia for the 
 circumference, but does not state his authority. Posch infers 
 that it was taken from the Chaldean value, since Ptolemy gives 
 a Chaldean mile equal to ; stadia, and ; times 24,000 = 
 180.000. In 827 an Arabian caliph imposed upon his astrono- 
 mers the task of measuring an arc, and of deducing from it 
 the length of the circumference of the earth.
 
 HISTORIC SKETCH. 3 
 
 Abulfeda in 1322 gave the following description of the 
 method employed by them : There were two parties ; one start- 
 ing from a fixed point measured a line due north with a rod, 
 the other party going due south ; both continuing until the ob- 
 served latitudes were found to differ by one degree from that 
 of the starting-point. The first party found 56 miles and the 
 second 56! miles for a degree. The latter result was accepted, 
 its equivalent being approximately 71 English miles. 
 
 This was a great improvement upon the methods of the 
 Grecians, who estimated their distances by days' marches of so 
 many stadia a day. 
 
 Fernel in 1525 made a measurement for the determination 
 of the length of a degree by counting the number of revolu- 
 tions made by a wheel of known circumference in going from 
 Paris to Amiens. He applied a correction to reduce the broken 
 line to a straight one, and the latitude observations were made 
 with a 5-foot sector, giving for a degree 365,088 English feet. 
 A few years later Father Riccioli made an arc-determination 
 in Italy, but it was too short to be of any importance. The 
 first attempt to determine the size of the earth by means of 
 triangulation was by Willebrord Snellius in 1615. He measured 
 a base-line with a chain between Leyden and Soeterwood, and 
 connected it by means of triangles, 33 in number, so as to com- 
 pute the distance from Alcmaar to Bergen-op-Zoom. This 
 distance he reduced to its equivalent along a meridian, giving 
 an arc of i ii' 05" amplitude, from which he found 55,074 
 toises for a degree (a toise being equal to 6.3946 English feet). 
 Kastner has shown that the neglect of spherical excess in the 
 reduction of these triangles causes an error of nearly a toise. 
 In 1722 the measurements were repeated, using for the angle- 
 determinations a sector of 5 feet radius; this second reduction 
 gave 57,033 toises for a degree. One can scarcely conceive of 
 the amount of labor such an undertaking necessitated at a 
 time when there were no logarithmic tables to lighten the work.
 
 A GEODETIC OPERATIONS. 
 
 Norwood in 1635 measured with a chain the distance from 
 London to York, obtaining for a degree 57,424 toises. 
 
 In the measurement of angles Snellius had sights attached 
 to his sector, making a close reading impracticable. 
 
 While the telescope was made use of as early as 1608, no 
 one had thought of putting it on an angle-reading instrument 
 until Picard, in 1669, placed in the focus of a telescope spider- 
 lines to mark the optical axis, which, according to some au- 
 thorities, had already been done by Gascoigne in 1640. He 
 measured a base-line nearly 7 miles long, and with a sector of 
 10 feet radius, to which was attached a telescope, the angles 
 were carefully read, until Malvoisine and Amiens were con- 
 nected by a chain of triangles. This gave an arc of 1 22' 58", 
 from which he computed 57,060 toises as the length of a 
 degree. At this time the effect of aberration and nutation 
 were unknown, which, if allowed for, would have shortened 
 his arc by 3". However, when his unit of linear measure was 
 more accurately compared with the standard it was found to 
 be too short, so that when Lacaille revised the work he ob- 
 tained the identical result that Picard had previously an- 
 nounced. 
 
 The uncertainty of ascertaining the circumference of the 
 earth from so short an arc was so keenly felt at this time that 
 the extension of this arc both northward and southward was 
 undertaken by the Cassini, father and son, Lahire, and Maraldi, 
 carrying it from Paris to Dunkirk, and from Paris to Perpignan, 
 the entire arc being about 8 31'. 
 
 The published results of Picard's work were rendered famous 
 by endorsing Newton's hypothesis of universal gravitation. 
 Newton had attempted to prove this theory by comparing the 
 force of gravity on a body at the moon's distance with the 
 power required to keep her in her orbit. He used in his com- 
 putations the diameter of the earth as somewhat less than 
 7000 miles. The result failed to show the analogy he had con-
 
 HISTORIC SKETCH, 5 
 
 ceived ; so he laid aside his theory, so brilliant in conception, so 
 lacking in verification. But twenty years later, when Picard's 
 length of a degree was made known, increasing the diameter of 
 the earth by about a thousand miles, Newton was able to show 
 that the deflection of the orbit of the moon from a straight 
 line was equivalent to a fall of 16 feet in one minute, the same 
 distance through which a body falls in one second at the sur- 
 face of the earth. The distance fallen being as the square of 
 the time, it followed that the force of gravity at the surface of 
 the earth is 3600 times as great as the force which holds the 
 moon in her orbit. This number is the square of 60, which 
 therefore expresses the number of times the moon is more dis- 
 tant from the centre of the earth than we are. If with the 
 rude means employed by Picard his errors had not eliminated 
 one another, or if their extent had been discovered without 
 knowing their compensating character, the undemonstrated 
 law of gravitation would have remained as an hypothesis, ce- 
 lestial mechanics would have been without the mainspring of 
 its existence, and we would now be groping in the darkness of 
 an antecedent century. 
 
 Newton also maintained that, owing to the greater centrifugal 
 force of the particles at the equator, a meridian section of the 
 earth would be an oblate ellipse ; that is, the equatorial axis 
 would exceed the polar. If such were the case, the radius of 
 curvature would increase in going from the equator towards 
 the pole ; and as the co-latitude is the angle formed by the 
 normal with the polar axis, if the normal increases, the arc of a 
 constant angle must become larger, therefore the oblate hy- 
 pothesis requires for verification that the degrees increase in 
 going from the equator towards either pole. Consequently the 
 results of Cassini's long arc determination were awaited with 
 impatience, until 1718, when the announcement was made 
 that the northern degree was shorter than the southern ; 
 this pleased the French, as it gave them an opportunity to
 
 6 GEODETIC OPERATIONS. 
 
 again say that the country across the Channel was a " Naza- 
 reth from which no good thing could come." A degree of 
 the northern arc gave 56,960 toises, and of the southern 
 57,098 toises, from which it appeared that the earth was pro- 
 late. 
 
 Huygens in 1691 published his theory regarding centrifu- 
 gal motion, describing experiments that proved that a rotat- 
 ing mass like the earth would have its greater axis perpendicu- 
 lar to the axis of rotation. Hence the terrestrial degrees 
 increase northward. It was a part of Newton's theory that as 
 the polar diameter is less than the equatorial, the force of 
 gravity must increase in going towards the pole, and therefore 
 a clock regulated by a pendulum would lose time when carried 
 towards the equator. When Richer returned in 1672 from the 
 Island of Cayenne, where he had been sent to make astronomic 
 observations, he found that his clock while at the island lost 
 two minutes a day when compared with its rate at Paris, and, 
 furthermore, the length of his pendulum beating seconds was 
 l^ lines shorter than the Paris seconds pendulum, showing that 
 Cayenne was farther than Paris from the centre of the earth. 
 A portion of this difference in the lengths of the pendulums 
 was supposed to be due to increased counteracting effect of 
 centrifugal force nearer the equator, but Newton showed that 
 the discrepancy was too great for a spherical globe. Varin 
 and Des Hays had a similar experience with pendulums taken 
 to points almost under the equator. 
 
 Under the excitement occasioned by this sharp controversy, as 
 well as from a desire to know the truth, the French Academy 
 decided to submit the problem to a most crucial test by meas- 
 uring one arc crossing the equator, and another within the 
 polar circle. Knowing the fierce criticism that would be 
 brought to bear upon every feature of the work, the partici- 
 pants determined to use the most refined instruments and 
 most approved methods. In May, 1735, an expedition consist-
 
 HISTORIC SKETCH. 7 
 
 ing of Godin, Bouguer, De la Condamine, and Ulloa set out for 
 Peru. The base was selected near Quito at an elevation of 
 nearly 8000 feet above sea-level. Its length was 7.6 miles as 
 deduced from a duplicate measurement, made by two parties 
 working in opposite directions. The measuring-rods were of 
 wood, twenty feet in length, terminated at either end in copper 
 tips to prevent wearing by attrition. 
 
 They were laid approximately horizontal, the deviation 
 therefrom being estimated by a plummet swinging over a 
 graduated arc. A comparison with a field standard was made 
 each day, this standard being laid off from the toise taken from 
 Paris, which afterwards became the legal unit in France, and is 
 known as the Toise of Peru. The angles of the 33 triangles 
 were measured on quadrants of 2 and 3 feet radius ; these 
 were so defective, however, that great care was necessary in de- 
 termining the instrumental errors and applying them to each 
 angle-determination. Twenty observations were made at dif- 
 ferent stations for ascertaining the azimuths. 
 
 The amplitude of the arc was found from simultaneous lati- 
 tude-observations made at the terminal stations on the same 
 star. Realizing that great uncertainties would arise from a 
 faulty determination of the amplitude, the latitude-observa- 
 tions were made with sectors 12 and 8 feet radius, on the sup- 
 position that the larger the sector the more accurate would be 
 the results. But the instability of the supports allowed such 
 great flexure that they were almost wholly reconstructed on 
 the field. 
 
 A southern base was measured as a check near Cotopaxi 
 at an elevation of nearly 10,000 feet above sea-level. Its 
 length, 6.4 miles, as measured, differed from the value com- 
 puted from the northern base by only one toise, and the entire 
 arc was but ten toises longer according to Condamine than 
 found by Bouguer. The amplitude as deduced by Bouguer 
 was 3 7' i", giving for the length of a degree reduced to sea
 
 8 GEODETIC OPERATIONS. 
 
 level 56,753 toises the mean of the two computations just 
 quoted. The field-work occupied two years, but the results 
 were not published until the beginning of 1746. 
 
 Von Zach revised the calculations, finding the arc to be 71 
 toises shorter; and Delambre recomputed the latitudes, from 
 which he found the amplitude increased by a little more than 
 2 seconds. According to the former, a degree would have at 
 that latitude a length of 56,731 toises, while the latter would 
 give 56,737 toises, a value indorsed by Arago. 
 
 The polar party, consisting of Maupertuis, Clairault, Camus, 
 Le Monnier, Outhier, and Celsius, Professor of Astronomy at 
 Upsal, reached its destinaton May 21, 1736. The river Tornea, 
 flowing south, with mountains of greater or less elevation on 
 each side, afforded in its valley a suitable location for the base, 
 and the mountains, points for the triangle stations. The 
 signals were built of trees stripped of their bark, in the shape 
 of a hollow cone. The angles were measured with a quadrant 
 of 2 feet radius provided with a micrometer, each angle being 
 read by more than one person, the average of the means of the 
 individual results being taken. Great care was exercised in cen- 
 tring the instrument and in checking the readings by observ- 
 ing additional angles whose sums or differences would give the 
 angles wanted. 
 
 Latitude observations were made by determining the differ- 
 ence of zenith distances of two stars with a sector consisting of 
 a telescope 9 feet long, which formed the radius of an arc 5 30'. 
 This arc was divided into spaces of f 30", which were subdi- 
 vided by a micrometer. From the observations corrected for 
 aberration, nutation, and precession, the amplitude was found 
 to be 57' 26^.93 according to Outhier, 57' 28*75 according to 
 Maupertuis, and 57' 2 8*.5 as given by Celsius. The base was 
 measured during the winter over the frozen snow and ice on 
 the river Tornea, the terminal points only being on land. The 
 measuring-bars were of wood, each 30 feet long, as determined
 
 HISTORIC SKETCH. 9 
 
 by comparison with an iron toise carried from Paris. Daily 
 comparisons were made by placing the rods between two iron 
 nails, previously driven at a distance apart just equal to the 
 length of one of the rods on the first day. It was found that 
 they had not changed in length during the work. 
 
 There were two parties, each having four rods, which they 
 placed end to end on the snow. In this manner the entire 
 base was measured twice, both parties laying the same number 
 of bars each day giving a daily check. The total difference in 
 the two results was only 4 inches in a distance of 8.9 miles, a 
 degree of accuracy that is quite remarkable when it is consid- 
 ered that the average temperature was 6 degrees F. below 
 zero. From this arc a degree cut by the polar circle was ascer- 
 tained to be 57,437 toises. While many precautions were taken, 
 the disagreement in the astronomic reductions, and some in- 
 strumental errors that were afterwards discovered, caused some 
 doubt as to the reliability of the work. If correct, a degree at 
 this point would be 377 toises longer than a degree at Paris, a 
 difference greater than the theorists had calculated, and more 
 confirmatory of the oblate hypothesis than was wanted. 
 
 Cassini, De Thuri, and Lacaille revised the French arc pre- 
 viously measured by J. and D. Cassini, and, comparing the 
 northern with the southern portion of the arc, they declared 
 that the earth was oblate; this was announced in 1744. In 
 1743, Clairaut, reasoning that the earth, instead of being of uni- 
 form density, each particle being pressed down by all that is 
 above, those near the centre must be denser than those nearer 
 the surface. Starting with the hypothesis that the density is 
 a function of the distance from the surface, he declared that 
 the earth was oblate, but not to the extent that Newton had 
 supposed. 
 
 Let us, in review, contemplate the condition of this problem 
 at this period : Newton, in 1687, from a theoretic analysis, said 
 the earth was oblate ; this explained the behavior of Richer's
 
 IO GEODETIC OPERATIONS. 
 
 clock in 1672. Huygens, in 1691, revolved a hollow metallic globe, 
 and saw it protrude at the centre ; hence, from analogy, he ac- 
 cepted the oblate hypothesis. Cassini's arc of 1718 declared 
 the theorists wrong. The Lapland labors of Maupertuis, nine- 
 teen years later, negatived Cassini's conclusion. Clairaut, in 
 1743, endorsed Maupertuis, but failed to show so great an ob- 
 lateness. In 1744, Lacaille, repeating the work of Cassini, 
 changed the results until they conformed to theory ; and hardly 
 a year later came the fruit of the ten years' labor in Peru to as- 
 sert that Newton, Huygens, and Clairaut were all right, in dif- 
 ferent degrees. 
 
 Lacaille, in 1750, went to the Cape of Good Hope to deter- 
 mine the moon's parallax, and while there he measured an arc 
 of [^degrees in south latitude 33 iS', from which he deduced 
 57,037 toises as the length of a degree. The short time de- 
 voted to this work, and the inferior quality of his instruments, 
 caused this determination to be lightly regarded. The next 
 triangulation was executed by Boscovich in 1751-53, in latitude 
 43 N., where an arc of 2 gave 56,973 toises as the length of 
 a degree. In 1768 Beccaria found 57,024 toises for a degree 
 in latitude 44 44' N. Zach revised this work and found a dif- 
 ference of 15 toises in the length of the arc, and numerous 
 errors in the angle-measurements. Also the proximity of the 
 northern terminus of the arc to the mountains suggests that 
 the unnoticed deflection of the plumb-line gave to the arc a 
 wrong amplitude. 
 
 In connection with Liesganig, the indefatigable Boscovich 
 measured an arc of 3, giving for the northern portion in lati- 
 tude 48 43', 57,086 toises for a degree, and for the southern 
 part they found a degree to be 56,881 toises a difference too 
 great to give to the work much confidence. 
 
 The surveyors Mason and Dixon (1764-68), in locating the 
 boundary-line between the properties of the Penn family and 
 Lord Baltimore, a portion of which afterwards became the
 
 HISTORIC SKETCH. II 
 
 boundary-line between Pennsylvania and Maryland, saw that 
 that part of the line separating Maryland from Delaware was 
 located on low and level land, almost coinciding with a merid- 
 ian. For this reason they concluded that it would be suitable 
 for measuring the length of a degree. The Royal Society of 
 London voted them money for the work. The whole distance 
 was measured with wooden rods 20 feet in length ; contact was 
 carefully made with rods level, and thermometric readings 
 made to correct for expansion. Latitude was ascertained 
 from equal zenith-distance observations, and azimuth meas- 
 ured from a meridian mark determined from astronomic obser- 
 vations. 
 
 The amplitude of the arc was i 28' 45", and the length as 
 measured gave for a degree 56,888 toises. 
 
 In 1783 the proposition was made on the part of the French 
 geodesists to unite Paris and Greenwich by, triangulation. 
 General Roy was placed in charge of the operations on the 
 English side of the Channel, and Count Cassini, Mechain, and 
 Legendre attended to that part of the work that fell within 
 France. In this work every precaution was taken to secure 
 good results, and all refinements at that time devised were 
 utilized. For the first time Ramsden's theodolite with a circle 
 of 3 feet in diameter was employed in measuring the angles. 
 
 This circle was divided into 15-minute spaces, and was read 
 at three points by micrometers rigidly connected with one 
 another. The telescope had a focal length of three feet, and of 
 sufficient power to render visible a church-tower at a distance 
 of forty-eight miles across water. The history of this the- 
 odolite would form a large part of the history of the English 
 triangulation. Sir Henry James, in speaking of it in 1863, 
 said : " When it is considered that this instrument has been 
 in use for the last seventy-five years, and that it has been 
 placed upon many of our very highest mountains, on our most 
 distant islands, and on the pinnacles of our loftiest churches, the
 
 12 GEODETIC OPERATIONS. 
 
 perfection with which this instrument was made, and the care 
 with which it has been preserved, is truly remarkable." Also 
 Colonel Clarke, in 1880, remarks that it is as good as when it 
 left the workshop. 
 
 The triangulation in England rested. upon the Hounslow 
 Heath base. The first measurement of this base was made in 
 June, 1784, with a steel chain of 100 feet in length, giving for 
 the length of the line, corrected for temperature, 27,408.22 feet. 
 A second determination was made using wooden rods, termi- 
 nating in bell-metal tips, the entire length being 20 feet 3 inches. 
 In the course of the work it was noticed that the rods were 
 affected by moisture so as to render the results, 27,406.26 feet, 
 unreliable. At the suggestion of Colonel Calderwood, it was 
 decided to measure the line with glass tubes. These were 20 
 feet long, supported in wooden cases 8 inches deep, and con- 
 tact was made as in the slide-contact forms. In the reduction 
 of the length of the base a carefully determined coefficient of 
 expansion, .0000043, was employed, giving for the length of the 
 base 27,404.0137 feet. 
 
 Another measurement made with a steel chain, using five 
 thermometers for temperature-indications, gave a result differ- 
 ing from the last by only 2 inches. This length was the equiva- 
 lent reduced to sea-level a correction being applied for the 
 first time in the history of geodesy. 
 
 In the French work nothing new was introduced except the 
 repeating-circle. This was constructed on a principle pointed 
 out by Tobias Mayer, Professor in the University of Gottin- 
 gen, which was thought to eliminate errors of graduation that 
 had at that time become a source of fear, owing to the imper- 
 fect means for graduating. By the method of repetition it 
 was supposed that if a number of pointings be made with 
 equal care, and the final reading be divided by the number of 
 pointings, the error of graduation as affecting the angle so re- 
 peated would be likewise divided, and hence be too small to
 
 HISTORIC SKETCH. 13 
 
 be appreciable. If all the parts of the instrument were rigid, 
 and if the circle or telescope could be clamped in place without 
 the one in its motion moving the other, the theory might be 
 endorsed in practice. However, these conditions have never 
 been definitely secured, nor is it likely that a clamp can be de- 
 vised that will not give in its working a travelling motion. 
 These obstacles did not present themselves with sufficient 
 force to cause the French to abandon this form of angle-read- 
 ing instruments until it had mutilated their labors covering 
 a half-century. 
 
 Barrow, in 1790, measured an arc of i 8' in East Indies, ob- 
 taining for a degree in latitude 23 18", 56,725 toises. 
 
 The year 1791 carries with it the honor of having witnessed 
 the inception of the most majestic scheme ever devised for ob- 
 taining and fixing a standard unit of measure. Laplace and 
 Lagrange, with the support of the principal mathematicians of 
 that period in France, proposed to the Assembly of France 
 that the standard linear unit should be a ten-millionth part of 
 the earth's quadrant, to be called a metre ; the length of this 
 quadrant to be determined by the measurement of an arc of 
 9 40' 24*, of which nearly two thirds was north of the 45th 
 parallel, the northern terminus being Dunkirk, and the south- 
 ern, Barcelona. Delambre was in charge of the work from 
 Dunkirk to Rodez, and Mechain completed that portion ex- 
 tending from Rodez to Barcelona. 
 
 Two base-lines were measured, one at Melun, near Paris, and 
 the other at Perpignan, each about seven and a quarter miles 
 long. The measuring-bars were four in number, each com- 
 posed of two strips of metal two toises in length, half an inch 
 in width, and a twelfth of an inch in thickness. The two 
 metal strips were supported on a stout beam of wood, the 
 whole resting on iron tripods provided with levelling-screws. 
 
 One of the strips was made of platinum ; the other, resting on 
 this, was copper, shorter than the platinum by about 6 inches.
 
 !4 GEODETIC OPERATIONS. 
 
 At one end they were firmly fastened together, but free to 
 move throughout the remainder of their lengths ; so that by 
 means of a graduated scale on the free end of the copper and 
 a vernier on the corresponding end of the platinum, the vary- 
 ing lengths owing to the different expansions of the two 
 metals could be determined, and hence the temperature known. 
 This was the invention of Borda, and is now known as the 
 Borda scale, or metallic thermometer. The bars were compared 
 indirectly with the toise of Peru by their maker, and No. I of 
 this set afterwards became a standard of reference. The angles 
 were measured with repeating-theodolites, and azimuth was 
 determined at five principal stations by measuring the angle 
 between another station and the sun, mornings and even- 
 ings. Latitudes were computed from zenith-distance observa- 
 tions at the termini and at three intermediate points. A com- 
 mission was appointed to review all the calculations : they 
 combined this arc with the Peruvian, deducing the length of a 
 quadrant whose legalized fractional part is the present metre. 
 
 Nouet, while astronomer to the French expedition to Africa 
 in 1798, measured a short arc, from which he found a degree to 
 be 56,880 toises. The disagreement between the computed 
 and observed azimuths obtained by Maupertuis amounting to 
 34* in the terminal line caused considerable suspicion to attach 
 to the entire work. The Stockholm Academy of Sciences de- 
 cided to have the stations reoccupied, and consequently, in 
 1801, sent Svanberg. Palander, and two others to Lapland for 
 that purpose. They did not recover all of the previously oc- 
 cupied stations, nor did they use the same terminal points, but 
 deduced as an independent value for a degree 57,196 toises. 
 
 Major Lambton measured an arc of i 33' 56" in India in a 
 mean latitude of 12 N. in 1802. After his death, in 1805, it 
 was continued by Colonel Everest with such vigor that by 1825 
 an arc of 16 was completed. 
 
 The French gave the English an impetus to push forward
 
 HISTORIC SKETCH. 1 5 
 
 geodetic work by their co-operation in the connection already 
 referred to, so that while in England a trigonometric survey was 
 being prosecuted, the requisite care was bestowed upon it to 
 make it of value in degree-determinations. From 1783 to 1800 
 this survey was under the direction of General Roy. Mudge 
 continued the triangulation for two years, completing an arc of 
 2 50', from which he found for the length of a degree in lati- 
 tude 53, 57,017 toises, and in 51, 57,108 toises; therefore the 
 degrees shorten towards the pole. 
 
 Mechain wished to carry his arc south of Barcelona to the 
 Balearic Isles, but was prevented by his unfortunate death. 
 However, the energetic mathematicians who made that period 
 of the French history so brilliant would not allow such a fea- 
 sible project to remain incomplete. So Biot and Arago spent 
 two years, beginning in 1806, in extending the triangulation 
 from Mt. Mongo, on the coast of Valencia, to Formentera, giv- 
 ing a complete arc of 12 22' 13*44. 
 
 The latitude of Formentera was determined from nearly 
 4000 observations on a and ft Ursae Minoris, but owing to the 
 fact that they were all made on stars on one side of the zenith, 
 erroneous star-places would introduce serious errors in the re- 
 sulting latitude, as demonstrated by Biot in 1825, when he ob- 
 tained for that station a latitude differing by 9" from the first. 
 The length of a degree as published in 1821 was 57,027 toises 
 in latitude 45 N. Bessel, using the corrected latitude of For- 
 mentera, found 56,964 toises ; and in 1841 Puissant discovered 
 another error which changed the degree's length to 57,032 
 toises. In the reduction of this work the principal of least 
 squares was used for the first time in adjusting the triangulation 
 in conformity with the geometric conditions, as will be explained 
 in a future chapter. 
 
 The errors already referred to in the reduction of this work 
 show the fallacy of accepting any determination of the earth's 
 quadrant as an unvarying quantity from which a standard, if
 
 1 6 GEODETIC OPERATIONS. 
 
 lost or destroyed, could be definitely restored with a length 
 identical with the previous one. Even if the earth be perfectly 
 fixed and stable in its size and shape, of which there is great 
 doubt, and the ten-millionth part of a quadrant always the 
 same, the uncertainties in obtaining the same value for this 
 quadrant twice in succession outweigh the utility of the plan 
 and the majesty of its conception. This is not intended as an 
 argument against the decimal feature, or the readiness with 
 which units of weight can be obtained from those of volume. 
 In this respect the metric system is superior to all others now 
 in use, and these advantages alone warrant its universal adop- 
 tion, while the fixity of the standards preserved by the Inter- 
 national Bureau of Weights and Measures is sufficiently certain 
 to dispel all doubts as to the change of length of the metre, 
 without feeling the necessity of frequently comparing it with 
 a physical law or mass supposed to be immutable. 
 
 Prussia began geodetic work in 1802 with the measurement 
 of a base-line near Seeburg by von Zach. This line was care- 
 fully measured and the end-points fixedly marked by inclosing 
 in masonry iron cannons with the mouth upwards. In the 
 mouth a brass cylinder was fastened by having lead run around 
 it ; the cross-lines on the upper surface of the cylinders denoted 
 the end of the line. The triangulation began in 1805, but was 
 stopped by the war with France in 1806, although Gotha, the 
 province in which the work was being prosecuted, remained 
 neutral. After the battle of Jena the people of Gotha, fear- 
 ing that the French would not regard their neutrality lasting, 
 especially if they should be suspected of harboring concealed 
 weapons, caused these cannons to be dug out and carefully 
 hid, thus sacrificing some accurate work to allay a foolish 
 fear. 
 
 Under Napoleon I. the importance of faithful maps for war 
 purposes at least was keenly felt, and to secure men trained 
 for the preparation of such maps the Ingenieur Corps was or-
 
 HISTORIC SKETCH. \"J 
 
 ganized, also the Ecole Polytechnique and the Ecole Speciale 
 de Ge"ode"sie. The basis of an accurate cartographic survey 
 must be a triangulation, and degree-measurements had such a 
 strong hold upon the mathematicians that the advisability of 
 giving to the triangulation the requisite accuracy to make it 
 useful for such determinations was never questioned. 
 
 Switzerland and Italy were to join their work to that of 
 France, to give an arc of parallel from the Atlantic Ocean to 
 the Adriatic Sea. This was begun in 1811, and continued by 
 one or more of the countries until its completion in 1832, giv- 
 ing an arc of 12 59' 4". Owing to serious discrepancies be- 
 tween the observed and computed values, this work received 
 but little credit. In one instance the difference in azimuth 
 was 49". 5 5, and in longitude the difference between the geo- 
 detic and astronomic was 51". 29. 
 
 The French expedition to Lapland for the purpose of an arc- 
 measurement incited the first astronomer of the St. Petersburg 
 Academy, De 1'Isle, to make a similar determination in Russia. 
 In 1737 he measured a base-line on the ice between Kronstadt 
 and Peterhof, and occupied several stations during that and 
 the two following years. However, it came to an end very 
 abruptly without leaving any definite results by which to re- 
 member it. 
 
 The first geodetic work in Russia that deserves the name 
 was begun in 1817 under the patronage of Alexander!., with 
 Colonel Tenner and Director Struve at the head. Tenner 
 began in the province of Wilma and continued until 1827, by 
 which time he had completed an arc of 4^, using a base 
 measured with an apparatus of his own devising, consisting of 
 two parallel bars of iron firmly fastened together. The angles 
 were read on a i6-inch repeating theodolite. Struve did not 
 receive his instruments until 1821, but in the ten years follow- 
 ing he finished an arc of 3^. 
 
 There was now a gap of about 5^ between the Russian and
 
 ,8 GEODETIC OPERATIONS. 
 
 the Lapland arcs which it was desired to close up. In this 
 work Struve was assisted by Argelander. They measured a 
 check-base with Struve's apparatus, completing the entire task 
 in 1844. In the mean time Tenner had added 3 25' to his arc. 
 Just here it might be of interest to remark that Bessel had 
 communicated to Tenner his discussion regarding the figure 
 and size of the earth. This was appended to Tenner's manu- 
 script record and placed in the care of the St. Petersburg 
 Academy in 1834, three years before it was published by Bes- 
 sel in the Astronomische Nachrichten, No. 333. 
 
 Permission was obtained from the Swedish authorities to 
 continue this arc across Norway and Sweden. This also was 
 placed under the direction of Struve, with the assistance of 
 Selander and Hansteen. The former finished his share of the 
 triangulation with a measured base in 1850. Hansteen com- 
 pleted the Norwegian portion, checking on a base of 1155 
 toises. The Russian parties, together with their co-laborers, 
 by 1855 had completed a meridional arc of 25 2O / 9 / '.29, ex- 
 tending from the Danube to the North Sea. Of this there 
 were two great divisions the Russian, with 8 bases and 224 
 principal triangles and 9 latitude-determinations; and the 
 Scandinavian, with 2 bases, 33 principal triangles, and 4 astro- 
 nomic stations. Prior to 1821 the principle of repetition was 
 exclusively used on horizontal circles in its original form. 
 Struve then decided that the periodic errors noticed when the 
 simple method of repetition was employed could be partially 
 eliminated by reversing the direction of rotation ; but he soon 
 abandoned this, and in 1822 began to measure angles a number 
 of times on different parts of the circle. 
 
 The test of the accuracy of this work is in the difference in 
 the lengths of junction-lines as computed from different bases. 
 From an examination of ten of these differences, I have found 
 that the average is 0.1718 toise, with 0.0179 as the minimum 
 and 0.4764 for a maximum. The values found for a degree
 
 HISTORIC SKETCH. 19 
 
 were: 57,092 toises in latitude 53" 20', 57,116 in 55 34', 57,121 
 in 56 32, 56,956 in 57 28', and 57,125 in 59 14'. The utility 
 of this arc for degree-measurements is not proportionate to its 
 immensity, because of the fewness of the astronomic deter- 
 minations only one in every two degrees of amplitude. 
 
 General von Muffling in 1818 connected the Observatory of 
 Seeburg with Dunkirk, and determined the amplitude of the 
 arc by measuring the difference in time between the stations 
 two by two. This was done by recording in local time the ex- 
 act instant at which a powder-flash set off at one station at a 
 known local time was seen at the other. The amplitude of 
 this arc, embracing 8 determinations of this kind in its chain, 
 was 82i / 1 8". 
 
 Between 1818 and 1823 Colonel Bonne connected Brest with 
 Strasburg, with a base near Plouescat. It is interesting to 
 note that in this work angles were measured at night, using 
 as a signal a light placed in the focus of a parabolic reflec- 
 tor. Differences of longitude were determined by powder- 
 flashes. 
 
 Gauss began the trigonometric survey of Hanover in 1820, 
 measuring an arc of 2 57', from which he found for a degree 
 57,126 toises in the same latitude in which Mudge in England 
 obtained for a degree 57,016 toises, and Musschenbroeck, in 
 Holland, 57,033 toises. It was while engaged upon this work 
 that Gauss first used the heliotrope that has since borne his 
 name. 
 
 Schumacher at the same time commenced the Danish trian- 
 gulation with the advice and assistance of Struve. His arc 
 of i 31' 53" gave for a degree 57,092 toises in latitude 54 
 
 8' 13". 
 
 In 1821 Schwerd concluded from his measurement of the 
 Speyer base that a short line most carefully measured would 
 give as good results as a longer one on which the same time 
 and labor would be expended. From his base of 859.44 M. he
 
 20 GEODETIC OPERATIONS. 
 
 computed the length of Lammle's base of 19,795.289 M., giving 
 a difference of only 0.0697 M. 
 
 Colonel Everest was appointed to succeed Colonel Lambton 
 in the direction of the great trigonometric survey of India in 
 1823. During the following seven years he measured three 
 bases with the Colby apparatus as checks to the triangulation 
 which he extended from 18 3' to 24 f. To Colonel Everest 
 is due the credit of introducing greater care in all the linear 
 and angular determinations. In the latter he employed 
 the method of directions in greater number than did his pred- 
 ecessors. 
 
 In 1831 Bessel and Baeyer undertook a scheme of triangula- 
 tion that was to unite the chains of France, Hanover, Den- 
 mark, Prussia, and Bavaria with that of Russia, and at the 
 same time serve for degree-measurements. It was oblique, so 
 that, by determining the direction and amplitude, degrees of 
 longitude as well as latitude could be found. The base-line 
 near Fuchsburg was measured with a slightly modified form 
 of the Borda apparatus now known as Bessel's apparatus, of 
 which there is now an exact copy in use in the Landes Trian- 
 gulation of Prussia. The length of this base was 934.993 
 toises when reduced to sea-level. The ends were marked by a 
 pier of masonry inclosing a granite block, in whose top was set 
 a brass cylinder carrying cross-lines indicating the end of the 
 base. Just above this was built a hollow brick column high 
 enough for the theodolite support, with a larger square stone 
 for a cap-stone. In the centre of this there was a cylinder 
 coaxial with the one below, so that the instrument could be 
 placed immediately over the termini of the base. The theodo- 
 lites had 12- and 1 5-inch circles, read by verniers, and the angles 
 were read by fixing the zeros coincident, and then turning to 
 each signal in succession with verniers read and recorded for 
 each. After completing the series, the signals were observed 
 in inverse order, the means of the two readings giving a set of
 
 HISTORIC SKETCH. 21 
 
 directions. The zero would then be shifted to another position, 
 and all the signals sighted both in direct and inverted order, 
 until a desired number of sets were secured. The method of 
 reduction is given on page 99. 
 
 Two kinds of signals were used ; one consisted of a hemi- 
 sphere of polished copper placed with its axis vertically over the 
 centre of the station. The sun shining on this gave to the ob- 
 server a bright point, but not in a line joining the centres of 
 the stations observing and observed upon ; consequently a cor- 
 rection for phase, as explained on page 144 had to be applied. 
 The other form consisted of a board about two feet square, 
 painted white with a black vertical stripe ten inches wide down 
 the centre. This board was attached to an axis made to coin- 
 cide with the centre of the station, so as^to permit the board 
 to be turned in a direction perpendicular to the line of sight as 
 different stations were being occupied. 
 
 The astronomic determinations were made at three stations 
 with the greatest possible care ; while the reduction of the tri- 
 angulation was a monument to the methods devised by Gauss 
 for treating all auxiliary angles as aids in finding the most 
 probable corrections to be applied to those angles absolutely 
 needed in the computation. The amplitude of the arc was 
 i 3O / 28".97. Using the two parts into which the arc was di- 
 vided by Konigsberg, the difference between the terminal 
 points taken as a whole, and the sum of the two parts was only 
 0.973 toise, which is an evidence of the great accuracy attained 
 in this work. The report of this triangulation was published 
 in Gradmessung in Ostpreussen, und ihre Verbindung mit Preus- 
 sischen und Russischcn Dreiecksketten, Berlin, 1838; and while 
 now nearly half a century has elapsed since its appearance, not 
 only its influence is still felt, but the operations then for the 
 first time described are now in use. 
 
 There is not a geodesist of the present time who is not in- 
 debted to this work for information as well as assistance, and
 
 22 GEODETIC OPERATIONS. 
 
 as long as exact science receives attention men will turn to 
 this fountain-head. My greatest inspiration comes from two 
 sources both perhaps sentimental, but none the less effica- 
 cious. My copy of the above book was presented to Jacobi 
 by Bessel, as shown by the latter's superscription. This is be- 
 fore me in reality ; the other remains in memory as the cordial 
 greetings and encouragement of Baeyer, with whom I worked 
 in the Geodetic Institute. 
 
 From 1843 to 1861 Sir A. Waugh, who succeeded Sir George 
 Everest, added nearly 8000 miles to the Indian chains. After 
 him came General Walker's administration, and during the 
 following thirteen years he completed 5500 miles of triangle 
 chains, occupied 55 azimuth stations, and determined 89 
 latitudes. 
 
 In this work the triangle sides are from 15 to 60 miles in 
 length. In those cases where it was necessary to elevate the 
 instrument masonry towers were erected, some as high as 50 
 feet. Luminous signals were used heliotropes by day, and 
 Argand lamps at night. The amplitude of the greatest Indian 
 arc is 23 49' 23"-54, but its exact value has been questioned, 
 owing to the uncertainties of the effect of local attractions in the 
 neighborhood of the Himalayas upon the latitudes and azi- 
 muths, as well as the negative attraction along the shore of the 
 Indian Ocean as pointed out by Archdeacon Pratt. When 
 the computed effects of these attractions are applied, there is 
 still a discrepancy. 
 
 A meridional arc of about 30 has been completed, but owing 
 to the impracticability of ascertaining the difference of longi- 
 tudes its amplitude is not accepted as sufficiently accurate to 
 warrant its use in degree-determinations. 
 
 The purpose of this great trigonometric survey was to fur-, 
 nish a basis for topographic maps ; consequently the chains of 
 primary triangles are parallel at such a distance- apart as to 
 allow the intervening country to be easily covered with
 
 HISTORIC SKETCH. 2$ 
 
 secondary triangles with the primaries for checks on each side 
 of the chasm. There are 24 chains running north and south, 
 and 7 east and west. 
 
 Between 1847 and I ^5 I the Russian chain was connected 
 with the Austrian, having 12 sides in common ; the greatest 
 discrepancy being o.ioi toise, and the least o.oi toise. 
 
 About the same time the junction of the Lombardy and 
 Swiss chains showed a difference of 0.31 and 0.34 metre. 
 
 In 1848 the astronomer Maclear revised Lacaille's Good 
 Hope arc, extending it to an amplitude of 3 degrees, from 
 which he deduced for I degree, in latitude 35 43', 56,932.5 
 toises. Comparing this with the French arc in approximately 
 the same northern latitude, we find a difference of only 48 
 toises in a degree. 
 
 In 1831 Borden devised a base-apparatus with which he 
 measured a base and began a triangulation over the State of 
 Massachusetts, making the commencement of geodetic work 
 in the United States. Borden read his angles with a 12-inch 
 theodolite, using the method of repetition. Latitudes were 
 determined from circumpolar altitude observations at 24 
 points. 
 
 Recently many of his stations have been re-occupied, intro- 
 ducing greater care in all features of the work and affording a 
 check on Borden's results. Comparing the two sets of values 
 for the geographical positions of the stations that are common, 
 it appears that there is a systematic increase in the errors, being 
 the greatest in the eastern part of the State, that being the 
 furthest from the base-line. The average discrepancy in the 
 linear determination is 1:11000, or somewhat less than 6 inches 
 in a mile. 
 
 The United States Coast Survey, organized in 1807, had 
 primarily for its object the survey of the coast, but this ne- 
 cessitated a carefully executed triangulation of long sides to 
 check the short triangle sides whose terminal stations were
 
 24 GEODETIC OPERATIONS. 
 
 sufficiently near one another for the coast topography and off- 
 shore hydrography. It soon became apparent that but little, 
 if any, additional care was needed to secure sufficient accuracy 
 to make this trigonometric work a contribution to geodesy. 
 By 1867 an arc of 3 23' was completed, extending from Farm- 
 ington, Maine, to Nantucket, with two base-lines, seven latitude 
 stations, and ten determinations of azimuth. 
 
 Summing the six arcs into which the whole naturally divides 
 itself, it was found that a degree in latitude 43 and longitude 
 70 20' was 111,096 metres, or 57,000.5 toises. 
 
 By 1876 the Pamlico-Chesapeake arc of 4 3i'-5 was com- 
 pleted, embracing in its chain of triangles six bases and fourteen 
 astronomic stations. The latitude of each of these stations was 
 computed from the one nearest the middle of the arc, and the 
 difference between this and the observed values, called station- 
 error, attributed to local deflection. This in no case exceeded 
 3^ seconds; and in general it was in accord with a uniform 
 law disclosed by the geology of the country over which the arc 
 extends. 
 
 From an elaborate discussion of the sources of error in this 
 arc, Mr. Schott concludes that the probable error in its length 
 is not in excess of 3^ metres. The length of a degree in lati- 
 tude 37 16' and longitude 76 08' is 56,999.9101568. 
 
 The triangulation is being continued southward, and in a 
 very short time it is hoped that the entire possible arc of 22 
 will be reduced and the results announced. An arc of parallel 
 is also under way, keeping close to the 39th. 
 
 Of this great arc of 49 about three fourths is completed. 
 This is the longest arc that can anywhere be measured under 
 the auspices of a single country. Consequently, considering 
 the great advantage to be derived from perfect harmony of 
 methods, it is no wonder that scientists in all parts of the 
 world are anxiously awaiting the completion of this important 
 work. Also, when done, it will be well done. The high stand-
 
 HISTORIC SKETCH. 2$ 
 
 ard of excellence introduced into this service at its beginning 
 f" makes the first results comparable with the most recent. 
 
 In 1857 Struve advocated the project of connecting the 
 triangulations of Russia, Prussia, Belgium, and England, giving 
 an arc of 69 along the 52d parallel. Bessel had already made 
 the Prussian -Russian connection, and in 1861 England and Bel- 
 gium joined with tolerable success, finding in their common 
 lines discrepancies amounting to an inch in a mile. 
 
 The Prussian and Belgium chains are not yet satisfactorily 
 united ; neither are the longitudes determined. 
 
 While a topographic map of Italy was begun in 1815, no 
 special interest was taken in geodesy until 1861, except in 
 rendering some slight assistance in that part of the French and 
 Austrian triangulation that overlapped. In this year Italy re- 
 sponded to the suggestion of Baeyer, adopted by the Prussian 
 Government, to form an association of the European powers to 
 measure a meridional and a parallel arc. 
 
 The Italian Commission was formed in 1865, and at once 
 elaborated plans for future work. It was decided to have six 
 chains of triangles, and for every twenty or twenty-five a care- 
 fully measured base ; also to connect Sicily with Africa ; direc- 
 tion-theodolites of 10- and 12-inch circles to be used. The base- 
 apparatus with which the first three bases were measured was 
 of the Bessel pattern. The base of Undine was measured with 
 the Austrian, and the next two with a Bessel equipped with 
 reading-microscopes for reading the divisions on the glass 
 wedges. 
 
 The numerous observatories are connected with the trigono- 
 metric stations, and one or two are to be erected in the merid- 
 ian of the arc to determine its deflection. 
 
 The geodetic work in Spain began with the measurement of 
 the Madridejos base in 1858. The apparatus used in this work 
 was specially designed for it, and the precision introduced 
 into the measurement of the base, as well as in the depend-
 
 26 GEODETIC OPERATIONS. 
 
 ing triangulation, has given to the Spanish work great confi- 
 dence. 
 
 This is especially fortunate, as it will form an important link 
 in the chain extending from the north of Scotland into Africa, 
 and in the oblique chain from Lapland to the same point. In 
 addition to the central base first measured, three others were 
 found necessary to check the system. 
 
 The general plan resembles that pursued in the India Survey 
 in having parallel chains at such a distance from one another 
 that the intervening country can be readily filled in with sec- 
 ondary triangles for the topographic purposes. 
 
 There are three of these meridional chains with amplitudes 
 of about six, seven, and seven and a half degrees, and an arc of 
 parallel of twelve degrees. 
 
 Likewise the Swedish coast-triangulation was begun in 1758 
 for the purpose of checking the coast-charts, and in 1812 an- 
 other triangulation embracing fifty stations and five base-lines, 
 measured with wooden rods, was started for a similar end. 
 However, it was not until the announcement of Bessel's results 
 that Sweden took an active interest in accurate work. 
 
 In 1839 the Alvaren base was measured with Bessel's appara- 
 tus, and again in the following year with the same bars, giving 
 a difference of 0.0145 metre in the two results. 
 
 So far the work was purely cartographic, and it was the in- 
 fluence of Baeyer that caused a partial transformation in the 
 methods, making them conformable to the system of the Per- 
 manent Commission for European Degree-measurements. 
 
 Three bases have been measured with a modified Struve ap- 
 paratus, giving excellent results ; in one instance the difference 
 between the two measurements being only 0.0029 metre, and 
 twenty-nine stations occupied, using Reichenbach and Repsold 
 theodolites. 
 
 Under the auspices of this commission the following coun- 
 tries are prosecuting geodetic work : Austria, Bavaria, Belgium,
 
 HISTORIC SKETCH. 27 
 
 France, Hesse, Holland, Italy, Portugal, Prussia, Russia, Sax- 
 ony, Spain, Switzerland, and Wiirtemberg. 
 
 LITERATURE OF THE HISTORY OF GEODESY. 
 
 Verhandlungen der allgemeinen Conferenz der Europaischen 
 Gradmessung. 
 
 Roberts, Figure of the Earth, Van Nostrand's Engineering 
 Magazine, vol. 32, pp. 228-242. 
 
 Comstock, Notes on European Surveys. 
 
 Baeyer, Ueber die Grosse und Figur der Erde. 
 
 Posch, Geschichte und System der Breitengradmessungen. 
 
 Merriman, Figure of the Earth. 
 
 Baily, Histoire de I'Astronomie. 
 
 Wolf, Geschichte der Vermessungen in der Schweiz. 
 
 Clarke, Geodesy. 
 
 Westphal, Basisapparate und Basismessungen. 
 
 Klein, Zweck und Aufgabe der Europaischen Gradmessung.
 
 GEODETIC OPERATIONS. 
 
 CHAPTER II. 
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 
 
 THE perfection of an instrument is the result of corrected 
 defects, and in the development of geodesy or degree-meas- ' 
 urements improved methods were closely followed by better 
 instruments. So that while discussing the progressive steps 
 of one, the other cannot be wholly neglected. 
 
 For the uncultured peoples, distances can be given with suf- 
 ficient accuracy as so many days' journey, and nothing but the 
 necessity to carry on record some measured magnitude would 
 call for a unit that could be readily attained. The first such 
 unit of which there is any authentic information is the Chaldean 
 mile, which was equal to 4000 steps of a camel ; the next was 
 the Olympian race-course, giving to the Greeks their unit 
 the stadium. The rods with which the Arabians measured the 
 two degrees already mentioned known as the black ell 
 have been lost, and not even their equivalent length known. 
 
 Fernel, in using the wagon-wheel for a measuring unit, found 
 it quite constant in length and of a kind easily applied, advan- 
 tages that are appreciated to this day by topographers, who 
 frequently measure meander lines by having a cyclometer at- 
 tached to a wheel of a vehicle. 
 
 When Snellius devised the method of triangulation there 
 were needed two forms of instruments one for linear measure- 
 ments, and another for angle-determinations. At this time 
 angles were measured with a quadrant to which sights were 
 attached ; a rectangle with an alidade and sights pivoted to one 
 of the longer sides, the other being divided into degrees ; a
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 2Q 
 
 square with the alidade in one corner and all four sides gradu- 
 ated ; a compass with sights ; a semicircle with alidade or 
 compass at the centre. Also for navigators there was the as- 
 trolabe, an instrument devised by Hipparchus for measuring 
 the altitude of the sun or a star. 
 
 Defects in graduation were early detected, and efforts to 
 avoid them made by increasing the radius of the sector, the 
 smallest used by the first astronomers being of 6 and 7 feet 
 radius ; and it is said that a pupil of Tycho Brahe constructed 
 a sector of 14 feet radius; while Humboldt says the Arabian 
 astronomers occasionally employed quadrants of 180 feet ra- 
 dius. In the case of large circles, or parts of circles, the divi- 
 sions that could be distinguished would be so numerous as to 
 render the labor of dividing very great, and the intermediate 
 approximation uncertain. 
 
 Nunez, a Portuguese, in 1542 devised a means of estimating 
 a value smaller than the unit of division. He had about his 
 quadrant several concentric circular arcs, each having one divi- 
 sion less than the next outer, so that the difference between an 
 outer and an inner division was one divided by the number of 
 parts into which the outer was divided. This differs from our 
 present vernier, first used by Petrus Vernierus in 1631, in which 
 the auxiliary arc is short and is carried around with the zero- 
 point. 
 
 A great impetus was given to applied mathematics by the 
 construction of logarithmic tables, according to the formulae of 
 Napier (1550-1617), and Briggs (15 56-1630), especially in facili- 
 tating trigonometric computations, which had now become the 
 basis of degree-measurements. 
 
 The first person to use an entire circle instead of a part was 
 Roemer in 1672, who deserves our thanks for having invented 
 the transit also. Auzout in 1666 made the first micrometer, 
 and Picard was the first to apply it, and a telescope with cross- 
 wires, to an angle-reading instrument. The results obtained
 
 30 GEODETIC OPERATIONS. 
 
 with this instrument were so satisfactory that Cassini used it 
 in his great triangulation begun eleven years later. The angles 
 in Peru were measured with quadrants of 21, 24, 30 and 36 
 inches radius, each provided with one micrometer. These 
 gave very fair results the maximum error in closure of a tri- 
 angle being 12 seconds, spherical excess not considered. This 
 would give an error of one unit in 5000 in the length of a de- 
 pending line a value ten times better than any obtained dur- 
 ing the preceding century. 
 
 With such close reading of angles the discrepancies between 
 measured and computed lines were quite naturally attributed 
 to the unit of measure, the method of its use, or its comparison 
 with a standard. As early as the Peruvian work the uncer- 
 tainty in the varying length of wooden rods because of damp- 
 ness, and of metal rods on account of heat, was appreciated ; 
 and in the measurement of these bases an approximate average 
 of 13 R. was assumed for the mean temperature. This hap- 
 pened to be the temperature at which the field standards had 
 been compared with the copy before leaving Paris, hence the 
 reason for legalizing this temperature for that at which the 
 toise of Peru is a standard. 
 
 In 1752 Mayer announced the advantages to be derived 
 from repeating angles, and a repeating-circle was constructed 
 upon this principle by Borda in 1785, for the connection of the 
 French and English work. The first dividing engine was made 
 by Ramsden in 1763, and a second improved one in 1773, 
 which did such good work that his circles soon became deserv- 
 edly famous. In 1783 this maker furnished an instrument to 
 the English party engaged upon the work just mentioned, 
 this was the first to be called theodolite. It had a circle three 
 feet in diameter, divided into ten-minute spaces, read by two 
 reading micrometer microscopes. One turn of the micrometer- 
 screw was equal to one minute, and the head was divided into 
 sixty parts, so that a direct reading to a single second could
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 31 
 
 be made, and to a decimal by approximation. It was also 
 provided with a vertical circle of 10.5 inches diameter, read by 
 two micrometers to three seconds. The success attained in 
 the use of this instrument, giving a maximum error of closure 
 of three seconds, was regarded as truly phenomenal. 
 
 Reichenbach began the manufacture of instruments, in 
 Munich, in 1804, f such a high grade of workmanship that it 
 was soon considered unnecessary to send to Paris or London in 
 order to secure the best. He fortunately furnished Struve with 
 a theodolite, putting a good instrument in the hands of one of 
 the most skilful observers who has ever lived, which contrib- 
 uted no little to his reputation. His circles were almost wholly 
 repeaters, a class of instruments exclusively used on the Con- 
 tinent, but not at all in England. 
 
 Littrow, at the Observatory of Vienna, was the first to aban- 
 don the method of repetition, in 1819; and Struve, in 1822, was 
 the next to follow. 
 
 The inconvenience attending the use of large circles was 
 very great, besides the irregularities produced from flexure on 
 account of unequal distribution of supports. This led to the 
 attempt to make a smaller circle with good graduation, and 
 reading-microscopes. This end was achieved by Repsold, who 
 made a ten-inch theodolite for Schumacher in 1839, with which 
 it was definitely demonstrated that as good results could be se- 
 cured with a ten or a twelve inch instrument as with a larger one, 
 and with less expenditure of time and labor, not considering 
 the difference in the first cost. So that now we find the effort 
 heretofore spent in constructing enormous circles given to per- 
 fecting the graduation, and, while using the instrument, to pro- 
 tect the circle from sudden or unequal changes of temperature. 
 Mr. Saegmuller's principle of bisection in dividing a circle 
 keeps the errors of graduation within small limits, and the new 
 dividing engines leave but little to be desired in the construc- 
 tion of theodolites.
 
 32 GEODETIC OPERATIONS. 
 
 In England and India eighteen-inch circles are now used 
 in place of those of twice that size formerly employed. 
 Struve had a thirteen-inch theodolite. In the U. S. Coast and 
 Geodetic Survey the large instruments have given way to those 
 of twelve inches. In Spain twelve- and fourteen-inch circles are 
 found to be the best, while the excellent work of the U. S. Lake 
 Survey was done with theodolites having circles of twenty and 
 fourteen inches in diameter the latter having the preference. 
 
 To describe the various forms of theodolites now in use 
 would necessitate a number of illustrations, and in the end be 
 tedious and unprofitable; the same general features being com- 
 mon to all, they only will be referred to. The end sought in 
 the construction of theodolites is to get an instrument with 
 parts sufficiently light to insure requisite stability, with circles 
 large enough to allow close readings, with the telescopic axis 
 concentric with the circle, a reliable means for subdividing the 
 divisions on the circle, and a circle so graduated as to be free 
 from errors, or to have them according to a law readily dis- 
 tinguished and easily allowed for. While every one concedes 
 that the foregoing requisites are imperative, in respect to some 
 there is a great difference of opinion as to when they are at- 
 tained. 
 
 The illustration appended shows an eight- to twelve-inch 
 theodolite of the form suggested by the experience of the skilled 
 officers of the U. S. Coast and Geodetic Survey. In its construc- 
 tion hard metal is employed, and as few parts used as possible. 
 The frame is made of hollow or ribbed pieces in that shape 
 that gives the greatest strength for the material. The bearings 
 are conical ; clamps of a kind that avoid travelling motion ; the 
 circle is solid, and of a conical shape to prevent flexure. The 
 focal distance is diminished so as to admit of reversal of tele- 
 scope without removing it from its supports, and the optical 
 power is increased to insure precision in bisecting a signal. 
 They are made as nearly symmetrical as possible, and when
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 
 
 there is no counterpoise provided, one of the proper weight is 
 put in place. They are furnished with three foot-screws for 
 levelling, resting in grooves converging towards the centre. 
 Sometimes a circular level is set in the lowest part of the 
 branching supports, and in other cases a single tubular level is 
 made use of. The optical axis is marked by having in the 
 principal focus spider-lines called a reticule, or a piece of very 
 thin glass on which fine lines are etched. The arrangement of 
 the lines is various, the forms depicted in the annexed cut be- 
 ing the ones most frequently found. 
 
 The instrument shown in Fig. I is one of directions in which 
 the circle is shifted for new positions. With a repeater the 
 only difference is the addition of a slow-motion screw to move 
 the entire instrument in accordance with the method of repe- 
 tition as explained on page 98. 
 
 The adjustments of a theodolite must be carefully attended 
 to and frequently tested. They may be described in general 
 as follow : 
 
 To Adjust the Levels. When the tripod or stand is placed in 
 a stable condition and the instrument mounted, bring it into a 
 level position, as indicated by the level, by turning the foot- 
 screws. Turn the instrument 180 degrees, correct any defect, 
 one half by means of the screws attached to the level, and 
 the rest by the foot-screws. Place the instrument in its first 
 position, repeat the corrections as before until no deviation is 
 noticed when the circle is turned. If there is a second level, it 
 is to be adjusted in the same manner. 
 z
 
 34 
 
 GEODETIC OPERATIONS. 
 
 FIG. i.
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 35 
 
 To Adjust the Spider-lines of the Telescope. (i) Place the 
 threads in the focus of the eye-piece, point to a suspended 
 plumb-line when the air is still, and see if the vertical thread 
 coincides with the plumb-line. If there is any deflection, loosen 
 the four screws holding the diaphragm and move it gently till 
 there is a coincidence, then tighten the screws and verify. (2) 
 If the level is correct, place the circle in a horizontal position 
 and sight to some clearly defined object ; move the instrument 
 sideways by means of the tangent screw and notice if the hori- 
 zontal thread traverses the point throughout its entire length, 
 if not, correct as in the above case. 
 
 To Adjust the Line of Collimation of the Telescope. When 
 the horizontal axis of the telescope can be reversed, point the 
 instrument to some clearly denned object, then reverse the 
 telescope and see if the pointing is good. If not, half the dif- 
 ference is to be corrected in the pointing and the other half 
 by moving the entire diaphragm to the right or left, as the case 
 may be. Continue this course until the pointing remains per- 
 fect after reversal. If the instrument does not admit of this 
 reversal, it must be turned in its Y's ; and if the reading is more 
 or less than 180 degrees from the first reading, correct as be- 
 fore, until there is just 180 degrees between the readings before 
 and after reversal. 
 
 The horizontality of the axis of the telescope is tested by 
 placing on the axis a portable level that is in good adjustment. 
 If a defect is apparent, it must be corrected entirely by raising 
 or lowering the movable end. 
 
 After completing these adjustments, it is well to repeat the 
 tests to see if any have been disturbed while the other ad- 
 justments were in progress. When large instruments with 
 reading-microscopes are used, the corrections for runs and 
 eccentricity must be determined. The former can be readily 
 ascertained as follows: Turn the micrometer in the direction 
 of the increasing numbers on its head till the movable cross-
 
 36 GEODETIC OPERATIONS. 
 
 wire bisects the first five-minute space ; call the reading a. 
 Reverse the motion and continue to the preceding five-minute 
 space ; call this b. Suppose 
 
 a = 45 40' + 4' 46".4, b = 45 40' -f 4' 44"- 2, 
 
 r = a - b = + 2". 2, m = ^~~ = A-' 45"-3- 
 
 Since the five-minute space contains 300 seconds, the correction 
 to a = r .a -f- 300 = 2".i ; correction to b = r(b 300) 
 _i_ 300 = -j- ".i i ; correction to m \(a -\- b 300)^ -=- 300 
 = o".88, in which a and b represent minutes and seconds of 
 the above readings. The corrected reading is therefore, 
 
 45 44' 45 ".3 - .88 =4544'44".42. 
 
 Occasionally the average error of runs is determined and a 
 table computed from the formula just given for a -f- b from 5 
 
 d 
 
 to 10 seconds. But in very accurate work the correction for 
 runs is made for each reading by recording the two micrometer- 
 readings just mentioned for each pointing. They are recorded 
 as forward and backward, as seen on page 101. 
 
 The eccentricity is owing to the centre of the axis carrying 
 the telescope not coinciding with the centre of the graduated 
 circle. As each point on the plate carrying the telescope must
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 37 
 
 return to its former position after each complete revolution, 
 there must be a point at which there is a maximum deflection 
 as well as a point at which there is no deflection, and at the 
 same time the intermediate positions have eccentric errors be- 
 tween these limits ; therefore it is necessary to examine the 
 whole circle. This can be done in connection with an exami- 
 nation of the two verniers. The difference in the reading of the 
 two verniers may, however, be due to other causes: the con- 
 stant angular distance between them may be more or less than 
 1 80 degrees, or it may be owing to errors of graduation, or 
 errors of reading, or to the eccentricity referred to. 
 
 Let c be the centre of the limb, 
 m that of the telescope, 
 6 = angle amb, 
 0' = angle deb, 
 E the difference, or error, 
 e = cm = the linear eccentricity, 
 oo = dcm, 
 
 r = radius of the circle, 
 d = cdm, 
 b = cbm, 
 dm = + b Q'+d; therefore, EQ-Q'^d-b. 
 
 As cm is never very large, we can put mb = r : in the tri- 
 igle cdi 
 we have 
 
 angle cdm, we have sin d = - sin GO, and in the triangle bcm, 
 
 sin b - sin bcm = - sin (GO 6'). 
 
 
 
 Also, since d and b are small, we can write for sin b, b. sin i", 
 and for sin d, t/.sin \" , so that we have,
 
 38 GEODETIC OPERATIONS. 
 
 
 
 ;7 [sin oo - sin (GO - 
 
 By expanding sin (oo 0'), and putting for the entire angles 
 their values in terms of the half-angles, we find, 
 
 E = ' 7sin * 6 ' ' C S Q? ~ 
 
 This expression is made up of two factors, and becomes o 
 when either factor becomes o, as e o, or cos (oo #') = o, 
 that is, when oo $0' 90, or 6' = 200 180, 
 
 Therefore when the verniers are 180 apart, the errors of 
 eccentricity are eliminated. Likewise E is a maximum when 
 cos (oo $0'} = -|-i, that is, when GJ \ff = o, or 2oo == 6'. 
 
 In accord with the principle that errors of eccentricity are 
 avoided when the angle is read from two points 180 apart, 
 circles are provided with two verniers that distance from each 
 other. Instead of verniers, however, we may have two micro- 
 scopes. 
 
 The practical difficulty of placing the zero-points just 180 
 apart makes it necessary to examine each circle to see what 
 the angular distance between them is. This is best accom- 
 plished by setting one vernier, say A, on each 10 mark, and 
 reading and recording vernier B. If a represent the amount 
 by which the angular distance differs from 180, and b the effect 
 of eccentricity on this distance, we will have B A 180 
 -\-a-\-b, and when the verniers change places b will have a 
 contrary effect, so that B A = 180 + a b ; therefore if we 
 take the mean of the differences B A for positions that are 
 just 180 apart, we will have the angular distance unaffected by 
 eccentricity. We so arrange our readings as to have on the 
 same line those that are 180 apart. We also place under B 
 A the first difference, and on the same line the second dif- 
 ference, the mean will be the average of the two, or 180 -+- a,
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 39 
 
 and the average of these means will be the mean distance be- 
 tween the verniers. 
 
 FIRST. 
 
 SECOND. 
 
 B -A. 
 
 A. 
 
 B. 
 
 A. 
 
 B. 
 
 ISt. 
 
 2d. 
 
 Mean. 
 
 00' 00" 
 
 1 80 00' 5 " 
 
 1 80 00' 00" 
 
 00' 00" 
 
 t 5" 
 
 O 
 
 + 2 ". 5 
 
 IO 
 
 10 
 
 190 
 
 05 
 
 + 10 
 
 + 5 
 
 + 7.5 
 
 20 
 
 05 
 
 200 
 
 00 
 
 + 5 
 
 o 
 
 -M .5 
 
 30 
 
 IO 
 
 210 
 
 05 
 
 + IO 
 
 4- 5 
 
 4-7 -5 
 
 40 
 
 55 
 
 2 2O 
 
 oo 
 
 - 5 
 
 o 
 
 - 2 .5 
 
 50 
 
 00 
 
 230 
 
 05 
 
 o 
 
 4- 5 
 
 + 2 .5 
 
 60 
 
 05 
 
 240 
 
 IO 
 
 + 5 
 
 + 10 
 
 + 7 -5 
 
 70 
 
 05 
 
 250 
 
 05 
 
 + 5 
 
 + 5 
 
 + 5 
 
 80 
 
 10 
 
 260 
 
 oo 
 
 + 10 
 
 o 
 
 4-5 
 
 90 
 
 05 
 
 270 
 
 10 
 
 4- 5 
 
 +10 + 7 .5 
 
 100 
 
 oo 
 
 280 
 
 55 
 
 o 
 
 - 5 - 2 .5 
 
 no 
 
 55 
 
 2 9 
 
 00 
 
 - 5 
 
 
 
 -2.5 
 
 1 20 
 
 55 
 
 300 
 
 05 
 
 - 5 
 
 + 5 
 
 o 
 
 130 
 
 05 
 
 310 
 
 05 
 
 4- 5 
 
 T 5 
 
 4-5 
 
 140 
 150 
 
 05 
 05 
 
 320 
 
 330 
 
 55 
 oo 
 
 ! 
 
 5 
 o 
 
 o 
 
 4-2 .5 
 
 160 
 
 05 
 
 340 
 
 oo 
 
 4- 5 
 
 o 
 
 4-2 .5 
 
 170 
 
 05 
 
 350 
 
 05 
 
 4- 5 
 
 + 5 
 
 4-5 
 
 Therefore the angular distance = 180 + 3".!. Mean = 3".!. 
 
 Now, knowing the angular distance between the two verniers, 
 the difference between it and the mean of B A will be the 
 errors of eccentricity and graduation, or b -\- g. 
 
 Angle dcA = m-\-A, therefore A 
 dcA m. If we call afthe reading on the 
 limb which is on the line of no eccentricity, 
 that is on the line drawn through the 
 centre of motion and centre of graduation, 
 and n any angle read by the verniers, 
 then n d will be the angle between the 
 vernier and line of no eccentricity, or dcA. 
 In the triangle Acm, sin Acm : sin A :: 
 Am : cm, but sin Acm = sin dcA = sin (n d), and Am = r, 
 nearly, making these substitutions: 
 
 FIG. 3.
 
 4O GEODETIC OPERATIONS. 
 
 . e.sin(nd) 
 
 sin (n d):smA::r'.e, or sin A = -- . 
 
 A being small, we can put for sin A, A . sin i", and the angu- 
 lar value for e to radius r, e.s'm l" ; then write for A in 
 seconds, A = e.s'm (n d), and for the two verniers, 6 = 
 2e. sin ( d\ A reading b' at 180 from the former will 
 have the same error, but with an opposite sign, b' = 2e 
 . sin (n d}. If we tabulate the differences between the 
 mean in our first table and the various readings for B A, 
 placing on the same line those that differ by 180 from one an- 
 other, they should be equal with opposite signs were it not for 
 errors of graduation; let these differences be D and D', then 
 =/>, and*' +*=/>', 
 
 2e sin (n 
 
 2e sin (n d)-\-g = D' 
 
 Subtracting, 
 
 4/? sin ( - d) = D - D', 2e sin ( - d) = %(D - D'} b, 
 
 or a value for b freed from errors of graduation. This will give 
 18 equations involving e and n. 
 Placing 8 = $(D- D'\ we have ; 
 
 ?, = 2e sin (10 -d) = 2*(sin 10 cosd cos 10 sin d}\ 
 r w = 2* sin (170 - d) = 2t(s'm 170 cos d cos 170 sin d).
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 41 
 
 Professor Hilgard's method for solving these equations with 
 respect to 2e cos d and 2e sin d, by least squares, is to multiply 
 each equation through by cos , and sum the resulting equa- 
 tions ; then each through by sin , and sum the results : this 
 will give us two normal equations of this form ; after factoring 
 2e cos d, and 2e sin d, 
 
 [#! sin o -f- $1 sin 10 . . . S lt sin 170] 
 
 = 2^cos^[sin s o + sin* i O . . . sin" I/O ] 
 
 2e sin */[sin o cos o+sin 10 cos io+. . . sin 170 cos 170]; 
 
 [tf, cos o + d, cos 10 ... d 18 cos 170] 
 
 = 2e cos d [cos o sin o . . . cos 170 sin 170] 
 2e sin d?[cos 8 o + cos 2 io a . . . cos 8 170] 
 
 sin o cos o = o, also for sin 10 cos 10 we can put sin 20 
 and so on with all the products of sines times cosines ; and we 
 find that this will give us pairs of angles that make up 360, 
 whose sines are equal but with opposite algebraic signs, so the 
 products reduce to zero. Again, we can arrange the second 
 powers so that all angles above 90 can be written 90 -f- n ; 
 sin* (90 -f- n) = cos 2 n, this added to sin 2 n = I, for example ; 
 sin 8 o = o, sin 8 10 + sin 8 100 = sin 2 10 + sin 2 (90 + 10) = 
 sin 2 10 + cos 2 10 = i. 
 This will give us half as many unities as we have terms less 
 
 N 
 two for the pairs, and sin 2 90 = i gives us 9 = . The nor- 
 
 mal equations will then reduce to 
 
 sin ) = Ne cos d, 
 cos n) = Ne sin d ;
 
 GEODETIC OPERATIONS. 
 
 tt. 
 
 First 
 6 + f- 
 
 Second 
 
 ISt-2d t 
 
 sin . 
 
 cosn. 
 
 S sin n. 
 
 5 cos n. 
 
 a 
 
 
 
 4- 2. 
 
 - 2.9 
 
 4 -5 
 
 O.OO 
 
 I.OO 
 
 o".oo 
 
 4 2". 50 
 
 IO 
 
 4 7- 
 
 + 2.1 
 
 4 -5 
 
 17 
 
 .98 
 
 4 -43 
 
 42 -45 
 
 20 
 
 4 2. 
 
 2.9 
 
 + - 5 
 
 34 
 
 94 
 
 4 o .85 
 
 42 .35 
 
 30 
 
 4 
 
 4 7- 
 
 7. 
 
 + 2.1 
 2.9 
 
 4 -5 
 -5 
 
 50 
 .64 
 
 .87 
 .76 
 
 + i .25 
 I .60 
 
 42 .17 
 - i .90 
 
 50 
 60 
 
 - 2. 
 4- 2. 
 
 -j-2.1 
 
 4 7-1 
 
 -5 
 - -5 
 
 76 
 .87 
 
 .64 
 50 
 
 - I .90 
 
 - 2 .I? 
 
 - i .60 
 - i .25 
 
 70 
 
 T 2- 
 
 4 2.1 
 
 
 94 
 
 34 
 
 o .00 
 
 o .00 
 
 80 
 
 
 - 2.9 
 
 + .0 
 
 .98 
 
 17 
 
 44 -90 
 
 40 .85 
 
 9 
 
 4 2. 
 
 4 7-1 
 
 5 
 
 I.OO 
 
 .00 
 
 2 .50 
 
 o .00 
 
 100 
 
 2. 
 
 7-9 
 
 4 -5 
 
 .98 
 
 - -17 
 
 42 -45 
 
 -o .43 
 
 no 
 
 - 7- 
 
 2.9 
 
 - .5 
 
 94 
 
 -34 
 
 -2 .35 
 
 40 .85 
 
 120 
 
 7> 
 
 4 2.1 
 
 - 5-0 
 
 .87 
 
 - -50 
 
 -4 -35 
 
 + 2 .50 
 
 130 
 140 
 
 -- 2. 
 -- 2. 
 
 4 2.1 
 
 - 7.9 
 
 o 
 4 5-0 
 
 .76 
 .64 
 
 - .64 
 - -76 
 
 o .00 
 43 .20 
 
 .00 
 
 -3 -80 
 
 150 
 
 160 
 
 -- 2. 
 -- 2. 
 
 2.9 
 - 2.9 
 
 42.5 
 
 4 2.5 
 
 50 
 34 
 
 - -87 
 - -94 
 
 tl -25 
 o .85 
 
 - 2 .I? 
 - 2 .35 
 
 170 
 
 4 2. 
 
 4 2.1 
 
 o 
 
 17 
 
 - .98 
 
 o .00 
 
 o .00 
 
 
 
 
 
 2(8 sin n) 
 
 40 .31 
 
 
 
 
 
 
 2(8 cos n) = 
 
 
 40 .I? 
 
 tan </=/ = tan 151 15' 40"; 
 
 e 
 
 0.17 
 
 iSsin 151 15' 40 
 
 77 = O".O2. 
 
 The line of no eccentricity is that passing through 151 15' 
 40"; the sign of e being minus, we know that the centre of mo- 
 tion is in the opposite direction from the centre of graduation 
 towards the reading d. In this case it is too small to be con- 
 sidered. To determine the error of graduation, we compute 
 the values of 2e sin (n d} = b; subtracting these results 
 from those in the last table marked ^-f-^in the first column, 
 we will have g. It is necessary to compute b for every 10 
 space, only up to 180, since b has the same value for 180 
 -{- n that it has for n, with the opposite sign, then subtract 
 these values from the second b -\- g.
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 43 
 
 n. 
 
 n-d. 
 
 it sin ( d}. 
 
 n. 
 
 '* 
 
 . 
 
 f- 
 
 O 
 IO 
 
 - 151 
 141 
 
 + 0.017 
 + -025 
 
 
 10 
 
 4- 2". 083 
 
 4-7 -075 
 
 1 80 
 I 9 
 
 - 2.883 
 4- 2.125 
 
 2O 
 30 
 
 121 
 
 4- -030 
 
 + -034 
 
 20 
 
 30 
 
 +2 .070 
 
 + 7 -066 
 
 20O 
 
 210 
 
 2.870 
 + 2.134 
 
 40 
 
 Ill 
 
 + -037 
 
 40 
 
 - 7 -937 
 
 2 2O 
 
 - 2.863 
 
 50 
 
 IOI 
 
 4- -038 
 
 50 
 
 - 2 .938 
 
 230 
 
 4- 2.138 
 
 60 
 
 9 1 
 
 4- .040 
 
 60 
 
 +2 .060 
 
 240 
 
 + 7-I40 
 
 70 
 
 - 81 
 
 + -038 
 
 70 
 
 4-2 .062 
 
 250 
 
 + 2.138 
 
 80 
 
 71 
 
 + -037 
 
 80 
 
 + 7 -063 
 
 260 
 
 - 2.863 
 
 90 
 
 - 61 
 
 4- -034 
 
 90 
 
 +2 .066 
 
 270 
 
 + 7-134 
 
 IOO 
 
 51 
 
 4- .030 
 
 IOO 
 
 - 2 .930 
 
 280 
 
 - 7-870 
 
 no 
 
 - 41 
 
 4- .026 
 
 no 
 
 - 7 .926 
 
 2 9 
 
 - 2.874 
 
 120 
 
 
 4" -O2O 
 
 1 20 
 
 7 .920 
 
 300 
 
 -\- 2.I2O 
 
 130 
 140 
 
 21 
 II 
 
 4- .014 
 4- -007 
 
 130 
 140 
 
 -f- 2 .086 
 + 2 93 
 
 310 
 320 
 
 4- 2.II4 
 
 - 7 893 
 
 150 
 1 60 
 170 
 
 OI 
 + 09 
 + 19 
 
 -j- .000 
 
 - .006 
 .013 
 
 150 
 
 160 
 170 
 
 4-2 .100 
 
 +2 .106 
 4-2 .113 
 
 330 
 340 
 350 
 
 2 .900 
 2.906 
 
 4- 2.087 
 
 .The sum of the squares of the 36 values for g give 714.7617, 
 therefore the probable error in any one is A / - = 
 
 4".6 ; this divided by the square root of two gives the prob- 
 able error of the reading of one vernier, owing to errors of grad- 
 uation and accidental errors of reading = 3".2. 
 
 If an angle is the mean of five repetitions, the probable error 
 of the average will be one fifth of 3 // .2 = o".64. 
 
 If the effect of eccentricity be considerable, the correction to 
 each angle should be computed by the equation b = 2e sin 
 ( d\ The probable error in graduation and reading is 
 used only in computing the probable error in a chain of tri- 
 angles, as will be seen later. If the instrument has two read- 
 ing-microscopes the procedure is essentially the same, but dif- 
 fers slightly when there are three. In this case every 5 or 10 
 space can be examined and the three microscopes read ; as be- 
 fore, we shall call the reading of the zero-point , and the 
 microscopes A, B, and C.
 
 44 
 
 GEODETIC OPERATIONS. 
 
 
 n. 
 
 *+!*>. 
 
 + 240*. 
 
 00' 00" 
 
 + 01" 
 
 + 02" 
 
 
 120 00 01 
 
 oo 
 
 01 
 
 Sum 
 One third 
 
 240 oo 02 
 
 04 
 
 oo 
 
 + 03 
 
 - 02.6 
 
 - 00.9 
 
 03 
 + 03-3 
 
 -j- OI.I 
 
 +' 
 
 0.6 
 
 O.2 
 
 = + I, average = + 0.3. 
 
 The first line gives the readings when the zero-point is n, 
 the order of the microscopes is A, B, and C\ in the next, zero 
 is at n -J- 120, and the order is C, A, and B; in the third, zero 
 is at + 240, ancl tne order is B, C, and A. The fourth line 
 contains the sums, and the continuation the average ; and by 
 subtracting the sums from this average we have the fifth line 
 containing three times the errors of trisection at this point. 
 
 Eccentricity is first determined : " Suppose ac n , fi n , and y n be 
 the observed errors of trisection corresponding to n, n-\- 120, 
 and n + 240, also [o^cos ], [/?cos(-|- 120)], [<* sin ] . . . 
 etc., be the sums of all the a n cos n, a n sin n, etc., then d t the 
 line of no eccentricity, 
 
 = _ [an cos n\ + [/?, cos (n + 120)] + Q cos (n + 240)] 
 [a n sin ] + [/? sin ( + 120)] + [y n sin ( + 240)] 
 
 also e" 
 
 _ __ [^ sin ] 4- [/? sin (+ 120)] -f \y n sin ( -f 240)] 
 
 where ^V = number of trisections. 
 
 " The correction for eccentricity is b e sin (n d\ then if 
 <**', P*', Yn errors of trisections freed from errors of eccen- 
 tricity, we will have :
 
 INSTRUMENTS AND METHODS OF OBSERVATION. 45 
 
 a n ' = a n e sin (n d) ; 
 
 /?' = /3 n - e sin (n + 120 d] ; 
 
 Yn = Yn e sin ( + 240 d). 
 
 Knowing ', /?', y n f , the residuals are squared, and the prob- 
 able error of graduation and reading found as in the preceding 
 case." 
 
 Considering that the determination of latitude, longitude, 
 and azimuth forms a part of practical astronomy, the only in- 
 struments that remain to be described are the base-apparatus 
 and heliotrope. The former is referred to in the chapter on 
 base-measuring, and the latter can be dismissed with a few 
 words. 
 
 The first heliotrope was used by Gauss in 1820. It was 
 somewhat complicated, consisting of a mirror attached to the 
 objective end of a small telescope. This mirror had a narrow 
 middle-section at right angles to the rest of it ; this was in- 
 tended to reflect light into the tube, while the remainder re- 
 flected the sun's rays upon the object towards which the tele- 
 scope was pointed. Bessel devised a much simpler form that is 
 still in use in Prussia. It has a small mirror, with two motions, 
 fastened to one end of a narrow strip of board, while at the 
 other end there is a short tube whose height above the board 
 is the same as the axis of the mirror. In this tube cross-wires 
 are stretched, and a shutter can be dropped over the end op- 
 posite the mirror. To use it, one fastens the screw that is at- 
 tached to one end in a suitable support and then by means 
 of a levelling screw at the other end, raises or lowers that 
 end until the centre of the mirror, the cross-wires and the 
 object towards which the light is to be reflected are in line. 
 The mirror is then turned so that the shadow of the cross- 
 wires falls upon their counterpart that is marked on the shutter 
 when the light can be seen at the desired point. Perhaps the 
 most convenient of all is the heliotrope that finds employment
 
 46 GEODETIC OPERATIONS. 
 
 in the U. S. Coast and Geodetic Survey. It can be seen in 
 Fig. 4. First of all, there is a low-power telescope provided 
 with a screw for attachment to a tree or signal. On one end 
 of the tube is a fixed ring of convenient diameter, say one 
 and a half inches, while at the other end is a mirror of two 
 inches in diameter, and at an intermediate point, nearer the 
 mirror, is another ring of the same height and size as the other, 
 but clamped to the tube, admitting of a motion around it. 
 
 To describe its use we will suppose it in adjustment. After 
 having screwed it to a post, the telescope is turned until the 
 cross-wires approximately coincide with the point to which the 
 light is to be shown ; then turn the mirror so that the shadow 
 of the nearer ring exactly coincides with the other ring. Then 
 as the earth revolving places the sun in a different relative 
 position, it will be necessary to continually move the glass in 
 order to keep the shadow of the back ring on the front one. 
 If the sun is behind the heliotrope an additional mirror will be 
 needed to throw the light upon the glass. 
 
 To effect the adjustment, it is necessary to have in the con- 
 struction the centres of the rings and the mirror at the same 
 distance from the optical axis of the telescope. Bisect some 
 clearly defined point, then sight over the tops of the mirror 
 and rings, turning the movable one until they are all in line 
 with the object bisected by the telescope. Owing to the large 
 diameter of the sun, a slight error in adjusting will not affect 
 the successful use of this kind of a heliotrope. 
 
 When the observed and observing stations are within twenty 
 miles of one another, the light spot may be too large to be 
 easily bisected ; then it is best to place between the glass and 
 rings a colored glass (orange is preferable), so as to reduce the 
 light as seen to a mere spot. A code of signals can be adopted 
 and messages exchanged between observer and heliotroper, 
 such as " Correct your pointing," " Stop for the day," " Set 
 on new station," " Too much light," " Not enough light," by
 
 INSTRUMENTS AND METHODS OF OBSERVATION. tf 
 
 cutting off the light with a hat or small screen ; a long stoppage 
 standing for a dash, and a short one for a dot, when the words 
 can be spelled out by the Morse code. 
 
 The maximum distance at which a heliotropic signal can be 
 seen depends upon the condition of the atmosphere. Perhaps 
 the greatest was on the " Davidson quadrilateral," where a 
 light was seen at a station 192 miles away. 
 
 A very convenient form of heliotrope, especially for recon- 
 noissance, is one invented by Steinheil, and known by his name. 
 It differs from all others in having only one mirror and no 
 
 FIG. 4. 
 
 rings, making it so simple in use and adjustment as to form 
 a valuable instrument. The glass has but one motion, but 
 the frame has another at right angles to it. 
 
 As can be seen from the illustration, the entire instrument 
 can be attached to an object by means of a wood screw, and 
 clamped in any position by other screws. In the centre of 
 the mirror the silvering is erased, making a small hole through 
 which the light of the sun can pass ; also in the centre of 
 the frame carrying the mirror there is an opening fitted with 
 a convex lens, and behind the lens is a white reflecting sur- 
 face usually chalk. To use the heliotrope, turn the glass so 
 that the bright point caused by the sun shining through the
 
 4 8 
 
 GEODETIC OPERATIONS. 
 
 hole coincides with the opening in the frame. This will give 
 in the focus of the lens an image of the sun, which will be re- 
 flected back- through the hole in the glass. Now, 
 if the entire instrument be turned so as to bring 
 this image upon the point at which the light is 
 to be seen, the rays falling upon the mirror will 
 be reflected in the same direction. 
 
 To see the fictitious sun, as the image is called, 
 one must look through the hole from behind the 
 glass, and as it is always small and quite indis- 
 tinct, some practice will be needed to recognize 
 it. This can best be acquired by turning the 
 image upon the shaded side of a house, then it 
 will be seen as a small full moon. The reflect- 
 ing surface can be moved in or out by a screw 
 from behind, and the only adjustment that is ever needed 
 is to have this surface at that distance that gives the best 
 image of the sun. After having placed the heliotrope in 
 the correct position, it should be clamped, and then the only 
 labor is simply to occasionally turn the glass so as to bring 
 the bright spot into coincidence with the opening in the frame. 
 In the Eastern States, through air by no means the clearest, a 
 light from a Steinheil heliotrope has been observed upon at a 
 distance of 55 miles. 
 They are made by Fauth of Washington. 
 
 FIG. 5.
 
 104 
 
 BA SE-MEA S UREMENTS. 49 
 
 CHAPTER III. 
 
 BASE-MEASUREMENTS. 
 
 As the foundation of every extended scheme of trigonomet- 
 ric surveys must be a linear unit, it is essential that the length 
 of this base should be determined with the utmost degree of 
 care. 
 
 But the labor and expense of measuring a base of favorable 
 length are so great as to preclude repeated measurements. In 
 order, therefore, to secure results at all comparable with the 
 precision desired, an apparatus of great delicacy is needed. 
 This becomes apparent when we consider that an apparatus of 
 convenient length is repeated from one to two thousand times 
 in the measurement of a base, and that even a small error in 
 the length of the measuring unit will be multiplied so as to 
 seriously affect the results. 
 
 And this error in a short line will be increased proportion- 
 ally in the computed lengths of the long sides of the appended 
 triangles. The figure and magnitude of the earth are deter- 
 mined from extended geodetic operations, and the elements so 
 determined are conditionally used in the re-reduction of trian- 
 gulation data, securing in this way a more probable expression 
 for the shape of our planet. 
 
 From this it may be seen that all of our errors are of an ac- 
 cumulative character, and seriously affect the results unless 
 fortuitously eliminated by a principle of compensation. 
 
 Since geodesy first received attention, the subject of most 
 important consideration has been the construction of a base- 
 apparatus that would secure good results without sacrificing 
 4
 
 jo GEODETIC OPERATIONS. 
 
 time and expense. The first form consisted of simple wooden 
 bars, resting on stakes previously levelled, and placed end to 
 end. When the configuration of the ground made it necessary 
 to make a vertical offset, it was done by means of a plumb-line. 
 Another form similar to this had a groove cut in the under 
 side to rest upon a rope drawn taut from two stakes of equal 
 elevation. In place of laying the rods on stakes or on a catenary 
 curve, it was once found convenient to place them on the ice, as 
 when Maupertuis measured the base in Lapland in 1736. This 
 line was measured twice, each time by a different party ; the 
 difference between the two results was four inches. This was 
 close work in a measurement extending over a distance of 8.9 
 miles. The rods used in this case were thirty-two feet long, 
 made of fir and tipped with metal to prevent wearing by attri- 
 tion. The Peru base measured at about the same time gave a 
 difference of less than three inches in the two measurements 
 in a distance of 7.6 miles. The wooden rods were found to 
 be affected by changes in the hygrometric conditions of the 
 atmosphere. This change was diminished by painting them. 
 Finally wood was abandoned as the material, and glass tubes 
 substituted. Of course with glass there was a continual change 
 in length due to expansion or contraction by thermal varia- 
 tions, that was not perceptible in the case of wood, but know- 
 ing the rate of expansion, the absolute length at any tempera- 
 ture can be theoretically computed. The temperature of each 
 tube during the entire measurement was ascertained by the 
 application of a standard thermometer, and the length of the 
 whole base was reduced to a temperature of 62 Fahr. The 
 difficulty of determining the temperature of the tubes was 
 considerable, since the thermometer reading gives the temper- 
 ature of the mercury in the thermometer, or, at best, that of 
 the external air, which will always differ from the temperature 
 of the measuring-bar. In the case of a sudden change of tem- 
 perature, the thermometer will respond more quickly than the
 
 BA SE-MEA SUREMENTS. 5 1 
 
 tubes, and its reading could not be taken as the reading of the 
 tubes. This trouble suggested the construction of an appara- 
 tus that would serve to indicate change in temperature as a 
 metallic thermometer. On this principle, Borda made four rods 
 for the special committee of the French Academy in 1792. 
 The rods were made of two strips of metal one of platinum, 
 and the other of copper overlying the former. They were 
 fastened together at one end, but free at the other and through- 
 out the remaining length. The copper was shorter than the 
 platinum by about six inches. It carried a graduated scale, 
 moving by the side of a vernier attached to the platinum ; the 
 reading of the scale indicated the relative lengths of the two 
 strips, and hence the length and temperature of the platinum. 
 The strips rested upon a bar of wood the entire apparatus 
 being six French feet in length. Contact was made by a slide, 
 the end of which was just six feet from the opposite end of the 
 platinum strip when the zero-mark on the slide coincided 
 with one on the end of the strip to which it was attached. 
 The rods rested upon iron tripods with adjusting-screws for 
 levelling, and the inclination was ascertained from a sector 
 carrying a level. It is interesting to note that the length of 
 the metre was first determined from the length of the quad- 
 rant computed from the base measured with this apparatus. 
 Borda's compensating apparatus in some form has been used 
 ever since it first came into notice. The principal varieties 
 are : Colby, Bache-Wurdeman, Repsold, Struve, Bessel, Hos- 
 sard, Borden, Porro, Reichenbach, Baumann, Schumacher, 
 Bruhns, Steinheil. 
 
 In these varietjes named after their inventors or improvers 
 the essential features sought for are : 
 
 i. The terminal points used as measuring-extremities must, 
 during the operation, remain at an unvarying distance apart, 
 or the variations therefrom must admit of easy and accurate 
 determination.
 
 52 GEODETIC OPERATIONS. 
 
 2. The distance between these extremities must be com- 
 pared with a standard unit to the utmost degree of accuracy, 
 and the absolute length determined. 
 
 3. In its construction provision must be made to secure 
 readiness in transportation, ease and rapidity in handling, 
 stability of supports and accuracy in ascertaining exact con- 
 tact and inclination. 
 
 The above conditions were secured in a great degree in the 
 Bache-Wurdeman apparatus, as used in the U. S. Coast and 
 Geodetic Survey since 1846. The description given by Lieu- 
 tenant Hunt in 1854 will be found quite explicit. For the 
 benefit of those who cannot consult the report which contains 
 this description the following abstract is given : the apparatus 
 sent to the field consists of two measuring-tubes exactly alike, 
 each being packed for transportation in a wooden box ; six 
 trestles for supporting the tubes three being fore trestles and 
 three, rear trestles each of which is packed in a three-sided 
 wooden box ; eight or more iron foot-plates on which to place 
 the trestles, and a wooden frame is afterwards made to serve 
 as a guide in laying down the foot-plates ; a theodolite for 
 making the alignment, and for occasionally referring the end 
 of the tube to a stake driven in the ground for the purpose; a 
 standard six-metre bar of iron in its wooden case, and a Sax- 
 ton pyrometer for effecting a comparison. 
 
 The measuring-bar consists of two parts a bar of iron and a 
 bar of brass, each less than six metres in length. 
 
 These are supported parallel to each other ; at one end are 
 so firmly connected together by means of an end-block, in 
 which each bar is mortised and strongly screwed, as to preserve 
 at that point an unalterable relation. The brass bar, which has 
 the largest cross-section, is sustained on rollers mounted in 
 suspended stirrups ; the iron bar rests on small rollers which 
 are fastened to the iron bar, and run on the brass one. Sup- 
 porting-screws through the sides of the stirrups are adjusted to
 
 BASE-MEASUREMENTS. 53 
 
 sustain the bars in place, and also serve to rectify them. Thus, 
 while the two bars are relatively fixed at one end, they are 
 elsewhere free to move ; and hence the entire expansion and 
 contraction are manifested at one end. The difference in the 
 length of the two bars is read on a scale attached to the iron 
 bar by means of a vernier fastened to the brass bar. The 
 scale is divided into half millimetres, of which the vernier indi- 
 cates the fiftieth part, so that by means of a long-focus micro- 
 scope the difference may be read to the hundredth part of a 
 millimetre without opening the case. Since the compensation 
 (described further on) can be made correct within its thirtieth 
 part, it is evident that the true length of the compound bars 
 may be known at any time from the scale-reading, with an un- 
 certainty no greater than the thousandth part of a millimetre 
 or a micron. 
 
 The medium of connection between the free ends of the two 
 bars is the lever of compensation, which is joined to the lower 
 or brass bar by a hinge-pin, around which it turns during 
 changes of temperature. A steel plane on the end of the iron 
 bar abuts against an agate knife-edge on the inner side of the 
 lever of compensation. This lever terminates in a knife-edge, 
 turned outward at such a distance from the centre-pin and the 
 other knife-edge bearing, that the end edge will remain un- 
 moved by equal changes of temperature in the two bars. The 
 end edge presses against a steel face in a loop made in the 
 sliding-rod. This rod slides in a frame fastened to the top of 
 the iron bar, and passes through a spiral spring, which acts 
 with a constant force to press the loop against the knife-edge. 
 The outer end of the sliding-rod bears the limiting agate plane. 
 Thus tire end agate is not affected in position by the expan- 
 sions of the brass and iron, acting as they do at proportional 
 distances along the lever of compensation, measured from its 
 sliding-end bearing. The rates of expansion for iron and brass 
 may safely be taken as uniform between the extreme expan-
 
 54 GEODETIC OPERATIONS. 
 
 sions and contractions to which they are subject in practice, and 
 the compensating adjustment once made is permanent. 
 
 The stirrups sustaining the rollers on which the brass bar 
 runs are made fast to the main horizontal sheet of the iron 
 supporting and stiffening work. This consists of a horizontal 
 and a vertical plate of boiler-iron, joined along the middle line 
 of the horizontal sheet by two angle-irons, all being perma- 
 nently riveted. Circular openings are cut out from both plates 
 to lighten them as much as practicable. A continuous iron 
 tie-plate, turned up in a trough-form, connects the bottoms of 
 all the stirrups. At the ends, stiffening braces connect the 
 two plates. 
 
 We now pass from the compensating to the sector end of the 
 tube, at which extremity are arranged the parts giving the 
 readings, and for adjusting the contacts between successive 
 tubes in measuring, thus making it the station of the principal 
 observer. The sector-end terminates in a sliding-rod, which 
 slides through two upright bars, and at its outer end bears a 
 blunt agate knife-edge, horizontally arranged, which in measur- 
 ing is brought to abut with a uniform pressure against the 
 limiting agate plane of the compensating end of the previous 
 tube. At its inner end, this sliding-rod rests against a cylindri- 
 cal surface on the upright lever of contact, so mounted as at its 
 bottom to turn around a hinge-pin. At top, this lever rests 
 against a tongue, or drop-lever, descending from the middle of 
 the level of contact, which is mounted on trunnions.* The 
 sliding-rod, when forced against the side of the lever of contact, 
 presses its top against the tongue of the level, and thus turns 
 the level by overcoming a preponderance of weight given to its 
 farther end, to insure the contact being always at a constant 
 
 * The device of the level of contact is supposed to be due to the elder Repsold, 
 who applied it first to the comparing- apparatus used by Bessel, in constructing 
 the Prussian standards of length. A duplicate of that comparator was procured 
 for the Coast Survey, by F. R. Hassler, Superintendent, in 1842.
 
 BA SE-MEA SURE MEN TS. 
 
 55 
 
 pressure between the agates, the same force being always 
 quired to bring the bubble to the 
 centre. The arrangement at the 
 two ends is shown in Fig. 6. 
 
 The sector is a solid metal 
 plate, mounted with its centre of 
 motion in the line of the sliding- 
 rod, and having its arc graduated 
 from a central zero to the limits 
 of ascending and descending 
 slopes on which the apparatus is 
 to be used. A fixed vernier in 
 contact with the arc gives the 
 slope-readings. A long level and 
 bubble-scale are so attached and 
 adjusted to the face of the sector- 
 plate that the zeros of the level 
 and of the limb correspond to 
 the horizontal position of the 
 whole tube. If, then, on slopes, 
 the bubble be brought to the 
 middle by raising or lowering the 
 arc-end of the sector (a move- 
 ment made by a tangent-screw, 
 whose milled head projects above 
 the tin case of the tube), the 
 vernier will give the slope at 
 which the tube is inclined, and 
 the sloping measure is readily 
 reduced to the horizontal by 
 means of a table prepared for 
 the purpose. The level of con- 
 tact and the lever of contact, 
 with their appendages, are all mounted on the sector and par- 
 take of its motions. A knife-edge end of the sliding-rod presses
 
 56 GEODETIC OPERATIONS. 
 
 on the cylindrical face of the contact-lever, this cylinder being 
 concentric with the sector, and the sector can therefore be 
 turned without deranging the contact. In fact, the contacts 
 are made with the sector-level horizontal, thus insuring the ac- 
 curacy of the contact-pressure. The contact-lever is supported 
 at bottom by two braces dropping down from the sector-plate, 
 and a spring, acting on a pin in the lever, steadies it against an 
 adjusting screw-end. A bracket from the sector-plate receives 
 the trunnions of the contact-level. A small screw projects from 
 the end of the tube to clamp or set the lever and level of con- 
 tact against a pin in the sector for security in transportation. 
 
 What is called the fine motion, required for adjusting the 
 contacts between the successive tubes, is produced by means 
 of a compensating rod or tube, one end of which is attached to 
 the truss-frame by a bracket over the rear trestle, and the other 
 receives a screw terminating in a projecting milled head. This 
 screw turns freely in a collar, bearing, by a projecting arm, 
 against the cross-bar which joins the main brass and iron bars, 
 and its nut is in the end of the compensation-rod. By turning 
 the screw in one direction, the bars are pushed forward, and 
 the opposite turning permits a spiral spring, arranged for the 
 purpose, to push back the system of bars, which slides through 
 its supports. Thus the contact is made by turning the screw 
 until the contact-level is horizontal. The compensating-rod 
 is composed of several concentric tubes, alternately of brass 
 and iron, arranged one within the other, and fastened at oppo- 
 site ends alternately. Thus, when a contact has been made by 
 the fine-motion screw, changes of temperature will not produce 
 derangement, as would be the case if this rod were not com- 
 pensating. The arrangement permits the observer conveniently 
 to work the fine-motion screw, and to observe its action on the 
 contact-level. 
 
 The apparatus thus described is enclosed in a double tin tubu- 
 lar case, diaphragms being adapted for supporting and strength- 
 ening the whole. The air-chamber between the two cases, one
 
 BASE-MEASUREMENTS. 57 
 
 and a half inches apart, is a great check on heat-variations. 
 Three side-openings, with tin and glass doors in each tube, 
 permit observations of the parts and of inserted thermometers. 
 The ends are closed, only the sliding-rod ends projecting at 
 each extremity, exposing the agates. Brass guard-tubes pro- 
 tect these, and for transportation tin conical caps are screwed on 
 the tube-ends. The fine-motion screw, the sector-tangent screw, 
 and the contact-lever-clamp screw project beyond the case. 
 The tube is painted white, which, with the air-chamber and 
 thorough compensation, effectually obviates all need of a screen 
 from the sunshine, which has usually been deemed requisite. 
 
 The tube rests on a fore trestle and rear trestle, which are 
 alike, except in the heads. Each trestle has three legs, com- 
 posed of one iron cylinder moving in another by means of a 
 rack, pinion, and crank, so as to raise or sink the head-plate. 
 The levelling and finer adjustment are by means of a foot- 
 screw in each leg, by -working which a circular level on the 
 connecting-frame is adjusted. A large axis-screw, resting on 
 the connecting-frame, and rising into a tubular nut, is turned 
 by bevelled pinions worked by a crank, and thus raises or lowers 
 this tubular nut and the cap-piece which it supports at top. 
 The axis-screw, the leg-racks, and the foot-screws give three 
 vertical movements in the trestle, by which its capacity for 
 slope-measurements is much amplified. 
 
 In the cap of the rear trestle, a lateral and a longitudinal 
 motion are provided for, by means of two tablets arranged to 
 slide, the upper one longitudinally on the lower one, and the 
 lower laterally on the head-plate of the axis-screw tube. Long 
 adjusting screw-handles extend to the observer's stand from 
 these two plates and from the axis-screw, enabling him to raise 
 or lower, to slide forward or back, to the right or the left, the 
 rear end of the tube. The fore trestle is similar, except that 
 its head is only arranged for a lateral movement, and a second 
 observer makes its adjustments by a simple crank. 
 
 Four men can carry a tube, by levers passed through staples
 
 58 GEODETIC OPERATIONS. 
 
 in blocks strapped under the tubes. The principal observer 
 and an assistant make the contacts and rectifications, the first 
 assistant directs the forward tube, and another preserves the 
 alignment with a theodolite. A careful recorder notes down 
 the observations, and an intelligent aid places the trestles and 
 foot-plates. 
 
 This scale referred to, known as Borda's scale, was introduced 
 in Bessel's system, the only difference being that he used iron 
 and zinc in the place of copper and platinum, -and measured 
 the interval with a glass wedge. In this the iron is the longer, 
 and supports on its upper surface the zinc. 
 
 The zinc terminates at its free end in a horizontal knife-edge, 
 and the iron bar very near this has attached to itself a piece of 
 iron with a vertical knife-edge on each side in the direction of 
 the length of the bar. The distance between the end of the 
 zinc and this fixed point, changing with the varying tempera- 
 ture, is measured by means of a glass wedge, whose thickness 
 varies from 0.07 of an inch to 0.17 of an inch, with 120 divis- 
 
 FlG. 7 . 
 
 ions engraved on its face, the distance between its lines being 
 0.03 of an inch. The other vertical knife-edge, projecting 
 slightly beyond the end of the bar, is brought, in measuring, 
 very near the horizontal knife-edge in which the opposite end 
 of the bar terminates, and the intervening distance measured 
 with the same glass wedge. If the wedge in this case be care- 
 fully read and its thickness at each division accurately known, 
 this method eliminates some of the uncertainties in the method 
 of contact. A pair of Bessel bars, slightly modified, is now in 
 use in the Prussian Landes-triangulation.
 
 BASE-MEASUREMENTS. 59 
 
 The annexed cut shows the arrangement of the knife-edges 
 in the two ends of the Bessel bars. 
 
 The apparatus devised by Colby consists of a bar of brass 
 and one of iron, fastened at their centres, but free to move the 
 rest of their lengths. Each end of one of the bars is a fulcrum 
 of a transverse lever attached to the same end of the other bar, 
 the lever arms being proportional to the rates of expansion of 
 the bars. In this way the microscopic dots on the free ends of 
 the levers are theoretically at the same distance apart for all 
 temperatures. As the terminal points were the dots on the 
 lever arms, contact could not be made in measuring, so the in- 
 terval between two bars was determined by a pair of fixed micro- 
 scopes at a known distance apart. 
 
 In all forms of compensating-bars, the components having 
 different rates of heating and cooling, their cross-sections should 
 be inversely proportional to their specific heats, and should be 
 so varnished as to secure equal radiation and absorption of 
 heat. Struve's apparatus consists of- four bars of wrought iron 
 wrapped in many folds of cloth and raw cotton. 
 
 Contact is made by one end of a bar abutting against the 
 the lower arm of a lever attached to the other, while the upper 
 arm passes over a graduated arc on which a zero-point indi- 
 cates the position of the lever for normal lengths. The tem- 
 perature is ascertained from two thermometers whose bulbs lie 
 within the bar. 
 
 From these descriptions it can be seen that the Bache appa- 
 ratus was a combination of principles separately used before. 
 It had Borda's scale, Colby's compensation-arm, and Struve's 
 contact-lever; with this difference: the lever, instead of sweep- 
 ing over a graduated arc, acted upon a pivoted level. The 
 form used by Porro in Algiers consisted of a single pair of bars 
 attached at their common centre and free to expand in both 
 directions. Each end of one of the bars carried a zero-point, 
 while the corresponding end of the other had a graduated scale 
 like Borda's. In measuring, a micrometer microscope is placed
 
 60 GEODETIC OPERATIONS. 
 
 on a strong tripod, with an adjustable head immediately over 
 the initial point. The apparatus is then placed in position on 
 another pair of trestles, completely free from the microscope- 
 stands, and moved by slow-motion screws until it is in line 
 and the zero-point in the axis of the microscope. The scale is 
 then read by means of the micrometer ; at the same time another 
 similar microscope is being adjusted over the forward end and 
 read. The bar is then carried forward, placed in position so 
 that its rear end is under the second microscope, and the for- 
 ward end ready for a third microscope previously aligned. 
 
 And so the work progresses until a stop is to be made ; then 
 the bar is removed and a point established under the forward 
 end of the bar. Every precaution is taken to estimate flexure 
 and to avoid uncertainties of collimation and unstable micro- 
 scopes. In Ibafiez's apparatus the component bars are copper 
 and platinum, mounted upon a double T-iron truss. Flexure 
 is determined by resting a long level on the bars at several 
 points at equal distances apart. It differs from the preceding 
 in having the bars exposed. 
 
 The Baumann apparatus, recently constructed for the Prus- 
 sian Geodetic Institute, has platinum and iridium bars resting 
 on an iron truss, with its entire length open to the free circula- 
 tion of the air. Inclination is determined by a level of preci- 
 sion and flexure by a movable level. 
 
 There are six microscope stands, the same number of trestles 
 for the bars, and thirty sets of heavy iron foot-plates. 
 
 The latter are put in position, and remain half a day before 
 being used. For each microscope-stand there are two tele- 
 scopes one for aligning and one for reading the scales. Six 
 skilled observers and about thirty laborers are needed in meas- 
 uring. Only one base has been measured with this apparatus 
 up to the present time that of Berlin in 1884 but the results 
 are not yet known. 
 
 The Repsold differs from the Baumann apparatus only in a 
 few points, the chief being: the component bars are steel and
 
 BA SE-MEA S UREMENTS. 
 
 6l 
 
 zinc, and the two are suspended in a steel tube which is 
 wrapped in thick felt. The small probable errors deduced by 
 Ibafiez and the officers of the Lake Survey in the results ob- 
 tained with the metallic-thermometer principle appear to com- 
 mand its continuance in the construction of base-apparatuses. 
 But in the Yolo base authenticated temperature changes were 
 not always accompanied by corresponding indications of the 
 Borda scale. In short, the behavior of the zinc component was 
 so unsatisfactory that a new apparatus for the Coast and Geo- 
 detic Survey is under consideration, in which the scale-readings 
 will be omitted, and either a partly compensated pair of bars 
 or a single carefully protected bar of steel adopted instead, 
 with daily comparisons with a field standard. For additional 
 information on the various forms of base-apparatuses the au- 
 thorities cited at the end of this chapter may be consulted. 
 
 It is interesting to note the results of various measurements 
 under different auspices with the same or different forms of 
 apparatuses. The following list gives the most important: 
 
 Name of base. 
 
 Measured by. Apparatus. 
 
 .Length. I 
 
 'rob. error. 
 
 Dauphin Island.. 
 Bodies Island 
 Edisto Island .... 
 Key Biscayne 
 Cape Sable 
 Eppinsr Plains 
 Peach Ridge 
 Fire Ishnd 
 
 U.S. C.^and G. S 
 
 Bache-W 
 
 Hassler.... 
 Bessel 
 
 Colby ' '.". 
 Bache-W 
 
 Repsold .'.'.'.'.'.'.'.'. 
 Slide-contact . . . 
 
 6.66 mi 
 6-75 ' 
 6.66 
 3-6 
 4 
 5-4 
 
 d 
 
 5-5 
 2300 m 
 2480 
 8912.5 fe 
 
 f:I - 
 
 4.6 
 3-8 
 
 17486^51 me 
 2400.07 
 2440.29 
 3318.55 
 4061.34 
 859-44 
 2016.56910 
 
 es. 
 
 [ 
 
 res. 
 es. 
 
 res. 
 ;es. 
 
 410000 
 
 425500 
 418600 
 454400 
 409600 
 551600 
 561880 
 483980 
 22800 
 16949 
 
 22222 
 667000 
 
 833 r 
 530000 
 2089000 
 1148600 
 
 t$L 
 6000000 
 3500000 
 2090000 
 3700000 
 715000 
 1333065 
 903784 
 1577945 
 
 Kent Island 
 Beverloo 
 Ostend 
 Cape Comorin.... 
 Keweenaw 
 Minnesota 
 Chicago 
 
 Nerenberg 
 
 Eng. Trig. S ... 
 U.S. LakeS 
 
 Wingate 
 
 U. S. Geol. S 
 
 Yolo 
 
 U. S. C. and G. S 
 Ibafiez and Hirsch 
 Hirsch 
 Haffnerand Overgaard.. 
 Kalmer and Lehrl 
 Schwerd 
 Italian Government 
 
 Schott 
 Ibafiez 
 
 Swedish Acad'y.. 
 Austrian 
 Schwerd 
 Bessel 
 
 Aarberg 
 Weinfelden 
 Joederen 
 Ilidze 
 Speyer 
 Fosjsria 
 
 Naples 
 Axevalla 
 
 Stecksen ... 
 
 Wrede 
 
 340.224 
 I357-033
 
 GEODETIC OPERATIONS. 
 
 Perhaps a better idea can be obtained of the accuracy of 
 base-measurements when we give a comparison of the measured 
 length of a line with its length as computed from another base. 
 A few such comparisons are here given : 
 
 Epping measured 871 5-942 metres. 
 
 Computed from Massachusetts base 871 5.865 " 
 
 " Fire Island base 8715.900 " 
 
 Massachusetts base measured 17326.376 " 
 
 Computed from Epping base 17326.528 " 
 
 " Fire Island 17326.445 " 
 
 Combining the errors of preliminary measurements with the 
 computed error in the triangulation, the appended results are 
 obtained : 
 
 Probable error in junction-line. Due to base. To triangulat'n Both. 
 
 From Epping base 0.17 metre. 0.76 metre. 0.78 metre 
 
 " Massachusetts base 0.20 " 0.32 " 0.37 " 
 
 " Fire Island base 0.39 " 0.66 " 0.77 " 
 
 Considering the distance apart of these bases, it is safe to say 
 that if the errors are constant the maximum error in the length 
 of any line of the triangulation is not more than 0.22 of an 
 inch to the statute mile. The above are the results of measure- 
 ments by the Bache-Wurdeman apparatus, angles measured 
 with a thirty-inch repeating-theodolite, and the triangulation 
 computed by Mr. Schott. Simply with the purpose of com- 
 paring the results obtained by different apparatus, I make an 
 extract from the report of the U. S. Lake Survey : 
 
 Chicago base measured log. in feet 4.39 17929 
 
 " computed from Fond du Lac " " 4.3918010 
 
 Difference = 0.14 metre. 
 Distance from Chicago to Fond du Lac, 150 miles.
 
 BASE-MEASUREMENTS. 63 
 
 Olney base measured log. in feet 4.3349231 
 
 " " computed from Chicago " " 4.3349231 
 
 Difference = 0.06 metre. 
 Distance from Chicago to Olney, 200 miles. 
 
 The Madridejos base, measured by General Ibaflez with his 
 improved Porro apparatus, was divided into five segments; the 
 central one was about 1.75 miles long. This one was meas- 
 ured twice, and used as a base in computing the length of each 
 of the other segments. The relation between the measured 
 and computed values may be seen in the following table : 
 
 Segment. 
 
 Measured (metres). 
 
 Computed (metres). 
 
 Difference (metres). 
 
 I 
 2 
 3 
 
 4 
 
 5 
 
 Total 
 
 3077-459 
 2216.397 
 2766.604 
 2723.425 
 3879.000 
 
 3077.462 
 2216.399 
 2766.604 
 2723.422 
 3879.002 
 
 0.003 
 
 O.O02 
 
 + 0.003 
 
 O.OO2 
 
 14662.885 
 
 14662.889 
 
 O.OO4 
 
 The Wingate base, measured with a slide-contact apparatus, 
 was divided into three segments; the middle one was measured 
 twice to see if a discrepancy sufficiently great to warrant a re- 
 measurement existed. The two results were in sufficient ac- 
 cord to admit of the acceptance of the entire measurement as 
 correct. However, each segment was used as a base for the 
 computation of the other segments. 
 
 The length of the line was : 
 
 With measured first and computed 2d and 3d. . . 6724.5309 m. 
 
 " second " 1st and 3d. .. 6723.7132 " 
 
 " third " ist and 2d. .. 6723.8248 " 
 
 Measured value of the whole line 6724.0844 "
 
 64 GEODETIC OPERATIONS. 
 
 Giving these values equal weight, the length may be written 
 6724.0383 0.12 metres. 
 
 Colonel Everest, with the Colby apparatus, measured in India 
 three bases, and joined them in the scheme of triangulation, 
 measuring the angles with a thirty-six-inch theodolite. 
 
 Dehra Dun. Damargida. 
 
 Measured length in feet 39 l8 3-87 41578-54 
 
 Computed " " 39183-27 41578.18 
 
 Only instructions of the most general kind can be given for 
 the mechanical part of measuring. The details vary with each 
 form of apparatus. The location of the base is a matter of 
 prime importance, and must be considered in connection with 
 the purpose for which the base is needed. If for verification, 
 it should be suitably situated for connection with the chain of 
 triangles it is intended to check. 
 
 If it is intended to serve as an initial base, a favorable con- 
 dition for immediate expansion should be sought. As the 
 base will usually be from three to seven miles long, the points 
 suitable for the first triangle-stations should be somewhat 
 farther than that apart, permitting a gradual increase in the 
 lengths of the sides. The best initial figure is undoubtedly a 
 quadrilateral of which the base is a diagonal, giving an expan- 
 sion from either side, or from the other diagonal. 
 
 If this be impracticable, the base must be a side of a com- 
 plete figure. Of course the termini must be intervisible, and 
 at the same time visible from every point of the line. If the 
 ground is irregular, having slopes exceeding three degrees in 
 inclination, it must be graded to within that limit, with a 
 width of about twelve feet. The method of alignment varies 
 with the views of the person in charge. 
 
 A good plan is to select a point approximately at the mid- 
 dle of the line. Place a theodolite there, and direct the tele-
 
 BASE-MEASUREMENTS. 65 
 
 scope to the temporary signal at one end and read the angle 
 to the other end ; if it differs from 180, move the instrument 
 in the proper direction until the angle is just 180. Assistants 
 are then sent towards each end, and, from signals from the per- 
 son at the instrument, secure points in line : these should be 
 placed about a quarter of a mile apart. Considerable experi- 
 ence has shown that the best form of aligning signal is a piece 
 of timber of suitable size, 2x4 inches or 4 inches square, driven 
 in the ground and sawed off a few inches above the surface. 
 In the top of this, bore a hole at the central point for the 
 insertion of an iron pin, twice as long as the hole is deep. 
 Take a corresponding piece of timber six or eight feet long and 
 make a similar hole in its end. It can then be adjusted to the 
 stake in the ground, and made stable by two braces, after be- 
 ing made perpendicular by means of a plumb line or a small 
 theodolite. The advantage of this form of signal is that it can 
 be removed when the measuring reaches this point, and be re- 
 placed for a future measurement without going to the trouble 
 of making a second alignment. A plan of aligning differing 
 from this is to have the instrument carefully adjusted and 
 placed three or four hundred yards from the end. Direct the 
 telescope to the temporary signal at that point, turn it in its 
 Y's, or 180 in azimuth, and fix a point directly in line. Then 
 place the instrument over the point so selected and locate 
 another point in advance, and so on till the opposite end is 
 reached. This will only be possible when one terminus has 
 been decided upon and the general direction of the line. Each 
 terminus of the base is marked by a heavy pier of masonry of 
 secure foundation with upper surface eighteen inches or two 
 feet below the surface of the ground. In the centre of the large 
 stone forming a part of the top of the pier a hole is drilled ; in 
 this, with its upper face even with the top of the stone, is 
 placed, and secured by having poured around it molten lead, 
 a copper bolt or a piece of platinum wire. 
 5
 
 66 GEODETIC OPERATIONS. 
 
 On the upper end of this bolt or wire a needle-hole may be 
 drilled, or a pair of microscopic lines drawn, whose intersection 
 marks the end of the base. Immediately above this should 
 be placed a surface-mark to which the position of the theodo- 
 lite can be referred in the triangulation ; also a set of witnesses 
 consisting of four stones projecting above ground, so placed 
 that the diagonals intersect above the under-ground mark. 
 
 When both ends are marked in this way before measuring, 
 the distance from the end of the last bar to the terminal 
 mark, already fixed, is measured on a steel scale horizontally 
 placed. 
 
 The only advantage possessed by this method is, that both 
 monuments have an opportunity to settle before the distance 
 between them is determined. It is believed, however, that 
 greater inaccuracies will result from the uncertainty in this 
 scale and its use than from the irregular settling of the pier 
 placed after the measurement is finished. Before beginning the 
 accurate measurement it is advisable to make a preliminary 
 measurement with a steel tape or wire, marking every hun- 
 dred lengths of the apparatus to be used. This will serve as a 
 check upon the record as the final work advances ; and if the 
 line is to be divided into segments it will show where the in- 
 termediate monuments are to be erected. When these inter- 
 mediate stations are occupied the angle between the ends and 
 the other points should be measured with great care, so that, 
 if the line be found to be a broken one, the exact distance be- 
 tween the termini in a straight line can be computed. If the 
 required distance cannot be obtained without crossing a ra- 
 vine or marsh, the feasible parts can be measured, and the other 
 portion computed by triangulation. 
 
 The form of record will of course vary with the kind of ap- 
 paratus used, but too much care cannot be taken in keeping 
 the record. The principal data needed in the reduction may 
 be stated as follow :
 
 BASE-MEASUREMENTS. 67 
 
 1. The time showing the time at which each bar was placed 
 in position in order to form some idea of the average speed at- 
 tained in the work. 
 
 2. The. whole number of the bar. When a preliminary 
 measurement has been made as suggested, the hundredth bar 
 should end near the stake previously driven ; if not, a remeas- 
 urement must be made from the last authentic point. This 
 should be at the end of the even-hundred bar, and perhaps 
 more frequently, especially if the day should be windy, en- 
 dangering the stability of the bars, or if the ground should be 
 boggy or springy. The simple method for placing this point 
 is to set a transit or theodolite at right angles to the line and 
 at a distance of twenty-five or thirty feet from it. After level- 
 ling, fix the cross-wires of the instrument upon the end of the 
 bar; then, pointing the telescope to the ground, direct the 
 driving of a stake in a line with this and with the aligning 
 telescope. The height of the telescope should be half the 
 height of the bar, so that the focus need not be changed. 
 
 Then in the top of this stake a copper tack is driven, and on 
 its upper face are drawn two lines coinciding with the vertical 
 threads of the two instruments. If they are in good adjust- 
 ment the intersection of these lines will mark the end of the 
 bar. A record must always be made when a stub is thus 
 placed. It is also advisable to place a stub under the instru- 
 ment used for this horizontal cut-off, so that if it should be 
 necessary to begin work at this point the instrument would 
 occupy the same position that it occupied before, eliminating 
 by this means the error that would arise from not having the 
 transit at right angles to the line. 
 
 Probably a more accurate method is to have a metal frame 
 one inch wide and two inches long with screw holes admitting 
 of attachment to a stake. This frame has sliding inside of it 
 another that can be moved by a milled-head screw, with a set 
 screw to hold it in place. On the upper surface of this frame
 
 68 GEODETIC OPERATIONS. 
 
 is a small dot or hole. When the approximate position of the 
 end is determined by a plummet, a stake is driven in the ground 
 until only an inch or so remains above the surface : to this is 
 attached the outer frame ; then, with the theodolite previously 
 set upon the end of the measuring-bar, direct the movement 
 of the inner frame until the hole or dot is bisected by the 
 cross-wires, when the frame is clamped in place and verified. 
 When microscopes are used, the dot can be brought under the 
 micrometer-wire that marked the position of the zero-point on 
 the bar. 
 
 3. The designation of the bar as A, B, or I, 2, etc., so that 
 it may be known how many times each bar was used. Since 
 the two are never of the same length, the distance obtained 
 by each bar must be separately computed and the two values 
 added to get the entire length of the line. 
 
 4. Inclination. When going up-hill the inclination is recorded 
 plus, and minus when going down. However, as the correc- 
 tion for inclination is always subtracted, the sign is of small 
 consequence. 
 
 5. Columns for the sector-error and the corrected values for 
 the inclination. Before beginning work each day the rods 
 should be placed on their tripods and be made perfectly hori- 
 zontal by raising one of them. To determine this, set up a 
 carefully adjusted theodolite at such a distance that both ends 
 of the bar can be seen. Set the thread on one end of the bar, 
 revolve the instrument in azimuth, and see if the thread be 
 on the other end : when such is the case, bring the bubble of 
 the sector in the middle of the tube and see what the scale- 
 reading is ; if zero, then there is no error. This test should be 
 applied at the beginning and close of each day's work, and the 
 average error added to or subtracted from the reading of in- 
 clination for that day. With secondary apparatus this is un- 
 necessary, as the positive and negative readings will be about 
 equal, so that the number of readings that are recorded too
 
 BASE-MEASUREMENTS. 69 
 
 great will be corrected by those that are too small by the same 
 quantity. 
 
 6. Temperature. The thermometers should be read about 
 every ten bars, and in the Borda rods the scales more fre- 
 quently. When the temperature gets above 90 Fahr., it is 
 advisable to stop work, especially if the bars are not compen- 
 sated, as the adopted coefficients of expansion at that tem- 
 perature are unreliable. 
 
 The Repsold apparatus, as used on the Lake Survey, and 
 the Davidson, with which the Yolo base was measured, were 
 protected during measuring by a canopy made of sail-cloth 
 mounted on Wheels, so as to move along as the work advanced. 
 
 In all kinds of apparatus it is advisable to measure when the 
 bars indicate a rising temperature, and also during the time 
 required for them to fall through the same amount. 
 
 7. A column for corrections for inclination, computed from 
 a formula to be given. 
 
 8. A column for remarks, explaining delays,. stoppages, the 
 placing of stubs, etc. 
 
 GENERAL PRECAUTIONS TO BE TAKEN WHILE MEASURING. 
 
 The rear end of the bar must be directly over the marking 
 on the initial monument. 
 
 The inclination must never be so great as to endanger a 
 slipping of the bars forward or backward. 
 
 The trestles should be so firmly set that there can be no un- 
 equal settling after the bar has been placed on them. 
 
 A bar should not be allowed to remain more than a minute 
 in the trestles, lest its weight should change their position. 
 
 When a stoppage is made to allow the aligning-instrument 
 to advance, a transit should be set up, as already described, 
 and its cross-wires firmly clamped on the end of the bar; then, 
 before resuming work, the position of the bar can be restored 
 if from any cause it has changed. 
 
 When the end has been transferred to a temporary mark, as
 
 JQ GEODETIC OPERATIONS. 
 
 when a stop is made for night or dinner, in resuming work it 
 is best to place the bar that the work closed with in the same 
 position it had before stopping ; then the new day's work goes 
 on as though there had been no break. If this plan is not 
 adopted, either in the transferrence to the ground or from it, 
 the end sighted will be more than the standard length from 
 the other end, being held out by the spiral spring that keeps 
 the agate beyond its proper distance, rendering it necessary to 
 record an index-error for every transferrence ; whereas in the 
 plan suggested there can be no danger of omitting to record 
 this index-error, nor of recording an erroneous value. 
 
 This precaution refers to that species of apparatus which 
 consists of a pair of bars, one abutting against the other, and 
 not where only one bar is used, as in the Repsold, Baumann, 
 and others. 
 
 The alignment must be made with precision, for all errors of 
 this kind are of the same character and do not cancel one 
 another. 
 
 Before beginning actual work the party should measure a 
 short distance several times, by way of practice, until the dis- 
 agreement between two measures is made very small. 
 
 COMPUTATION OF RESULTS. 
 
 In order to know the horizontal distance between the two 
 ends of the base it is necessary to know the number of times 
 the measuring-unit was used, and its exact length each time 
 that it was employed. To this must be added index-errors, 
 and the amount by which the last bar fell short of the ter- 
 minus. Also, there are to be subtracted the quantities that 
 were needed to reduce each length to its horizontal projection, 
 and those negative errors that could not be obviated. 
 
 A carefully kept record will show how often the bars were 
 used ; but to ascertain their length is a more difficult problem, 
 depending upon : (a), a knowledge of the exact length of the 
 adopted standard ; (&), a known relation between the measuring-
 
 BASE-MEASUREMENTS. /I 
 
 bar and the standard at a certain temperature ; (c\ a knowledge 
 of the temperature of the bars each time used, and the coeffi- 
 cients of expansion. 
 
 The Committee Metre is the standard of linear measures 
 now in use, and with a certified copy of this, all our units are 
 compared. This comparison can be described only in outline. 
 We have two firmly built pillars at a convenient distance apart 
 for the bars that are to be compared. On one is an abutting- 
 surface, and on the other is a comparator. In general, this 
 comparator consists of a pin held out by a spiral spring but 
 capable of being withdrawn by a micrometer-screw. This pin 
 works a lever on whose longer arm is a point that is to be 
 brought into coincidence with a fixed zero-mark. Between 
 these two pillars is a carriage rigidly constructed but com- 
 pletely isolated from them. On this carriage are placed the 
 standard and the bar that is to be compared. The former is 
 placed between the abutting-surface and the micrometer-pin, 
 the screw is turned until the zero-marks coincide, and the turns 
 and division recorded. 
 
 The carriage is then moved along until the bar is brought 
 into place and the micrometer is again read. The difference 
 in the readings will correspond to the difference in lengths in 
 terms of micrometer turns and divisions the value of a turn 
 and a division being found by measuring with the screw the 
 length of a standard centimetre. In very accurate comparisons 
 the bars are immersed in glycerine, which can be readily kept at 
 the same temperature for a long time. 
 
 The temperature is ascertained from three thermometers 
 one at each end, and one at the middle. Also, to eliminate 
 accidental errors, a number of readings are made with the bars 
 reversed, turned over, taken in different order, and at different 
 temperatures. The average difference will be the difference 
 in the lengths of the standard and the bar at the average tem- 
 perature, supposing that the coefficients of expansion remain 
 constant. Then knowing the temperature at which the stand-
 
 72 
 
 GEODETIC OPERATIONS. 
 
 ard is correct and its coefficient of expansion, its true length 
 can readily be computed for this average temperature. To 
 this, add the average difference just referred to and we have the 
 exact length of our bar at this mean temperature. To illus- 
 trate: let M be the standard, A the bar under comparison, p 
 the difference in microns, which is obtained by multiplying the 
 turns and divisions of the micrometer by the previously ascer- 
 tained value of one turn. 
 
 Temp. 
 
 A - M. 
 
 57.58 
 52.60 
 55-29 
 55-16 = /. 
 
 + ?'- 5 
 
 + 7-5 
 + 9-8 
 + 8.27 
 
 Therefore, A M-\- 8.27/1 at 55. 16. Suppose e be the co- 
 efficient of expansion for M, and T the temperature at which 
 Mis correct, then we have A=M +Me($$.\6 T) + 8.27/1. 
 
 To determine e we must have the pillars of the comparator 
 at a fixed distance apart, and then measure this distance with 
 a bar at different temperatures. In order to insure the bar 
 being at the same temperature, it is best to place it in glycer- 
 ine previously heated, and leave it there for half an hour. Let 
 D be the difference between the constant distance and the dis- 
 tance as observed at various temperatures, / the average, and 
 t the observed temperatures. 
 
 t. 
 
 D. 
 
 ,-, ,-A, 
 
 > F. 
 99.08 
 
 441-5 
 
 + 28.13^ = 
 
 178.9 
 
 83.68 
 
 342.9 
 
 12.73^ = 
 
 80.3 
 
 72.08 
 
 268.2 
 
 1.07^ = 
 
 5-6 
 
 57-58 
 
 175-9 
 
 - 13-37'= - 
 
 86.7 
 
 42-39 
 
 084.5 
 
 28.56^ = 
 
 178.1 
 
 70.95 = /, 
 
 262.6 = Z? 
 

 
 BA SE-MEA SUREMENTS. 
 
 73 
 
 Forming the normal equations by multiplying each equation 
 by the coefficient of e in that equation, and taking the sum of 
 the resulting equations, we get 1948.92^ 12,306.4^, or 
 e = 6.315 /*. Substituting this value of e, we have for A, 
 A := J/ + 6.315/^(55. 16 T) + 8.2;//. There is a probable 
 error in this determination which can be carried through the 
 future computations. 
 
 The way in which the temperature-observations are utilized 
 depends upon the accuracy desired ; ordinarily the average 
 temperature of each bar in a segment is employed. So that if 
 we have n lengths of a four-metre bar with the above coeffi- 
 cient of expansion, a length equal to A at 55.i6, and the aver- 
 age temperature t in that segment, we shall have the distance 
 = n[A + 4X0.000 006315 (/ 55.i6)]. When greater ac- 
 curacy is required, the length of each bar can be computed in 
 the same manner, and the aggregate length obtained by sum- 
 mation. 
 
 When a Borda scale or metallic thermometer is used, it is 
 necessary to know how much in thermometric scale a division 
 is equal to. The scale is usually divided into millimetres, and 
 read by a vernier or microscope to o.oi mm. 
 
 Temp. 
 
 *-/ 
 
 Scale = S. 
 
 JS. 
 
 o p 
 
 109.41 
 94.11 
 
 + 31-79 
 + 16.49 
 
 8.60 
 
 8.17 
 
 + 0.92 
 + 0.49 
 
 79-21 
 
 + 1-59 
 
 7-74 
 
 -f 0.06 
 
 61.16 
 
 - 16.46 
 
 7.16 
 
 0.52 
 
 44.22 
 
 - 33-40 
 
 6.72 
 
 0.96 
 
 77-62 = / 
 
 
 7.68 = S, 
 
 
 By letting;? be the quantity representing the differential ex- 
 pansion of the component bars, and as it varies with the tem- 
 perature, we may take the values of t / as the coefficients 
 of x and solve by least squares. The normal equation will 
 give 2671.54*- = 78.05^, or x 0.02922^ = 29.22^.
 
 74 GEODETIC OPERATIONS. 
 
 That is, a change of one degree Fahr. is represented by 
 0.029 division, or the smallest value that can be estimated 
 on the vernier, o.oid = iF.; consequently the scale-readings 
 can be readily converted into degrees of temperature and the 
 reduction for length made as in the preceding case, or the 
 change in length may be found directly in terms of scale-read- 
 ings. If we have a four-metre bar with the coefficient of ex- 
 
 pansion just found, o.o\d~ ^ ^ J* = 8.64^. 
 
 Then if S be the scale-reading at w r hich M is a standard, 
 and S any other reading during the measurement or the aver- 
 age, A = M-\- 8.64/^(5 ), and the entire line 
 
 = n[M+ 8.64X5 -5.)]. 
 
 Correction for inclination : if R represent the length of a 
 bar, h its horizontal projection, and the angle of inclination, 
 it is apparent that h = R. cos 0, then d the correction R h 
 = R - R . cos = R(i cos 0) = 2R. sin 2 $6. As 6 is small 
 sin 2 %0 = -J sin 2 6 (nearly), so we may write 
 
 Rsin'O sin" i' sin 2 i' 
 
 d = - - - = -R6*', log = 2.626422. 
 
 Having determined by comparison the average length of the 
 bars, a table should be computed for each, giving the values 
 for d for each fractional part to which the sector can be read, 
 and within the limits observed. Then from this table correc- 
 tions for inclination can be taken and inserted in the record- 
 book. If there are any index-errors, as stated might occur in 
 the transferrence of the end to the ground, they must be added 
 to the computed length. 
 
 Probable error. This may be derived 
 
 I. By measuring the base a number of times, then deducing
 
 BASE-MEASUREMENTS. 75 
 
 the probable error in accordance with the principle of least 
 squares. 
 
 2. By dividing the line into segments and computing the 
 other segments from each one as a base by triangulation. 
 
 3. By checking one base from another in the chain of trian- 
 gulation, and determining the probable error in the second 
 from that of the first and of the measurement of the angles in 
 the triangulation. 
 
 4. From all known sources of error in measurement. 
 
 The fourth method is the only one that needs expansion at 
 this point. The principal sources of error in measurement 
 are: 
 
 1. In determining the length of the bar. 
 
 2. Backward pressure. 
 
 3. Error of alignment. 
 
 4. In transferring end to the ground. 
 
 5. In the determination of inclination. 
 
 6. Personal errors of the observers. 
 
 These are determined as follows : the first is obtained from 
 repeated comparisons with the standard, and is made up of 
 two parts uncertainty in the expansion of the bars, and acci- 
 dental errors in comparing. Of these the former is found from 
 the residuals in the series of determinations of the coefficients 
 of expansion. Calling this rf, we have for the entire n bars 
 n.r,'. Likewise the error from comparison is found in a simi- 
 lar manner from the series of comparisons, if we designate this 
 r t ', the entire error r t = n.rj. 
 
 The error of contact depends upon the force with which the 
 agate is held out beyond its proper position. When a bar is 
 in its right place, and the next bar brought into contact with 
 it, the pressure necessary to bring it to its place forces the rear 
 bar backward ; and when the rear bar is taken away the for- 
 ward bar, being relieved of this pressure, moves back by the 
 same amount. Consequently the total backward movement is
 
 76 GEODETIC OPERATIONS. 
 
 double the effect of pressure. This must be determined by 
 experiment in various positions of the bar. As every bar ex- 
 cept the first and last are doubly affected, these each bein 
 changed only once by this pressure, the total correction wL. 
 be twice the displacement multiplied by one less than the 
 number of bars. Usually this is too small to be considered, 
 and applies to those bars only that are used in pairs one bear- 
 ing in contact against the other. 
 
 By (3) is not meant the uncertainty of having the line as a 
 whole straight, but in placing the bar exactly in that line. The 
 aligning instrument is placed in front at distances varying from 
 50 to 900 feet, and the alignment is effected by bringing the 
 agate of the bars into coincidence with the vertical thread of 
 the telescope ; or when the bars are provided with a vertical 
 rod immediately over their centres, this is sighted to. It is 
 apparent that the bisection of this may not be perfect ; and, in 
 fact, when the light falls unequally upon the object sighted to, 
 the illuminated spot is bisected, which may be altogether to 
 one side of the centre. 
 
 However, the error of bisection cannot be greater than the 
 radius of the agate or aligning-rod, and its effect upon the true 
 length of the line will depend upon the distance to the transit. 
 The nearer the transit, the less is the likelihood of making an 
 erroneous bisection. By placing a scale directly under the 
 agate, and having the person at the transit direct the mov- 
 ing of the bar until he considers it in line, make a note of the 
 scale-reading, and after a number of trials the average variations 
 may be taken as the error most likely to be committed at 
 that distance. Suppose it was found that the errors were a 
 for the maximum, and b for the minimum distances, the an- 
 gular variations might be written : a times one second divided 
 by the length of the bar, call this m, and similarly for b, which 
 we will call n. The correction for this deviation will be the 
 difference between the length of the bar and the vertical pro-
 
 BASE-MEASUREMENTS, 77 
 
 jection for this angular deviation. As already shown, this is 
 
 R . sin 4 /// R . sin 2 n 
 equal to , and . 
 
 Only the first and last few bars of each segment will need 
 to have this total lateral correction applied ; for the remaining 
 bars it will be sufficient to take the average of m and n, in t^e 
 formulae just given. As the total correction from this cause 
 will never amount to a tenth of an inch, it is usually omitted, 
 and its probable error is never considered. 
 
 The error from the fourth source is determined from experi- 
 ment, as in the preceding case. Suppose it is 0.082 mm.; 
 as there is a double transfer, the entire error will be 0.082 
 \/~2 mm. = o.i i mm., and the total for n bars will be o.ii 
 Vn. mm. = r 3 . 
 
 The fifth source of error is quite apparent. The sector that 
 shows the inclination usually reads to single minutes, some- 
 times to ten seconds. As it is impracticable to obtain more 
 than one reading for each inclination, there is an uncertainty 
 as to its correctness. This will vary with the skill of the ob- 
 server and the character of the sector used. The probable 
 error of a single determination should be ascertained as fol- 
 lows : place the bar firmly in its trestles and make several 
 readings of the scale when the bubble of the level is in the 
 same position. From a number of such scale-readings the 
 probable error is deduced in the usual manner. 
 
 To determine the effect of this error on the computed cor- 
 rections for horizontal projections, the average observed in- 
 clination must be approximated. Suppose this to be 2, the 
 probable error of inclination 30", and the length of the bar R. 
 It has already been shown that the correction for inclination d 
 = R(i cos 9}. As 6 in this case is taken as 2, an approxi- 
 mate value for the change in d by a mistake of 30" in Q 
 can be computed by getting d' when 6 = 6 30"; d' = 
 R\\ cos(ft-|- 30")], and the probable error in any one deter-
 
 78 GEODETIC OPERATIONS. 
 
 mination wijl be the difference between d and d' or r t ', r t = 
 e Vn where n = the number of bars and e = d d'. 
 
 To recapitulate : those errors that are known to exist and 
 the direction of whose effect is unmistakably determined can be 
 applied in the reduction of the length of the base, while those 
 that are merely probable must be used simply in obtaining the 
 probable error of the measurement as a whole. The value for 
 the length of the base must be diminished by the amount of 
 backward pressure, errors of alignment, and errors of inclina- 
 tion ; but the remaining errors having a double sign must be 
 regarded as probable ; if individually they be represented by 
 *V ?v r r n, and the total error by R, we will have 
 
 R = 
 
 As the sides of the triangulation are at different elevations 
 and the base and check-base not on the same plane, it is neces- 
 sary to know their lengths at some common-datum plane. 
 This by common consent is the half-tide level of the ocean. 
 
 bg, height above half-tide = h ; 
 
 ae, the half-correction for reduction = - ; 
 
 ae : ed : : ab : be ; 
 
 ae = 
 
 ed.ab 
 
 be 
 
 2ab 
 
 B 
 
 2ae = c = h . 
 
 bgis so small in comparison with eg 
 that it may be omitted, and we write ;
 
 BASE-MEASUREMENTS. 79 
 
 h.B Jh V\ 
 
 = B -7^ , where R = radius of 
 
 radius of curvature \/? 
 
 curvature at the mean latitude of the base. 
 
 From the corrected value for the length of the base c is to 
 be subtracted. If the elevation of the base was found by dif- 
 ferent methods, or from different bench-marks, an uncertainty 
 may arise in the value of h, giving a probable error for c. 
 
 REFERENCES. 
 
 U. S. Coast and Geodetic Survey Reports as follows: 1854, 
 pp. 103-108; '57, pp. 302-305 ; '62, pp. 248-255 ; '64, pp. 120- 
 144; '73, pp. 123-136; '80, pp. 341-344 ; '81, pp. 357-358; '82, 
 pp. 139-149; also pp. 107-138 ; '83, pp. 273-288. 
 
 Clarke, Geodesy, pp. 146-173. 
 
 Report of U. S. Lake Survey, pp. 48-306. 
 
 Zachariae, Die Geodatische Hauptpunkte, pp. 79-110. 
 
 Jordan, Handbuch der Vermessungskunde, vol. ii. pp. 
 
 73-113- 
 
 Experiences Faites avec 1'Appareil a Mesurer les Bases. 
 
 Compte Rendu des Operations de la Commission pour e"ta- 
 lonner les Regies employes a la Mesure des Bases Geodsiques 
 Beiges. 
 
 Westphal, Basisapparate und Basismessungen. 
 
 Gradmessung in Ostpreussen, pp. 1-58.
 
 80 GEODETIC OPERATIONS. 
 
 CHAPTER IV. 
 
 FIELD-WORK OF THE TRIANGULATION. 
 
 SUPPOSING that a base has been carefully measured, or the 
 distance between two stations previously occupied accurately 
 known, the next thing to be done is to lay out a scheme of tri- 
 angles covering the desired territory. Their arrangement into 
 figures depends upon : 
 
 1. The special purpose of the work. 
 
 2. The character of the country over which the system is 
 to be extended. 
 
 If the object is to measure arcs of a meridian or of a par- 
 allel, for the purpose of determining the figure of the earth, 
 great care should be exercised in selecting triangles that are 
 approximately equilateral ; for if in the computation a very 
 long side is to be computed from a short one, an error in the 
 latter will be greatly magnified in the former. If the purpose 
 is simply to meet the wants of the topographer, the stations 
 should be selected with special reference to his needs and 
 without regard to the character of the figures thus formed. 
 In an open prairie where signals have to be erected without 
 any assistance from natural eminences, their arrangement may 
 be made in strict accord with theoretical preference. 
 
 The plainest system of the composition of triangles into 
 figures is a single string of equilateral triangles which possess 
 the advantages of speed and economy of time and labor. Hex- 
 agonal figures are preferred by some, but the general prefer- 
 ence is for quadrilaterals with both pairs of diagonal points in- 
 tervisible. This system covers great area and insures the 
 greatest accuracy.
 
 FIELD-WORK OF THE TRIANGULATION. 8 1 
 
 Equilateral triangles will furnish nine conditions. 
 
 Hexagons, with one side in common, twenty-one conditions. 
 
 Quadrilaterals, twenty-eight conditions, covering the same 
 area (approximately). 
 
 Signals. After deciding upon the positions of the stations, 
 the next subject for consideration is the kind of signals to be 
 used. In short sights, the best form is either a pole just large 
 enough to be seen, or a heliotrope fixed on a stand or a tripod 
 carefully adjusted to the centre of the station. As the helio- 
 tropers are usually persons with but little experience, range- 
 poles should be previously set, enabling them to point their 
 instruments with some degree of precision. 
 
 (For a description of the heliotrope, its adjustments, and use, 
 see page 45.) 
 
 Owing to the fact that there are so many days during which 
 it is impossible to use the heliotrope, and also the additional 
 trouble that frequently when the sun is shining the air is so 
 disturbed that the object sighted is too unsteady to bisect with 
 any certainty, the effort is constantly being made to devise 
 some form of night signal to take the place of day signals. 
 
 The great obstacle to the successful solution of this problem 
 has been the dimness or expense of the lights that have been 
 tried, such as oil-lamps, magnesium, or electric lights. In June, 
 1879, Superintendent Patterson of the U. S. Coast and Geo- 
 detic Survey directed Assistant Boutelle to make an exhaustive 
 series of observations with the various methods of night signals, 
 with a view to determine the most effective method to be used 
 in triangulation. 
 
 The special points to be considered were : 
 
 1. Simplicity and cheapness. 
 
 2. Adaptability to the intelligence of the men usually em- 
 ployed as heliotropers. 
 
 3. Ease of transportation to heights. 
 
 6
 
 82 GEODETIC OPERATIONS. 
 
 4. Penetration, with least diffraction and most precision of 
 definition. 
 
 5. The best hours for observation. 
 
 6. Lateral and vertical refraction, and the extent to which 
 the rays are affected by the character of the country over 
 which they pass. 
 
 An accurate account of the various experiments made by 
 Captain Boutelle are given in Appendix 8 of the C. and G. S. 
 Report for 1880. I shall take the liberty of quoting his con- 
 clusions ; they are : 
 
 " The experience of the past season enables me to state with 
 some precision the cost of the magnesium light, so much supe- 
 rior to every other yet tried. 
 
 " The success in two instances of burning the light by a time- 
 table established that method as perfectly practicable. 
 
 " It reduces the time of burning it to twenty minutes per 
 hour, or to eighty minutes for four hours' observation. With 
 a delivery of ribbon of fifteen inches per minute, the cost will 
 be two dollars per night for each light used. The average 
 number of primary stations observed upon at any one station 
 is six, of which three would require the magnesium light, 
 making the expense six dollars per night. The nights when 
 observation would be practicable and the lights burned may be 
 taken as averaging three in a week, or seven at each station. 
 
 "Apart from the first cost of apparatus, we should therefore 
 have as the additional outlay for night observation for a pri- 
 mary triangulation : 
 
 " I. Additional pay of six heliotropers $3-OO 
 
 " 2. cost of burning three magnesium lights 
 
 every other night 3.00 
 
 "3. " cost of kerosene-oil for three lamps... 0.20 
 "4. " cost per day for supplies, etc 0.80 
 
 " Whole additional daily cost t $7.00
 
 FIELD-WORK OF THE TRIANGULATION. 8$ 
 
 " To offset this additional party expense there will be : 
 
 " i. The shortening of time required in occupation of each 
 station by the addition of four hours of observing each clear 
 day after sunset. The average time of observation each day 
 being two hours, this time will be tripled on each clear day 
 and night. 
 
 " 2. Necessity for encamping at many stations may be avoid- 
 ed, where now the probabilities of a long detention and the 
 lack of any decent quarters within a reasonable distance require 
 the transportation and use of equipage. 
 
 "The conclusions to which the experiments and results have 
 led me may be generally summed up as follows : 
 
 " i. That night observations are a little more accurate than 
 those by day, but the difference is slight so far. 
 
 " 2. That the cost of apparatus is less than that of good 
 heliotropes. 
 
 " 3. That the apparatus can be manipulated by the same 
 class of men as those whom we employ as heliotropers. 
 
 " 4. That the average time of observing in clear weather 
 may be more than doubled by observing at night, and thus 
 the time of occupation of a station proportionally shortened. 
 Hazy weather, when heliotropes cannot show, may be utilized 
 at night. 
 
 " 5. That reflector-lamps, or optical collimators, burning coal- 
 oil, may be used to advantage on lines of 43.5 miles and under. 
 But for longer lines the magnesium lights will be best and 
 cheapest, as being the most certain. 
 
 " 6. That for the present we should keep up both classes of 
 observation, both by day and night ; and that the observers in 
 charge of the various triangulations should be informed of 
 the progress already made, and encouraged to improve on the 
 methods and materials thus far employed in night observa- 
 tions." 
 
 At this time many of the parties in charge of triangulation-
 
 84 GEODETIC OPERATIONS. 
 
 work, under the auspices of the Coast Survey, make night ob- 
 servations. The wisdom of this plan is duly appreciated by 
 all who 'have observed in the Eastern or Middle States. 
 
 It might be safely said that more time is spent in waiting for 
 suitable weather than in reading the angles, and any means for 
 diminishing this waste will be gladly adopted, especially by 
 those who have had their patience taxed by having to wait day 
 after day for the haze to pass by. 
 
 For short sights or for secondary triangulation a reflecting- 
 surface, such as a tin cone, will be sufficient. Still better is a 
 contrivance made of tin, in the shape of the children's toy, that 
 is made to revolve by a current of air, and fixed on an axis in 
 the top of a pole or tree. If it is of the proper shape, in turn- 
 ing it will catch the sun's rays at the right angle to send a re- 
 flection to the desired point, except when the sun is on the 
 opposite side from the observer. In lines still shorter a simple 
 pole, supported by a tripod, or a straight tree will answer. 
 Care must be taken, however, to have the pole or tree no larger 
 than is necessary to render it visible, as large bodies are diffi- 
 cult to bisect. A diameter of 6 inches will subtend an angle 
 of one second at a distance of 20 miles ; for 40 miles, 12.3 inches ; 
 and at 60 miles, 18.5 inches. Sights have been made upon 
 a tree 12 inches in diameter at a distance of 55 miles. 
 
 Much time can be gained and accuracy secured by making 
 the observations at the most favorable time. For instance, if 
 a pole is to be sighted, the proper time is in the morning when 
 looking towards the east, and in the evening when looking 
 westward. If a reflecting object is used, the opposite rule to 
 the above must be followed. 
 
 It is frequently necessary to elevate the instrument and ob- 
 server in order to obtain a longer length of line, or to overcome 
 some impediment. Fig. 9 will give an idea of the form that 
 has been found most convenient. When it is to be constructed 
 on a hill or mountain, it will be found advisable to cut the
 
 FIELD-WORK OF THE TRIANGULATION. 8$ 
 
 FIG. 9.
 
 86 GEODETIC OPERATIONS. 
 
 timbers at the bottom, in order to save the transportation of 
 useless materials. 
 
 In order to secure the requisite stability, and to prevent 
 shaking of the instrument by the observers moving around, it 
 is necessary to have a double structure one for the theodolite, 
 and one to support the platform for the party observing. For 
 a low structure the form used by the Prussian Geodetic Insti- 
 tute will be found sufficiently firm. It is a vertical piece of tim- 
 ber to support the instrument, braced by a tripod, the whole sur- 
 rounded by a quadrangular platform. But when a height of 
 more than twenty feet is needed, the kind devised by Mr. 
 Cutts, and improved by Captain Boutelle, will be found more 
 satisfactory. 
 
 I have worked on several of this pattern, and can vouch for 
 their rigidity ; and when an awning is attached to the legs of 
 the scaffold to shade the tripod, the unfortunate results of 
 " twist " from the action of the sun's rays are avoided. 
 
 From a glance at Fig. 9 it will be seen that the signal con- 
 sists of two parts a tripod and a square scaffold. It is the 
 average experience that a safe signal, strong enough to with- 
 stand the heaviest winds we have, should be built of timbers 
 6 by 8 inches, with diagonal braces 2 by 2 and 3 by 3. The 
 size of the base is a function of the altitude, a good ratio 
 being one foot radius for every eight feet of elevation. The 
 legs of the tripod should be set three feet in the ground, and 
 would, if continued, meet at a point four feet above the plat- 
 form. So that for a signal whose scaffold is to be eighty feet 
 above the station-surface we would have eighty-seven feet for 
 the vertical height of the tripod, and the radius of the base 
 would be ^--(-0.67 ft. = 1 1. 54 ft. 
 
 To lay out the base, drive a stub in the ground at the cen- 
 tral point, and with a radius equal to that computed describe 
 a circle ; mark off on this circumference points with a chord 
 equal to the radius, and the alternate points will be the places 
 for the feet of the tripod. With a level, or an instrument that
 
 FIELD-WORK OF THE TRIANGULATION. 8? 
 
 can be used as a level, the bottom of the holes for the tripod 
 can be placed on the same plane, by marking on a rod a dis- 
 tance that is equal to the height of the axis of the instrument 
 and three feet more, then the holes are to be dug until this 
 mark coincides with the cross-wires of the telescope when the 
 rod is in each hole. 
 
 The tripod, being the highest and the innermost structure, 
 should be raised first. The plans adopted for this differ with 
 different persons ; some frame two legs with their bracing, raise 
 them with a derrick, guy their tops, raise the third and brace 
 it to the other two. A platform is built on the top of this on 
 which the derrick is placed, another section is then lifted into 
 place as before, the derrick again moved up until the top is 
 reached. Then the blocks are attached to the top of the tripod, 
 which is well guyed, and the sides of the scaffold raised as a 
 whole or in sections. 
 
 If the station is wooded, one or two large trees may be left 
 standing and the blocks attached to their tops for raising the 
 timbers. Signals ninety-four feet high have had their sides as 
 a whole put in place, held there with guys until the opposite 
 pair was raised and the whole braced together. This can also 
 be done in the case of the tripod, by laying the single piece 
 down with its foot near its resting-place, and the pair lying in 
 the same direction framed together ; then with ropes rigged to 
 a tree left standing, or to a derrick, the pair is raised until it 
 stands at the right inclination, and held in place with ropes 
 until the single piece is brought into position. To keep the 
 feet from slipping, an inclined trench can be made towards the 
 hole, or they can be tied to trees or a stake firmly driven into 
 the ground. I have put up tripods in a way still different. 
 By framing one pair, and attaching between their tops the top 
 of the third by means of a strong bolt, the whole stretched out 
 on the ground in the shape of a letter " Y," with the feet of 
 the pair fastened near their final resting-place. The apex is 
 lifted and propped as high as possible, then a rope is passed
 
 88 
 
 GEODETIC OPERATIONS. 
 
 through between the legs of the pair and attached to the leg of 
 the single one near its lower end. It will be seen that as 
 this leg is drawn towards the other two the apex is hoisted up. 
 
 I have erected a high signal in this way by hitching a yoke 
 of oxen to the single leg and hauling it towards the other two. 
 If a tree should be in a suitable place, a rope passing through 
 a block, attached as high up as possible in the tree, will be of 
 great service in hoisting the apex. 
 
 A good winch will be of great use, and plenty of rope will 
 be needed, and marline for lashing. If all the timbers are cut 
 and holes bored ready for the bolts, the labor of erection will 
 be of short duration. Captain Boutelle's tables, enabling one to 
 cut the timbers for a signal for any height, are inserted here : 
 DIMENSIONS IN FEET. 
 
 TRIPOD. 
 
 SCAFFOLD. 
 
 Vertical 
 
 
 
 Three feet below 
 
 
 
 Three feet below 
 
 S& 
 above 
 station 
 point. 
 
 Vert, 
 length. 
 
 Slant 
 length. 
 
 station-point. 
 
 Vert, 
 length. 
 
 Slant 
 length. 
 
 station-point. 
 
 Rad.+ o.6 7 
 
 Side of eq. 
 triangle. 
 
 One half 
 diagonal. 
 
 Side of 
 square. 
 
 32 
 
 39 
 
 39-31 
 
 5-54 
 
 9.60 
 
 38 
 
 38.52 
 
 14-33 
 
 2O.26 
 
 48 
 
 55 
 
 55-43 
 
 7-54 
 
 13.06 
 
 54 
 
 54-75 
 
 17.00 
 
 24.04 
 
 64 
 
 7i 
 
 71-55 
 
 9-54 
 
 16.52 
 
 70 
 
 70.97 
 
 19.66 
 
 27.80 
 
 80 
 
 87 
 
 87.68 
 
 "54 
 
 19.99 
 
 86 
 
 87.19 
 
 22.32 
 
 31-57 
 
 96 
 
 103 
 
 103 . 80 
 
 13-54 
 
 23-45 
 
 102 
 
 103.41 
 
 25.00 
 
 35-35 
 
 DIMENSIONS OF TRIPOD. 
 
 Slant dist. 
 from top. 
 
 Vert. dist. from 
 
 top = L. 
 
 R = radius. 
 = j + 0.667. 
 
 Length of hor. 
 brace = 1.732^. 
 
 Length of 
 diagonal 
 braces. 
 
 Size of 
 braces. 
 
 Feet. 
 
 
 Feet. 
 
 Feet. 
 
 Feet. 
 
 Inches. 
 
 O 
 
 o.oo 
 
 0.667 
 
 
 
 
 5 
 
 4.96 
 
 1.287 
 
 2.229 
 
 
 
 8 
 
 7-94 
 
 1.659 
 
 2.873 
 
 
 
 13 
 
 12.90 
 
 2.279 
 
 3-947 
 
 6. 02 
 
 3 by 2 
 
 20 
 
 19.85 
 
 3.148 
 
 5-452 
 
 8-39 
 
 3 by 2 
 
 2 9 
 
 28.78 
 
 4.264 
 
 7-385 
 
 11.00 
 
 3 by 2 
 
 40 
 
 39-69 
 
 5.628 
 
 9.748 
 
 13-87 
 
 3 by 2 
 
 53 
 
 52.59 
 
 7.240 
 
 12.540 
 
 17-05 
 
 3 by 3 
 
 63 
 
 67.48 
 
 9-102 
 
 15-765 
 
 20.55 
 
 3 by 3 
 
 85 
 
 84.35 
 
 II. 212 
 
 19.420 
 
 24.40 
 
 3 by 3 
 
 103 . 80 
 
 103 . oo 
 
 I3.542 
 
 23-455 
 
 25.00 
 
 3 by 3
 
 FIELD-WORK OF THE TRIANGULATION. 
 
 Ground 
 Bottom of holes 
 
 Vertical 
 length 
 from top. 
 
 Slant 
 length 
 along 
 outside 
 edge. 
 
 Feet. 
 3-05 
 19.27 
 35-49 
 5I-7I 
 67.93 
 84.15 
 100.38 
 103.42 
 
 Slant 
 length 
 along 
 centre 
 post. 
 
 Halfdiag. 
 from 
 station- 
 point to 
 outside 
 edge. 
 
 Hori- 
 zontal 
 braces 
 side of 
 square. 
 
 Size of 
 hori- 
 
 braces. 
 
 Inches. 
 3 by 4 
 3 by 4 
 3 by 4 
 4 by 4 
 4 by 4 
 4 by 4 
 
 Diag- 
 braces. 
 
 Size of 
 diag- 
 onal 
 braces. 
 
 Inches. 
 
 3 by 3 
 3 by 3 
 3 by 3 
 3 by 4 
 3 by 4 
 3 by 4 
 
 Feet. 
 3 
 19 
 35 
 51 
 67 
 83 
 99 
 
 102 
 
 Feet. 
 
 Feet. 
 8.49 
 II-I5 
 13 82 
 16.49 
 I9-I5 
 21.82 
 24.50 
 25.00 
 
 Feet. 
 12. OO 
 15-77 
 19-54 
 23.32 
 27.08 
 30.86 
 34- 6 5 
 35.36 
 
 Feet. 
 
 21.27 
 23-90 
 26.81 
 29.91 
 21.66 
 *22.23 
 
 
 
 
 67.50 
 83.60 
 99.70 
 102.72 
 
 
 
 ' 
 
 
 *One foot from ground. 
 
 The floor of the scaffold should be twelve feet square, giving 
 room for a tent large enough for the observers to move around 
 in, and sufficient space outside to pass around while fastening 
 the tent to the railing. 
 
 A good shape for an observing tent is hexagonal, four and 
 a half feet across, and six and a half high, one side opening for 
 its entire length for exit and entrance, and the other sides hav- 
 ing a flap that opens from the top to a little below the height of 
 the instrument. This will keep out the sun, and also, by open- 
 ing only that part that is needed, the tendency of the wind to 
 cool the sides of the circle unequally can be diminished. A 
 corner post will be needed at each vertex, and the top can be 
 supported by a rafter running from each corner of the platform 
 meeting over the centre. To determine the size of the base 
 of the scaffold, we find the ratio of the half diagonal to the ver- 
 tical height and add to this the half diagonal of the top. One 
 foot in six has been found to give stability to the signal, so that 
 for a scaffold 80 feet high with 3 feet in the ground, we have 
 for the half diagonal of the base - 8 / -|- the half diagonal of the 
 top = 23 feet, and the side of the square 27.6 feet. The slope 
 can be found by trigonometry, tan. of slope = vertical height 
 divided by half the difference of the upper and lower diag-
 
 90 GEODETIC OPERATIONS. 
 
 onals. It is well to brace the signal by wire guys running from 
 each length of timber in the scaffold legs. 
 
 Probably the highest signal ever erected was built by Assist- 
 ant Colonna of the U. S. Coast and Geodetic Survey in Cali- 
 fornia. 
 
 A large red-wood tree was cut off 100 feet from the ground 
 and a twofold signal built, a platform fastened to this high 
 stump, and a quadripod from the ground for the support of the 
 instrument. The total height was 135 feet. The observers 
 were hoisted up in a chair attached to a rope passing through 
 a fixed pulley at the top, and hauled by a winch on the ground. 
 
 When the country is approximately level, the curvature of 
 the earth will obstruct a long line of sight, unless the instru- 
 ment be elevated or a high signal erected. When we know 
 the distance within a mile or two between the points on which 
 it is desired to establish stations, the problem is to find how 
 high the signals or scaffolds must be in order to be intervisible. 
 Also, when two suitable points of known altitudes are chosen, 
 with an intervening hill of known elevation, the problem is to 
 find how high one must build to see over it. 
 
 Let h = height in feet ; 
 
 d = distance of visibility 
 
 to horizon in feet ; 
 R = average radius of cur- 
 vature in feet, 
 log R = 7.6209807. 
 
 The distance d being a tangent, it is a mean proportional be- 
 tween the secant and the external segment, that is, h : d :: d 
 \h + 2R, but h is so small compared with 2R that it can be 
 omitted, and we have k=o.66?2d\ This is to be increased by 
 its o.07th part for terrestrial refraction, making h =0.7139^',
 
 FIELD-WORK OF THE TRIANGULATION. 9! 
 
 If we wish to know how far above the horizon the line of 
 sight passes from two points of known elevation, we find the 
 distance to the point of tangency. 
 
 Let D the whole distance ; 
 d = the shorter distance ; 
 a the height above the tangent ; 
 m = the coefficient of d* in the above expression. 
 
 k a = md\ Ha- m(D - d}* = mD 1 
 
 by subtraction 
 
 Hh = mD* - 2mDd, or 2mDd = ml? (H - K)\ 
 
 therefore, d 
 
 This gives the distance from the lower point to the point of 
 tangency ; then the height at which this tangent strikes either 
 station can be found by the above formula, h = 0.7 139^/ 2 , or 
 a h 0.714^'. 
 
 If there is an intervening hill, we first compute the point of 
 tangency of the line from the higher station ; then, how high 
 up the intervening hill this tangent strikes. To this add the 
 amount by which the lower hill exceeds this tangent plane : if 
 this be more than the height of the intervening hill, it can be 
 seen over ; if less, the difference will show how much must be 
 added to the height of the terminal stations. 
 
 If the intermediate hill be so heavily timbered as to render 
 it impracticable to have it cleared, the height of the trees must 
 be added to the elevation of the hill ; and at all times it is best 
 that the line of sight should pass several feet above all inter- 
 mediate points. The following table gives the difference be- 
 tween the true and apparent level in feet at varying distances :
 
 9 2 
 
 GEODETIC OPERATIONS. 
 
 Dis- 
 tance, 
 miles. 
 
 Difference in feet for 
 
 Dis- 
 tance, 
 miles. 
 
 Difference in feet for 
 
 Curvature. 
 
 Refraction. 
 
 Curvature 
 and 
 Refraction. 
 
 Curvature. 
 
 Refraction. 
 
 Curvature 
 and 
 Refraction. 
 
 I 
 
 0.7 
 
 O.I 
 
 0.6 
 
 34 
 
 771-3 
 
 108.0 
 
 663.3 
 
 2 
 
 2-7 
 
 0.4 
 
 2-3 
 
 35 
 
 817.4 
 
 114.4 
 
 703.0 
 
 3 
 
 6.0 
 
 0.8 
 
 5-2 
 
 36 
 
 864.8 
 
 121. 1 
 
 743-7 
 
 4 
 
 10.7 
 
 i-5 
 
 9-2 
 
 37 
 
 9I3-5 
 
 127.9 
 
 785-6 
 
 5 
 
 16.7 
 
 2-3 
 
 14.4 
 
 38 
 
 963.5 
 
 134-9 
 
 828.6 
 
 6 
 
 24.0 
 
 3-4 
 
 20. 6 
 
 39 
 
 1014.9 
 
 I42.I 
 
 872.8 
 
 7 
 
 32-7 
 
 4.6 
 
 28.1 
 
 40 
 
 1067.6 
 
 149-5 
 
 918.1 
 
 8 
 
 42.7 
 
 6.0 
 
 36.7 
 
 4i 
 
 II2I.7 
 
 157-0 
 
 964.7 
 
 9 
 
 54.0 
 
 7-6 
 
 46.4 
 
 42 
 
 II77.0 
 
 164.8 
 
 IOI2.2 
 
 10 
 
 66.7 
 
 9-3 
 
 57-4 
 
 43 
 
 1233-7 
 
 172.7 
 
 1061.0 
 
 ii 
 
 80.7 
 
 "3 
 
 69.4 
 
 44 
 
 1291.8 
 
 lSO.8 
 
 IIII.O 
 
 12 
 
 96.1 
 
 13-4 
 
 82.7 
 
 45 
 
 I35I.2 
 
 189.2 
 
 1162.0 
 
 13 
 
 112. 8 
 
 15-8 
 
 97-0 
 
 46 
 
 1411.9 
 
 197.7 
 
 1214.2 
 
 14 
 
 130.8 
 
 18.3 
 
 112.5 
 
 47' 
 
 1474.0 
 
 206.3 
 
 1267.7 
 
 15 
 
 150.1 
 
 21.0 
 
 129.1 
 
 48 
 
 1537-3 
 
 215.2 
 
 1322 I 
 
 16 
 
 170.8 
 
 23-9 
 
 146.9 
 
 49 
 
 1602. o 
 
 224.3 
 
 1377-7 
 
 17 
 
 192.8 
 
 27.0 
 
 165.8 
 
 50 
 
 1668.1 
 
 233-5 
 
 1434-6 
 
 18 
 
 216.2 
 
 30-3 
 
 185.9 
 
 5i 
 
 1735.5 
 
 243.0 
 
 1492.5 
 
 IQ 
 
 240.9 
 
 33-7 
 
 207.2 
 
 52 
 
 1804.2 
 
 252.6 
 
 1551.6 
 
 20 
 
 266.9 
 
 37-4 
 
 229-5 
 
 53 
 
 1874-3 
 
 262.4 
 
 1611.9 
 
 21 
 
 294-3 
 
 41.2 
 
 253-1 
 
 54 
 
 1945.7 
 
 272.4 
 
 1673-3 
 
 22 
 
 322.9 
 
 45-2 
 
 277-7 
 
 55 
 
 2018.4 
 
 282.6 
 
 1735-8 
 
 23 
 
 353-0 
 
 49.4 
 
 303-6 
 
 56 
 
 2092.5 
 
 292.9 
 
 1799.6 
 
 24 
 
 384-3 
 
 53-8 
 
 330.5 
 
 57 
 
 2167.9 
 
 303-5 
 
 1864.4 
 
 25 
 
 417.0 
 
 58.4 
 
 358.6 
 
 58 
 
 2244.6 
 
 314.2 
 
 1930.4 
 
 26 
 
 451- 1 
 
 63.1 
 
 388.0 
 
 59 
 
 2322.7 
 
 325.2 
 
 1997-5 
 
 27 
 
 486.4 
 
 68.1 
 
 418.3 
 
 60 
 
 2402.1 
 
 336.3 
 
 2065.8 
 
 28 
 
 523-1 
 
 73-2 
 
 449-9 
 
 61 
 
 2482.8 
 
 347-6 
 
 2135.2 
 
 29 
 
 561.2 
 
 78.6 
 
 482.6 
 
 62 
 
 2564.9 
 
 359-1 
 
 2205.8 
 
 30 
 
 600.5 
 
 84.1 
 
 516.4 
 
 63 
 
 2648.3 
 
 370.8 
 
 2277.5 
 
 31 
 
 641.2 
 
 89.8 
 
 551-4 
 
 64 
 
 2733-0 
 
 382,6 
 
 2350.4 
 
 32 
 
 683.3 
 
 95-7 
 
 587.6 
 
 65 
 
 2819.1 
 
 394-7 
 
 2424.4 
 
 33 
 
 726.6 
 
 101.7 
 
 624.9 
 
 66 
 
 2906.5 
 
 406.9 
 
 2499.6 
 
 The following example will illustrate its use: Suppose we 
 have a line of 14 miles from A to B, and at B it is convenient 
 to build a signal 21 feet high. By lobking in the table in the 
 fourth column, we find that the line of sight will strike the 
 horizon at 6 miles, leaving 8 miles to be overcome at A. Op- 
 posite 8 in the first column we find 36.7 feet in the fourth, 
 therefore at A we will have to build 37 feet to see B. 
 
 To illustrate the second problem :
 
 FIELD-WORK OF THE TRIANGULATION. 93 
 
 Let h' = height of higher station = 1220 feet; 
 h = height of intervening hill = 330 feet ; 
 h" = height of lower station = 700 feet; 
 d = distance from h to h" = 24 miles ; 
 d' = distance from h to k' = 40 miles ; 
 d-\- d' = distance from h' to h" = 64 miles. 
 
 700 feet strikes the horizon at 34.9 miles, 64 34.9 = 29.1 
 miles from that point to the other station. By looking in the 
 table at 29.1 miles, the tangent strikes the other station at 486 
 feet, 1220 486 = 774 feet, the distance the top is above the 
 tangent, and 29.1 24 = 5.1 miles that the point of tangency 
 is from the intervening hill, and hence strikes it at 15 feet. 
 Now, if we conceive a line to be drawn from the top of the 
 higher to the top of the lower, we will have with the tangent 
 a right-angle triangle in which the elevations at the higher 
 and intervening hills above the tangent are proportional to 
 their distances from the lower ; or, 24 : 64 : : x : 774, x 290.2 ; 
 that is, this sight-line strikes the intervening hill at 290 feet 
 above the tangent, and the tangent strikes it at 15 feet, or the 
 sight-line hits the intervening hill at 305.2 ; as this is 330 
 305.2 = 24.8 feet below the top, the two stations are not inter- 
 visible. 
 
 The lower station being the nearer the intervening hill, it 
 would be the one to build on. To determine the height of 
 the necessary signal, we have the following proportion : 
 
 40 : 64 : : 24.8 : x, or x = 39.6 feet. 
 
 In determining the altitude of stations, or intervening hills, 
 an aneroid barometer will give a result sufficiently accurate. 
 If the barometer is graduated to inches and decimals, the fol- 
 lowing table, giving heights corresponding to readings of bar- 
 ometer and thermometer, will be useful in estimating the 
 height :
 
 94 
 
 GEODETIC OPERATIONS. 
 
 Ba- 
 rom- 
 eter. 
 
 Mean of Observed Temperatures, Fahrenheit. 
 
 32. 
 
 42. 
 
 52. 
 
 62. 
 
 72. 
 
 82. 
 
 92. 
 
 30.0 
 
 
 
 
 
 
 
 
 29.9 
 
 87.5 
 
 89.4 
 
 91.4 
 
 93-3 
 
 95-3 
 
 97-2 
 
 99-2 
 
 29.8 
 
 175-3 
 
 179.2 
 
 183.1 
 
 187.0 
 
 190.9 
 
 194.8 
 
 198.7 
 
 29.7 
 
 263.4 
 
 269.3 
 
 275-1 
 
 280.9 
 
 286.8 
 
 292.7 
 
 298.5 
 
 29.6 
 
 351-8 
 
 359-6 
 
 367-4 
 
 375-2 
 
 383-0 
 
 390-9 
 
 398.7 
 
 29.5 
 
 440.5 
 
 450-3 
 
 460.0 
 
 469.8 
 
 479-6 
 
 489.4 
 
 499.2 
 
 29.4 
 
 529.5 
 
 541.3 
 
 553-0 
 
 564-7 
 
 576.5 
 
 588.2 
 
 600. i 
 
 29.3 
 
 618.8 
 
 632.6 
 
 646.3 
 
 659-9 
 
 673.7 
 
 687.4 
 
 701.3 
 
 29.2 
 
 708.4 
 
 724.2 
 
 739-9 
 
 755-4 
 
 771-3 
 
 787.0 
 
 802.8 
 
 29.1 
 
 798.3 
 
 816.1 
 
 833-8 
 
 851-3 
 
 869.2 
 
 886.9 
 
 904.7 
 
 29.0 
 
 888.5 
 
 908.2 
 
 927.9 
 
 947-6 
 
 967.4 
 
 987.2 
 
 1007.0 
 
 28.9 
 
 979.0 
 
 1000.7 
 
 1022.4 
 
 1044 . 2 
 
 1065.9 
 
 1087.8 
 
 1109.6 
 
 28.8 
 
 1069.9 
 
 1093-5 
 
 III7-3 
 
 1141.1 
 
 1164.8 
 
 1188.8 
 
 I2I2.6 
 
 28.7 
 
 1161.1 
 
 1186.7 
 
 1212.5 
 
 1238.3 
 
 1264.1 
 
 1290.0 
 
 I3I5-9 
 
 28.6 
 
 1252.5 
 
 1280.3 
 
 1308.1 
 
 1335.9 
 
 1363-8 
 
 I39I-6 
 
 I4I9-5 
 
 28.5 
 
 1344-3 
 
 1374-2 
 
 1404.0 
 
 1433.8 
 
 I463-7 
 
 1493-6 
 
 I523-5 
 
 28.4 
 
 1436-4 
 
 1468.4 
 
 1500.2 
 
 1532.1 
 
 1563-9 
 
 1595-9 
 
 1627.9 
 
 28.3 
 
 1528.5 
 
 1562.9 
 
 1596-8 
 
 163). 7 
 
 1664.5 
 
 1698.6 
 
 1732.7 
 
 28.2 
 
 1621.5 
 
 1657-7 
 
 1693.7 
 
 1729.6 
 
 1765-6 
 
 1801.7 
 
 1837.9 
 
 28.1 
 
 1714.6 
 
 1752.8 
 
 1790.9 
 
 1828.9 
 
 1867.0 
 
 1905-2 
 
 1943-4 
 
 28.0 
 
 1808. I 
 
 1848.3 
 
 1888.5 
 
 1928.6 
 
 1968.8 
 
 2009 . o 
 
 2049.3 
 
 27.9 
 
 1901.9 
 
 1944.2 
 
 1986.4 
 
 2028.6 
 
 2071.0 
 
 2113.2 
 
 2155-6 
 
 27.8 
 
 1996.0 
 
 2040.4 
 
 2084.7 
 
 2128.9 
 
 2173-5 
 
 2217.8 
 
 2262.3 
 
 27.7 
 
 2090.5 
 
 2136.9 
 
 2183.4 
 
 2229.6 
 
 2276.3 
 
 2322.7 
 
 2369.3 
 
 27.6 
 
 2185.2 
 
 2233.8 
 
 2282.4 
 
 2330 7 
 
 2379-4 
 
 2428.0 
 
 2476.7 
 
 27-5 
 
 2280.3 
 
 2331-1 
 
 2381.7 
 
 2432-2 
 
 2482.9 
 
 2533-6 
 
 2584.5 
 
 27.4 
 
 2375-8 
 
 2428.7 
 
 2481.4 
 
 2534-1 
 
 2586.8 
 
 2639.6 
 
 2692.7 
 
 27-3 
 
 2471.6 
 
 2526.7 
 
 2581.3 
 
 2636.2 
 
 2691.1 
 
 2746.0 
 
 2801.3 
 
 27.2 
 
 2567.8 
 
 2625.0 
 
 2681.9 
 
 2738.9 
 
 2795-9 
 
 2852.9 
 
 2910.3 
 
 27. i 
 
 2664.3 
 
 2723-6 
 
 2782.6 
 
 2841.8 
 
 2901.0 
 
 2960.2 
 
 30I9-7 
 
 27.0 
 
 2761.2 
 
 2822.6 
 
 2883.9 
 
 2945 I 
 
 3006.5 
 
 3067.9 
 
 3I29-5 
 
 When the station is on some hill, the name should be the 
 popular designation of the elevation, or the name of the person 
 who owns the property on which it is situated. 
 
 It is a great mistake to attempt to bequeath to posterity the 
 name of one of the party locating the signal. When no name 
 can be found to attract attention to the locality of the station, 
 a number will answer the purpose of a name. As soon as a 
 station is ready for occupancy it will be found advisable to 
 write a description of the signal, its position, and the way to
 
 FIELD-WORK OF THE TRIANGULATION. 95 
 
 reach it from some well-known thoroughfare, to be sent to 
 headquarters. This would be of especial service in case the 
 work should cease before the completion of the observations, 
 to be resumed at some future time by another party. 
 
 PORTER. 
 
 " This point is at the head of Blue Lick, a tributary of the 
 Left Fork of Twelve Pole, in Wayne County, W. Va. It is 
 on the farm of Larkin Maynerd. The signal is built in the 
 form of a tripod, and stands on the highest point of a large 
 field. 
 
 " A wagon can be taken up the Twelve Pole from Wayne 
 C. H., and up Blue Lick to the signal. From this point can 
 be seen Scaggs, Pigeon, Williamson, Vance, Runyan, and Rat- 
 tlesnake." 
 
 "Station No. 24: Ford County, 111.; corner of sections, 14, 
 15, 22, 23 ; township, 23 ; range, 10 east." 
 
 Each station should be provided with an underground mark, 
 consisting, when accessible, of a stone pier with a hole drilled 
 in the top and filled with lead bearing a cross-mark on its upper 
 surface, the intersection forming the centre of the station. 
 The top of the stone should not be within eighteen inches of 
 the surface of the ground, so as to be below the action of 
 frost, and any disturbance likely to arise from a cultivation of 
 the soil. Occasionally it has been found convenient to build 
 above this another pier to a height of eighteen or twenty inches 
 above ground, to serve as a rest for the instrument when the 
 station is occupied, and a stand for the heliotrope when the 
 station is observed upon. 
 
 When large stone cannot be had, a section of an earthen- 
 ware pipe four inches in diameter may be used by filling it 
 with cement and broken stone. The upper surface can be 
 marked with lines before the cement sets, or a nail driven in 
 while it is plastic.
 
 96 GEODETIC OPERATIONS. 
 
 When it is impracticable to dig a hole of any depth for 
 a masonry superstructure on account of a stone ledge immedi- 
 ately underlying the soil, it will be found sufficient to drill in 
 the top of the rock a hole and fill it with lead. Whatever form 
 of underground or permanent station-marks is used, it is essen- 
 tial to have surface, or reference-marks. 
 
 These are usually large stones set N., E., S., and W. of the 
 centre, and at such distances that the diagonals joining those 
 at the corners will intersect directly over the centre. 
 
 For immediate use it is well to place over the centre a sur- 
 face-mark, so that should anything happen to the signal before 
 it is finished with, it can bq replaced without disturbing the 
 permanent mark. 
 
 When the signal is a high tripod, or when it is necessary to 
 raise the instrument at the time the station is occupied, the 
 relative position of the centre of the station and the centre of 
 the instrument must be tested at frequent intervals, as an un- 
 equal settling of the signal would deflect it from or towards the 
 centre. The quickest way to determine this relative position 
 is to set up a small theodolite at a convenient distance from 
 the centre, and fix the intersection of the cross-wires on the 
 centre of the underground mark, or reliable surface-mark ; then, 
 by raising the telescope, determine two points on opposite 
 sides of the top of the signal. Then repeat this operation from 
 a position approximately at right angles to the first position. 
 Draw a string from each pair of points so fixed, and the inter- 
 section of these strings will indicate the centre. If possible, 
 the instrument should be placed directly over this point; if 
 not, then the distance to the point, and its direction referred to 
 one of the triangle-sides, should be carefully measured and re- 
 corded. 
 
 Sometimes it happens that, in the case of a very high signal 
 situated on a sharp point, no position can be found from which 
 both the top and the station-mark can be seen. To meet just
 
 FIELD-WORK OF THE TRIANGULATION. 97 
 
 this difficulty, Mr. Mosman has devised an instrument with a 
 vertical axis resting on levelling-screws, and so adjusted that 
 when freed from errors the telescope revolves around an 
 imaginary axis passing through the intersection of the cross- 
 wires. 
 
 The optical features are such as to admit of focusing it on 
 objects at distances varying from a few inches to 150 or 200 
 feet. To use it, you place it on a support over the centre of 
 the station, the support, of course, having a hole through it. 
 After levelling the instrument, move it until the cross-wires 
 coincide with the station-mark ; then, by simply changing the 
 focus, a point can be found in the intersection of these wires. 
 This operation should be repeated with the telescope in dif- 
 ferent positions; and as different points are obtained, the centre 
 of the figure formed by joining these points will be the one 
 desired. 
 
 The reverse operation can also be successfully performed 
 with this instrument. 
 
 It sometimes occurs that a straight tree is used as a signal, 
 in which event it is necessary to occupy an eccentric station. 
 This must be marked with as much care as though it were the 
 true station. The method for reducing the observed angles 
 will be given in full on page 143. 
 
 In the record-book must be kept a description of the mark- 
 ings of the stations ; and when an eccentric position is occupied, 
 the distance and direction already referred to are to be carefully 
 entered. 
 
 The method of observing horizontal angles must depend 
 upon the accuracy desired and upon the kind of instrument 
 used. Regarding the maximum error in closing primary tri- 
 angles to be three seconds, or six for secondary, a number of 
 precautions must be taken. The principal ones may be classed 
 under the following heads : 
 7
 
 g8 GEODETIC OPERATIONS, 
 
 1. Care in bisecting the object observed upon. 
 
 2. Stability of the theodolite-support. 
 
 3. Elimination of instrumental errors. 
 
 4. Preservation of the horizontality of the circle. 
 
 5. Rapidity of pointings. 
 
 6. Observations at different times of the day. 
 Conditions I and 2 are self-apparent, and the best means of 
 
 compliance therewith will readily suggest themselves to the 
 observer. 
 
 The instructions for eliminating instrumental errors have 
 already been given (see Chap. II.). 
 
 When the theodolite is placed in position and levelled, see 
 that the adjustments have not been disturbed before beginning 
 a set of readings. If, while observing, the level shows a change 
 in the horizontality of the circle, do not disturb it until the 
 set is finished. But if the deflection be considerable, the read- 
 ings must be thrown away. 
 
 The advantage of pointing rapidly is the greater certainty 
 of having the same state of affairs when sighting to all of the 
 signals in the circuit, since it diminishes the interval during 
 which there can occur unequal expansion of the circle ; twist 
 in the theodolite-support, changes in the illumination of the 
 different signals, or flexure of the circle from any cause. 
 
 By making observations at different times of the day, errors 
 arising from lateral refraction may be diminished because of 
 the changes in the condition of the atmosphere. 
 
 There are two principal classes of theodolites repeating- and 
 direction-instruments. The former gives a number of readings 
 in a short time, but a new source of errors is introduced by the 
 repeated clamping and unclamping. However, if the clamps 
 do not produce what is called travelling, the principle of repeti- 
 tions renders it possible to obtain a large number of readings 
 on all parts of the circle, and thus tends to free the average 
 from the effect of errors of graduation, for if the divisions on
 
 FIELD-WORK OF THE TRIANGULATION. 99 
 
 one side of the circle are too far apart, there will be other 
 parts on which the divisions are too close. In measuring an 
 angle with a repeater, it is best to set the circle at zero ; 
 point on the first station on the left, bisect the signal, see that 
 the circle is clamped, and then turn to the next station. Read 
 and record both verniers, turn the entire instrument back to 
 the initial point and bisect ; then unclamp the telescope and 
 point to the second station, clamp, and turn back to the first. 
 Repeat this operation until the whole circle has been passed 
 over; divide the last readings by the number of pointings, and 
 the quotient will be the value to adopt as the average for the 
 two verniers. The advantage of recording the first reading is 
 that it serves as a check on the number of degrees and minutes 
 in the final result. 
 
 If there are several angles at a station, it is advisable to read 
 them individually and in all combinations. Calling the angles in 
 their order, I, 2, 3, 4, and 5, we read and repeat I, then 2, 
 3, 4, and 5 ; afterwards I, 2, as one angle; then I, 2, 3, as one ; 
 I, 2, 3, 4, as one ; and I, 2, 3, 4, 5 ; also 2, 3 ; 2, 3, 4 ; and 2, 
 3, 4, 5 ; then 3, 4; 3, 4, 5 ; and, finally, 4, 5, this closing the 
 horizon. The advantage of this can be seen when we take up 
 the adjustment of the angles around a station. When an in- 
 strument can be reversed in its AT's, it will be found desirable 
 to make a similar set with the telescope reversed, and record 
 these as R. 
 
 With a direction-instrument, it is not necessary to make 
 these combinations. The plan is to make 5, 7, II, 13, 17, 19, 
 or 23 series, by dividing the circle into such a number of parts; 
 as each one is prime to two or three reading-microscopes, no 
 microscope can fall on the same part of the limb twice in 
 measuring the same angle. 
 
 Suppose we decide to make eleven series, we first find the 
 initial pointing for each set of the series. One eleventh of 
 360 = 32 43' 38".2, two elevenths = 64 27' i6".4, etc.
 
 IOO GEODETIC OPERATIONS. 
 
 We set the circle approximately on zero, and turn the entire 
 instrument upon some arbitrary point that can be readily bi- 
 sected, cla.mp on this, and make bisection perfect by moving 
 tangent-screw of telescope if necessary. Read and record all 
 the microscopes, taking both forward and backward microme- 
 ter-readings as already explained. Turn the telescope until the 
 next signal is bisected ; read and record as before ; continue 
 around to the last. After recording this last reading, see if the 
 signal is bisected ; if so, record the same values as just read for 
 the first on the return set. When the initial point is reached, 
 reverse the telescope and make a set as before, recording this set 
 as R. Then set on the second position and make a complete 
 set, continuing in this way until all of the positions are used. 
 
 It is desirable to sight on all of the signals every time, and 
 also on the azimuth-mark if one has been erected ; but if one 
 should become indistinct, while all the others show well, this 
 one can be omitted and supplied afterwards. It will be seen 
 at once that by this method we get an angle as the difference 
 of two directions ; hence the probable error of an angle will be 
 \^2 times the probable error of a direction. The record-book 
 should be explicit, giving the time of each pointing, position 
 of circle at the initial point, the position of the telescope, D or 
 R, appearance of signal (the latter is of importance in weighting 
 angles); also, if a tin cone is sighted, the time and direction of 
 the sun referred to the cone must be recorded as data for cor- 
 recting for phase. 
 
 If the triangulation is for general topographic purposes, it 
 will be found advisable to read angles to prominent objects 
 that may be in view, since if one is seen from two well-deter- 
 mined stations its position can be approximately located. 
 Preliminary computations should be carried along in the field, 
 so as to apply reduction from eccentric stations to centre and 
 deduce the probable errors of the angles.
 
 FIELD-WORK OF THE TRIANGULATION. 
 
 101 
 
 If this falls beyond the predetermined limit, or t if the tri- 
 angles do not close after allowing for spherical excess within 
 the limit prescribed, the angles should be remeasured. 
 
 The example here given is taken from record-book just as it 
 came from the field d is the forward, and d' the backward 
 micrometer-reading : 
 
 HORIZONTAL DIRECTIONS. 
 
 Station: Holmes, W. Va. 
 Observer: A. T. M. 
 
 Instrument: 114. 
 
 Date: Sept. 7, 1881. 
 Position: n. 
 
 Series 
 and 
 No. 
 
 Objects 
 Observed. 
 
 Time. 
 h. tn. 
 
 Tel. 
 
 Mic. 
 
 ' 
 
 ' 
 
 d. 
 
 d'. 
 
 Mean 
 d. 
 
 Remarks. 
 
 io 
 
 Table Rock. . . . 
 
 5 ' io 
 
 D 
 
 A 
 
 32 
 
 44 
 
 48 
 
 48 
 
 
 Weather clear, at- 
 
 
 
 
 
 B 
 C 
 
 
 
 25 
 25 
 
 *4 
 
 32.50 
 
 mosphere moder- 
 ately clear. 
 
 
 Somerville 
 
 5 = 8 
 
 D 
 
 A 
 B 
 
 319 
 
 44 
 
 74 
 
 5" 
 
 73 
 49 
 
 
 Wind S.W., light. 
 Ther. 97 . 5 . 
 
 
 
 
 
 C 
 
 
 
 53 
 
 53 
 
 58.66 
 
 
 
 Somerville .... 
 
 5 - '3 
 
 R 
 
 A 
 
 139 
 
 44 
 
 73 
 
 7 
 
 
 Reversed. 
 
 
 
 
 
 B 
 
 
 
 40 
 
 ; } 
 
 
 
 
 
 
 
 C 
 
 
 
 48 
 
 47 
 
 52.83 
 
 
 
 Table Rock.... 
 
 5 : 20 
 
 R 
 
 A 
 
 g 
 
 212 
 
 44 
 
 5 
 
 5 
 
 
 
 
 
 
 
 C 
 
 
 
 30 
 
 ag 
 
 33-o 
 
 
 X1 
 
 Table Rock.... 
 
 5 : 35 
 
 R 
 
 A 
 
 13 
 
 44 
 
 49 
 
 48 
 
 
 
 
 
 
 
 B 
 
 
 
 
 IO 
 
 
 
 
 
 
 
 C 
 
 
 
 3 
 
 28 
 
 32.16 
 
 
 
 Piney 
 
 5 : 33 
 
 R 
 
 A 
 
 301 
 
 18 
 
 64 
 
 3 
 
 
 Heliotrope. 
 
 
 
 
 
 B 
 
 
 
 
 34 
 
 
 
 
 
 
 
 C 
 
 
 
 45 
 
 43 
 
 47-33 
 
 Reversed. 
 
 In addition to the determination of the geographic positions 
 of various points by triangulation, it is also possible to obtain 
 with some precision their elevation, for, since we compute the 
 distances between the stations, the only remaining term is the 
 angle of depression or elevation from the station occupied to 
 each that can be seen. This necessitates a determination, by 
 levelling, of the height of the initial point only. 
 
 The field-work can be easily described as consisting of a
 
 I0 2 GEODETIC OPERATIONS. 
 
 number of readings of the angles from the zenith to each sta- 
 tion. In the computation given on page 90, it will be seen 
 that vertical refraction affects this angle ; but if the zenith-dis- 
 tances be measured from each station to the others at the 
 same time, supposing the refraction to be equable throughout 
 the intervening space, the uncertainty caused by the unknown 
 deflection of the sight-line will be eliminated. 
 
 But as it is not always feasible to have all the points oc- 
 cupied at the same time, the zenith-distances can be meas- 
 ured on different days, and when possible, under such varying 
 atmospheric conditions as to secure the same average relative 
 refraction. The best time is between the hours 10 A.M. and 
 3 P.M. The height of the theodolite above ground must be 
 known, as well as that of the signal sighted. 
 
 In 1860, Assistant Davidson organized a series of experi- 
 ments to obtain a comparison of the various methods of deter- 
 mining altitudes. He used a Stackpole level, a rod carefully 
 compared with a standard and levelled in both directions. The 
 measures of zenith-distances were reciprocal. They were 
 made seven times daily for five days. The barometric series 
 consisted of hourly readings during five days of a mercurial 
 barometer, attached, detached, and wet-bulb thermometers. 
 The differences in the altitudes are : 
 
 As determined by levelling, 598-74 metres. 
 
 " " zenith-distances, 598.64 " 
 
 " " atmospheric pressure, 595.26 " 
 
 REFERENCES. 
 
 U. S. Coast and Geodetic Survey Reports, 1876, pp. 238- 
 401 ; '80, pp. 96-109; '82, pp. 151-208. 
 Puissant, Gode"sie, vol. i. pp. 350-376. 
 Bessel, Gradmessung in Ostpreussen, pp. 59-128.
 
 FIELD-WORK OF THE TRIANGULATION. IO3 
 
 Ordnance Survey, Account of the Principal Triangulation, 
 pp. 1-61. 
 
 Struve, Arc du Mridien, vol. i. pp. 1-35. 
 
 Publications of the Prussian Geodetic Institute, especially 
 " Das Hessische Dreiecksnetz" and " Das Rheinische Dreiecks- 
 netz," II. Heft.
 
 IO4 GEODETIC OPERATIONS. 
 
 CHAPTER V. 
 
 THEORY OF LEAST SQUARES. 
 
 WHEN in the various measurements of a magnitude a num- 
 ber of results are obtained, it is a matter of great importance 
 to know which to regard as correct. That all cannot be cor- 
 rect is apparent, and that some one is true may safely be 
 assumed. Errors which render a magnitude too great are 
 called negative errors, and those which make it too small 
 are positive errors. Should, for instance, the true length of a 
 line be 73.45 chains, and its length found by measurement to 
 be 73.44 chains, the error would be -j- O.OI chain ; while if 
 the measurement show 73.46 chains the error would be o.oi 
 chain. 
 
 It may be accepted as a general rule that positive and nega- 
 tive errors are equally probable ; also, that small errors are 
 more likely to occur than great ones, since the tendency to 
 commit a great error would be readily detected before record- 
 ing it, while those smaller could not be easily distinguished 
 from the value afterwards found to be correct. 
 
 Let the angle x be measured n times with equal care, so 
 that in each result there is the same liability for an error to 
 occur; let the individual values obtained be v v v v ...v n . 
 Since x is the true value, the errors will be : x #,, x z/ 2 , 
 . . . x v n ; these we will denote by dx r dx v . . . dx n , and, 
 from what has just been said, some are positive and some are 
 negative. As there exists the same probability for the posi- 
 tive as for the negative errors, and since the individual errors
 
 THEORY OF LEAST SQUARES. 10$ 
 
 are nearly equal to each other, their sum will nearly amount 
 to zero, and we may put, 
 
 dx, + dx, + dx* + dXi+... + dx n = o, 
 or (x - v,} + (x - v t ) . . . + (x - v n } = o, 
 
 whence nx = z/, -f- v^ -f- v t -f- . . . -f~ v n . Hence : 
 
 (i) x = (v l -f- z> 3 + ^3 + v) -T- , which is simply the 
 arithmetical mean of the n terms. This, however, gives no infor- 
 mation as to the value of the errors. If for each positive error 
 we had committed an equal negative error,the arithmetical mean 
 would give the correct value, but this fortuitous elimination 
 can only be expected in an infinite number of observations ; 
 even then it will not enable us to form any definite opinion as 
 to the degree of accuracy attained in the individual observa- 
 tions. In order to accomplish this, we must find some means 
 for preventing the positive errors from destroying the negative 
 ones. Gauss found the way by taking into account not the 
 errors themselves, but their squares, which are positive, and 
 hence cannot eliminate one another. 
 
 For brevity we will write [zy] for the series of terms involv- 
 ing v, as v t , v y , v a , . . . v n , and S[v M ~] for the sum of such a series. 
 Hence we may put for the sum of the squares of the errors, 
 S\d, t x~^, or S[;tr vjf. The value of x will approach the 
 nearest to its correct value when the arithmetical sum of the 
 errors is the smallest, or when the sum of the squares of the 
 errors is a minimum ; that is, when 5 [^^ 2 ] is a minimum. 
 
 Let y 
 
 then by differentiation,-^- = 2S[d n x~\.
 
 IO6 GEODETIC OPERATIONS. 
 
 As this is to be a minimum, we place the first differential co- 
 efficient = o, 
 
 or S\d n x\ = o, or S[(x *<)] = o. 
 
 If x v, + x v^ + x v 3 . . . -f x v n o, 
 
 or 
 
 a result identical with the one previously obtained. 
 
 The converse can also be demonstrated; that is, the arithmet- 
 ical mean gives to the square of the residuals the minimum, 
 
 \d n x\ = 
 
 = (* - v,} + (x - Vf ) + (x - v^ . . . + (x - VH ) = o ; 
 
 squaring this, 
 
 r = (* - v>r + (* - ,) 
 
 but x = *2S* an d 
 
 substituting these values, 
 
 'vj? = ff.T JJLL . (0 
 - " J L " J w k ; 
 
 Suppose we now take some other value, JF,, so that d^x^ d^x^ 
 . . . d n x l represent the residuals, then
 
 THEORY OF LEAST SQUARES. 
 
 substituting in this equation the value for [z/] z derived from 
 
 therefore, [<W = \d rt xj -f n(x x$ ..... (2) 
 
 Since (x x$ is always positive, \d n x^ is greater than 
 that is, the square of the residuals when any value other than 
 the arithmetical mean is used, is greater than when the arithmet- 
 ical mean is taken. From this it is seen that the arithmetical 
 mean is the most probable value ; but the correct value might 
 be a little more or a little less than this mean. 
 
 When the individual results are nearly the same, we might 
 be satisfied with any one ; but when there is a great range, we 
 accept the average even with some trepidation. Now, if we 
 had some term that depended upon the residuals, the magni- 
 tude of this term might be taken as the measure of precision : 
 this is what is known as the probable error. 
 
 The development of an expression for the probable error 
 has been undertaken by many persons, and prosecuted in vari- 
 ous ways, but all attaining the same end. The discussion that 
 follows is taken principally from Chauvenet. 
 
 Let us recapitulate what are known as the theorems of the 
 theory of probabilities.
 
 108 GEODETIC OPERATIONS. 
 
 1st. Equal positive and negative errors are equally probable, 
 
 "\, in a large series of observations, are equally frequent. 
 
 2d. There is a limit of error which the greatest accidental 
 errors do not exceed ; if / denote the absolute magnitude of 
 this limit, all the positive errors will be comprised between o 
 and -j- ^ ar >d all the negative errors between oand /, so that 
 the errors are contained within 2/. 
 
 3d. Small errors are more frequent than large ones. 
 
 So that the frequency of the error may be considered a func- 
 tion of the error itself. If A be an error of a certain magnitude, 
 and its frequency cpA, this function will be a maximum when 
 A = o, and be o when A = /. If we denote the probability 
 of an error A by y, we have y = (pA, an equatioij of a curve 
 in which A is the abscissa and y the ordinate; as A has equal 
 values with contrary signs, the curve is symmetrical with re- 
 spect to the axis^, and for_^ = o, A = +00. 
 
 We shall therefore consider A as a continuous variable, and 
 cpA as a continuous function of it. 
 
 If there are n errors equal to A, n' = A' . . . , and the entire 
 number equal to m, the respective probabilities are 
 
 n n 
 
 a>A = , q>A =. , etc., 
 m m 
 
 therefore, <pA -\- <pA' -J- <pA" -j- . . . = i. 
 
 However, the continuity of the curve requires that the suc- 
 cessive values of A shall differ from one another by an infini- 
 tesimal, so that the number of values for q>A is infinite.
 
 THEORY OF LEAST SQUARES. IOQ 
 
 Let us take the smallest unit of magnitude in the observa- 
 tions as I, then the probability of the error A maybe regarded 
 as the same as the probability that the error falls between A 
 and A -\- i, and the probability of an error between A and 
 A -\- i will be the sum of the probabilities of the errors A, 
 d _|_ It A 4- 2 . . . A 4- (i i). By making i small the proba- 
 bility of each of the errors from A to A -\- i will be nearly the 
 same as A, and their sum will approximate i<pA. When the 
 interval between the successive errors approximates an infini- 
 tesimal, the expression becomes more nearly exact, and for i 
 we may put dA, and write (pA . dA as the accurate expression 
 for the probability that an error falls between A, and A -f- dA. 
 Hence the probability that an error falls between the limits 
 -f- oo and oo is the sum of the elements of the form cpA . dA, 
 
 or the integral / cpA . dA = i. 
 
 Suppose the quantity M be a function of x, y, z, etc., A, A', 
 A" , be the errors, and <pA, cpA', <pA" , their respective probabili- 
 ties. 
 
 Since the probability of J/will be the product of the proba- 
 bilities of the quantities of which Mis a function, we may put 
 P = cpA . (pA' . cpA" . . . 
 
 From preceding principles we know that the most probable 
 system of values of the unknown quantities x, y, z . . . will be 
 that which makes Pa. maximum ; therefore we obtain the dif- 
 ferential coefficient of P with respect to each variable and 
 place it equal to o. Log P varies with P, and as P is a func- 
 tion of x, y, 2, . . . , the differential coefficients of P with 
 respect to x, y, z, . . . , must separately = o ; or 
 
 I dP I dP I dP 
 
 ' = ' --=> '=' 
 
 But, log P = log (pA -f log <pA' -\- log (pA" ... (2)
 
 HO GEODETIC OPERATIONS. 
 
 dP dcpA . dtpA' . dcpA" 
 Therefore, _ = -^ + -^- + /T . . . (3) 
 
 Divide (3) by dx, dy, and dz, ... 
 
 (A) 
 
 dP 
 
 ~ q>Adx 
 dcpA 
 
 dcpA' } dcpA" 
 
 P.dx 
 
 dP 
 PTTy 
 
 dP 
 
 TTdz 
 
 cpA'dx (pA"dx 
 d(pA' dcpA" 
 
 ~ cpAdy 
 d(pA 
 
 d(pA' 
 
 dcpA" 
 
 etc., 
 
 <pA'dz 
 
 \ <pA"dz 
 etc.; 
 
 since the first members are equal to zero, from (i). 
 
 In (A) let us place for -: , cp'A. dA; for -~T, <P'A' . d A'. 
 tpA q)A 
 
 Then they will become 
 
 etc., etc. 
 
 If * be the correct value of M, M, M", etc., 
 A = M-x, A'=M'-x, A" =
 
 THEORY OF LEAST SQUARES. Ill 
 
 dA dA' dA" 
 
 from which -7- = r- = r . . . = i. 
 
 dx dx dx 
 
 The first equation at (B) becomes 
 
 i 
 -x) + 9 '(M' -x) + cp'(M" -*)... = a (4) 
 
 Now, if in this equation we suppose M' = M" . . . = M 
 mN, where m represents the number of observations, and 
 since the arithmetical mean is the most probable value of x, 
 
 4-M" ... 
 
 since there will be m I terms after the first, each equivalent 
 to M mN, 
 
 or x = ~(M + mM - nfN - M+ mN) 
 
 = -(mM - m 
 
 = M mN + N 
 = M (m i)N; 
 
 or M x = (m i)N, 
 
 and M' - x - M" - x = M - mN - X-,
 
 H2 GEODETIC OPERATIONS. 
 
 but x = M mN + N\ 
 
 therefore 
 
 = M - mN - M + mN - N = - N. 
 Substituting these values for x in (4), we get 
 
 <p' [(m - i)N] + (m - !)?'(- N) = o, 
 
 since there are m I terms after the first each equal to N. 
 Transposing, 
 
 dividing by N, 
 
 cp'(- N) 
 
 (m- \}N - N ' 
 
 This is a true expression for all values of m, or (m i), or 
 N(m i). As the second term is not affected by changes 
 in m, the expression is a constant. By putting A = (m i)N, 
 
 cp'A 
 we will have - = a constant which is called K. 
 
 If - = K, cp'A =
 
 THEORY OF LEAST SQUARES. 113 
 
 dcpA 
 But we supposed cp' A = y - (page 1 10); 
 
 theref re 
 
 - 
 cpA 
 
 Integrating this, log cpA = \KA* -f- log c, 
 or <pA = C (* Kl ^, 
 
 in which e is the Naperian base. 
 
 Since (pA must decrease as \K increases, it must be negative. 
 Placing h* for \K, we have 
 
 likewise <p A' = ce ~ h ^'\ 
 
 cpA" = ce-w\ 
 etc. 
 
 We found that P <pA . <pA' . <pA" . . . ; 
 
 therefore P = c(e~ h *<* -\- e~ *' A " + <r *'* . . .), 
 
 />=<,--'; ...... V* . (5) 
 
 which represents the probabilities of all errors from A to A* 
 inclusive. 
 8
 
 H4 GEODETIC OPERATIONS. 
 
 To determine the constant c, we will take the integral on 
 page 109, 
 
 f cpA 
 and substitute for (pA, ce~ *' A * , 
 
 We will write ? = f?A\ and A = v ; 
 
 /+ / 
 
 then we obtain / ce~ fl d-r\ 
 
 J- k 
 
 factoring | , j m e-*dt=i. 
 
 Let m=f die-**; 
 
 / 00 
 
 then, since this integral is independent of the variable, we may 
 also put 
 
 m = f due- '; 
 
 J-o> 
 
 by multiplication, 
 
 ***.&*-&+*> ..... (6)
 
 THEORY OF LEAST SQUARES. 11$ 
 
 If we integrate between the limits oo and o, then between 
 the limits o and oo their sums will be the value of the defi- 
 nite integral between the limits -j- oo and oo. 
 
 We shall now place u tv, and du = tdv ; then (6) becomes 
 
 wt = dv .dt.te- 
 
 v being regarded as the variable, and t the constant, 
 
 =SJ dv iF^ = Ktan " T " tan " I 
 
 w 
 
 Without changing notation : 
 
 or the total integral, _^ dte~*= Vx 
 
 But
 
 Il6 GEODETIC OPERATIONS. 
 
 therefore r VTC I, c1/n = h y c= =-. 
 n \ n 
 
 Placing this value for c in tpA = ce ~ h "^ 
 
 we have <pA = ;=e ~ ASA * . 
 
 This method of obtaining the value for the definite integral 
 is taken from Laplace, Mfchanique Celeste. 
 
 Identical results are obtained by different methods by Pois- 
 son and Airy. 
 
 The probability that the error falls between A and A -\- dA 
 
 h 
 is =*-**\W, and that it falls between the limits o and a is 
 
 Vn 
 
 h f* = * 
 
 - / e- h *^dA, as already explained. 
 
 t/A = o 
 
 Let t hA, then A T ; and a? A y , then t = ah. 
 h li 
 
 Substitute these values in the last integral ; it becomes 
 
 after multiplying by two, since the sum of negative errors is 
 considered equal to the positive errors. 
 
 This integral has been computed for values of t. A table of 
 these is given in Merriman's " Least Squares." 
 
 From this table it is found that the error which occupies the 
 middle place in the series of errors, arranged in the order of
 
 THEORY OF LEAST SQUARES. IT/ 
 
 their magnitude, has the same number of errors above as be- 
 low ; therefore, the error satisfying this condition is that for 
 which the value of the integral is . If we designate the cor- 
 responding value of / by p, we find from the same table that 
 
 p = 0.4769. 
 
 If r be the error in a series of observations whose precision is 
 
 P , P 
 T h - 
 h r 
 
 P P 
 
 h, we can put p = hr, r = j, h = . 
 
 MEAN OF THE ERRORS. 
 
 If we have a series of m errors A, A' . . .\ a positive, and a 
 negative each equal to A, or 2a in all, the probabilities are that in 
 
 all there will be , . . . The mean of these errors, suppos- 
 m m 
 
 ing each repeated a number of times proportional to the proba- 
 
 . 2aA + 2a'A' + 2a"A" 
 bihty of its occurrence, is . . . 
 
 = 2 J + 2A' 
 m m 
 
 The probability of an error A has been shown equal to <pA . dA. 
 So that the above expression for an infinite number of terms 
 approximates a series of terms of the form 2 A . cpA . dA. But 
 on page 1 16
 
 Il8 GEODETIC OPERATIONS. 
 
 multiplying by 2 A . dA, we have 
 
 . dA = -,Ae- 
 
 ' 7t 
 
 If 17 be the mean error, 
 
 h\ therefore rj = - = - 1=- I.i82or; 
 
 r P - 
 
 
 or r = 0.84537, since p 0.4769 
 
 As was stated elsewhere, it is not feasible to obtain the 
 mean of the error, since the negative and positive errors being 
 theoretically equal, their sum will become zero. So we take 
 the sum of the squares of these errors, and the square which is 
 the mean of these squares is the square of the mean error. 
 
 That is, if s be the mean error, 
 
 = -1; 
 
 2/*" 
 
 r 
 
 hV2 P_ 
 
 r 
 r = 0.6745 s. 
 
 = i.4826r ; 
 
 This value of r is the probable error of anyone of the observed 
 values of the unknown quantity, x.
 
 THEORY OF LEAST SQUARES. 1 19 
 
 Let us now look for an expression for the value of the 
 arithmetical mean, r a . 
 
 Equation (5), P <pA -j- tpA' -J- cpA" . . . 
 
 The most probable value of the observed quantity is that 
 which makes Pa. maximum, or that makes A* -j- A'* -|- A"* . . . 
 a minimum. 
 
 But it has been shown that the arithmetical mean renders 
 the sum of the squares a minimum ; therefore P represents the 
 probability of the arithmetical mean when A, A', A" . . . repre- 
 sent the residuals referred to this mean. The probability of 
 any other value of x, as x -\- dx, will be 
 
 but A*= ms\ P' = 
 \A\ being the sum of the errors, = o. 
 PSB*** -**-***, 
 
 since d = o, when x = x^ 
 
 P\P'\\ 
 P\P' :: 
 
 dividing the second ratio by the third term. 
 If m=i, P: P' :: i :*-*
 
 I2O GEODETIC OPERATIONS. 
 
 In this single observation the probability of zero-error, as in 
 the arithmetical mean, is to the probability of error, d, as 
 I : e - AV2 . 
 
 As h is the measure of precision of a single observation, /i* 
 will be the square of this measure. 
 
 In the expression for the error of the arithmetical mean we 
 find for the square of the measure of precision of m observa- 
 tions K'm ; therefore the measure of precision of the arithmetical 
 mean of m quantities is h \ r m. That is, the measure of pre- 
 cision of the mean increases as the square root of the num- 
 ber of observations. 
 
 r. = =-, r - 0.67456, r. = .6745*. ; 
 \ m 
 
 therefore = . 
 
 Vm 
 
 If v, i\, z/ a . . . be the observed values of a quantity whose 
 mean is x, the residuals will be u, u /t , or x v, x v 
 x v n . . . If x were the correct value, x v, x v t , x v u 
 would be equal to A, A' , A" . . . 
 
 and m? = [//'] = ['] ; 
 
 However, this does not consider the mean errors of the resid- 
 uals. Suppose A u d, A' = ' d, A" = u" d, . . .
 
 THEORY OF LEAST SQUARES. 
 
 s ] = me = (u <Tf -f (' - </)' 4- (" - 
 
 = [*] _ 2 []^ + *</' 
 
 = M + w ^* since [] = o. 
 
 121 
 
 * may be taken = ' = 
 
 so that 
 
 transposing 
 
 r = 0.6745 
 
 ,.= 0.6745 
 
 To determine the probable error of the arithmetical mean, 
 we find the difference between each individual result and the 
 mean, square these quantities, and divide their sum by
 
 122 
 
 GEODETIC OPERATIONS. 
 
 m (in _ i) f where m represents the number of individual 
 results. Extract the square root of this quotient, and mul- 
 tiply by 0.6745 ; the product will be the probable error of the 
 arithmetical mean. 
 
 The whole operation can be performed by logarithms. 
 
 To illustrate : 
 
 Angle. 
 
 u. 
 
 u 1 . 
 
 66 - 54' ~ 
 
 12. "5 
 13. 5 
 ii 3 
 
 4- i.i 
 
 + 0.1 
 
 + 2.3 
 
 I. 21 
 .OI 
 5-29 
 
 
 16. 5 
 
 2.9 
 
 8. 4 I 
 
 
 12. 3 
 
 + 1-3 
 
 1.6 9 
 
 
 IS- 5 
 
 - i-9 
 
 3-6i 
 
 Average 
 
 13- 6 
 
 
 20.22 = [ s ] 
 
 The probable error of a single determination, 
 
 The probable error of the arithmetical mean, 
 '20.22 
 
 ;= 0.6745 
 
 = o".54- 
 
 If the probable errors of the means of different sets of deter- 
 minations have been found, their relative weights may be 
 readily ascertained. Let h, /* A, ... be the measures of pre- 
 cision, and r, r v r, . . . the probable errors. 
 
 Suppose we compare our individual observations with a fic- 
 titious standard whose mean error is e, ; and the actual observa-
 
 THEORY OF LEAST SQUARES. 123 
 
 tions with a mean error s, need w in number to reduce the mean 
 
 error of their arithmetical mean to ; this gives 7=^, or 
 
 \ w 
 
 Wf = f, s . Likewise any other set would give Wj /3 = ,*, or 
 w^'* = TV? ; that is, w l : w : : f 2 : /a . We call w, w, . . . , 
 etc., the weights ; they are reciprocally proportional to their 
 probable errors. 
 
 The arithmetical mean of n 1 observations of weight w, a of 
 weight w v etc., would be 
 
 _ 
 
 \nw] 
 or x = -F =;-, and 
 
 where f 1 is the mean error of unit weight. 
 
 Let v, = ! x, v^ 7/ 2 x, v t n 3 x, 
 
 /3 = v* -)- f 2 ; but W" = * = w,V^ + W^*, 
 
 and * = wj>?+ iv^*, etc. 
 
 If m be the sum of such terms,
 
 124 GEODETIC OPERATIONS. 
 
 this, substituted in the value of Q , gives 
 
 r_\wv^___ 
 
 F - V \w\(m - i) 
 
 In figure- or station-adjustment, if the number of repetitions, 
 or some other well-established reason, does not afford weights 
 for the averages used, the reciprocals of their probable errors 
 can be used. While in the development of the foregoing 
 formulae there were a number of assumptions, and some ap- 
 proximations to cause cautious persons to distrust the absolute 
 rigor of the results, it will be apparent to all that the arithmet- 
 ical mean deserves a confidence that varies with different cases. 
 
 Suppose in measuring an angle ten results are obtained in- 
 dividually differing considerably among themselves. In an- 
 other measurement of the same angle ten other results are 
 secured with a very small range ; now, if the average be the 
 same in these two cases, the latter would be more readily ac- 
 cepted, as the residuals are individually smaller. So we need 
 some exponent of confidence that is a function of the residuals ; 
 and if our accepted value of the probable error is not absolutely 
 correct, it will afford us some information as to the agreement 
 of the individual results with the arithmetical mean, and in a 
 number of different determinations it gives us all the relative 
 information we need. 
 
 I shall add just here, without demonstration, other formulae 
 in general use in determining probable errors: 
 
 Probable error of a single observation,
 
 THEORY OF LEAST SQUARES. 125 
 
 Probable error of the arithmetical mean, 
 r. = 0.6745, 
 
 If m = number of observed angles ; 
 
 r = number of conditions in a chain ; 
 
 probable error of an adjusted angle = /* / - times prob- 
 able error of an observed angle, supposing the weights ap- 
 proximately equal (Walker). 
 
 If an angle be determined by a direction-instrument, its value 
 will be the difference of two directions ; so that if a is the 
 probable error of a direction, a V 2 will be the probable error 
 of the angle. 
 
 If r J9 r v r 3 . . . be the probable errors of different segments 
 of a base-line, the probable error of the line as a whole, R = 
 Vr? + r; + r?... 
 
 We have now shown how to determine the probable error 
 of an angle or a base-line. The next subject to consider is to 
 what extent these errors in an angle or in a base will affect the 
 computed parts. As the errors just referred to are small in 
 comparison with the magnitudes themselves, we may omit in 
 all the discussions into which they enter all products, and pow- 
 ers above the first. All geodetic computations are based upon 
 formulae relating to triangles, so we will investigate those ex- 
 pressions which are of most frequent occurrence. Denoting 
 the sides of a plane triangle by a, b, and c, the corresponding 
 opposite angles by A, B, and C, and the errors with which they 
 may be affected by da, db, dc, dA, dB, and dC, we can find by
 
 126 GEODETIC OPERATIONS. 
 
 computation the value of any three if the values of the other 
 three be known (provided one be a side). The following 
 formulae are those most frequently used : 
 
 (1) a. sin. B = & .s'm A ; 
 
 (2) c = a . cos B -f- b . cos A ; 
 
 (3) A+B+C=iSo. 
 
 As da represents the correction to a, a-\-da will be the cor- 
 rect value of a, or a -f- da, b -J- db, c -|- dc, A -\- dA, B-\-dB, 
 C -\-dCj will be the true values. Substituting these inequa- 
 tions (i), (2), (3), we shall have : 
 
 (4) (a + da) sin (B -f dB) = (6 + db} sin (A 
 
 (5) (c + dc) = (a + da) cos (B + dB)+(b + db) cos (A + dA) ; 
 
 (6) A+dA+B-\-dB+C+dC= 180. 
 
 But sin (B + dB) = smB-\-dB. cos B, 
 
 since sin dB = dB, and cos dB = i ; 
 
 also sin (A -|- dA) = sin A -j- dA cos A ; 
 
 cos (B -f <B) = cos B dB sin B ; 
 cos (^4 + dA) = cos A dA sin A 
 
 Introducing these values in (4), (5), (6), and omitting all 
 products of da . db, etc., we shall get
 
 THEORY OF LEAST SQUARES. 12? 
 
 a . sin B + da . sin B + a . dB cos B b . sin A -\- db . sin A 
 
 -f- b . dA cos A ; 
 
 c -f- afc = a . cos B -\-da. cos 5 # . <sZ? . sin 2? + b . cos ^4 
 
 . cos A b . dk4 .sin ^4; 
 
 4- _{_ c+ dA + </ -f dC= 1 80. 
 
 Subtracting equations (i), (2), (3), from these just given, we 
 obtain: 
 
 (7) da . sin B + . dB cos 5 = db . sin ^4 + # ^ cos ^ ; 
 
 (8) fc = ^ . cos B a . dB sin B + db . cos ^4 . dL4 sin 
 
 (9) dA+dB + dC= o. 
 
 In a similar way, expressions can be obtained for the other 
 parts, as : 
 
 (10) dc . sin A -f- c . dA cos A = <afo. sin C-f-rt; . */*. cos f; 
 (i i) <# . sin C -f * . dC . cos C = dc . sin 5 + c . dB . cos B ; 
 
 (12) <z = dc.CQsB c.dBsinB + db.coz Cb.dCsin. C; 
 
 (13) d# = da . cos C a . dC sin C-\- dc . cos A c .dA sin A. 
 
 Suppose that in a triangle c, A, and B are given, and by 
 means of them the values a, b, and C are computed. Know- 
 ing the limits of errors with which these quantities are affected, 
 it is required to find the limits of the errors with which the 
 computed quantities are affected ; that is, knowing dc, dA, and
 
 128 GEODETIC OPERATIONS. 
 
 dB, we are to determine da, db, and dC. From equations (7) 
 and (8) we have : 
 
 (14) dC=-dA-dB-, 
 
 (i 5) da . sin B db . sin A = a . dB cos B -f b . dA cos A ; 
 
 (16) da . cos B + <# . cos ^ = afc -f a . dBsin B + b . dA sin .4. 
 
 Multiplying (15) by cos A, (16) by sin A and adding, we get: 
 
 (17) da(s'm A . cos B -f- cos ,/4 . sin B) = dc . sin .4 -f- . dA 
 
 a .d(cos A cos B sin ^ sin B}, 
 
 or, <fe . sin (A + -5) = dc . sin ^4 -f b . dA a . dB . cos (A 
 
 " 
 
 sin 
 
 Also multiplying (15) by cos B, (16) by sin B, and subtracting, 
 we get : 
 
 db . sin (A + .#) = d&r . sin B + ^ . dB b . dA . cos (<4 + B}, 
 
 -V j_ > 
 
 ~ ^ 
 
 _ 
 sin (^ + ^) sin (A+B)~ sin (^ + B)' 
 
 Since sin (^4 -J- -5) = sin C, and cos (^4 + ^) = cos C, we can 
 write : 
 
 . smA 6.dA . 7r> cos C 
 
 da = dc- -^ -\- -. -^ -4- a . dB- ^, 
 
 sin C ' sin C ' sin C
 
 THEORY OF LEAST SQUARES. 12$ 
 
 ,smB a.dB 
 
 db dc- 
 
 sin C ' sin C ' sin C 
 
 And, again, since sin A : sin C\\a : c, sin B : sin C \\ b : c, and 
 for cos C : sin C we may put cot C, the equations then reduce 
 to the following very simple form : 
 
 da = dc a - -f -r 1 -^ -\-a.dB cot C, 
 <: ' sin (7 
 
 db = dc . f- .' ^ + b . dA cot C ; 
 c sin 6 
 
 or, obtaining the relation between da and a, db and , 
 da dc . b .dA 
 
 a c <* . sn 
 
 db dc a. dB 
 
 . cot C, 
 
 . cot 
 
 Suppose ^r = 564.8, ^4 = 61 12' 12", B = 74 16' 30", and 
 that the error in c referred to c, or , be less than o.oooi, and 
 the maximum error in A and B is i". 
 
 It is required to compute , -j-, and dC. 
 
 log b = 2.8894998 log a = 2.8487359 
 
 log dA = 4.6855749 log sin C = 9.8458288 
 
 7.5750747 2.6945647 
 
 2.6945647 
 
 4.8805100 = log of second term.
 
 130 
 
 GEODETIC OPERATIONS. 
 
 log dB 4.6855749 
 log cot C = 0.0072518 
 
 4.6928267 = log of third term. 
 
 First term 
 
 Second term 
 
 Third term 
 
 log a 
 logdB 
 
 log cot C 
 log dA 
 
 First term 
 
 Second term 
 
 Third term 
 
 o.oooiooo 
 0.0000076 
 0.0000049 
 
 0.0001125 = error of a proportional to the 
 
 side a. 
 
 2.8487359 log b = 2.8894978 
 
 4.6855749 log sin C = 9.8458288 
 
 7.5343108 2.7353266 
 
 2.7353266 
 
 4.7989842 = log of second term. 
 
 0.0072518 
 4.6855749 
 4.6928267 = log of third term. 
 
 o.oooioooo 
 0.00000629 
 0.00000493 
 
 O.OOOIH22 = error of b proportional to b. 
 
 The discussion of these formulae will develop some very in- 
 teresting facts concerning the best-shaped triangles to make 
 use of in prosecuting accurate geodetic work. Upon inspect- 
 ing the equations it will be seen that the denominators of each 
 term of the second members is sin (A -f B}, consequently 
 when A -\- B is nearly 180, or when C is very small, those 
 terms involving sin C or sin (A -f- B) as a denominator will be 
 made quite large, and will give to da or db a value unduly great.
 
 THEORY OF LEAST SQUARES. 131 
 
 Again, the second members will have the smallest value 
 when sin C has its greatest value or when C = 90; supposing 
 that C = 90, then placing sin C = sin 90 = I, the equations 
 reduce to the form 
 
 Should dA and */2? be of about the same value, and b be 
 greater than a, or b : a greater than I, and a : b be less than I, 
 we will have da : a greater than db\b\ or if b is less than a 
 we will have da : a less than db : b. 
 
 From which we can see that it will be best when a = b, 
 consequently when A = B. Remembering what has just been 
 said, we see that the right isosceles triangle is theoretically the 
 best form of triangle to make use of. 
 
 From a similar discussion it will be apparent that if b or a 
 were the given side, the smallest error in the other quantities 
 would occur when B or A = 90. 
 
 As all the angles cannot be each equal to 90, the best tri- 
 angle is the equilateral. A similar value can be obtained by 
 direct differentiation. 
 
 sin A , ism A \ Js\nA\ , ,,smA 
 
 ^.b, da = d(- . b] = d(- 5 J . b + db 
 sin B \sm B I \sm B' sir 
 
 4* 
 
 \S1 
 
 sin ' 
 fr.cosA .s\nB .dAb .s\nA cos .d 
 
 sin BJ ' sin 3 B 
 
 It.cosA.dA #.sin^ cos B 
 sin B sin B sin B 
 
 .dB.
 
 132 GEODETIC OPERATIONS. 
 
 b c b.cosA,dA 
 
 Since = j. we may write for : ^ , 
 
 sin B sm A' y sin B 
 
 a . cos A . dA 
 
 : = a cot A . dA. 
 
 sin A 
 
 &.s'mA cos B 
 Also, smce a = --, and = cot B, 
 
 b sin A cos B 
 
 ~~D~ - ~ 5 dB = a . cot B . dB ; 
 smB sm B 
 
 therefore, da ^db a . cot B . dB -f a cot A . dA. 
 sm r> 
 
 Or, by logarithms, 
 
 log a = log sin A -f log b log sin B ; 
 differentiating, 
 
 da _ cos A . dA db cos B . dB 
 a sin A b sin ^ 
 
 = cot A . <W + ~ - cot . dB ; 
 
 <fc = cot A . <4 -f ^ . <# cot B .dB ; 
 
 d; _ sin A 
 
 b ~ sin B '
 
 THEORY OF LEAST SQUARES. 133 
 
 hence, da = ^db a cot B . dB 4- a cot A . dA. 
 sin /> 
 
 Given in a triangle the values of c, A, and C ; required to 
 compute the values of , #, and B, and so find the limits of 
 errors of the latter, supposing the errors of the former are 
 known. 
 
 From the equations already given we find : 
 
 dB=-dA -dC\ 
 
 da . sin (A + ff) = dc . sin A -f b . dA a . dB . cos (A -\- B) ; 
 db . sin (^4 + B) - dc . sin B -j- a . </- b.dA. cos (4 + ). 
 
 But sin (A-\- B) = sin C, and cos (A + ) = cos C; 
 
 hence, da . sin C = dc . sin ^ -(- . dW -j- # . dB . cos 7 ; 
 db.smC=dc.smB -\-a-dB -{-b .dA . cos 7; 
 
 dfc . a b . dA a . dB . cos C 
 da = --- : -^ H -- : :* . 
 c sin 6 sin C 
 
 Since dB dA dC, we may write : 
 
 dC . cos C" 
 
 = -- \- - ' 
 c a . sin 6 
 
 since b a cos C -\- c cos A, we may put i: cos A for # a cos 7, 
 
 <&z ^ , c . cos ^( . dA - 
 
 then - --- ---- . r dC . cot C ; 
 
 # * ' . sin 6 
 
 but . sin C = c . sin A, 
 
 therefore = -- + dW . cot ^4 - dC . cot 67.
 
 134 GEODETIC OPERATIONS. 
 
 db dc.s\r\B t a.dB dA. cos C 
 Likewise, ,- = j. 7?- -I- -r-. -^-\ -- : 7=; , 
 b bsmC b sin C sine 
 
 _ dc a .dA a.dC dA cos C 
 ~ c b sin C b sin C sin C ' 
 
 & cos C ~ adA a ' dC 
 
 ~~c tr.s'mC ~ b sin C 
 
 _dc c . cos B . dA a . dC 
 ~ c b sin C b . sin C' 
 
 dc a.dC 
 
 = --- dA . cot B -r^^- 
 
 c b . sin C 
 
 Let c = 450, A - 53 19' 16", C = 61 42' 32", we will find 
 by computation ^ = 64 -58' 12", a = 409.855, b = 463.05. 
 
 Suppose that c be reliable to within o.oooi of its entire 
 length, so that dc -v- c = o.oooi, and let dA = dC = 5" : 
 
 log dA = 5.3845449 log dC= 5.3845449 
 
 log cot A = 9.8720420 log cot C = 9.7309796 
 
 dA . cot A = 0.00001805 
 dc -T- = o.oooioooo 
 *#7 . cot C = o.oocoi 3047 
 
 da-^a 0.000131097 
 log *W = 5.3845449 
 log cot B = 9.6692660 
 
 log dA. cot ^ = 5.0538109 = 0.000011319 
 
 log a = 2.6126301 
 log dC = 5.3845449 
 
 = 7.9971750 
 log = 2.6656276 
 log sin C = 9.9447545 
 
 log (J.sin C) = 2.6103821, log (a . dC -5- . sin C) = 5.3867929
 
 THEORY OF LEAST SQUARES. 135 
 
 dA . cot B = o.ooooi 1319 
 a . dC -7- b . sin C = 0.000024366 
 
 Subtracting the sum of these two quantities from dc -f- c, 
 we get 
 
 db -r- * = 0.000064315, dB = ($" -+- 5") 10". 
 
 Since */" may be either positive or negative, we may select 
 the sign which will give the maximum value for the error 
 da -~ a. We have therefore added the expression dC . cot C. 
 
 In the following discussion, we will do the same, suppos- 
 ing the errors committed in measuring A and B were nearly 
 equal and of the same sign. 
 
 When they are nearly equal and of the same sign, according 
 to the equation already given, they will compensate one 
 another. But should dc have the same sign as dA and dC, it 
 would lessen the value of da when we have dC . cot C greater 
 than dA . cot A, since the amount of error committed in measur- 
 ing the angles is less than that of the measured side. 
 
 The value, therefore, of da -f- a will become the least when 
 C is less than A and acute. But if A be obtuse, consequently 
 cot A negative (assuming dA and dC to be positive), da -r- a 
 will become the least when C is acute, for then the last two 
 terms are to be subtracted from dc -=- c. 
 
 For A = 90, the third term will disappear entirely, 
 which circumstance will be advantageous with respect to da 
 -f- a. 
 
 In regard to the side a, A must, therefore, be either a right 
 angle or obtuse, and fas small as possible. 
 
 The same reasonings apply with respect to b, with the ad- 
 ditional circumstance that B should be also very small. 
 
 Therefore, in the present instance, a large value for A and 
 small values for B and C will produce the least errors.
 
 136 GEODETIC OPERATIONS. 
 
 Suppose we have the two sides and the included angle with 
 their limiting errors given, and wish to find the limiting errors 
 of the unknown or computed parts ; that, is having a, b, and C, 
 to compute the value of the errors in c, A, and B. 
 
 From equations (7), (8), and (9) we have 
 
 (1) d 
 
 (2) a . dB . cos B b . dA . cos A db . sin A da . sin B ; 
 
 (3) dc + a - dB . sin B -f- b . dA . sin A = db . cos A + da . cos B. 
 
 Substituting for dA, (dC+ dB}, (2) becomes 
 (4)0 . dB . cos B -f b . cos A (dB + dC}--da . sin B +db . sin A. 
 By expanding and transposing we get 
 
 (5) (a . cos B -f b . cos A)dB = da . sin B -f db . sin y4 
 
 b.dC.cosA. 
 
 Putting c for # . cos B -\- b . cos ^4, 
 
 (6) <: . dB = da . sin B + <# . sin A b . dC '. cos A, 
 likewise, 
 
 (7) c.dA = dA . sin B db . sin A a . dC . cos B. 
 By transposing (3), we have 
 
 (8) dc = da.cosB + db.cosAa.dB.s\r\B b.dA. sin A.
 
 THEORY OF LEAST SQUARES. 137 
 
 Multiplying this through by c and substituting torc.dAand 
 c .dB, the values given on page 136, we get 
 
 (9) c . dc c. da . cos B -f- c. db . cos A 
 
 a . sin B( da . sin B -j- ^ . sin ^4 b.dC. cos ^4) 
 
 . sin ^4 (d0 . sin B db . sin A a.dC . cos Z?). 
 = c .da.cosB -{-c .dBcosA 
 
 -f- ( . sin B b sin .4 )(</<& . sin B db sin A) 
 -}- a.b. dC(cos A . sin B -f sin A . cos ^). 
 
 The first factor = o, and the last is sin (A + B) or sin C; 
 therefore 
 
 (10) c . dc = c . da . cos B -\- c . db . cos y2 + a . # . d\ sin C. 
 By expansion and substitution, (9) becomes 
 
 (10) dc = da . cos B + db . cos ^4 -j- # . . dC . sin C -f- ^ ; 
 Dividing (6) and (7) by c, we get 
 
 (i i) dB = db . sin A -r- c da . sin B -f- c b . dC '. cos ^4 -=- ; 
 (12) dk4 = db . sin ^4 -=- c -j- dk . sin B -i- ^ a .dC. cosB -^-c. 
 
 In computation this formula is used like those already illus- 
 trated, so it will be needless to give a solution of an example 
 of this kind. 
 
 The only remaining case is when we have the three sides 
 with their limiting errors to find the limiting errors of the com- 
 puted angles. The discussion of this problem is of interest 
 simply from a theoretical point of view, since such a case will 
 never arise in any one's experience. 
 
 Rewriting equations already deduced (page 1 27), we start with 
 
 ( i) 6.sinA.dA+a.smJ3.dl? = da.cosJ8 -\-db . cos A dc ; 
 
 (2) b.cosA.dAa. cos B.dB da.sinB d 
 
 (3) dA + dB + dC=o.
 
 138 GEODETIC OPERATIONS. 
 
 Solving the first two equations with reference to dA and dB. 
 and substituting in the results the values of dA and dB in (3), 
 and putting sin C for sin (A -j- ), cos C for cos (A -f- B\ 
 we will obtain 
 
 (4) dA = da -j- b . sin C db . cos C -f- # . sin C dfc . cos j5 
 
 -=- . sin 7; 
 
 (5) ^/.Z? = da . cos 7 -j- a . sin T -j- db -~ a , sin C dc . cos A 
 
 -f- tf . sin (7; 
 
 (6) dC=da(b . cosC - a)+ a . & .sm C + db(a .cosC- b) 
 
 -r-a. &.sinC-{- dc(a . cos .Z? + . cos A) a . b . sin C. 
 
 Should we have da -~ a = db -h b = dc -j- c, the errors of the 
 sides would be proportional to the sides themselves. 
 
 The defective triangle would then be similar to the true 
 triangle, and the corresponding angles would be equal each to 
 each, and we would have dA = dB = dC '= o. 
 
 When the three angles of a triangle are observed, the differ- 
 ence between their sum, after subtracting the spherical excess, 
 and 180 is the total error of the triangle. 
 
 Let us call this E, the errors of the individual angles x, y, 
 and z, with respective weights, , v, w, 
 
 By preceding theory ux* -\- vy* + wz* = a minimum ; differ- 
 entiating with respect to x, y, and 2, we get ux = vy = tvz. 
 
 u w u w 
 
 wvy -\- wuy -\- uvy = wuE ; 
 _ wuE wvE uvE 
 
 ~ wv -\-wu-\- uv' ~ wv -f- wu -\- uv 1 Z ~ wv-\-wu-\-uv
 
 THEORY OF LEAST SQUARES. 139 
 
 The limits of errors may be found in a similar manner for 
 all combinations of triangles; hence a polygon may be decom- 
 posed into triangles and the limits of error found by the 
 methods just described. 
 
 This method is not altogether satisfactory, since in the com- 
 putation of the error in each triangle we use the errors of only 
 two of the angles, ignoring the third. 
 
 TT sin A A 
 
 From trigonometry, we have a = -. . b\ 
 
 by differentiation, we have 
 
 da=acotA.dAacotB.dB, . . . . (i) 
 
 b being a constant. 
 
 Let of, fi, and y be the measured angles, and A, B, and 
 C the correct values. The triangle error, after correcting for 
 spherical excess, is a-\- ft -j- y 180, one third of which may 
 
 be attributed to each angle, so the error in a = , 
 
 and the correct value 
 
 also, B = p-- = -, (3) 
 
 iSo _ 2 y-/3-a + 180 
 C-y-~ -. (4) 
 
 As A and B depend upon a, fi, and y, the total error in the 
 side a, or e a will be a function of a, ytf, and y;
 
 140 GEODETIC OPERATIONS. 
 
 In which f a) f /s, and f y are the probable errors in a, ft, and y, 
 and in (i) da = aco\.A.dA a.cotB.dB., 
 
 so we must obtain dA from (2), in terms of ex, ft, and y. 
 Likewise, dB from (3), or we may write (i) 
 
 I2a 13 v+ i8o\ 
 da = a cot A . d\ ! J 
 
 ,(2/3 a y-\- i8o\ 
 acotB.d{ - J. ... (6) 
 
 Differentiating (6) with respect to a, ft, and y, we have 
 da 2 
 
 (7) 
 
 ^ 
 
 ^ -- a cot A -a.cota\ - cot A -- Cot2?]|(8) 
 -f t 
 
 Squaring (7), (8), and (9), 
 
 = ia' cot 8 ^ + *-(? cot ^ . cot B + -#' cot 5 B ; 
 
 =-a* cot 9 ^ + a cot ^ . cot B + z 2 cot 2 B ; 
 -a* cot 2 ^4 - -- 2 cot A . cot ^ + -a 2 cot 2 .5 ;
 
 THEORY OF LEAST SQUARES. 141 
 
 therefore, supposing a = fp = f y = e, we have 
 (da V , (da V 
 
 = '[fa' cot 2 A 4- ftf 2 cot A . cot B + f fl 9 cot 5 E\ 
 = f V(cot' ^4 + cot ^ . cot + cot 2 B) ; 
 
 = fa 1/|(cot a ^4 -f cot A . cot .5 + cot 2 E). 
 
 As a is small, the second member can be converted into a 
 linear unit by writing it equal to its algebraic value times sin 
 
 l". 
 
 e a = ea. sin i" i/|(cot 2 ^ 4- cot A . cot .# + cot 2 B). (10) 
 
 This is a rigorous expression for the probable error of a side, 
 as computed from a base supposed to be free from error. 
 The side a of the first triangle may be regarded as the accu- 
 rate value of the base of the next triangle, and the probable 
 error of another side computed, and so on through the entire 
 chain. So we may put for (, the error of the last side, 
 
 = easin i" V |^(cot a A + cot^ . cot B + cot 2 B). (11) 
 
 In which 2 represents the sum. In determining the angles to 
 be used in this formula, it must be remembered that A is op- 
 posite the side whose error is being -determined, and B is op- 
 posite the side whose error was last computed.
 
 142 GEODETIC OPERATIONS. 
 
 To illustrate: 
 
 Starting with the base b, we first pass to u ; 
 
 in this case A is I, B, 3 ; 
 
 then to x, A is 4, B, 5 ; 
 
 then to #, A is 7, ^, 8. 
 
 Hence, 2 cot' ^ = cot 2 (i) + cot 2 (4) -f cot 2 (;) . . . et 
 
 In (n), is the average angle-error in the chain. If the proba- 
 ble error of the base be f/, this error will be carried through 
 the chain without augmentation or diminution, owing to inac- 
 curacies in the angles, but it will be increased in the ratio of 
 the length of the computed line to the base. In the first 
 
 & 
 
 computed side a, the error from this source will be &'. Sup- 
 pose this be /, then in the next triangle, if c is computed from 
 
 c c a c , 
 
 a, the error e y = . s x = . j- . e 6 = T e 6 , and so on through 
 a a b b 
 
 the entire chain; so if n be the last line, ' = r f t f - The total 
 
 o 
 
 error, , from both sources, will be E = Vf n ' 3 -f- ". If each 
 side of each triangle has been computed by two different 
 routes, the value for E must be divided by V^2, or, E =
 
 CALCULATION OF THE TRIANGULATION. 143 
 
 CHAPTER VI. 
 
 CALCULATION OF THE TRIANGULATION. 
 
 HAVING assumed in the field that we had a line of known 
 or approximately known length for a base-line, we measured 
 the angles of all the triangles of our net a sufficient number of 
 times to eliminate instrumental errors ; and now wish to com- 
 pute the distances of all the stations from one another, as far 
 as possible. 
 
 When the three angles of a triangle are measured with the 
 same care, it will be found that their sum will not equal 180 
 -|- spherical excess, and when two individual angles are meas- 
 ured separately and then as a whole, the sum of the two will 
 not equal the two when treated as a single angle; and, again, 
 the sum of the angles that complete the horizon will always 
 differ from 360. 
 
 The problem then is to find results from a number of ob- 
 served values that will approach the nearest to the truth, and 
 at the same time eliminate those discrepancies just referred to. 
 
 There are two classes of conditions that should be fulfilled : 
 
 (a) the sum of the individual angles should equal the meas- 
 
 ured whole ; 
 
 (b) the sum of all the angles completing the horizon should 
 
 equal 360. 
 
 The operation of filling these conditions is called station-ad- 
 justment. 
 
 (c) The three angles of a triangle should equal 180 ; 
 
 (d) The length of every side should be the same, regardless of 
 
 the route by which it is computed.
 
 144 
 
 GEODETIC OPERATIONS, 
 
 The filling of these conditions is called figure-adjustment. 
 These two adjustments are to be effected simultaneously, since 
 the same quantities occur in each. 
 
 The method of adjustment will depend on the way in which 
 the angles are read ; whether with a repeating-theodolite, or 
 direction-instrument. If with the former, the average value 
 obtained for each angle will be the quantity that enters the 
 equations formed by the expressed conditions. 
 
 Before writing these equations we must correct the angles 
 for run of micrometers, as already explained on page 35, and 
 for phase. 
 
 The latter is the effect of sighting to the illuminated por- 
 tion of a signal instead of the centre ; it is only appreciable 
 when a tin cone, or some large reflecting surface, is observed 
 on. This bright spot will be exactly in the line to the centre 
 when the sun is directly behind the observer, and furthest 
 from the centre when the sun is at an 
 angle of 90 with the cone and ob- 
 server. 
 
 Let 7be the centre of the signal and 
 O the the position of the observer, the 
 distance in the figure being greatly 
 shortened, proportioned to the size of 
 the signal. 
 
 The rays of the sun may be re- 
 garded as parallel and illuminating 
 half of the signal, as A SB. Of this 
 the observer sees only^S^F; this he 
 bisects, sighting to D instead of C. 
 This causes an error equal to the 
 angle COD. 
 
 Let SCG = x, EC = r, OC = D, 
 
 and COD = d. KF\s the projection of the visible arc, and 
 CD = \EK. AK, being perpendicular to EF, and FAE a
 
 CALCULATION OF THE TRIANGULATION. 145 
 
 right triangle, AK* = EK.KF = EK. 2EC (nearly), or, 
 
 but AK=EC.sinACE, 
 
 - 2EC' 
 hence, AK* = EC* . sin* ACE, 
 
 EC*, sin* ACE EC. sin* ACE 
 
 and EK = 
 
 2EC 
 
 As ACE Is small, we can write sin* ACE = 4 sin* % ACE, EK 
 = 2r . sin* $ACE, but ACE = GCS, both being complements of 
 
 f* fC 
 ACG, or, ACE = x ; also CD = - ; substituting, CD = rsin* 
 
 \x. In the right triangle OCD, sin0 = -^ = T Sl ^ **. As (9 is 
 small, sin 6 = 6. sin i", or, = S . 
 
 This correction is to be subtracted, when the sun is to the 
 right of the observer, and added when the sun is to the left. In 
 the case of independent angles, if both objects observed need a 
 correction for phase, the two individual corrections are to be 
 subtracted if they have opposite signs, and added when they 
 have the same signs. 
 
 In the principle of directions, each direction should be cor- 
 rected for phase, using only the average direction in applying 
 the correction, and at all times measuring the angle x about 
 the mean time of the series of observations. 
 
 A similar correction is to be applied when an eccentric sig-
 
 146 GEODETIC OPERATIONS. 
 
 nal was sighted ; in this case it is necessary to know the per- 
 pendicular distance from the centre of the signal to the line 
 joining the observed and observing stations. This will form a 
 
 right triangle in which sin 6 = -=-, or = -= : 77, in which 
 
 D Z/, sini" 
 
 6 = correction ; r, the perpendicular, and D, the distance be- 
 tween the stations. This correction is additive when the point 
 observed is within the angle formed by the centres of the two 
 stations, and subtractive when it falls without. 
 
 This is but a special case of reduction to centre, discussed 
 on page 196, and like the latter can be applied later as well 
 as at this point. 
 
 With the average angles corrected we wish to find those 
 values that will fulfil the required conditions and at the same 
 contain the largest element of truth. We have seen that the 
 arithmetical mean renders the sum of the squares of the resid- 
 uals a minimum, and that the most probable value of a num- 
 ber of disagreeing results is the one that makes the squares of 
 the errors a minimum. So we shall now look for that most 
 probable value which will fulfil the conditions. 
 
 Suppose we have a series of a observations, giving m for the 
 arithmetical mean, and a series of &, giving n for the mean ; the 
 relative value of these two means would be to each other as 
 a: b. 
 
 Consequently the larger number of equally good observa- 
 tions we have, the better relative value we will get for the 
 mean. If, therefore, the first arithmetical mean, v^ be obtained 
 from a series of a t , the second, z/ a , from a^ . . . the th, v n , from 
 a n , we will have for the most probable value of x,
 
 CALCULATION OF THE TRIANGULATION. 
 If an angle be measured 
 
 10 times with the result 18 18' 12", 
 8 times with the result 18 18' 2" ', 
 5 times with the result 18 18' 21", 
 4 times with the result 18 1 8' 30", 
 
 10 + 8+ 
 
 On page 123 it was shown that the residuals, or individual 
 errors, squared were multiplied by their weights, and these 
 products summed, to give the square of the probable error of 
 the observations as a whole. Then, since this probable error 
 is obtained by taking that value which reduces the residuals 
 squared to a minimum, the sum of the individual errors 
 squared, each multiplied by its respective weight, must assume 
 the form which renders it a minimum. 
 
 By way of illustration, let us take the following example : 
 Suppose A, B, and C, be three angles in a plane around a 
 point as a common vertex, and amounting to 360. Suppose 
 the measured values be A, B, and C\ a t , b t , and c t their true 
 values, and a, b, and c the errors of A, B, and C, so that we 
 have A -\-a = a,B-\-b = b, C -\- c = c/, also, a 4 + b, + c, 
 = 360. 
 
 From a set of 10 measurements A = 120 15' 20" ; 
 From a set of 12 measurements B = 132 16' 30" ; 
 From a set of 15 measurements C = 107 28' 19".
 
 148 GEODETIC OPERATIONS. 
 
 = 360 oo' 09" 
 ~ ^ oo/ " 
 
 a + + c = - 00 oo' 09" (i) 
 
 Taking the sum of the squares of these errors, # a -j- b* + *, 
 and multiplying each by its respective weight, 
 
 I0rt'+ I2#'+ 15*:*. (2) 
 
 From (i) c = a b 9 ; 
 
 -f- iSa + 1 8^ ; 
 
 = 15(81 + rt 2 + & + 2^ + 1 8* + 1 
 = 1215 -f 150'+ 15^+ 
 
 substituting this value for 15** in (2), we get 
 
 270^+ 1215. (3) 
 
 According to principles already explained, we obtain the dif- 
 ferential coefficient with respect to a and b, and place each re- 
 sult equal to zero ; 
 
 $oa + 30^ + 270 = o ; 50 + 3^ + 27 = o ; 
 270 = 0; 50 + 9^ + 45 =o; 
 
 therefore, b + 3" = o, b = 3", a = 3".6; substituting in 
 
 (l),<r=:-2". 4 . 
 
 The same result may be obtained by using an indeterminate 
 coefficient, and afterwards eliminating it, 
 
 2 9 (a
 
 CALCULATION OF THE TRIANGULATION. 149 
 
 we take 2cp to avoid the use of fractions. Differentiating this 
 with respect to a, b, and c, and placing the results equal to 
 zero, we get 
 
 2oa -f- 2<p = o or loa -\- cp o ; 
 24&-{-2<p=o i2#-f-0 = o; 
 
 30^ -\- 2cp = o 1 5^ + ^ o. 
 
 Eliminating q> from two of these equations, we get 
 
 loa I2b = o; 
 
 I2b i$c = o; 
 
 also, a-\-b-\-c-\-g = o. 
 
 By the simple elimination, we get 
 
 =-3".6, *=-3" ^=-2"4- 
 
 A = 120 1 5' 20" s".6 = 120 15' i6".4; 
 B = 132 16 30 3 .o = 132 16 27 .o; 
 C = 107 28 19 2 .4 = 107 28 16 .6 ; 
 a i J r j )i J r c i = 360 oo x oo". 
 
 The above is the simplest case in practice ; that is, when only 
 one condition is to be fulfilled. Let us pass to a more com- 
 plicated case, or when several conditions are to be fulfilled. 
 
 Suppose, in Fig. 13, we have from repeated measurements 
 the following results : 
 
 (1) MON= 68 37' i " with the weight 5; 
 
 (2) MOP = 140 2 19 with the weight 10 ; 
 
 (3) NOQ = 134 15 41 with the weight 20 ;
 
 GEODETIC OPERATIONS. 
 
 (4) 
 (5) 
 
 (6) 
 
 (7) 
 (8) 
 
 NOR = 211 56 10 with the weight 15 ; 
 
 FOR = 140 30 40 with the weight 12 ; 
 
 MOQ = 202 52 46 with the weight 18 ; 
 
 NOP = 71 25 38 with the weight 16 ; 
 
 QOR = 77 40 6 with the weight 20. 
 
 Upon inspection it will be seen that the following conditions 
 should be fulfilled: 
 
 (I) 
 (2) 
 
 (3) 
 (4) 
 
 (2) -(i) = (7): 
 (4) -(3) = (8): 
 
 (5) + (7) = (4) 
 (6) - (3) = (i). 
 
 (A) 
 
 Denoting the corrections to the angles by a, b, c, . . . k, 
 
 (1) [(2) + J] - [(i) + a] = [(7) + ] ; ^ 
 
 (2) [(4) + /] - [(3) + c\ = [(8) + *] ; I 
 
 (3) [(5) + e] + [( 7 ) +] = [(4) + </] ; f 
 
 (4) [(6) +/] - [(3) + c\ = l(i) + a]. J 
 
 (B)
 
 CALCULATION OF THE TRIANGULATION. 151 
 
 Substituting in these equations the angles designated by (i), 
 (2), . . . (8), they reduce to 
 
 (C) 
 
 These are the relations that must exist between the correc- 
 tions that are to be applied to the different angles. 
 
 Squaring each symbolic correction and multiplying each by 
 its respective weight, we have 
 
 (I) 
 
 b a g= 
 
 20" 
 
 (2) 
 
 d- c -h= 
 
 23 
 
 (3) 
 
 e+g-d= 
 
 8 
 
 (4) 
 
 f-c-a= 
 
 4 
 
 $0* + io' -f 20^' + i$d* + 12*' + i8/ 3 + i6g* + 2oh\ (D) 
 From the equations at (C) we obtain 
 
 (i), = + -20; (2), c = d /* 23 ; 
 (3), ' = <*- 
 
 Substituting these in (D), 
 
 - 2o) 3 + 2o(^/- h - 23)' + i5^ z 
 - A - 19)'+ 
 
 must be a minimum. 
 
 The square, omitting constants, gives : 
 
 ' -f- iotf 2 + ic^" 2 + 2cwg- 400^ 400^+20^" 4- 
 
 920^-f920/2+ 15^'+ I2^ a + I2^ 2 
 
 - 192^-+ i8 2 + i$d* + 18^ -f- 
 
 684^ 684^ + 684/2 + j6- 2
 
 152 GEODETIC OPERATIONS. 
 
 Differentiating this and placing the coefficient of each equal to 
 zero, we have 
 
 330 i& + 10^- i< = 542 ; 
 180 -f- 6$d I2g 38/2 = 706 ; 
 
 $a - 6d + 19^ =' 148 ; 
 
 ga + 19^ 297* = 401. 
 
 The solution of these equations gives a = 6.4.6, d = 5.89,^ = 
 7.95, and h = 7-97, which values substituted in C, give b = 
 5.59, c 9-14, e = 5-94> an d f = 1.32. Since the 
 errors are to be obliterated in applying the correction, each 
 correction must have the opposite sign to its error ; so that if 
 the above represent the errors, they are to be applied with 
 contrary signs to the respective angles, which reduce the 
 angles to : 
 
 (i)= 68 36' 54 ".54 ; 
 
 (2) = 140 2 24 .59 ; 
 
 (3) = 134 15 50 .14; 
 
 (4) = 211 56 4 .11 ; 
 
 (5) = 140 30 34 .06 ; 
 
 (6) = 202 52 44 .68 ; 
 
 (7) = 71 25 30 .05 ; 
 
 (8) = 77 40 13 .97. 
 
 H.K In order to furnish practice, 
 
 the following observed angles 
 
 B.K >>^ \ / are taken from the author's 
 
 record-book. The corrected 
 results are given, so that those 
 adjusting them can verify their 
 
 C.T ---- """ work. This, however, can be 
 
 FlG - I4 done by seeing if the condi- 
 
 tions are fulfilled when the corrected values are taken. The 
 weights are equal, and so can be omitted.
 
 CALCULATION OF THE TRTANGULATION. 
 Observed. Corrected. 
 
 (1) CTto BK= 3 624 / 23 // .25 = 36 24' 22". 73 ; 
 
 (2) CTto HK= 49 53 49 .36 = 49 53 51 .61 ; 
 
 (3) CT to C = 95 06 40 .80 = 95 06 39 .05 ; 
 
 (4) BKtoHK= 13 29 31 .11 = 13 29 28 .86; 
 
 (5) BKto C = 58 42 14 .55 = 58 42 16 .30. 
 
 B.B 
 
 C 
 
 153 
 
 (I) 
 
 (2) 
 
 (3) 
 (4) 
 (5) 
 (6) 
 (7) 
 (8) 
 
 Observed. Corrected. 
 
 BB to C = 26 44' 50". 5 7 = 26 44' 57".82 ; 
 BB to /f = 62 55 56 .14 = 62 55 47 .315 
 BB to AT = 85 08 27 .43 = 85 08 29 .005 
 C to H = 36 10 41 .57 = 36 10 49 .495 
 C to BK = 58 23 31 .86 = 58 23 31 .185 
 H to BK =22 12 40 .79 = 22 12 41 .69; 
 H to CT = 53 09 ii .98 = 53 09 10 .18; 
 BK to CT = 30 56 26 .69 = 30 56 28 .49. 
 
 In a large number of condition equations the above opera- 
 tion may be considered long and tedious, so that one of the 
 following methods may be found preferable. 
 
 Suppose we have, as the result of the same number of related 
 quantities x, y, and 2, the values N^ N v and N 3 , giving the 
 equations :
 
 154 GEODETIC OPERATIONS. 
 
 ,* + *,.? + *,*... = JVi;i 
 
 tf + + '.'.'. = N,\ \ ' ' ' ' (A) 
 
 *!.,:=: jJ 
 
 in which the coefficients are known. As the number of un- 
 known quantities is less than the number of equations, a direct 
 solution is impossible. 
 
 Designating the errors by u, we can write equations at (A), 
 
 - N n = u n . 
 
 By the principles already stated, the most probable values 
 for these various quantities are those which render the sum of 
 the squares of the errors, z/, 2 -f it* . . . -\-n,?, a minimum. Plac- 
 ing all terms but those depending upon x, equal to M lt M t 
 . . . M M , equations at (B) will take the form 
 
 a^x -f- MI = u l ; -j 
 
 a t x -f M * = , ; I / 
 
 Taking the sum of the squares of both members of the equa- 
 tions at (C), we obtain 
 
 J... + (a n x + 3f.) = /+ , a . . . .'. 
 Differentiating this with respect to ^r, and placing the first dif-
 
 CALCULATION OF THE TRIANGULATION. 155 
 
 ferential coefficient equal to zero, we have, after dividing by 2, 
 '/ " 
 
 From this, we see that to form the most probable value for x, 
 we multiply each equation by the coefficient of x in that equa- 
 tion, add these products, and place the sum equal to zero. By 
 doing this with y and z we will obtain one equation for each 
 unknown quantity, from which each can be found by the or- 
 dinary methods of elimination. 
 
 To illustrate : x-\-2y -f- 2z 2 = o ; (i) 
 
 2x + y+ ,?+ 4 = 0; (2) 
 
 3* + y- ,3- -3 = 0; (3) 
 
 x 2y -f- 2z 8 = o. . . . . . . (4) 
 
 Multiply (i) by I, x -\-2y-\-2z 2=0; 
 
 multiply (2) by 2, 4* 2y 2z 8=0; 
 
 multiply (3) by 3, gx -f- 37 3-3- 9 = 0; 
 multiply (4) by i, x 2y -f- 2z 8=0; 
 
 by adding, 15-^+ y 3 27 = o (5) 
 
 Multiply (i) by 2, 2;tr -f- 47 + 4# 4 = o ; 
 
 multiply (2) by i, 2^- -(- jj/+ <sr + 4 = 0; 
 multiply (3) by i, 3^+J ^ 3=o; 
 
 multiply (4) by 2, 2.x -f- 4JK 4^ + 16 = o ; 
 by adding, x-{-ioy -|- 13 o. ... (6) 
 
 Likewise by multiplying (i) by 2, (2) by I, (3) by I, and (4) 
 by 2, and adding, we get 
 
 X+ 102- 13 = (/)
 
 1 5 6 
 
 GEODETIC OPERATIONS. 
 
 Eliminating (5), (6), and (7), which are called the normal 
 equations, by the usual algebraic method, we find x = 2, y = 
 1.5, and z 1.5. 
 
 To further illustrate this method, we 
 H.K will take another example : 
 
 C.T 
 
 FIG. 16. 
 
 (i) 
 
 = 87 47' 42".s ; 
 
 (a) H.KtoH = 144 17 47 .5; 
 
 (3) C to 77 = 56 30 09 .o ; 
 
 (4) 7 to C.T = 148 04 22 .5; 
 
 (5) H to C.T = 91 34 14 -SI 
 
 (6) C.TtoH.K=i24 07 29.5. 
 
 In this figure there are three condi- 
 tions to be fulfilled : 
 
 (3) + (5) = (4), (O + (3) = (2), and (2) + (5) + (6) = 360. 
 
 As the changes in these values will not likely affect any- 
 thing beyond the seconds, suppose we designate the seconds 
 of the angles by a, b . . . f, so that we will write the angles : 
 
 (i)= 87 47' + *"; 
 (2) = 144 17 +b ; 
 
 (6) = 124 07 +/ . 
 
 (3) + (5) = (4), (3)- 56 
 
 (5)= 9i 34'+'; 
 (3) + (5) = H8 04'+ c + e, (4) = 148 04' + d ; 
 
 therefore,
 
 CALCULATION OF THE TRIANGULATION. 157 
 
 Also (I) + (3) = (2), (i) = 8747'+ a ; 
 (3)= 5630'+^ 
 (i) + (3) = I44i7'+ * + c, (2) = 144 i/ + b ; 
 
 therefore, a -j- = b. 
 
 (2) + (5) + (6) = 360, (2) = 144 17'+ b ; 
 
 (5)= 9i 34 + *; 
 (6)= 124 07+/; 
 (2) + (5) + (6) -359 58' + + *+/= 360; 
 
 therefore, + ^ +/= 120". 
 
 By observation, # = 42".5 ; 
 
 b = 47"-5 J 
 
 ' = 09" ; 
 
 ^ = 22".5 ; 
 
 * = i4".s ; 
 
 /= 29".5. 
 From condition, c -{-e = d\ 
 
 Substituting in observation equations the values of d, and b 
 as determined by the conditional equations, we can write 
 
 a= 42".$; 
 
 a + c= 47".$ ( a + c) = 6: 
 
 c= 9" ; 
 
 c + e= 22".$ ( c + e) = d; 
 
 e= i 4 ". 5; 
 
 /= 29".5; 
 
 = I20 7 '.
 
 158 GEODETIC OPERATIONS. 
 
 From these we obtain the following normal equations : 
 
 2a + c = 90 ; 
 = 120; 
 = 79; 
 
 =i57; 
 fi + e + 2f = 149.5. 
 
 Solving for a, b, c t e, and/", by ordinary method of elimina- 
 tion, we find a= 4i"-i25, b = 48".8;5, = 7".75, ^=<r + ^ 
 = 49"-375> ' = 4i"-625, and/ = 2 9 ".5. 
 
 This gives for the angles the following as the most probable 
 values : 
 
 (i)= 87 47' 41 ". 1 25; 
 
 (2) = 144 17 48 .875 ; 
 
 (3) = 56 30 7 75 ; 
 
 (4) = 148 04 49 .375 ; 
 
 (5) = 9 1 34 4i -625 ; 
 
 (6) = 124 07 29 .5. 
 
 Observations with different weights can be adjusted by this 
 method. Since we do not use in this case the square of the 
 error, or some quantity involving the error squared, but only 
 the first power, we must therefore multiply the error, or 
 quantity involving the error, by the square root of the weight. 
 The weight can be determined from the probable error as ex- 
 plained on page 123, if not taken directly from the number of 
 measurements. 
 
 When it is desired to make use of this method for adjusting 
 observations of different weights, the outline of the method 
 may be given as follows. 
 
 For each of the observations write an observation equation.
 
 CALCULATION OF THE TRIANGULATION. 159 
 
 For each condition write a conditional equation. 
 
 From the conditional equations obtain as many values as 
 possible for one unknown quantity in terms of others, and sub- 
 stitute in the observation equations. Multiply each observa- 
 tion equation by the square root of its weight. Form the nor- 
 mal equations and solve as in ordinary cases. 
 
 While normal equations will afford an excellent solution for 
 any number of observation and conditional equations, the 
 labor becomes quite great when we have a large number of 
 equations, or large quantities to handle. 
 
 In such cases the method of correlatives as developed by 
 Gauss will afford the readiest solution. This method pertains 
 to equations of condition only, and in terms of corrections that 
 are to be applied to the various quantities in order to make 
 them fulfil the required conditions. 
 
 Suppose a, ft, y . . . represent the corrections, and the con- 
 ditional equations expressed in terms of these corrections with 
 coefficients whose values are known, as well as the absolute 
 term ; for instance, in the last example we had the condition 
 (3) + (5) - (4), but in reality (3) + (5) = (4) + i", or (3) + (5) 
 
 (4) = i". So if a-, /?, and y represent the corrections ap- 
 plied to (3), (4), and (5), their algebraic sum should equal 
 
 i", to counteract the error -{- i"; that is, a -j- y /3 = i". 
 In this case the known coefficients are i, and the absolute 
 term \" . So that, in general, we may express the condi- 
 tional equations in terms of known coefficients, and absolute 
 terms, with the corrections as the unknown quantities ; as 
 
 a v a + a,fi -f a,y . . . = M, ; 
 
 ,a -f n$ -f n 3 y . . . = M n .
 
 I6O GEODETIC OPERATIONS. 
 
 Since the most favorable results are obtained by making the 
 sum of the squares of the errors a minimum, if we take a^ 
 + a*0* + a *Y* a n9^ = M, and differentiate it with respect 
 to each variable, and making the first differential equal to zero, 
 we will have, after dividing by 2, 
 
 jlfi -{- a^dy . . . + a n dg> = o ; 
 
 jy ... + 
 
 m^dft + m^dy . . . -f- m n d<p o. 
 
 Also, a + f? + Y* + <P* = a minimum, or 
 
 a .da.-^- ft . d/3 -j- y . dy . + q> dg> = o. . . (B) 
 
 As the number of equations is less than the number of un- 
 known quantities, a part, as M, can be found in terms of the 
 others ; with these values substituted in equations (A), we will 
 have M less than originally, and each of these may be made 
 equal to zero. Chauvenet accomplishes this result in the fol- 
 lowing way: multiply the first equation at (A) by ,, the 
 second by k v the third by & 3 , and the nth by k n and equation (B) 
 by I ; then add these products. Now, supposing ,, k^ . . . 
 etc., are such that M of the differentials disappear, the final 
 equation will contain M' M (calling M' the original number) 
 differentials with M' equations. Making them severally equal 
 to zero, we get 
 
 c,k 3 . . . m,k m a = o ; 
 
 a n k n -f b n k n + c n k n . . . m n k n - cp = o. 
 Now, by multiplying the first by a v the second by a^ etc., and
 
 CALCULATION OF THE TRIANGULATION. 
 
 l6l 
 
 adding the products, expressing the sum of like terms by 2, 
 we get 
 
 Likewise, multiplying the first equation by b lt the next by 
 ^ . . . b n y we have 
 
 , -f 
 
 This will give as many normal equations as there are unknown 
 quantities k v etc.; so that we obtain a, ft, y, etc., in terms 
 of k v etc. While the theory of this is quite complicated 
 and involves a knowledge of differential equations, in practice 
 it is exceedingly simple, as the appended example will show: 
 
 44; 
 .01 ; 
 .46; 
 
 (1) B.RioC= 75 3i' 53' 
 
 (2) B.R to R = 144 36 49 
 
 (3) B.R to = 239 35 03 
 
 (4) C to R 69 05 oo .57 ; 
 
 (5) C toG = 164 02 51 .52; 
 
 (6) R to G 94 58 05 .44. 
 
 The conditions to be fulfilled are : 
 
 (i) + ( 4 )-(2)=o; 
 -(3)=o; 
 
 B.R 
 
 FIG. 
 
 However, we find that 
 
 (I) + (4) -(2)= 5" 
 (2) + (6) -(3)=- 9 ".oi 
 (4) + (6) -(5)= i4"-49
 
 162 
 
 GEODETIC OPERATIONS. 
 
 and the corrections necessary to neutralize these errors will be 
 5, -J-Q.OI, and 14.49- Indicating the corrections by the 
 same symbols we have used for the angles, and transposing the 
 constant needed, we will write the above equations, 
 
 (i) + (4) -(2)+ 5 =o, (a) 
 (2) + (6) -(3)- 9.01-0, (b) 
 (4) + (6) -(5) +14.49=0, (c) 
 
 Now we rule as many vertical columns as there are con- 
 ditions in this case, three, and as many horizontal ones as there 
 are quantities to correct, in this case six. 
 
 In the first condition we have -j- (i) 
 -f- (4) (2), so we write -|- k^ opposite 
 I, -\- k l opposite 4, and k l opposite 
 
 2. 
 
 The second condition has (2) + (6) (3); so we put -j- &, 
 opposite 2, and 6, and &, opposite 3. 
 
 The third condition involves (4) -\- (6) (5); so we put -f- k* 
 opposite 4 and 6, and s opposite 5. 
 
 The first equation of correlative is to contain the contents of 
 the 1st and 4th horizontal columns, and minus the contents of 
 the 2d ; this is determined by that equation having (i) -(- (4) 
 
 1st column contains -J-^,; 
 
 2d column contains -j- ^ ~~ &v [signs changed as it is (2)] 
 
 4th column contains -f~ , -f- &,; 
 
 ist correlative contains 3^ , -f- s .
 
 CALCULATION OF THE TRIANGULATION. 
 
 163 
 
 In the second conditional equation, we have (2) -j- (6) (3), 
 so we take the contents of the 2d and 6th horizontal columns 
 and the 3d with signs changed. 
 
 2d contains k r -f- , ; 
 
 6th contains -(- k^-\-k^ 
 
 3d contains (sign changed) -(- &, ; 
 
 (2) + (6) - (3) contains - k, + & + k v 
 
 2d correlative equation. 
 
 Likewise for the 3d we get 
 
 Placing these correlatives in the equations involving the cor- 
 rections, (a), (t>), and (c), we get 
 
 3^, -,+ ,+ 5 =0; 
 
 , + 3/&, + , 9- 01 = o , 
 
 .+ 4 + 3^ + 14-49 = o. 
 
 By ordinary process of elimination, we find , = 3 // -37, k^ = 
 
 Angle. 
 
 ISt. 
 
 2d. 
 
 3d. 
 
 Correction. 
 
 Corrected Angles. 
 
 I 
 
 +.-A 
 
 
 
 + 337 
 
 75 3 1 ' 56".8i 
 
 2 
 
 4^*i 
 
 + ^ a 
 
 
 - 3.37 + 687 
 
 144 36 52 .51 
 
 3 
 
 
 - ^2 
 
 
 - 6.87 
 
 239 34 56 .58 
 
 4 
 
 -fi, 
 
 
 + /&S 
 
 + 3-37 - 8.24 
 
 69 04 55 .70 
 
 5 
 
 
 
 ->&3 
 
 + 8.24 
 
 164 02 59 .76 
 
 6 
 
 
 ** 
 
 + 4i 
 
 + 6.87-8.24 
 
 94 58 04 .07 
 
 These corrections are determined in this way : 
 
 (1) is in the first condition and positive, so it is affected by 
 
 -{-*.; 
 
 (2) is in the first and second, negative in the first, and posi- 
 
 tive in the second ; therefore it is affected by k, and
 
 164 GEODETIC OPERATIONS. 
 
 (3) is negative in the second so it is corrected by k t ; 
 
 (4) is positive in the first and third, so its correction will be 
 
 ^M-MM 
 
 (5) is negative in the third, therefore its correction is k t ; 
 
 (6) is positive in the second and third, so it will be corrected 
 
 by +*, + *,. 
 
 These values of ,, k v and k^ applied as just indicated to the 
 observed angles, will give the most probable values for the 
 angles that will make them conformable to the conditions. 
 
 It may be noticed that the method of forming the equa- 
 tions of correlatives is the same as forming normal equations. 
 To illustrate, let us take (a) of the conditional equations ; the 
 coefficient of (i) is -{- I, of (4) is -}- I, and of (2) is i. 
 
 Multiply horizontal column i by -f i, = -j- A 
 multiply horizontal column 4 by -f- I, = -}- k, 
 multiply horizontal column 2 by i, = -\- k l 
 
 (I) + ( 4 ) -(2) = 3^ 
 
 therefore 3^, &, -f- &, + 5 = O. 
 
 This is the better plan when the coefficients are not unities, 
 When the observations have different weights, the operation 
 is somewhat complicated and can 
 be best explained by solving an 
 example ; 
 
 (1) Cto P = 107 53' oo". 07 weight 5; 
 
 (2) Cto A = 171 42 02 .18 weight 4; 
 
 (3) Cto B = 198 10 28 .22 weight 6; 
 (4)Pto A = 63 49 05 .86 weight 2; 
 ($)Pto = 90 17 16 .02 weight 3; 
 (6) A to B = 26 28 04 .54 weight i.
 
 CALCULATION OF THE TRIANGULATION. 
 The conditional equations are : 
 
 i6 5 
 
 ) + (5) -(3) -12.13-0; 
 -( 5 )- 5-62=0; 
 
 designating the corrections by the same symbols as the angles. 
 If the equations on page 159 had been weighted before dif- 
 ferentiation, a*, (?...(}? would have been multiplied by the 
 respective weights of the observation to which they were to 
 form corrections. These weights, say w lt >... /, being con- 
 stant factors, would remain in the differentials ; so that the 
 equations just referred to would have for their last term w,or, 
 iv^ft ... w n <t>. Then afterwards, when multiplied by # a v 
 etc., before summing the products, in order to get or, ft . . . <p 
 freed from factors, since we only know the values of these 
 errors unaffected by their weights, we must divide the first 
 equation by w lt second by w v etc. 
 
 We make the arrangement as though 
 there were no weights so far as the 
 position and signs of the correlatives 
 are concerned, but take the reciprocal 
 of the weight of the angle as the co- 
 efficient. 
 
 Angle. 
 
 *,. 
 
 *,. 
 
 *,. 
 
 I 
 
 + t*i 
 
 + i^ 3 
 
 
 2 
 
 i>j 
 
 
 
 3 
 
 
 ^2 
 
 
 4 
 
 + i*. 
 
 
 + i^s 
 
 5 
 
 
 ~f~ "i"^2 
 
 i^3 
 
 6 
 
 
 
 + 1*3 
 
 To illustrate : the first condition equation involves a correc- 
 tion to be applied positively to (i) and (4), and negatively to 
 (2). And since the reliability of these angles is proportional 
 to 5, 2, and 4, it is apparent that the corrections they should 
 receive would be in the inverse proportion, or ^, , and .
 
 1 66 GEODETIC OPERATIONS. 
 
 Therefore for the correction, this equation suggests that for 
 (l) we should write ,; for (4), /,; and for (2), \k^. 
 
 Second conditional equation involves corrections to(i)-|- 
 (5) (3). These angles are in point of accuracy proportional to 
 their weights, 5, 3, and 6; therefore the corrections will have 
 the inverse proportion, \, ^-, and \. So we write the correc- 
 tions ; the second conditional equation suggests for(i), -f- \k^ 
 for (5), +R: and for - (3), - \k v 
 
 Likewise in the third, for (4), -\-\k^ for (6), -f- \k t \ and for 
 
 Now, to form the equations, the first condition requires the 
 sum of the quantities in the first and fourth horizontal col- 
 umn, and the negative of the second. 
 
 (i) contains j 
 
 (4) contains 
 
 (2) contains \k^ 
 
 (i) + (4) - (2) contains (| + i + *)*, + \k % + 
 
 The second condition requires (i) + (5) (3). 
 
 (i) contains ^k l -\- \k t 
 
 (5) contains -|~ %k % f^ 3 ; 
 
 (3) contains -J- %k t ; 
 
 (0+ (5) ~ (3) contains ^ + (| + ^ + ^ s - ^,. (/) 
 
 Likewise, (4) contains , 
 
 (6) contains + &,; 
 
 (5) contains , -f \k 
 
 (4) + (6) - (5) contains -R - \k % + (i + i + t)^,- (^) 
 
 Clearing equations (e\ (/), and (g) of fractions and substi-
 
 CALCULATION OF THE TRIANGULATION. 
 
 167 
 
 luting them for the values of the corrections in (#), (), and (c), 
 we get 
 
 375=o; .... (A) 
 --^- 12.13 = 0; . . . 
 
 5-62-0; ... 
 
 (0 
 
 Reducing 57^, -f- I2&, + 3o s + 225.00 = o ; 
 6k, -j- 2i, - io s - 363-9 = 5 
 3^, - 2& 1 +ii, 33.72 = 0. 
 
 Eliminating by the usual process, we find that 
 
 k, i6".so, 4 = 28".o8, , = I2".6S. 
 The plan for applying these values can be best exhibited : 
 
 Angle. 
 
 ISt. 
 
 zd. 
 
 3d- 
 
 Correction. 
 
 Corrected Angle. 
 
 2 
 
 -tt 
 
 i^ 3 
 
 
 - 3-30+ 5-6l 
 - (-4-12) 
 
 107 53' 02". 38 
 171 42 06 .30 
 
 3 
 
 
 -M, 
 
 
 - 4.68 
 
 198 10 23 .54 
 
 4 
 
 i*. 
 
 
 |>&3 
 
 - 8.25+6.34 
 
 63 49 03 .95 
 
 5 
 
 
 Mi 
 
 -** 
 
 9.36 4.22 
 
 90 17 21 .16 
 
 6 
 
 
 
 *, 
 
 12.68 
 
 26 28 17 .22 
 
 (1) is corrected by , and \k v or 3.30 + 5-6i ; 
 
 (2) is corrected by , or \( 16.50) =4.12, etc. 
 
 In the case of weighted observation, the method of correla- 
 tives is far the simplest. 
 
 While station adjustment is of somewhat frequent occur- 
 rence, yet the angles regarded as forming parts of a triangle 
 more frequently require attention. The geometric require- 
 ment that the three angles of a triangle equal 180 furnishes a 
 condition to begin with ; likewise, these angles as individuals
 
 1 68 GEODETIC OPERATIONS. 
 
 may form a part of a station condition. In this case which is 
 the rule we combine station-adjustment with what is known 
 as figure-adjustment; that is, bringing the angles into con- 
 formity with the geometric requirements of the figure. 
 
 The first geometric condition is that the angles of the tri- 
 angles equal i8o |- spherical excess, or A -\- B -\- C = 
 180, in which, A, B, and 7 are the measured angles of the 
 triangle, and e = spherical excess. To find what the errors 
 are in the case of each triangle, it is necessary to determine 
 the value of . By geometry we know that the three angles of 
 a spherical triangle bear the same relation to four right angles 
 that its area bears to a hemisphere ; that is, : 27r::area : 27rr a , 
 
 area area 
 
 f = r-. being small, in seconds = . sin I . = -^ : .-.. 
 
 r r 3 sin i" 
 
 As the triangle is small compared with the surface of the 
 sphere, it may be regarded as equivalent to a plane triangle = 
 
 ^a.b . sin C, hence = i : ;, in which a, b, and C represent 
 
 the two sides and included angle, and r the radius of a sphere. 
 r can be considered a mean proportional between the radius 
 of curvature of a meridian and the normal of a point whose 
 position is the centre of the triangle. 
 
 On page 207, R -, , . , , XJ> N = i rci r * = 
 
 (i f sin* L}* (I ** sin* )* 
 
 N . R = -, _ a ~ a y-\i I dividing by I ? and neglecting terms 
 
 ^i 
 involving powers of e above the fourth, r" = 
 
 _ 2 f sin 1 L 
 
 d & 
 
 . ., J-N = ; a F- Substituting this val- 
 
 2 sm'Z) I + e cos 2L 
 
 f a.b&m 
 
 ue for r\ = j-r T , -. The factor . 
 
 2a sm I 2 a sin i" 
 
 varies with 2Z,, and can be computed with L as the variable 
 for every 30' and tabulated ; calling this term , =a . b . sin C . n.
 
 CALCULATION OF THE TRIANGULATION. 
 
 [69 
 
 The most elaborate spheroidal triangle-computation for 
 spherical excess shows that the result obtained by using the 
 above formula will differ from the correct value, only in the 
 thousandth part of a second. For preliminary field computa- 
 tion the excess may be taken as i" for every 200 square kilo- 
 metres, or 75.5 square miles ; and when the sides are 4 miles or 
 under, it can be disregarded. The following table contains n 
 for L, from 24 to 53 30', based upon Clarke's spheroid. The 
 table must be entered with the average latitude of the tri- 
 angle approximately. 
 
 Latitude. 
 
 Log*. 
 
 Latitude. 
 
 Log*. 
 
 Latitude. 
 
 Log*. 
 
 24 oo' 
 
 I.4059 6 
 
 34 oo' 
 
 1.40509 
 
 44 oo' 
 
 1.40410 
 
 24 30 
 
 92 
 
 34 30 
 
 05 
 
 44 30 
 
 05 
 
 25 oo 
 
 83 
 
 35 oo 
 
 00 
 
 45 oo 
 
 00 
 
 25 30 
 
 84 
 
 35 30 
 
 495 
 
 45 30 
 
 395 
 
 26 00 
 
 80 
 
 36 oo 
 
 9i 
 
 46 oo 
 
 90 
 
 26 30 
 
 76 
 
 36 30 
 
 86 
 
 46 30 
 
 85 
 
 27 oo 
 
 72 
 
 37 oo 
 
 81 
 
 47 oo 
 
 80 
 
 27 30 
 
 68 
 
 37 30 
 
 76 
 
 47 30 
 
 75 
 
 28 oo 
 
 64 
 
 38 oo 
 
 7i 
 
 48 oo 
 
 69 
 
 28 30 
 
 1.40559 
 
 38 30 
 
 I . 40466 
 
 48 30 
 
 1.40364 
 
 29 oo 
 
 55 
 
 39 oo 
 
 61 
 
 49 oo 
 
 59 
 
 29 30 
 
 5i 
 
 39 30 
 
 56 
 
 49 30 
 
 54 
 
 30 oo 
 
 47 
 
 40 oo 
 
 5i 
 
 50 oo 
 
 49 
 
 30 30 
 
 42 
 
 40 30 
 
 46 
 
 50 30 
 
 44 
 
 31 oo 
 
 37 
 
 41 oo 
 
 4i 
 
 51 oo 
 
 39 
 
 31 30 
 
 33 
 
 41 30 
 
 36 
 
 51 30 
 
 34 
 
 32 oo 
 
 28 
 
 42 oo 
 
 3i 
 
 52 oo 
 
 29 
 
 32 30 
 
 24 
 
 42 30 
 
 26 
 
 52 30 
 
 24 
 
 33 oo 
 
 19 
 
 43 oo 
 
 20 
 
 53 oo 
 
 19 
 
 33 30 
 
 1.40514 
 
 43 30 
 
 I.404I5 
 
 53 30 
 
 1.40314 
 
 The spherical excess computed by this formula is for the 
 entire triangle ; and, unless there is considerable difference in 
 the lengths of the opposite sides, one third of the excess is to 
 be deducted from each angle of the triangle ; but this reduced 
 value is used only in the triangle condition, and not in the 
 station condition.
 
 170 
 
 GEODETIC OPERATIONS. 
 
 If in this figure we have measured the angles numbered and 
 have the averages, which we will designate by the numbers, it 
 will be seen that a great variety 
 of conditions may be written. 
 But upon examination it will be- 
 come apparent that some of th'e 
 angles are indirectly two or more 
 times subjected to the same or 
 equivalent conditions. For in- 
 stance, if (3) + (7) + (9) = 180, 
 and (ii) + (13) + (i) = 1 80, the 
 condition that (2) + (7) -f- (10) -f- 
 (13) = 360 is already fulfilled. 
 Also, if (i) + ( 3 ) -(2), and (2) + 
 (5) = (4), the condition (i) +(3) + ($) = (4) is unnecessary; 
 and, again, if (3) + (7) + (9) = i*) , (i i) + (13) + (0 = 180, 
 and (14) + (2) -f- (6) = 180, then (i) + (3) = (2) is unneces- 
 sary. If we have the most probable value for (6) and (7), their 
 difference will be (8) without involving (8) in the adjustment; 
 or if we have the best values for (8) and (6), their sum will 
 give (7) ; or if we have (i), (2), and (4), we can find by subtrac- 
 tion the most probable value for (3) and (5). 
 
 From this we learn that it is useless to involve whole angles 
 and all of their parts in different conditions. With such a fig- 
 ure the following conditions would be sufficient : 
 
 FIG. 19. 
 
 (2) + (5) -(4); 
 
 (3) +(7) + (9) -i 80; 
 
 (8) + (10) + (12) =180; 
 
 (2) + (7) + (10) + (13) -360; 
 
 (13) 
 
 Other combinations could also be used.
 
 CALCULATION OF THE TRIANGULATION. 
 
 In such adjustments the method of correlatives should be 
 used, as the labor does not increase rapidly in proportion to 
 the increased number of conditions. 
 
 The equations like those given so far are called angle or an- 
 giilar equations. The theorem in trigonometry that the ratio 
 of sides is equal to the ratio of the sines of the opposite angles 
 
 gves us -j-~ = 
 
 . . . 
 
 Since A, B, and C are fixed points, 
 
 the distances AB and AC are constant; therefore -^repre- 
 sents a constant quantity, so that if (3) and (7) are changed at 
 all, the sine of (3) and its correction 
 must have the same ratio to the 
 sine of (7) and its correction that sin 
 (3) has to sin (7). This involves 
 another condition, which will now 
 be elaborated. 
 
 From the theorem just referred 
 to, we obtain the following equa- 
 tions : 
 
 OB _ sin b t (M _ sin b, OD _ sin b t OC _ sin b l 
 OA sin a/ OD ~~ sin a,' OC ~ sin ,' OB ~ sin / 
 
 Multiplying these equations together, member by member, we 
 obtain 
 
 OB . OA . OD . OC _ _ sin b t . sin b, . sin b t . sin b, 
 OA . OD . OC. OB ~ ~ sin a, . sina a . sin a, . sin a A 
 
 or, sin #, . sin a, . sin a 3 . sin a t = sin b t . sin b a . sin b 3 . sin b^
 
 1/2 GEODETIC OPERATIONS. 
 
 But these are the values after correction, so we will put M^ 
 M v M v M t , for a lt a,, a v a;, N lt N v N tt and N. for b n , b v 
 and b;, and denote the necessary corrections by v lt v,, v^ v 4 , and 
 x^x v x 3 , and x 4 . Substituting these values in the last equa- 
 tion, we have 
 
 sin (M, + f ,) . sin (M 3 -f v,) . sin (M, + z\) . sin (J/ 4 + z/ 4 ) 
 = sin (N, + ^) . sin (N n + ^) . sin (TV, -f ^r 8 ) . sin (N 4 
 
 or, passing to logs, 
 
 log sin (M, + ?,) + log sin (^f, -f- v,) + log sin (M, + ,) 
 
 + log sin (M t -f- ^ 4 ) 
 = log sin (^ + *,) + log sin (^ 2 + x t ) + log sin (N 3 + ^r 3 ) 
 
 Since v v v v v 3 , v<, x x v x^ x^ are very small, we may develop 
 each of the above terms by Taylor's theorem, stopping with 
 the first power of the correction : 
 
 , (d log sin M.\ 
 log sin (M, + Vl ) = log sin M, + - ^ 1 ** '> 
 
 ,.,,... , sn . 
 
 log sin (M, + v.) = log sin M % + - *jg* 
 
 etc., etc.; 
 
 Id log sin 
 
 . ,,, I 
 
 log sin (N, + ^) = log sin N, + ( 
 
 etc., etc.; 
 
 in which ?>,, v v v t , z/ 4 , ^r,, ^r t , x 9 , x are expressed in seconds, so
 
 CALCULATION OF THE TRIANGULATION. 173 
 
 that -- -rrf - 1 is the log difference for one tabular unit for 
 
 the angle M v or the tabular difference for M t ; let us call this 
 difference d d v d t , d for M M v M v M tt and # tf a , tf s , # 4 , for 
 
 A7 AT 1 AT ]\f 
 *jj -< v a> ^ V 3> " 
 
 Substituting these values in the lr.st equation, we have 
 
 log sin M l -f- d?X -f- log sin M t -\~ dju^ -|- log sin J/ 3 
 
 -f- log sin J/ 4 
 = log sin N, + tf,^, + log sin 7V, -f d,^, -f- log sin 7V 3 + 6^ t 
 
 -j- log sin ^V 4 -[- ^ 4 ^r 4 . 
 
 When transposed, 
 
 log sin M l + log sin J/ 2 -(- log sin J/ 3 -f- log sin M< 
 log sin A 7 , log sin N^ log sin N 3 log sin N t 
 
 = to + to + to + to - <ta - te - ^3^ 3 - < 4 
 
 an equation in which the unknown quantities are the correc- 
 tions to the angles, or the same quantities that are sought in 
 the adjusting equations. 
 
 This gives directly an equation of condition, for since the 
 sum of the log sines of M t , M^ M 3 , and M t should equal the 
 sum of the log sines of N^ N^ N^, and N v the corrections d 1 v l 
 + df>i + d t v t + dp t should equal d lX , -\- 8^ -f d^ + djc c 
 But if the log sines of (M) differ from the log sines of (TV), then 
 that amount of difference must be corrected in ef ] v l -j- djt t . . . 
 etc. This is called the linear equation. 
 
 By way of illustration, suppose we have the appended fig- 
 ure with the average angles as given : 
 
 (i)= 5o3i'i3".68; 
 (2}= 14 51 47 .88; 
 (3) not needed ;
 
 174 
 
 GEODETIC OPERATIONS. 
 
 (4)= 
 (5)= 
 (6) = 
 (7) 
 (8) = 
 
 71 46' 16-36; 
 82 32 49 .52; 
 32 04 12 .49; 
 not needed ; 
 30 03 29 .39; 
 
 (9) =133 03 52 .48; 
 (10) = 67 23 18 .99; 
 (n) not needed; 
 (12) = 57 42 49 .56. 
 
 We first deduce the linear equa- 
 tion : 
 
 //#.sin(io) = HP. sin (8); 
 
 by multiplication 1 ,. 
 
 sin (2) . sifl (lo) . sin (4) := sin (6) . sin (g) sin (12)* 
 
 Writing for tabular difference <?(2), <?(lo), etc, and [2], [10], 
 etc. as the corrections for (2), (10), etc., we have 
 
 log sin (2) + 8(2} [2] + log sin (lO) -f J(iO) [10] 
 
 -{-logsin(4) + d(4)[ 4 ] 
 = log sin (6) + <J(6) [6] + log sin (8) -f T(8) [8] 
 
 + log sin (12) -f <?(i2) [12], 
 
 From the table of logs, we find : 
 
 log sin (6) = 9.72505722, 8 (6) = .00000336; 
 log sin (8) = 9.69978200, S (g) = 364 ; 
 
 log sin (13) = 9.92705722, 6(l2]= 133; 
 
 sum = 29.35189644.
 
 CALCULATION OF THE TRIANGULATION. 175 
 
 log sin (2) = 9.40910559, d (2) = .00000794; 
 
 log sin (4)= 9.97763813, * (4) = 69; 
 
 logsin(io) = 9-96526395, tf(io) = 87; 
 
 sum 29.35200767. 
 
 log sin (2) -J- log sin (4) -(- log sin (10) 
 
 = log sin (6) -f log sin (8) -f- log sin (i 2) -(- 0.00011123. 
 
 As the corrections are to neutralize this difference, we write 
 
 2] - O.OOOIII23. 
 
 Substituting for #(2), #(4), etc., their values, we have, after 
 multiplying by loooooo to avoid decimals, 
 
 7 . 94 [2] + .8 7 [io] 
 
 = I-33M 4- 3.6 4 [8] + 3-36[6] - 111.23. 
 
 Transposing and passing to our usual notation, 
 
 7.94(2) + .87(10) + .69(4) - 1-33(12) - 3-64(8) - 3-36(6) 
 
 -f- 111.23 = o. 
 
 The angle equations are those involving the angles that 
 will not be doubly adjusted. In the present case they will be, 
 when expressed in terms of their corrections, 
 
 (5) + (i)- (9) + 10.72-0; 
 (9) + (2)+ (6)- 7-15 = 0; 
 (i) + (4) + (12) +19.60 = 0; 
 (5) + (8) + (10) -22.10 = 0;
 
 176 GEODETIC OPERATIONS. 
 
 7.94(2) + .87(10) + .69(4) - 1.33(12) - 3-64(8) - 3.36(6) 
 
 + 111.23 = o. 
 
 In this (3) is omitted, since if (2) and (12) are known, (3) can 
 be found by subtraction. Likewise, (11) is the sum of (8) and 
 (4); also (7), the difference between (10) and (6). So we now 
 simply form the correlative equations from these five condi- 
 tional equations. 
 
 
 ISt. 
 
 zd. 
 
 3d. 
 
 4 th. 
 
 5th. 
 
 I 
 
 *i 
 
 
 * 3 
 
 
 
 2 
 
 
 * 2 
 
 
 
 7-94*6 
 
 4 
 
 
 
 k$ 
 
 
 . 6o>&5 
 
 5 
 
 *i 
 
 
 
 *4 
 
 
 6 
 
 
 *a 
 
 
 
 3 36^5 
 
 8 
 
 
 
 
 4 
 
 - 3^64*5 
 
 9 
 
 *i 
 
 *a 
 
 
 
 
 10 
 
 
 
 
 *4 
 
 .87*5 
 
 12 
 
 
 
 k3 
 
 
 - 1.33*6 
 
 The formation of the first four normal equations follows the 
 principles repeatedly given, but as something new may appear 
 in obtaining the fifth equation, it will be formed in detail. 
 
 7.94 times column 2 = 
 
 .87 times column 10 = 
 
 .69 times column 4 = 
 
 1.33 times column 12 = 
 
 3.64 times column 8 = 
 
 3.36 times column 6 = 
 
 Total, 
 
 7-94^, +63.04364; 
 
 .87*4+ -7569^; 
 
 .69^ + .4761^; 
 
 -i-33^ + 1.7689^.; 
 
 -3-64^+13.24964; 
 3-364 +11.28964; 
 
 4.584- .644-2.774+90.58474. 
 
 Barlow's table of squares will facilitate work, as the coeffi- 
 cients of the terms in the side equation are squared in finding
 
 CALCULATION OF THE TRIANGULATION. 
 
 177 
 
 the coefficient of the correlative corresponding to the equation 
 of condition formed by the side equation. In this case, the 
 fifth conditional equation is the side equation, and the coeffi- 
 cients of k 6 in the fifth normal equation are the squares of 
 7.94, etc. 
 
 The normal equations are : 
 
 3/fc, k t + , + k< + 10.72 = o; 
 
 - h + 3^ 3 + 4-58 6 - 7-15 =o; 
 
 + k, + 3 8 - .64^ + 19-60 = 0; 
 
 k, + 3/& 4 - 2.77^- 9.29 = o; 
 
 4.58^ - .64^, - 2.77k, -f 90.58^ + 111.23 = o. 
 
 The solution of these equations gives k l = "-53, &, = 4 // -4O, 
 , = 6". 66, ki = i ".94, k b = i ".43. 
 
 These values are applied to the various angles as indicated 
 in the table just given. For instance, (2) is to be corrected by 
 , and 7.94 times k b . 
 
 The best rule that can be given for the formation of side 
 equations is to regard one of the ver- 
 tices as the vertex of a pyramid, with 
 the figure formed by the other points 
 as the base, and take the product of 
 the sines of the angles in one direction, 
 equal to the product of the sines in the 
 opposite direction. 
 
 Take H as the vertex, and WPB as 
 the base ; then, 
 
 sin HWP. sin HPB . sin HBW 
 
 = sin HBP. sin HPW. sin HWB. ; 
 
 that is, sin (12) . sin (8) . sin (6) = sin (10) . sin (4) . sin (2), as was 
 otherwise obtained. The angles at the point used as the ver- 
 
 12
 
 1/8 GEODETIC OPERATIONS. 
 
 tex are not involved in this equation, so they must be involved 
 in a station adjustment, or in a triangle condition. 
 
 If one should find it difficult to conceive a pyramid con- 
 structed in this way, he can without trouble secure the side 
 equation in the manner made use of on page 174, in which we 
 started from HW '. sin (2) = HB . sin (6). 
 
 In the next equation obtain a value of HB. in another tri- 
 angle, as HB . sin (10) = HP. sin (8) ; then in terms of HP., as 
 HP. sin 4=ffW. sin 12. 
 
 This is as far as we can go, as we have returned to the start- 
 ing-point. Suppose we start from WP. 
 
 WP.sin(ii) = W#.sin(7); 
 
 WSsin (6) = Wtf.sin(g); 
 
 WH.sm(i) = WP.sin(4)' t 
 
 by multiplying, sin (i i) . sin (6) . sin (i) = sin (7) . sin (9) . sin (4). 
 The same can be obtained by taking Was the vertex, and 
 BHP as the base, the angles in one direction will give 
 
 sin WPH. sin WHB. sin WP=s'm WBH. sin WHP. sin WPB. 
 
 In writing down the equations to be used, a good plan is to 
 put down the sides emanating from the pole to all the other 
 points, putting the line first in the first member, and then in 
 the second ; as, 
 
 coming back to the first line used. Then we put in the angle 
 that is opposite the side in the other term ; as, (i i) opposite
 
 CALCULATION OF THE TRIANGULATION. 179 
 
 WB, (7) opposite WP, in accordance with the trigonometric 
 theorem. 
 
 The following rule, so frequently quoted, is taken from 
 Schott (C. S. Report, 1854). 
 
 The only choice in selecting the station to be used as the ver- 
 tex, or pole, as it is sometimes called, is to take that vertex at 
 which the triangles meet which form the triangle equations of 
 condition, and to avoid small angles, since the tabular differ- 
 ences, being large, will give unwieldy coefficients. It is some- 
 times difficult to determine the precise number of condition 
 equations that can be formed. 
 
 The least number of lines necessary to form a closed figure 
 by connecting/ points is/, and gives one angular condition. 
 Every additional line, which must necessarily have been ob- 
 served in both directions, furnishes a condition ; hence a sys- 
 tem of / lines between / points, I p-\-\ angle equations, 
 where it must be borne in mind that each of the / lines must 
 have both a forward and a backward sight. 
 
 When, in any system, the first two points are determined in 
 reference to one another by the measurement of the line join- 
 ing, then the determination of the position of any additional 
 station requires two sides, or necessarily two directions; hence 
 in any system of triangles between/ points, we have to deter- 
 mine / 2 points, which require 2(p 2) directions, or by 
 adding the first 2p 3. Consequently, in a system of / lines, 
 / (2p 3), or / 2/ -\- 3 sides are supernumerary, and give 
 an equal number of side equations. 
 
 We have, therefore, 
 
 / / -j~ I angle equations ; 
 / 2/> -|- 3 side equations ; 
 2/ 3^ + 4 in all. 
 
 It is apparent that each point may be taken as the pole, and
 
 1 80 GEODETIC OPERATIONS. 
 
 as many side equations formed as there are vertices. In a 
 quadrilateral, for instance, if four side equations are formed, the 
 fourth equation would involve the identical corrections con- 
 tained in the others. Since there are only 12 angles in all, 
 these can be incorporated in two equations, each of which con- 
 tains 6 angle corrections. 
 
 From the formulas just given, it will be seen that 4 condi- 
 tional equations will be sufficient in a quadrilateral ; I side 
 and 3 angle equations, or 2 side and 2 angle equations, but 
 never more than 2 side equations. 
 
 The method of station adjustment differs somewhat from 
 the foregoing when the values of the angles depend upon di- 
 rections. 
 
 In nearly all refined geodetic work angles are so determined ; 
 that is, the zero of the circle is set at any position, the tele- 
 scope is pointed upon the first signal to the left, and the mi- 
 crometers or verniers read ; the telescope is then pointed to 
 each in succession and the readings recorded. After reading 
 the circle at the last pointing, this signal is again bisected and 
 readings made, likewise with the others in the reverse order. 
 The telescope is reversed in its Y's and a similar forward and 
 backward set of pointings and readings made. These form a 
 set. The circle is then shifted into a new position and another 
 set observed, as already described. The average of the direct 
 and reversed readings of each series is taken as a single deter- 
 mination of a direction. 
 
 Let x be the angle between the 
 
 zero of the instrument and the direc- 
 tion of the first line, A, B, C, etc., 
 lg the angles the other lines make with 
 the first, whose most probable val- 
 FIG ues are to be determined, and let 
 
 / ,*, m* . . . be the reading of the 
 circle when pointing to the signals in order, of which x^ is the
 
 CALCULATION OF THE TRIANGULAT10N. l8l 
 
 most probable, and the errors of observation m l x^. Suppos- 
 ing no errors existed, we should have the following equations: 
 
 t l x^ A = o ; 
 * x^ C o. 
 
 The second series would give 
 
 m i x i I m * x i A = o ; 
 
 m* x^ B = o ; m t 3 x^ C = o ; 
 
 and the wth, m n x n = o ; win x n A = o ; 
 
 vi, i x n B = o ; win x n C = o. 
 
 The most probable values will be those the sum of the 
 squares of whose errors is a minimum. Also, the errors 
 squared must be multiplied by the corresponding weights,/,, 
 P\>P\ - ' Pv P* which will give 
 
 - *, - ^) a + A f - *. - 
 
 etc., etc. 
 
 Differentiating with respect to x^ x v x z . . . A, B, C . . . and 
 placing the differential coefficients separately equal to zero, we 
 shall have
 
 182 
 
 GEODETIC OPERATIONS. 
 
 V ;/ i + AX 1 + AX 2 + AX" 
 = (A +A 1 +A a + A' X+A 1 
 
 A', + AX 1 +AX 2 +AX 3 
 
 = (A +A 1 +A 2 +A 3 K+A'^+A'^+A 3 ^ - ; [ (A) 
 
 A*.+AX' +AX 2 +A 3 ^s 3 
 
 = (A + A 1 + A 9 +A 3 ;*.+A' 
 etc., etc.; 
 
 A'^i' H A 1 ^ 1 + AX 1 + 
 
 - (A 1 + A 1 +A' M + AX +AX +AX . . - 
 
 AX 1 + AX 1 +AX 1 + 
 
 = (A 2 + A 2 +A 2 )& +/.X +AX -f AX . . . 
 
 ' = (A 3 +A a +A 3 - )<^+AX +AX +AX . 
 
 In these equations x^ m lt x^ m v x^ m 3 . . . are the 
 errors of observation ; calling these ;r,, x v x^ . . . they will rep- 
 resent the corrections of the first, second, third . . . pointings 
 from the zero-mark usually a small quantity. 
 
 By multiplying out the parenthesis in the second member of 
 (A), and transposing all the terms from the first, we have 
 
 o = A*,- A^+AX-AX'+AX- AX'+AX- AX' 
 
 Introduce into each parenthesis #/, ; except the first, 
 
 JO. . . ',
 
 CALCULATION OF THE TRIANGULATION. 183 
 
 For .*, T#, substitute ^r,, and for aw, 1 ;;/, write ;,'; re- 
 membering that m t s , which is to take the place of mf m e , does 
 not mean the /th reading on the sth arc, as recorded, but the 
 recorded reading minus the reading of the zero on that arc. 
 
 This will reduce the last equation to 
 
 o = A-*'i +AX AX' + AX AX'+AX AX* 
 
 AX' + AX'+AX 
 = (A +A 1 +A 2 
 
 In the same manner the other equations (A) reduce to 
 
 AX'+AX'+AX'--. 
 
 - (A + A 1 + A 2 + A 3 K+ A 
 
 AX'+AX'+AX 3 -.. 
 
 - (A + A 1 + A 2 + A 3 K+ 
 
 Likewise, equations (B) reduce to 
 
 AX'+AX'+AX 1 - 
 
 = (A 1 + A 1 +A 1 "-)A + AX +AX + AX 
 
 AX 2 + AX 3 + AX 2 --- 
 
 - (A'+A 2 +A 2 ..'.)* +AX +AX +AX . 
 
 AX 3 +AX 3 +AX 3 .. 
 
 = (A 1 +A 3 + A 3 - )C + AX + AX + AX 
 
 (D) 
 
 When the signals observed upon are numerous, the solution 
 of equations (C) and (D) would be very laborious.
 
 1 84 GEODETIC OPERATIONS. 
 
 Captain Yollond, of the Ordnance Survey of Great Britain, 
 found the method of successive approximations sufficiently 
 accurate. 
 
 Suppose x^ x^ ^3 ... severally equal to zero in (D), from 
 which we find the first approximation : 
 
 A , _ AX' + AX 1 + AX 1 . . 
 _ AX 3 + AX* + AX' . 
 
 ~~ ~T5 I T~5 I Ta 
 
 _ AX 8 + AX 8 + AX 8 
 A 3 +A 3 +A 3 ..- 
 
 Substituting these values in (C), we obtain a new value for x^ 
 
 _p*(m*A')-\-p?(m? B') + p*(m* C) . . . 
 ** A+A 1 
 
 A + A'+A' + A 3 ... 
 
 Substituting these values in (D), we obtain the second approxi- 
 mation, or 
 
 
 A'+A'+A 1 .-.
 
 CALCULATION OF THE TRIANGULATION. 1 8$ 
 
 _ /.' - *.) + A 2 - *.) + A' -*.)..-. 
 
 A'+A'+A 1 .-. 
 
 _ A 8 - *.) + A' - *.) + A' - *.) 
 
 '' 1 
 
 The values can be further substituted in (C) and the result- 
 ing values of x^ x v x^ . . . placed in (D) for the third approxi- 
 mation for A, 13, C . . . However, the second has been found 
 sufficient in good work. 
 
 The weights for observed directions is unity, and zero for 
 any directions that could not be observed. The work can be 
 materially shortened by pointing on the first object on the left, 
 as the beginning of each series; and in each successive series 
 the readings of the first direction should be diminished by the 
 preceding direction, in this way taking as a zero the first direc- 
 tion of each series. 
 
 In the ordnance survey, the readings on the initial object 
 were made the same in the different series by adding to the 
 average readings of the microscopes on each signal such a 
 quantity, positive or negative, as to make the initial readings 
 the same. 
 
 Considering the weights unity, 
 
 m* + m, 1 -f- m, 1 . . . 
 
 si = - , 
 
 where n represents the number of series, or A' = the arithmet- 
 ical mean, say M^ in the same way we find 
 
 B' = M V C = M t ... 
 
 Substituting these values in the expressions for x v x^ , . . we 
 have
 
 1 86 GEODETIC OPERATIONS. 
 
 , = ~ -M,+ m? - M, + m? - M 9 . . . ), 
 
 and similarly for x^ x t ... we get jfcf, 1 , M 3 l . . . 
 Placing these values in the second approximation, 
 
 A" = -- MS + mJ-MJ + mS-M, 1 . . .) 
 
 B" = l -W- M; + <- M; + < -M; . . . ); 
 
 C" = l -(m? - Ml + < - ^,' + ' s s -'> 
 
 We have first obtained a constant reading for the initial di- 
 rection, either its angular distance from an azimuth-mark, of 
 by making the first direction zero. We then found the aver- 
 age of each direction, giving A' = M^ B' M^ ... or the 
 arithmetical mean as the first approximation. Next we sub- 
 tracted each average from each reading, giving a set of errors 
 the average of those in the same series giving M*, M* . . . 
 
 Afterwards these are taken from the readings of the corre- 
 sponding series, giving diminished values of each direction ; 
 and the average of these diminished directions gives the second 
 approximation. 
 
 A symbolic analysis can be seen in the appended table, fol- 
 lowed by an example taken from the Report of the Ordnance 
 Survey, 1858, page 65:
 
 CALCULATION OF THE TRIANGULATION. 
 
 I8 7 
 
 Initial Object. 
 
 A. 
 
 B. 
 
 C. 
 
 Averages. 
 
 m 
 m 
 m 
 
 ml 
 ml 
 
 ml 
 ml 
 ml 
 
 ml 
 ml 
 
 
 Average 
 
 Mi 
 
 Mt 
 
 M 3 
 
 
 
 m 1 M 
 
 ml - M, 
 
 
 Ml 
 
 
 ml - Mi 
 
 ml M, 
 
 ml - M 
 
 M 3 ' 
 Ml 
 
 
 ml Ml 
 ml - Ml 
 ml Ml 
 
 Ml - Ml 
 ml - Ml 
 ml - Ml 
 
 ml - Ml 
 ml - Ml 
 ml - Ml 
 
 
 Averages... . 
 
 A" 
 
 B" 
 
 C" 
 
 
 No. of 
 Series. 
 
 Initial O. 
 
 A - ii* 7'. 
 
 B = 37 34'- 
 
 C = 97" 54'- 
 
 D = 220 3 '. 
 
 Average 
 Errors. 
 
 2 
 
 42l'29".2I 
 29 .21 
 
 36' '.04 
 35 9 1 
 
 I4".07 
 
 47"-84 
 
 19". oo 
 
 18 18 
 
 
 3 
 
 29 .21 
 
 34 - 2I 
 
 ir .86 
 
 
 
 
 4 
 5 
 6 
 
 29 .21 
 29 .21 
 2Q 21 
 
 32 4i 
 
 10 .71 
 ii .91 
 
 46 .05 
 
 48 .30 
 
 16 .30 
 14 .17 
 18 *^Q 
 
 
 
 
 
 
 
 
 
 Average 
 
 29". 21 
 
 34"- 64 
 
 12". 14 
 
 47"-4 
 
 i7"-25 
 
 
 Errors. 
 
 oo".oo 
 
 ,OO 
 
 + i"-40 
 
 4*- I 27 
 
 + i"-93 
 
 4- o". 44 
 
 + i"-75 
 
 + O Q7 
 
 -f- i".io 
 
 
 .00 
 
 .00 
 
 - o .43 
 
 - 2 .23 
 
 o .28 
 - i -43 
 
 - i -35 
 
 - o .95 
 3 08 
 
 o .23 
 - I .19 
 
 
 oo 
 
 
 
 
 + 1 34 
 
 4- o 67 
 
 
 
 
 
 
 
 
 
 28". 1 1 
 28 48 
 
 34"- 94 
 35 18 
 
 I2". 9 7 
 
 46". 74 
 
 17". 9 
 17 45 
 
 
 
 29 44 
 
 
 12 OQ 
 
 
 
 
 
 30 .40 
 29 .81 
 28 <u 
 
 33 -60 
 
 II .90 
 
 12 .51 
 
 46 .28 
 
 48 .90 
 
 17 .49 
 
 M -77 
 
 
 
 
 
 
 , 
 
 
 
 Average 
 
 29". 13 
 
 34"-54 
 
 I2". 3 7 
 
 47"-3i 
 
 I7".ii 

 
 1 88 GEODETIC OPERATIONS. 
 
 This gives the directions as follows : 
 
 Initial object = 4 2i / 29 // .i3 ; 
 direction A" II 7 34 .54; 
 direction B" 37 34 12 .37; 
 direction C" = 97 54 47 .31; 
 .direction D" = 220 3 17 .11. 
 
 The third approximation, obtained in the same way, gave. 
 omitting degrees and minutes: initial object = 29 // .i2, A'" 
 34".55, B"' 12". 40, C" = 47"-34, D"' = I7".o8, values dif- 
 fering from the above in the hundredths place only. 
 
 The angles depending upon these directions will be involved 
 in the figure-adjustment, so their corrected values should be 
 written A + (i), B + (2), +(3) ... in which (i), (2), (3) ... 
 are the corrections obtained in the figure-adjustment. In this 
 operation the directions obtained at different stations have not 
 the same weight ; however, this can be computed from the 
 
 formula already given on page 125, where we found p = ^ 
 i n* 
 
 n* 
 
 So we find the residuals by taking the difference between 
 the individual diminished measures and the average, and di- 
 vide the number of readings on that direction squared by 
 twice the sum of the squares of the residuals ; in the case of 
 
 16 
 = 
 
 To illustrate the formation of the equations of geometric 
 condition let us take an example. 
 The ajigles adjusted at stations are :
 
 CALCULATION OF THE TRIANGULATION. 
 
 I8 9 
 
 at T, M = oooo'oo" 
 
 F= 83 30 34 -866 + (i) 
 
 W 287 14 13 .822 + (2) 
 at M, T oo oo oo 
 
 F= 66 56 10 .619 + (4) 
 
 F= 293 57 16 .395+ (6) 
 at F, W = oo oo oo 
 
 J/ = 20 oo 09 .436 + (7) 
 
 T= 349 33 27 -528 + (i i) 
 at W, F oo oo oo 
 
 7- = 13 17 05 .983 -f (12) 
 M= 332 59 01 -843 +(15). 
 
 In the triangle MTF, we are to find the angles at each ver- 
 tex, as follows : 
 
 833o'34".S66+(i) = MTF; 
 
 66 243 .605 (6) =FMT, or 360 direction 
 
 Ftrom T; 
 
 302641 .9o8+(7)-(n) = MFT\ 
 
 i8ooo'oo' / .379+(i)-(6)+(7) (i i)= sum ; 
 
 1800000 .015 = 1 80 + spherical excess ; 
 
 o = o".364 + (i) - (6) + (7) - (i i). Equation (I) 
 
 To find MFT, we subtract the direction of T from W, from 
 360; this gives angle WFT; to this add the direction of M 
 from W, or the angle WFM. 
 
 In the triangle TMW, 
 
 7245 / 4 6 // .i78-(2) = MTW\ 
 
 66 56 10 .6i 9 +(4) = WMT\ 
 
 40 18 4 .I40+(i2) (15) =MWT; 
 
 i8ooo'oo".937-(2)+(4)+(i2)-(i5)= sum ; 
 
 1 80 oo oo .01 1 = i8o+spherical excess ; 
 
 = + // . 9 26-(2)+( 4 )+(i2)-(i5). Equation (II)
 
 igO GEODETIC OPERATIONS. 
 
 In the triangle WTF, 
 
 I 3 I7'05". 9 83+(I2) = TWF; 
 
 156 1621 .044+(i)-(2) = WTF\ 
 
 102632 .472 (11) = TFW; 
 
 i7959'59"499+(0-( 2 )-( II )+( 12 ) = sum '> 
 
 1800000 .009 = 1 80 -[-spherical excess; 
 
 o= -o".5io + (i)-(2)-(n)+(i2). Equation (III) 
 
 In the quadrilateral TMFW, the side equation is 
 
 _ sin TMW. sin FWT . sin TFM 
 ~ sin MWT. sin TFW. sin FMT' 
 
 sin TMW = 9.9638207,6 + 8.965(4) (8.965 = tab. dif.) ; 
 
 sin FWT = 9.3613403,1+ 89.174(12); 
 
 sin TFM = 9.7047600,1 + 35-824[(7) (n)] ; 
 
 29.0299210,8. 
 
 = 9.8107734,2 -f 24.826[(i2) - (15)] ; 
 sin TFW = 9.2582687,7 114.245(11); 
 sin FMT = 9.9608833,6 - 9.354(6) ; 
 29.0299255,5. 
 
 o = - 44-7 + 8.965(4) + 9.354(6) + 35-824(7) + 78.421(1 1) 
 
 4- 64. 348(1 2) + 24.826(1 5). Equation (IV) 
 
 These four equations are solved for the unknowns, which 
 are applied to the given directions with their proper signs, or 
 to the angles directly, as just deduced.
 
 CALCULATION OF THE TRIANGULATION. 19! 
 
 In an extended triangulation, the position of every point is 
 influenced to a certain extent by the directions at the adjacent 
 signals ; consequently, it is advisable to include in the equations 
 of condition as many directions as possible. The influence of 
 these directions upon an initial point diminishes with the dis- 
 tance, and finally becomes inappreciable, so that the triangula- 
 tion can be divided into segments, each containing a conven- 
 ient number of conditional equations. The corrections of the 
 first are computed, and, as far as they go, these corrected val- 
 ues are substituted in the equations of condition in the second 
 figure, and the sum of the squares of the remaining errors, each 
 multiplied by its corresponding weight, made a minimum. 
 
 The equations of condition (I), (II), (III), (IV) . . . may be 
 written 
 
 o = a -f a,x, 
 o = b + b,x, 
 o = c + c^ + cs. . . ; h (E) 
 
 If /,,/ ... be the weights, corresponding to the corrections 
 #,,.*:.,..., the requirement that the sum of the squares of the 
 errors be a minimum is 
 
 A*i* + A*.* +A*i* = a minimum. . . (F) 
 
 Differentiating (E) and (F), we have 
 
 o = a,dx v -f a,d 
 o = b,dx, -f- bji 
 o = c,dx, + cjx* + c t dx t . . . 
 
 o =
 
 192 GEODETIC OPERATIONS. 
 
 Solving these equations as explained on page 160, we have 
 
 . . . (G) 
 
 Substituting the values of x x v x^ ... as found in these equa- 
 tions in (E), we have 
 
 or, o = 
 
 In the same way, remembering that (a 2 ) is the sum of the 
 squares of quantities like a, as a?-{- a?-\- a* . . . and (ab) = a 1 6 1 
 
 (H) 
 
 / //,..., being auxiliary multipliers, have their values ob- 
 tained from (H) and substituted in (G), giving the numerical 
 values of x^, x x^ . . . 
 
 Instead of using f lt f t . . . the Roman numerals I, II, III . . . 
 will be found more convenient, especially when the conditional 
 equations are so numbered. The normal equations can be 
 more readily formed. 
 
 To illustrate, suppose we have the following equations of 
 condition :
 
 CALCULATION OF THE TRIANGULATION. 193 
 
 I, O - - I.4042-(2) + (5) - (7) + (8) ; 
 
 II, o = - 2. 7 737-(2) + (4) ; 
 
 III, o=-o. 9 59S-(8) + (n); 
 
 IV, o= - 1.2157 (3) + (4); 
 
 V, o = - o. 9 2o 4 -(3) + (5) - (7) + (10) ; 
 VI, o =- 0.8424-0) + (4); 
 VII,o=-o. 3 20i-0) + (5)-(7) + (9); I m 
 
 vin, o = +0.999 -(O + (3); 
 
 XXV, o= + 
 
 -6.188(7); 
 etc., etc. 
 
 (i), (2), (3) ... represent the corrections to directions of the 
 same number ; then we multiply the terms involving (i), (2), 
 by the reciprocals of their weights, giving 
 
 (i) = o.oSooVI o.oSooVII o.oSooVIII 
 
 ((i) occurs in VI, VII, and VIII, and 0.800 is the reciprocal of 
 
 its weight) ; 
 
 (2) = 0.2060! o.2o6oll 0.0000309XXV ; 
 ( 3 )=-o.i58oIV-o.i58oV + 
 
 (4) ^+o.33SoII+o.3 3 SoIV+o.338oVI+5.263302XXV ; 
 
 (5) =+0.2260! +o.226oV + o.226oVII- 3 .5i 9 22XXV ; 
 
 etc., etc. 
 
 (K) 
 
 These values of (i), (2), (3) ... are substituted in the equa- 
 tions of condition (I), giving numerical values for I, II, III . . . ; 
 13
 
 194 GEODETIC OPERATIONS. 
 
 then these values substituted in equations (K) give the val- 
 ues of the corrections (i), (2), (3) . . . , which, when applied to 
 the directions, will give their most probable values, satisfying 
 the geometric conditions. 
 
 For the various methods of adjustments, see : 
 
 Jordan, Handbuch der Vermessungskunde, vol. i., pp. 339- 
 
 34 6 - 
 
 Bessel, Gradmessung in Ostpreussen, pp. 52-205. 
 
 Clarke, Geodesy, pp. 216-243. 
 
 Wright, Treatise on the Adjustments of Observations, pp. 
 250-348. 
 
 Ordnance Survey, Account of Principal Triangulation, pp. 
 354-416. 
 
 C. and G. Survey Report for 1854, pp. 63-95. 
 
 Die Konigliche Preussische Landes-Triangulation, I., II., 
 and III. Theile. 
 
 When a number of normal equations are to be solved, it is 
 found, by some, desirable to eliminate by means of logarithms ; 
 but, as logarithms are never exact, there will always remain 
 small residuals when the corrections are applied. Direct elimi- 
 nation is preferable, unless the coefficients are large ; then the 
 logarithmic plan is somewhat shorter. We will illustrate with 
 an algebraic equation : 
 
 3 -f- x -\- 2y 222 = 0;. . . . (i) 
 
 4, r _ y _j_ 3(gr - 35 = ;. . . . (2) 
 
 4* + ix - 2.y -19 = 0;. . . . (3) 
 
 2U + 4 y -f 22 46 = O (4) 
 
 If the first equation were multiplied by ^, the coefficient of 
 u would be the same as in (3), and upon subtraction the u's 
 would disappear. To multiply by ^ is simply adding log 4 
 log 3 to the logarithms of the coefficients of (i), omitting 32* ; 
 we write, then, the logs of these coefficients :
 
 CALCULATION OF THE TRIANGULATION. 
 
 195 
 
 Dog of coef., o.oooo 
 log 4 log 3, 0.1248 
 add 0.1248 
 
 nat. numbers, 1.333 
 coef. of (2), 3 
 subtract 1.667 
 
 y- 
 
 z. 
 
 22. 
 
 
 0.3010 
 
 no.oooo 
 
 n i. 3424 = 
 
 o; 
 
 0.1248 
 
 0.1248 
 
 0.1248 ; 
 
 
 0.4258 
 
 #.1248 
 
 n i. 4672 ; 
 
 
 2.666 
 
 - 1-333 
 
 - 29.33 ; 
 
 
 2 
 
 + 
 
 - 19 ; 
 
 
 + 4.666 
 
 - 1-333 
 
 - 10.33. 
 
 (5) 
 
 Take a factor that will make the coefficient of u in another 
 equation equal to its coefficient in one of the other equations, 
 multiply (4) by 2, or add to the logs of the coefficient in (4), 
 the log of 2 = 0.3010. 
 
 Logs of coef. of (4), .... 
 
 log 2, 
 
 add 
 
 nat. numbers, .... 
 coef. of (3), 32 
 
 subtract 3 
 
 y- 
 
 z. 
 
 4 6. 
 
 0.6020 
 
 O.3OIO 
 
 #1.6627; 
 
 .03010 
 
 .030IO 
 
 .03010 ; 
 
 0.9030 
 
 O.6O2O 
 
 7*1.9637 ; 
 
 8 
 
 4 
 
 -92 ; 
 
 2 
 
 
 - 19 ; 
 
 
 
 4 
 
 -73- - - 
 
 (6) 
 
 Continue to eliminate the same quantity from all the remain- 
 ing equations until one equation remains with one unknown 
 quantity. 
 
 The only advantage that this method suggests is, that only 
 one quantity is used as a multiplier to make the coefficients 
 identical ; that factor is usually a fraction, whose log is simply 
 the difference between the logs of the numerator and denom- 
 inator. 
 
 Mr. Doolittle, of the Coast Survey, has developed another 
 method of elimination, which can be found in the Report for 
 1878, page 115.
 
 I 9 6 
 
 GEODETIC OPERATIONS. 
 
 REDUCTION TO CENTRE OF STATION. 
 
 the directions adjusted it is necessary, when an eccen- 
 tric position has been occupied, to reduce the corrected ob- 
 served dire<jons to their equivalents at the centre, before 
 cornputing tJje distances and co-ordinates, 
 
 FIG. 25. 
 
 = observed angle; 
 x = desired angle. 
 
 Angle from signal to A = a, to B = b, 
 Angle mA-\-x 0-}-B. 
 
 sin A
 
 CALCULATION OF THE TRIANGULATION. 
 
 197 
 
 As B and A are always very small, they may be regarded as 
 
 equal to B sm I ", and ^4 sin I ", 
 
 r sin (0 _|_ ) r sin n 
 
 ' - 
 
 hence, 
 Also, 
 
 rsin(0-f-) rs'mn 
 b sin i" # sin i"" 
 
 angle between B and C = /2 OC -f- 
 
 r . sin (fl-f-0-1-0 7 ) r sin . 
 
 . sm i 
 
 a sin i' 
 
 From the above equations it will be seen that all angles that 
 are read from the same initial point have for their corrections 
 the same last term ; so this term can be 
 computed for each initial direction and 
 applied to the various angles. In both 
 terms of the corrections there are two con- 
 stants for each station, rand sin i"; so the 
 work can be facilitated by tabulating their 
 values. The signs of the terms will de- 
 pend upon the sign of the sine function. 
 
 It will assist in the computation to take 
 the angle between the signal and the first 
 point to the right and continue in that 
 direction. 
 
 Signal 23 feet from instrument. Angle 
 between //"and C.T = 71. 
 
 C.T 
 
 Fzc. 26. 
 
 Log dist. //"to C.Tin M. = 4-7534757; 
 log dist. H to B.K in M. = 4.6503172 ; 
 log dist. //"to H.K\\\ M. = 4.8385482; 
 log dist. //"to 7 in M. = 4.6145537.
 
 198 
 
 GEODETIC OPERATIONS. 
 
 
 Sig. toC.2' + 
 C.Tlo B.K. 
 
 Sig. tocr+ 
 
 C.TloH.K. 
 
 Sig. to C.T+ 
 C.TloC. 
 
 Sig. to J3.K + 
 B.K to H.K. 
 
 SI WfSc + 
 
 Direction .... 
 
 107 24' I?" 
 9.9796466 
 0.8457389 
 5.3496828 
 5.3I4425I 
 1.4894934 
 30". 86 
 
 120 53' 55" 
 9.9335264 
 0.8457389 
 5.1614518 
 5-3I4425I 
 1.2551442 
 17". 99 
 
 166 06' 57" 
 9.3801384 
 0.8457389 
 5.3854463 
 5.3i4425t 
 0.9257487 
 8". 42 
 
 120 53' 59" 
 
 9.9335214 
 0.8457389 
 5.1614518 
 5.3144251 
 I-255I372 
 
 17". 99 
 
 166 06' 55" 
 9-3801555 
 0.8457389 
 5.3854463 
 5.3144251 
 0.9257658 
 8". 43 
 
 Log r(M). . . . 
 Co. log dist. . 
 Co. log sin i" 
 
 Cor 
 
 
 
 Signal to C. T. 
 
 Signal to B.K. 
 
 
 jl OO OO 
 
 107 24 17 
 
 
 9 9756701 
 
 
 Log ?(M) 
 
 0.8457389 
 
 0.8457389 
 
 Co. log dist 
 Co. log sin i" 
 
 Cor 
 
 5.2465243 
 5.3144251 
 1.3823584 
 
 24". 12 
 
 5.3496828 
 5.3144251 
 1.4894934 
 
 30". 87 
 
 
 
 
 CORRECTED ANGLES. 
 
 C.T.to B.K= 36 24' i7"+3o".86-24".i2=36 24' 2 3 // .7 4 ; 
 C.T.ioH.K= 49 53' 55"+i7". 9 9-24".i2=49 53' 4 8 // .8; ; 
 C.T to C. = 95 06' 57"+ 8".42-24 // .i2=95 06' 4i".3o; 
 = 13 29' 42 // -i-i7 / '. 9 9-3o // .87=i3 29' 29 // .i2 ; 
 C. = 58 42' 38"+ 8 // .42-3o // .87=s8 42' i5 x/ .55. 
 
 The distances used above were obtained from the observed 
 values of the angles, and are, therefore, only approximate. In 
 the case of refined work, it will be necessary to use these cor- 
 rected values and again compute the distances ; then, with the 
 correct distances, recompute the reduction to centre. 
 
 With the most probable value for all the directions, the 
 angles of all the triangles can be found by taking a given di- 
 rection from 360, or by adding or subtracting two or more 
 directions.
 
 CALCULATION OF THE TRIANGULATION. 199 
 
 Then with a base, measured, or previously computed, each 
 side can be found by the trigonometric formulae a = 
 
 b sin (A - -) b. sin (c - -] 
 
 ^ , and c = 5_, in which * is the 
 
 sin (B-'-) sin (B-'-) 
 
 computed spherical excess, as obtained from using approxi- 
 mate lengths and angles. 
 
 If there are more than one base in the triangulation-net, the 
 most satisfactory method is to compute each base from all the 
 others, and take the mean of the logarithmic values so found; 
 or, if the entire scheme is involved in a single figure, the abso- 
 lute term in the side equation can be made equal to the ratio 
 
 D 
 
 of the two bases, -~, instead of unity. 
 
 We can also find the length of any line as influenced by two 
 or more bases. Let B^ B v B 3 ... be the bases, x the most 
 probable value of any side in the triangulation, and r lf r v r 3 
 ... be the ratio of each side respectively to x\ the errors 
 then will be 
 
 Now, if A' A> A be the weights of the bases, then 
 
 p,(r,x - B$+pfrs - B$ + A(r,* - ,)' = a minimum. 
 
 Placing the differential coefficient with respect to x = o, we 
 find
 
 200 
 
 GEODETIC OPERATIONS. 
 
 From this we can find the most probable length of one base 
 from all the others. To do this we suppose x to be one of 
 the bases, say B then r, = I, 
 
 A+A'.'+A'.' 
 the correction will be x B lt or 
 
 " 
 
 A*. + A* 
 
 A 
 
 A 
 
 The adjustments so far considered affect the geometric con- 
 ditions, and in their operations may, by changing the direc- 
 tions of the lines, change the azimuth, making a greater or less 
 difference between the observed and computed azimuths. In 
 refined geodetic work, the azimuth is observed at least twice 
 in each figure, and sometimes twice in each quadrilateral. 
 
 FIG. 27. 
 
 Using Wright's figure and notation, we take PQ&nd TU as
 
 CALCULATION OF THE TRIANGULATION. 2OI 
 
 two lines whose azimuths have been observed with the simplest 
 and most approved connections. PQ, as a known line, enables 
 one to compute PR, and from PR we can go direct to SR, 
 thence to ST; so these lines are called sides of continuation. 
 Let A v AV A 3 . . . be the angles opposite the sides of con- 
 tinuation ; 
 2?,, B v J5 a . . . the angles opposite the sides taken as 
 
 bases ; 
 
 C v C v C 3 . . . the angles opposite those sides not used ; 
 Z Z v the measured azimuths of PQ and 777, 
 
 supposed to be correct, and therefore 
 subject to no change ; 
 
 Z' the computed azimuth of TU, reckoning from the 
 south around by the west. 7,, 7 2 , C 3 , C t , are the only angles 
 that enter into this computation ; and the excess, , of the 
 observed over the computed azimuth gives 
 
 = , . . Eq.(L) 
 
 in which (7,), (C y } . . . represent the corrections to ",, !,... 
 Now, since the triangles have had their angles adjusted to the 
 conditions imposed upon them, their total corrections must be 
 
 . . . . (M) 
 
 ;)=o.j 
 
 Also, the sum of the errors squared 
 
 (A^f -f- (#))" H~ (1)' a minimum.
 
 2O2 GEODETIC OPERATIONS. 
 
 The solution of these equations would give 
 
 A t = IE, A, = -.. 
 
 If there were n intervening triangles, we would find 
 
 'A,- E, A, = ----, 
 2n ' 2n 
 
 From which the following rule is deduced : 
 
 " Divide the excess of the observed over the computed azi- 
 muth by the number of triangles, and apply one half of this 
 quantity to each of the angles adjacent to the unused side, and 
 the total quantity with its sign changed to the third angle. In 
 each following triangle the signs are reversed." 
 
 The discrepancies between the observed and computed lati- 
 tudes and longitudes are very slight, and can be adjusted arbi- 
 trarily.
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 
 
 CHAPTER VII. 
 
 FORMULA FOR THE COMPUTATION OF GEODETIC LATITUDES, 
 LONGITUDES AND AZIMUTHS. 
 
 WHEN we know the geographical position of a point, and 
 the distance and direction to another, the co-ordinates of the 
 second can be computed from the data just named, by using 
 formulae for the difference in the latitudes, longitudes and azi- 
 muths. 
 
 In the above meridian section, let A be a point whose 
 latitude is L. By definition it is equal to the angle AN, 
 formed by the normal A N and the equatorial radius EC. 
 
 a* tf 
 AG = N, e 1 = 1 , in which a is the semi-major axis, 
 
 and b the semi-minor.
 
 2O4 GEODETIC OPERATIONS. 
 
 r* 
 
 The subnormal in an ellipse MN '= CM '.-,. 
 
 AM = NM . tan L = CM. - 3 . tan L, 
 a 
 
 squaring, AM* = CM* . - 4 tan 8 L. (i) 
 
 The equation of an ellipse gives 
 b\ . 
 
 therefore CM' . ~ t tan 4 L = - t (a* CM 9 ), 
 
 CM*.-, tan a L = a*- CM* \ 
 
 clearing of fractions and transposing, 
 
 CM\V tan' L + a*) = a 4 ; 
 
 hence CM* - 
 
 tan 3 Z -fa" 
 
 sin* L 
 substituting - -^j for tan* L,
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 2O5 
 
 -^j-^ a a* cos* Z a* cos 8 L 
 
 ~ ^s'm'L-}-a l cos r L = b\i cos 2 Z) +V cos 8 
 
 a: 4 cos 2 Z 
 
 of= , a ' cos 
 
 From definition V 2 = 2 ^ 2 , 
 
 ' cos Z- 2 cos 
 
 ai - sn 
 
 a 1 cos Z # cos L 
 
 4/0" _ 0V sin 2 L <*Vi (? sin 2 Z- ^l e* sin 2 Z' 
 which is the radius of a meridian. 
 
 AO AO a 
 
 In the triangle A GO, AG -. JT=T^ = / = 
 
 sin A GO cos ^ie'sirfL 
 
 = N, or the normal produced to the minor axis. 
 
 The ordinate AM can be found from the equation of the 
 ellipse, a 1 . AM* + & . ~CM* = c?b\ 
 
 <?V - VCM* a*V V a 3 cos'L 
 
 AM = r 
 
 a* ~ a* a\i e* sin 2 Z) 
 
 F cos 8 L _ b\i - sin 8 Z) 
 
 " i -* 8 sin 8 L~ ~ i -* 2 sin 2 
 
 y _ ^y sin 2 L-F + F sin 3 L 
 i e* sin 2 L
 
 2C6 GEODETIC OPERATIONS. 
 
 - ^ sin'Z(i-g') _ a* sin* Z(i 
 
 i e* sin 3 L i e* sin' 
 
 therefore 
 
 a(i <? 7 ) sin L 
 
 AM = 
 
 Vi - i sin' L 
 
 To find the normal AN, we take the triangle AMN, in 
 which 
 
 sm I/I ^ sin 8 
 
 The radius of curvature, R = 
 
 dx* 
 
 d*y V (dy\ Vx* 
 
 In the general equation, -7-5- = -- =-,, [r~l = -r- 
 
 dx* ay \dx' ay 
 
 substituting these values, 
 
 - L 1 + J g y w 
 
 
 + </ _ g y w + y^i _ c^y + wy 
 
 P ' V L a J " a 4 ? 
 
 In this expression we place for x the value we found for CM, 
 and for y that of AM; this gives
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 2O? 
 
 pz'(i e*y sin" L a*(i e*)* a"(i e*)* sin* 
 i e* sin 2 L I e" sin 2 L 
 
 R _ (i-O 
 
 (i / sin 2 Z)5* 
 
 The terminal points , b, c, d and ^ of the radii of curva- 
 ture form an evolute ; at the equator, 
 
 At the pole, 
 
 7 _ nn o 
 
 ~ 
 
 The above formulae are in terms of geographic latitude; 
 the geocentric latitude is equal to the angle formed at the cen- 
 tre by the equator and radius. In the figure it is the angle 
 A CM. Calling it 6, we have 
 
 _ AM _ a(\ e*) sin L a cosL 
 
 = MC = i^TUSTZ:"* "*" i -Ssm'L 
 
 b\ 
 
 -- = ( i e] tan L = -. tan L. 
 L a
 
 208 
 
 GEODETIC OPERATIONS. 
 
 It is always less than the geographic latitude, the difference 
 being greatest at those places where a* F is the greatest, or 
 at latitude 45, N. or S., where the 
 difference is about 11' 30". 
 
 In the adjoining figure, Pis the 
 pole, E the plane of the equator, 
 A and B two points on the earth's 
 surface whose latitudes are L and 
 L', co-latitudes A and A', and the 
 geodesic line AB is /. An and 
 Bn',-t\\Q normals, are N and N', 
 and R and R the radii of curva- 
 ture. 
 
 The azimuth is estimated from 
 the south around by the west; the 
 angle PAB = 180 Z, will be 
 designated x, and the angle be- 
 tween the two meridians AP and 
 BPis the difference of longitude, 
 dM. 
 
 In the spherical triangle APB, 
 
 FIG. 29. 
 
 cos A' = cos A cos l-\- sin A sin / cos x. 
 
 In relation to A and A', / is very small, so that a series involv- 
 ing / will converge, so we write A' =/(A -j- /). 
 Developing this by Taylor's formula, we have 
 
 1 ' 1 I 
 
 A. /v I -jrt 
 
 ' dl 
 
 2.3.<# s 
 
 In order to find these differential coefficients, some relation 
 must be established between A, A-f- d\ and dl. Taking these
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 2OO, 
 
 as sides of a differential spherical triangle having the angle x 
 between the first and third sides, we have 
 
 cos (A -J- dK) = cos d/cos A -j- sin A sin <//cos x. 
 
 Obtaining the differential coefficients of A with respect to /, 
 we substitute them in (i) ; this is accomplished by expanding 
 the last equation so as to have 
 
 cos A cos d A sin A . sin d\=. cos dl cos A 
 
 -j- sin A sin dl cos x ; 
 cos d\ = I, 
 
 and sin </A = d\. This reduces the equation to 
 
 cos A sin A d"k. = cos A dl-\- sin A d/cos x. 
 
 Cos A may be assumed equal to cos \.dl, and therefore 
 they eliminate each other, leaving jj = cos x. 
 
 </'A <TA dx dx 
 
 77- = sin x . dx Tn = sin x -77 ; but -77 = sin x . cot A ; 
 dl dl dl dl 
 
 -775- = sin' x cot A, also 777- = sin* x cos x(\ + 3 cot 1 A). 
 dl dl 
 
 Substituting L' and L for A' and A, and remembering that cot 
 A =r tan Z., we obtain from (i), 
 
 L'L = -I cos x i/ 8 sin 1 x tan L 
 
 + |/' sin 1 x cos x(\ + 3 tan 1 L), 
 14
 
 2IO GEODETIC OPERATIONS. 
 
 x = 1 80 Z, cos x = cos .2, sin x = sin Z, 
 L' - L = - ^/Z 
 
 - |/ 3 sin 2 ZcosZ(i + 3 tan'Z); 
 
 or, -*TZ:=/cosZ+i/ 2 sin'ZtanZ 
 
 - & 3 sin 2 ZcGsZ(l + 3 tan'Z). 
 
 The value of / has been considered as expressed in arc, 
 
 while in computation it will be given in linear measure. There- 
 
 >y- 
 
 fore / = -vv, where K is the length of the line, and N the ra- 
 dius of the imaginary sphere on which L is a point 
 
 /fcosZ , ^ 2 sin'ZtanZ JTsin'ZcosZ 
 __ + _ ____.. ___ 
 
 This needs a further transformation, to refer the formula to 
 an ideal sphere whose radius is the radius of curvature of the 
 middle meridian. This, however, cannot be known until L' is 
 computed ; however, we can start with the value of R for the 
 initial latitude, and apply a correction. The reduction is made 
 
 N 
 by multiplying by the ratio of -77; we also divide by arc i" to 
 
 convert the arc dL into a linear multiple of i' '. This gives 
 
 rd 77 sin Zcos z(
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 211 
 Denoting the radius of curvature of the mean meridian by 
 
 n _ n 
 
 R f>t , dL must be increased by ~ - . dL\ 
 
 ~ (i - ? sin 2 Zji (i - e 1 sin' L m )* 
 
 (i - e* sin* ) -(i-e* sin 9 
 ' I - *' sin 1 Z*(i - ^ sin a L 
 
 Expanding by binomial formula, 
 
 - * sn = i - sn ^ ^ sn 
 
 / sin 2 Zi = I - * 1 sin 2 4 4 
 
 subtracting = f^- 2 sin 2 L f^ 2 sin 2 L m . . . , 
 
 omitting higher powers of e\ or, |<? 2 (sin 2 sin 2 Z w ). 
 This can be reduced as follows: 
 
 sin (L Z. OT ) sin L cos L m cos L sin L m ; 
 
 sin (L -\- Lm) = sin L cos L m + cos L sin L m ; 
 
 sin (L L m ) sin (L-\-L m ) = sin 2 Z cos 3 L m cos 2 sin 2 L m 
 = sin 2 Z(i-sin 2 Z w ,)-(i-s 
 = sin 2 L sin 2 L m . 
 
 L m is the mean latitude between L and L -f- dL, 
 
 sin (Z - L m ) sin (Z + Z w ) = sin (L - L - \dL) sin (2L + \dL) 
 = dL sin Z, . cos Z, nearly.
 
 212 GEODETIC OPERATIONS. 
 
 R~R,n 
 
 Then ' 
 
 _ 3 f a(i e*)dL . sin L . cos L (i e 1 sin 2 L m )* 
 ~% e (i-S sin*L)i.(i ~ 
 
 dL.s'mL. cos Z, 
 
 As this is a small quantity, it can be converted into a linear 
 function by dividing it by arc i", giving 
 
 R - R, n _ dL* sin Z . cos L 
 
 ~ 
 
 - ^ 3 sin 3 )* arc 
 Introducing this into (2), we have, after placing 
 
 n- l r- tanZ 
 
 " 7 " ~ // ' 
 
 /f ^ 3 sin L cos 
 
 ~ x/ * C 
 
 R . arc i x/ * ~ (I - S sin 2 Z) arc i x/> 
 
 -flX = ^Tcos Z.B K* sin 2 Z. C- /WT sin 2 Z. 
 
 The last term was devised by Professor Hilgard, in 1846. 
 
 The factors B, C, D, E, are given in the last pages com- 
 puted for Clarke's (1866) Spheroid. 
 
 When the line is not more than fifteen miles, the third term 
 can be omitted, and 7* 2 put for (dL)*, giving as an abbreviated 
 formula 
 
 - dL = KcosZ. B+ K*sm*Z. 
 
 Francoeur has given a purely trigonometric method for de- 
 riving the formula just obtained.
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 21$ 
 
 Using the same figure, we write PA = 90 Z, PB = 90 
 -L'. 
 
 In the spherical triangle PAS, we know PA, AB = !, and 
 the angle PAB = 180 Z. 
 
 cos PB cos PA cos /-)- sin PA sin /cos PAS; 
 cos (90 Z 7 ) = cos (90 L) cos / 
 
 + sin (90 Z) sin /cos (180 Z) ; 
 sin L' = sin L cos / cos L sin / cos /?. 
 
 Subtracting both sides from sin L, 
 
 sin L sin L' sin Z sin Z cos / -f- cos L sin / cos Z 
 = sin Z( i cos /) -f- cos L sin /cos Z 
 = 2 sin L sin 4 / -f- cos L sin / cos Z. 
 
 sin Z - sin L' = 2 sin (Z L'} cos |(Z + L') ; 
 suppose Z - L = d, then Z -f L' = 2Z-(Z-Z') =2L-d, 
 and \(L - L'} = K ^(2 L-d) = L- \d, 
 
 therefore sin L sin Z' = 2 sind?cos(Z </) 
 
 = 2 sin <fcos Z cos <3?--sin L sin 
 
 that is, 2 sin ^ cos \d cos Z -{- 2 sin 2 ^ sin Z 
 
 = 2 sin Z sin 2 / -f- cos Z sin / cos Z. 
 
 Dividing by 2 cos L cos 2 ^, we obtain 
 
 sin \d sin 2 */sin L sin Z sin 2 ^/ cos Z sin IcosZ 
 cos </ ' cos" */ cos Z ~ cos Z cos 2 %d ~"" 2 cos Z cos"
 
 214 GEODETIC OPERATIONS. 
 
 tan L sin 2 l sin/cos^ 
 
 tan 
 
 tan \d(l + tan */tan L) = ^-y-^tan L sin a \l-\-^, sin / cos 
 
 Placing H for the last parenthesis, we may write 
 
 TT 
 
 tan \d(\ + tan ^ tan L) = - 5-=--^ =r #(i -f tan 2 
 
 tan ^ + tan 4 \d tan Z = H-\-ff tan 3 ^ ; 
 
 tan|^+(tanZ -H)tan*$d=H. . . . (i) 
 In the expression for H, /is small ; so we can write for sin / 
 
 r r 
 
 its serial value, sin / = / ^-, also sin 2 %/ = , which will 
 give 
 
 H = i/cos Z+ \r tan L - T y ' cos Z. 
 
 We must now solve (i) for \d\ for short we will put \d = 
 c, and tan L H = h, so (i) reduces to 
 
 TJ 
 
 tan c + tan a c . h = H, or tan c = r -, -- . (2) 
 
 I -)- h . tan c ^ ' 
 
 Neglecting h . tan c, we have tan c = H, as the value of first ap- 
 proximation. Substituting this value for tan c in the second 
 member of (2), 
 
 TT 
 
 tan c = , ?r = H /iff', by division.
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS, 21$ 
 
 Again, substituting this second approximation in (i), we have 
 as the third approximation 
 
 tan c = H- hH* + 2/?H 3 ; 
 by continuing to the fourth, we get 
 
 The development of an arc in terms of its tangent gives 
 
 c = tan c \ tan 3 c -\- ^ tan 6 c . . . 
 Placing in this the value of tan c, just found, we have 
 
 Resuming our notation, we have c = \d, and h = tan L 
 H\ this gives 
 
 h = tanZ, |/cosZ ^/* tan L + T V/'cosZ; 
 i^ = i/cos Z+ i/ 2 tan Z - -jV/ 3 cos Z 
 
 - (tan L- i/cos Z- /' tan Z+ T V/ 3 cosZ. . . ) 
 (|/ cos Z+ i/ 2 tan L T y 3 cos Z) 3 . . . 
 
 Multiplying this out and retaining terms of / to / 3 , 
 
 =!/ cos Z+i/' tan Z /" tan Zcos'Z- ^r cos Z 
 
 + T V/ 3 cos 3 Z i/ 3 tan 5 cos Z-f i/ 3 tan'Z cos 3 Z 
 = |/cosZ+i/ a tanZ(l - cos 3 Z) - 1 V^ 3 cosZ(i - cos'Z) 
 -i/ 3 cosZtan a Z(i cos a Z) 
 = i/cos Z + i/ a tan L sin" Z - T ^/ 3 cos Zsin' Z 
 
 -i/'cosZsin'Ztan'Z 
 = i/cos Z+ i/ J tan L sin s Z- ^ 3 cos Zsin'Z(i+3 tan'Z) ;
 
 2l6 GEODETIC OPERATIONS. 
 
 ar^/cosZ-f-i/'tanZsin'Z / 3 cosZsin'Z(i -f 3tan'.). 
 
 Remembering that d = L L r , we have here the identical 
 formula given on page 210. 
 
 There is still another form to which this can be reduced, in- 
 volving more factors that can be tabulated, and at the same 
 time occurring in the computation of longitude and azimuth. 
 
 K K(i -/sin'Z)* 
 
 Take (2) and substitute u" for -^^ - Tl * - r r/ - - ; 
 
 N sin l" tfsm \" 
 
 multiply this by -~-, it becomes - -5 ; reducing this frac- 
 tion and omitting terms above ^*, 
 
 = j _[_ / - f sin 8 L = i + e\i - sin 8 Z) = I + ^ cos a L. 
 The first term becomes 
 
 likewise the second term 
 
 -=(!+** cos' L) (u" sin Z) a tan L ^- ; 
 substituting, it becomes 
 L'=L-(i+e* cos a L)u" cos Z ^ sin I'^i+^cos' L) (u" sin Z) 9 . 
 
 This formula gives good results for distances of twenty miles 
 and under. 
 
 The algebraic sign of the different terms depends upon the 
 trigonometric functions of Z.
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 21? 
 
 Whenever the sides are more than a hundred miles long, this 
 method of difference of latitude will introduce some errors. 
 
 In that case, the method to be followed is to solve the 
 spheroidal triangle formed by the two points and the pole, in- 
 volving as trigonometric functions the sought co-latitude, azi- 
 muth and difference of longitude. 
 
 LONGITUDE. 
 
 Referring to the figure on page 208, and using the same no- 
 tation, we get in the triangle ABP, sin A' ; sin x : : sin / : sin dM. 
 
 Supposing the radius of the sphere to be Bn' = N, I =. -^, 
 and that /and dM are proportional to their sines 
 
 A" 1 
 
 dM=-= 
 
 A' and L' being complementary, sin A' = cosL'. 
 
 If very accurate geodetic computations are to be made, a 
 small correction must be applied, owing to the difference be- 
 tween the arcs and sines of small angles. This correction can 
 be taken from the table on page 274. 
 
 The quantity dM increases towards the west. The alge- 
 braic sign of the equation depends upon sin Z, which is -f- be- 
 tween o and 180. 
 
 If we place u" = . 77, dM = j- 
 
 cos L 
 
 K K" 
 
 this, however, supposes that -^ = ^=, which is only approxi- 
 mately so.
 
 . 
 218 GEODETIC OPERATIONS. 
 
 AZIMUTH. 
 
 The initial azimuth of the base or initial line being known, 
 that of any line emanating from either extremity can be known 
 by adding to or subtracting from the azimuth of this base the 
 angle between the two lines. 
 
 Let A B be the base, and C a point making angle m with 
 AD at A: if the azimuth of AB = Z, that of AC will equal 
 Z . m. But in order to determine the direction of a line ex- 
 tending from C, as CD, the azimuth of CA must be known. 
 If the earth's surface were a plane, Z' would equal 180 -f- Z ; 
 but the spheroidal shape of the earth complicates this as well 
 as all other geodetic problems. 
 
 Again referring to figure on page 208, in the triangle APB, 
 by Napier's Analogies, 
 
 tan \dM : cot \(x + x'} : : cos (A/ A) : cos i(A -f A'), 
 
 from which cot^O + x') = C S , > tan \dM '; 
 
 COS "$\A* ") 
 
 but x=iSo-Z, 
 
 therefore x -f x' = 180 - Z-f x' = 180 + (x 1 Z) ; 
 
 cot i[(i8o + (x' Z)~\ = tan \(x' Z), 
 but (x' Z} dZ, also \ = 90 L, and ^'=go L', 
 
 therefore A + A 7 = 180 - L - L' = 180 - (L + L'} ; 
 
 A 7 - A = 90 - L' - (go - L} = L - L' = dL ; 
 
 cos ![(i8o - (L + L'}} = sin \(L + //) - sin 
 the formula then reduces to
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 2 19 
 
 tan \dZ = - tan 
 
 COS 
 
 Supposing that tan \dZ : tan \dM : : dZ : dM, 
 
 j-r jut- sm A* 
 dZ = dM 
 
 This is not exactly correct; the correction can readily be found 
 by adding a term, say x, to the fourth term of the above pro- 
 portion and solving for the value of x. It will be found that 
 3 cos* L m sinZ.,,, sin 2 i" must be added to the above value 
 
 cos* L m sin L m sin 2 i" 
 of dZ. The factor ---- can be tabulated as 
 
 factor F, a table of which is appended ; the expression then 
 becomes 
 
 - dZ = 
 
 i 
 
 cos dL 
 
 The algebraic sign of dZ will depend upon dM. As the azi- 
 muth is estimated by common consent from the south around 
 by the west, so long as the initial azimuth is less than 180, 
 the reverse azimuth Z' = Z + 180 + dZ '; but if more than 
 1 80, Z' = Z - 180 -f dZ. 
 
 A table of values is given for cos \dL for lines of twenty 
 miles and under. The term involving f can be omitted, and 
 the value of dM deduced above substituted in its place, giving 
 
 Z Z 1 80 -j -- jj- . sin L m . It has also been found suf- 
 
 ficiently accurate to omit cos \dL and write dZ dM sin L m . 
 In accurate work the azimuth should be determined at least 
 once in every figure by astronomic observation. This opera- 
 tion is fully described in works on practical astronomy.
 
 220 GEODETIC OPERATIONS. 
 
 L. M. Z. FORM FOR PRIMARY TRIANGULATION. 
 
 Z 
 
 L 
 7, 
 & 
 
 /.' 
 
 Mount Blue to Mount Pleasant 
 Mount Pleasant and Ragged (. 
 Mount Blue to Ragged (360 - 
 
 
 26 19 
 85 35 
 300 44 
 + 50 
 301 34 
 121 34 
 
 * 
 
 i-34 
 3-7i 
 5-05 
 5-os 
 
 \. is to the left of M. P.) 
 
 (8 5 - 35' etc. - 26 - 19' . . . ) 
 
 
 Ragged to Mount Blue (Z + d 
 
 Z 180") 
 
 
 L 
 dL 
 L 1 
 
 44 
 44 
 
 43 
 3 
 
 40.121 
 
 5S-978 
 44-M3 
 
 Mount B 
 i 10740. 6 A 
 Ragged. 
 
 ue 
 
 i 
 
 M 
 
 iM- 
 M' 
 
 70 
 69 
 
 20 11.921 
 II 27.659 
 08 44.262 
 
 /, log 5. 0443070 
 
 K 
 cos Z 
 B 
 
 h 
 
 5.0443070 
 9.7084622 
 8.5.104895 
 
 A- 
 sin 8 Z 
 C 
 
 3d term 
 4th term 
 
 K 
 sinZ 
 
 A> 
 cos' 
 ar. co. 
 
 dM 
 
 10.08861 
 9-86854 
 1.39991 
 
 (dL)* 
 D 
 
 6-5372 
 2-3933 
 
 h 
 
 K* sin" 
 
 
 3.2632 
 9-9571 
 6.2069 
 
 3.2632587 
 
 1.35706 
 
 8.9305 
 
 9.4271 
 
 ist term 
 2d term 
 
 dL 
 3 d and 
 3 4 th 
 
 -dL 
 
 L m 
 
 1833.406 
 22-754 
 
 0.085 
 
 - 0.267 
 
 5.0443070 
 9 9342721" 
 8.5090158 
 0.1446254 
 
 arg. 
 K 
 
 dM 
 cos 
 
 + 317 
 + 99 
 
 (dM)* 
 
 dM 
 
 sin L m 
 cos \dL 
 ar. co. 
 
 dZ 
 
 2d term 
 
 10.896" 
 7.840 
 
 1856.160 
 - .182 
 
 8.736 
 3-6322302 
 9.8454305 
 0.0000040 
 
 1855-978 
 44-28-12.13 
 
 99 
 3.6322302 
 4287.757 
 
 3.47766 4 7" 
 3003.76 
 - S 3003.71 
 
 Notes upon the Computation. The angle Mount Pleasant 
 and Ragged is recorded minus, since the second point is to the 
 right of the first contrary to the graduation of the instrument. 
 1 80 is subtracted, since the general direction is east. In the 
 sixth column, 218 and 317 correspond to the correction due 
 to the supposition that the arc and sine are equal ; the value, 
 99, is added to dM ; dM\s negative, since sin Z is minus. In 
 the azimuth-computation, the second term, .05, is the antiloga- 
 rithm of 8.736 ; this is negative, therefore .05 is subtracted. 
 
 The data here used were taken from the Coast Survey rec- 
 ords. There the Z, M and Z were computed, using Bessel's 
 constants: the results are, Z B = O5".55, Z c $".0$, L B 
 = 43"-955. L c = 44"i43, M = 43".578, M? = ^".262. 
 
 In using the abbreviated formula, the third and fourth terms
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 221 
 
 would be omitted in latitude, but in their place should be in- 
 serted VD. 
 
 In longitude, the correction for the ratio of sine to arc is not 
 inserted. Also, for azimuth-computation cos \dL, and (dM^F 
 are insignificant, and consequently left out. The terms that 
 are disregarded could not affect the result beyond the tenth 
 of a second, in lines less than a hundred miles in length. A 
 very convenient form in use in the U. S. Geological Survey is 
 appended, employing the abbreviated formula already given : 
 
 Names of 
 Stations. 
 
 Position. 
 
 Observed 
 
 Angles. 
 
 Correc- 
 tion by 
 L.S. 
 
 Correc- 
 tions 
 arbi- 
 trary. 
 
 Spheri- 
 cal 
 Angles. 
 
 Spheri- 
 cal 
 Excess. 
 
 Final Plane 
 Angles. 
 
 Big Knob 
 
 Sought, 
 
 '44 17 55-62 
 
 " 
 
 .4 
 
 55-22 
 
 -x.8 3 
 
 '44 17 53-39 
 
 Holston 
 
 Right^H 
 
 c *z, 
 
 III 
 
 13 29 28.86 
 
 
 " 
 
 28.76 
 
 -1.83 
 
 13 29 26.93 
 
 High Knob.. . Left, 
 
 22 12 41.69 
 
 
 .18 
 
 41.51 
 
 -1.83 
 
 22 12 39-68 
 
 Comput- 
 Letter. 
 
 Logarithms 
 of their Sines. 
 
 Calculation of the Sides. 
 
 Sides in 
 Yards. 
 
 Designation. 
 
 5. 
 
 9.7660909 
 
 log J? = 4.839*933 
 a. c. log sin S = 0.2339091 
 
 4.8780609 
 
 Holston High Knob. 
 
 
 
 
 
 
 R. 
 
 
 log LS = 4.4409976 
 
 safefc*h- -*- 
 
 log sin Z, = 9.5775136 
 
 4.4798652 
 
 Big Knob High Knob. 
 
 L. 
 
 
 log/?.9 - 4.6506160 
 
 4.6894836 
 
 Holston Big Knob. 
 
 The column marked correction by L. S. is for the correc- 
 tions obtained in figure-adjustment. When it is not possible 
 to make this adjustment, the error, after deducting spherical 
 excess, must be distributed arbitrarily. If the angles are ap- 
 proximately equal, one third the error should be applied to
 
 222 
 
 GEODETIC OPERATIONS. 
 
 each angle ; if not equal, the distribution should be propor- 
 tional to the size of the angles. If one signal should be dim, 
 or uncertain, it may be best to give to the angle between it and 
 the other point the bulk of the error. Occasionally the angle 
 deserving the greatest correction can be determined by ex- 
 amining the individual readings. If they vary considerably, 
 showing a wide range, the inference is that the average is 
 somewhat uncertain, and that the principal source of error in 
 the triangle is at this point. Such evidence as this, and the 
 appearance of the signals from each other should have some 
 weight in distributing the error. The most convenient form 
 of blank for computation is to have three or four sets of the 
 upper slip printed on the left side, and the same number of 
 the lower, on the right side of a book. 
 
 Names of Stations. 
 
 LATITUDES. 
 
 L'= L-u" (i-f " cos 2 L) cos Z - i sin i" sin* Z" 2 (i-f e cos" L) tan L. 
 
 Holston 
 
 M 
 
 
 
 1 
 
 .M 
 
 4 
 
 M 
 
 i I 
 
 4 
 
 ii 
 
 ^ H 
 High Knob. . . 
 
 Authority, 
 Latitude (L). . 
 
 log K (yds) 
 
 U. S. Geo. Survey. 
 .....'=3627 27.41 
 
 ^ sin i" 
 
 . . 4 3845448 
 
 . 4 8780609 
 
 IQO- 
 
 
 2 log sin Z. . 
 
 
 10g ^sini"" 
 
 log" = 3-3484585 
 log(i + * cos' 2 /,)-.= 0.0018710 
 log cos Z. ( )= 9.8460032 
 
 . . 6 69691701 
 
 
 . .= 0.0018710! 
 ..= 9.8685368 
 
 log tan L . . . 
 log 2d term . 
 
 2d term .... 
 L-\-L' . 
 
 log ist term. . 
 
 3. 1963327 
 
 .= 0.6576932 
 
 = 4"-55 
 
 . . 73 21 01.84 
 
 
 
 2d term 
 SL 
 
 ..(-)= 4-55 
 
 1567.02 
 (-(-) 26 07 02 
 
 L 
 
 36 27 27 41 
 
 Latitude (L'). 
 
 
 L + L' 
 
 2 
 
 .=364030.92 
 
 =365334.43
 
 GEODETIC LATITUDES, LONGITUDES AND AZIMUTHS. 22$ 
 
 LONGITUDES. 
 
 AZIMUTHS. 
 
 REMARKS. 
 
 M' 
 
 sinZ 
 
 L+L> 
 
 ~*~ co&L' ' 
 
 
 Authority 
 Longitude 
 
 log sin Z. 
 log u" . 
 
 U. S. G. S. 
 M = 82 04 38.17 
 
 
 U. S. G. S. 
 
 134 32 39-47 
 
 180 
 
 
 Azimuth Z 
 
 Z (-M 1 80 
 
 (+) = 9-8529118 
 3 3484585 
 
 314 32 39.47 
 
 . Z + Z' 
 
 log sin ... = 
 
 ...(-L) 
 
 log cos L 
 log (&W). 
 
 SM ... 
 
 
 3.2013703 
 
 9.7761771 
 3.2984111 
 
 (-)-) _ 9.9029592 
 
 (+) = 3-2984111 
 <+) = 1987.98 
 ("{") 33 7 98 
 
 log SZ (-) = 
 <5Zin seconds.. . = 
 SZ ( ) 
 
 3.0745882 
 1187.38 
 
 19 47.38 
 314 32 39.47 
 
 M 
 
 82 04 38 17 
 
 Z(+)i8o = 
 Azimuth Z' 
 
 M' 
 
 
 
 314 12 52.09 
 
 
 
 
 The latitude blank should occupy the left, and the longitude 
 and azimuth the right side of a book. Two forms should be 
 on each page, the second serving as a check computation, by 
 determining the third point of the triangle from the other end 
 of the base. For example: in triangle ABC, suppose L. M. 
 Z. of A and B is known, C can be determined from A, and 
 also from B. The average of these values is to be taken, to 
 be used in connection with A or Z? in determining D, etc. . . .
 
 224 GEODETIC OPERATIONS. 
 
 CHAPTER VIII. 
 
 FIGURE OF THE EARTH. 
 
 WITH the geographical positions of the termini of a line and 
 its length known, it is possible to find an equivalent for its 
 length along a meridian or a parallel, thus obtaining a value 
 for a degree in that latitude. 
 
 Assuming that the earth's meridian section is an ellipse of 
 small ellipticity, we can develop a formula giving the length of 
 an arc in terms of the terminal latitudes, the semi-axes, and 
 ellipticity. Also the problem almost the converse, by which 
 the values of the axes and ellipticity can be found. 
 
 Let L, L', and / represent the terminal and middle latitudes 
 of an arc whose amplitude is /I ; a, b, and f, the semi-major, 
 semi-minor axes of the meridian-section, and the ellipticity ; S, 
 the length of the arc ; r, the radius vector, and 6, the geocen- 
 tric latitude. 
 
 The equation for the ellipse is 
 
 - 
 
 x = r cos 6, y r sin 6, substituting in (i), 
 
 r> cos" 6 r* sin 8 8 _ 
 * ~ l > 
 
 cos' , sin" 8 I 
 divide by r 1 , -- + - = (2)
 
 FIGURE OF THE EARTH. 
 
 On page 207 we found 
 
 tauten A or 
 
 
 f , ~ 
 
 from which sin* 6 = 
 
 , 
 os 2 6 a'cos*L' 
 
 b* sin* L cos* 
 
 . 
 a cos" L 
 
 Substituting for cos* 6, i sin' 6, and solving, we get 
 
 t ft _ b* sin*Z 
 
 "Vcos'Z + ^sin'r 
 
 By a similar process we get 
 
 t a _ _ dcos* L 
 ~ 4 ' 
 
 Placing these equivalents in (2), 
 
 a* cos 4 L -f- 6 4 sin 4 L 
 
 (3) 
 
 In the ellipse, tf = a*(i ff in which e is the ellipticity ; sub- 
 stituting this in (3), 
 
 f)*sin* L I 
 
 4 cos 4 L -\-a\\ f) 4 sin^L ~ P' 
 
 Dividing out ^", and writing i sin* L for cos* L, after reduc- 
 tion, we have 
 
 = a\i 4 sin * -f- 6e* sin* Z- 4^' sin* Z. + * 4 sin*Z) ; 
 
 omitting terms involving powers of f above the second, 
 15
 
 226 GEODETIC OPERATIONS. 
 
 r = a(i sin 8 L). 
 The formula for rectifying a polar curve is, 
 
 ds I W dr* dr . _ _ 
 
 -jyr =r A / r --jfi -f- -Tr a. ~/T = 2d: Sin L COS A 
 
 -rp = I 2 -}- 4 sin 8 Z. 
 
 This is obtained by differentiating the equation 
 
 a" tan = V tan Z, or tan = (i )" tanZ; 
 
 dZ 
 
 j * sin 2 L cos 8 L 
 
 a(i 2s -\- 3 sin" Z.) ; 
 
 omitting in the above all terms involving above the second 
 power before extracting the square root. 
 
 ds = a(i 2+3 sin 2 L)dL = a(i - |- cos 2ZyZ, 
 
 placing sin 8 Z = (i cos 2Z). 
 
 Integrating the above between the limits Z and L', we have 
 s = a[(i |f) (Z - Z') - f (sin 2Z - sin 2Z')], 
 
 a b 
 b (i ), from which = . Substituting this in the 
 
 above equation,
 
 FIGURE OF THE EARTH. 227 
 
 = (lf) (Z - i-) - 
 
 f(# ) (sin Z cos L sin U cos Z'). . (4) 
 
 Z-Z' = A, and 
 
 sin A. = sin (Z- Z') sin Z cos Z' sin Z'cosZ; 
 cos 2/ = cos (Z -j- Z') = cos Z cos Z' sin Z sin Z'; 
 sin A cos 2/ = sin Z cos Z cos 2 L' sin" Z . sin Z' cos Z' 
 
 sin L cos'Z cos Z' + sin" L' sin Zcos Z. 
 
 Substituting in this 
 
 sin* L' = i cos* Z', also sin a Z = I cos* Z, 
 it reduces to 
 
 sin A cos 2/ = sin Z cos Z sin L cos Z'', 
 
 which is the same as the last term in (4) ; therefore 
 
 s - \(a + ff)l - f (a - b] sin A cos 2/. (5) 
 
 This requires a particular ellipsoid from which to obtain the 
 value of a and b, but it gives a means of finding a and b^'\i all 
 the other terms are known, which is the problem geodesy at-
 
 228 
 
 GEODETIC OPERATIONS. 
 
 tempts to solve. Suppose s, A, /, s', A', /' be the lengths, am- 
 plitudes, and mean latitudes of two arcs, we will have 
 
 s = \(a -f- )A |(0 b) sin A cos 2/; 
 s' = \(a + b)\' - \(a - b) sin A' cos 2/' ; 
 
 (a 
 solving for --, and 
 
 a -f- b s' sin A cos 2/ j sin A' cos 2/' 
 2 ~ A sin A' cos 2/' A' sin A cos 2/ 
 
 a b 
 
 s'\ sV 
 
 3 ' A' sin A cos 2/ A sin A' cos 2/" 
 
 from which a, b, and f can be found, s and s' are the distances 
 between parallels, whereas in practice our lines make an angle 
 with the meridian, so that its projection upon the meridian 
 must be found. 
 
 To find the effect of errors in the values s and s' upon a 
 and b, we would differentiate the above equations, regarding 3 
 and s' only as variables. In the result the 
 denominators would remain ; consequently 
 the minimum error would occur when the 
 denominator is a maximum, that is, when 
 2/' = o, and / = 90, or when one arc is at 
 the equator and the other near the pole. 
 
 Let P be the pole of the spheroid, PM 
 and PN two meridians passing through the 
 points M and TV, whose geographic and 
 geocentric latitudes are Z,, L', 0, and 0' '. 
 PM = 90 - 0, and PN = 90 - 6', from 
 FlG - 3 - which NE = S 6', which we will call x ; 
 
 also the line NM= s, a known quantity.
 
 FIGURE OF THE EARTH. 229 
 
 In the spherical triangle MPN, by Gauss's formulae, 
 
 sin$(PN- PM} cos$MPN = sin \MN sin \(PMN - PNM); 
 cos (PN PM ) cos \MPN = cos pflV sin \(PMN+ PNM). 
 
 Dividing the first by the second, 
 
 = 180 - Z, PNM = Z - 1 80 
 hence i(/>3flV - /WJ/) = i( 3 6o - (Z4- Z 7 )) 
 
 and %(PMN+ PNM) = \(Z - Z). 
 
 Substituting these values, 
 
 sin^CZ+Z) , ^ j 
 
 Placing h = in i/y__ ^-y we have tan j = * tan - ; writ- 
 ing for tan - and tan - their developments, 
 
 Solving this equation for x in terms of s, by approximation,
 
 230 GEODETIC OPERATIONS. 
 
 we have for the first value x sk ; substituting this for x* and 
 x', we have, after transposing, 
 
 x_sh sVi_sW ^__^! _ sh shtf - 
 
 ~2 ~ ~2 ' 24 ~~ 24 ' 240 ~~ 240 ' ' 2 ~*~ 2 \ 12 
 
 for the second approximation ; and this value of x, substituted 
 in the first equation, gives 
 
 for the (A) third approximation. 
 
 - si 
 _ s 
 
 i(i ~ cos (Z' - Z) - j(i - cos (Z+Z) 
 sitf^Z' Z) 
 
 sin Z sin Z'
 
 FIGURE OF THE EARTH. 23! 
 
 Regarding the earth's meridian section as an ellipse, we 
 know from the properties of an ellipse that 
 
 x = a cos , and y = b sin u, 
 
 in which u is the eccentric angle, or reduced latitude. 
 Differentiating the above, 
 
 dx = a sin udu, dy = b cos udu, 
 
 If we consider this point, whose co-ordinates we have just 
 written, to be in latitude L, and an element of the elliptic 
 curve to be ds, it will be the hypothenuse of a right triangle, 
 in which 
 
 dx = ds sin L, and dy = ds cos Z., 
 or dx a sin udu ds sin L, dy b cos udu = ds cos L. 
 Dividing, 
 
 -r tan u tan Z-, or a tan u =. b tan Z, (i) 
 
 a sn w , . 
 
 and (fr= ___. (2) 
 
 The value found for * in (A) was for a spherical surface ; to 
 transform to an ellipsoid it will be necessary to pass to a dif-
 
 232 GEODETIC OPERATIONS. 
 
 ferential triangle on. each. In the figure on page 228, suppose 
 we call PNM a differential triangle on an ellipsoid, in which 
 EN '= dL, NM = do-, and the angle PNM a, then da cos a 
 = dL. To convert dL into arc measure, we multiply it by the 
 radius of curvature of the meridian, or 
 
 ds cos a = RdL. (3) 
 
 Likewise, if we conceive the same triangle to be on a sphere 
 of radius a, then as will be the length of the arc MN, then 
 
 ado- cos OL = adu, or do- cos a = du. (4) 
 
 ds RdL 
 Dividing (4) by (3), - = - 
 
 substituting in this R = 
 
 also from (i), we find sin L = - ( 5 ; 
 
 ds 
 
 _ f sin' Z* ' I - ^ cos 9 
 
 f _ iH^*L_ r ' J 
 
 L 1 ~^i-^cos a J 
 
 e* cos* u 
 
 i e* cos 
 
 8 u ' Fi ^ a cos" u e* sin a 
 
 I - ? COS 8 jT~
 
 FIGURE OF THE EARTH. 233 
 
 a(l e')* I 
 
 e* cos a u ' 
 
 r *-^ 7 
 
 Li *" cos 2 uJ 
 
 
 or = ^i - <* cos' , (5) 
 
 which gives the relation between an infinitesimal length on a 
 sphere to a corresponding length on an ellipsoid. 
 
 To integrate this, Jordan takes a spherical triangle with 
 sides equal to u, u\ and cr, and angle opposite u 1 = a 1 - then 
 
 sin u = sin u 1 cos <r -\- cos u 1 sin cr cos or 1 , 
 placing the serial value for cos <r and sin <r, 
 
 sin u = sin 'f i . . .j + (<r . . .) cos u 1 cos a 1 , 
 
 omitting all powers of <r above the second. 
 Squaring this equation, 
 
 sin" u sin* u l (i a 3 } -j- a* cos" w 1 cos 2 a 1 
 
 -{- 2<r sin u 1 cos ' cos a\ 
 
 For sin* u, write I cos 8 u, then transpose, change signs, mul
 
 234 GEODETIC OPERATIONS. 
 
 tiply by e\ subtract from I, and extract the square root; this 
 gives 
 
 r I e* cos 2 u = i -- cos 2 u 1 -{- e'er sin ' cos u l cos or 1 
 
 -j -- o- 2 (cos 2 ' cos 2 or 1 sin 2 w 1 ). (6) 
 
 Placing this in (5), and integrating with respect to dv, we 
 find 
 
 s t ? A , ^V . 
 
 = <rl I -- cos u I -| sin cos 
 
 s 2 l cos" or 1 sin" w 1 ). (7) 
 
 cos a 
 
 This can be written, including e\ 
 
 s ( f , A 
 = <rl i -- cos u ) 
 a \ 2 I 
 
 r f* e* "I 
 
 I -j <r sin u 1 cos u 1 cos a 1 -j- ^-cr^cos 4 u 1 cos a or 1 sin" ') . 
 
 If we place a 1 = o, we have <r = ', then the last equa- 
 tion becomes, after placing 5 for s, 
 
 S e 1 
 
 - =(u ')(i -cos'w 1 )
 
 FIGURE OF THE EARTH. 235 
 
 From sin u = sin '( i j -f- cr cos l cos #', 
 
 \ve get by transposition 
 
 cr s 
 sin u sin w 1 = cr cos ' cos a 1 sin #'. (9) 
 
 But we had 
 
 # w 1 ;r, or u = u l -\-x; hence sin u= sin (u 1 -\- ^r). 
 
 Developing this by Taylor's formula, 
 
 sin u = sin u 1 -(- ;r cos w l sin w 1 ; (10) 
 
 or sin u sin u 1 = x cos u 1 sin u 1 
 
 a* 
 = o- cos ' cos a 1 sin w 1 . (l i) 
 
 Solving this equation by approximation, 
 
 x cos u 1 = <T cos u 1 cos a 1 , or ^r = cr cos a 1 . 
 Substituting this in (il), 
 
 . . , i a ' i 
 
 x cos u 1 - cos" a sin u = (r cos cos a sin ;
 
 236 GEODETIC OPERATIONS. 
 
 dividing by cos u\ 
 
 <r 2 <r 9 
 x cos 2 a 1 . tan u 1 = a cos a 1 tan u 1 . 
 
 2 2 
 
 a* o- 1 
 
 x = <r cos a 1 tan u l + cos 2 or 1 tan a 1 
 
 = <r cos J tan u l (l cos* or 1 ) 
 
 = cr cos a 1 tan u l sin' a 1 = u u l ; (12) 
 
 substituting this in (8), 
 
 - = (u u l }( i - cos 2 u 1 }} i -f- -" sin ' cos u 1 cos a 1 
 
 (I \ 2 /I 2 
 
 -j ( sin" u 1 -}- 2 cos 2 u 1 cos 2 a 1 -f- sin 2 ' cos' a 1 ) I. (13) 
 
 Dividing this equation by (7), we have 
 
 5 u uT So* ~l 
 
 - = i (3 sin 2 u 1 -f- 2 sin 2 ' -f- sin 2 u 1 cos 2 a 1 I 
 
 = ^r-[_ l ~ ~^(~ sin!> u * + sin ' wl ( x ~~ sin * al )J
 
 FIGURE OF THE EARTH. 237 
 
 If we had taken u l as the unknown side in our spherical tri- 
 angle, giving 
 
 sin u l = sin u cos <r -\- cos u sin <r cos a, 
 
 we would have found, by pursuing a course similar to the 
 above 
 
 5 _'/ fa* . , \ 
 
 - = \i- sm usm a], 
 
 from which we could obtain 
 
 sin 9 u 1 sin" a 1 = sin a sin" or, or sin w 1 sin a 1 = sin # sin or ; 
 
 hence we can involve both the direct and reverse azimuth as 
 well as the terminal latitudes by writing for sin" u 1 sin 2 a 1 , sin 
 u 1 sin a 1 sin u sin a, so that (14) will become 
 
 . 
 
 I I --- sm u sin a sin u sin <*). (15) 
 12 
 
 Resuming the former notation, we will put a = Z, and or 1 = 
 
 Z 1 1 80; also, remembering that I If = -- . , , . -- ^r, 
 
 sin 3 ^(Z 1 Z}' 
 
 we can for smZ sin Z 1 write sin'-K-Z" 1 ^) (i #*), and 
 substitute in (15) the value of u u 1 in (A), which gives 
 
 = h ( i --- sin w 1 sin u sin Z" 1 sin Z} 
 
 S \ 12 
 
 [i - ~ (i - 70 + ^ (i - ^ s )(2 - 3^')} (16)
 
 238 GEODETIC OPERATIONS. 
 
 This still involves s and cr, so there is needed a relation be- 
 tween them. To attain this we take equation (7), using only 
 two terms, 
 
 \ 
 
 = <Tl I COS U COS '). 
 
 Squaring this, and omitting terms in *? 4 , 
 
 = o" a (i ^ s cos cos l ), 
 j 
 
 Or <7* = -YT 5 r> (17) 
 
 ar (i r cos cos ) 
 Writing /" = i + e* cos u cos w 1 , 
 
 (16) becomes 
 
 5 = sh\ (l -^ sin ' sin u sin Z 1 sin Z) 
 
 ^V" / si n Z sin Z 1 \ jV si n Z sin Z 1 . 
 
 /I _i_ __l 1 (2 ^A ^ 
 
 v ' i2a*\sin l i(Z l Z)/ 240(1* sin* %(Z l Zy J y j' 
 
 sin|(Z+ Z 1 ) 
 in which h = -. rr^i ^r- 
 
 This is substantially the same formula as given by Bessel in 
 Astronomische Nachrichten, No. 331, pp. 309-10, except Z 1 is 
 within the polar triangle, which gives 
 
 - Z 1 )'
 
 FIGURE OF THE EARTH. 239 
 
 sin Z sin Z* 
 
 and 
 
 or approximately 
 
 Then, writing 
 
 /" = i + *" cos u cos u\ and ^ = i -|- e* cos (u -f- '), 
 
 Bessel's formula becomes 
 
 sinZsin Z ' 
 
 _ ^-^ 
 
 sin ZsinZ 1 
 
 In both of these formulae it is to be remembered that 
 tan # = 4/1 ^ 2 tan Z, and tan u l = Vi e* tan U. 
 
 Also, if the line deviates but little from the meridian, the 
 first term will be sufficient. 
 
 When a long arc has been measured, it has been found best 
 to divide it into several sections, from each of which data can 
 be obtained for finding the axes of the earth, and the ellipticity. 
 When these arcs are small, the method given on page 228 will 
 give fair results. But Clarke's solution is perhaps the best ; it 
 is, in the main, as follows: 
 
 Let R, x, and y be the radius of curvature of an ellipse, and 
 
 co-ordinates of the point whose latitude is L, then -, -f- ^ = I ; 
 but we have shown that 
 
 x = a cos u, tan u = Vi e* tan L,
 
 240 GEODETIC OPERATIONS. 
 
 from which 
 
 x a cos L(i e* sin 3 L) - *, 
 
 also 
 
 j/ = sin u = #sin L(\ e*)(i e 1 sin 5 L} - * ; 
 
 tf=*(i-^)(i-*'sin'Z)-. 
 
 Expanding, and neglecting e', 
 
 2 sin 4 L + J/^ 4 sin 4 Z). 
 Substituting I cos 2 L for sinZ 3 , 
 
 Writing A=a(i-&- -&e<), B = - a(&+ &e<), C = 
 
 we have R A + 2B cos 2L + 2^ cos 4^, (B) 
 
 This is an ellipse if ^B 1 = 6AC. 
 
 Now, if 5" be the length of an arc of an elliptic meridian, it 
 was shown on page 231 that 
 
 , c . r . , , c ^sin udu 
 
 aS sin L = a sin udu, as = - : j-~. 
 
 sin L,
 
 FIGURE OF THE EARTH. 241 
 
 From a tan u b tan L, we found 
 
 du l/i - 
 
 therefore 
 
 dL ~ i - f sin' D 
 ds a sin u Vi 
 
 dL sin L(\ e* sin' L)' 
 But from the preceding relation 
 
 sin L V i 
 Sin U = 4 
 
 Substituting this, we have 
 
 3Z 
 
 by integration, 
 
 5 = ^Z + ^sin2Z \C sin 4^+ a constant. 
 
 If L be the mean latitude of an arc whose amplitude is A, 
 and the above expression be integrated between the limits 
 L ^A. and L -(- A., we will obtain 
 
 5" = A\. -\- 2B cos 2L sin A. -)- 6* cos 4^ sin 2/1. (19) 
 
 In this ^4, B, and Care the only unknown quantities, so that 
 if S, L, and A be free from errors, three equations would be suf- 
 ficient for determining A, B, or C, and, consequently, a, e, and 
 16
 
 242 GEODETIC OPERATIONS, 
 
 b. But every arc is affected with an error, in length as well as 
 middle and terminal latitudes, so that from a number of dis- 
 cordant results we must find the most probable values for A, 
 B, and C by the principles of least squares. 
 
 Suppose the terminal latitudes have a small error in each of 
 x, and xf, so that the amplitude would be A -j- x* x^ and 
 the latitudes L A +* and L A -f- *, x . 
 
 Placing these corrected values in (19), 
 
 s 2L sin (A + */ - *,) 
 
 C cos 4^ sin 2(A -f- .ar, 1 jr,). (20) 
 
 In expanding this we treat x? ;r, as a single term, and 
 being small, cos (x* x^ = I, and sin (x* x^) = x* x^ so 
 
 sin (\-\-x ^ x^) = sin A -)- cos A(jtr/ x^} ; 
 
 sin 2(A.+;r 1 I -* 1 )= 2 sin [A + (jr/ - ^)] cos [A -f (** - ^)] 
 
 = 2[sin X-l-cosA^, 1 ^^[cos A sin A(^/ ^)] 
 = sin 2 A -}- 2(jr,' -r- x^ cos 2A ; 
 
 substituting these expressions in (20), 
 
 S=A(l+ */ - ^) + 2,5 cos 2Z[sin A + cos Afo 1 - JT,)] 
 
 + Ccos 4^[sin 2A -}- 2(^, 1 #,) cos 2A]. 
 
 Solving for ^r, 1 ^r, 
 
 cos 2A) = 5 ^A 
 2Bcos2L sin A (Tcos^Zsin 2A. (21) 
 
 If we write A-\-2B cos A cos 2Z, 2(7 cos ^L cos 2A = , (2 1) 
 will reduce to
 
 FIGURE OF THE EARTH. 243 
 
 IS _\ 2BH . 
 ', x^ Irs- A J,u sin A cos 2L 
 
 f- sin 2! cos 4Z. (22) 
 
 Expressing ^r, 1 , .* and A in seconds we approximate the 
 length of a second of latitude by assuming the average radius 
 of curvature to be 20855500 ft. we must write 
 
 v= 2085 5 500 sin i". 
 
 Then we assume three auxiliary quantities, u, v, and Z, and 
 place 
 
 A 20855500\ 
 
 _ - _i_ v 
 
 A nr^~ ~\ 
 
 A 2OO ' IOOOO 
 
 C Z 
 
 A loooo" 
 Substituting these, (22) becomes 
 
 IOOOOU 
 
 _ 1 _U sin ^ cos 2 ^ 
 
 2 oo sin i" 
 
 sin A cos 2Zz/ sin 2A cos 
 "~ ~ioooo sin i" "^ loooo sin i" ' ^ ^ 
 
 Again we assume 
 
 IS sin A cos 2L
 
 244 GEODETIC OPERATIONS. 
 
 sin \ cos 2L^i sin 2\ cos 
 
 
 10000 sin i"' ' IOGOO sin i" ' 
 
 fj. = I -f- -g-^-jj- cos A. cos zL. 
 
 Then (23) can be written 
 
 x* x^ = m -\- au -\- bv -f- ^.Z, 
 or #,' = ;r 1 + a4-<< + ^ + c ^- ( 2 4) 
 
 For each arc or partial arc we will have an equation like (24), 
 which is to be solved by the principles of least squares, by 
 making the sum of the squares of the errors a minimum ; then 
 equating the differential coefficients of the symbolic errors 
 with respect to u, v, z, x?, etc., to zero, there will be as many 
 equations as there are unknown quantities to be solved by al- 
 gebraic methods. Knowing u, v, and z, we find A, B, and C, 
 which substituted in (B) give R. 
 
 To determine the axes and ellipticity, we take the equations 
 on page 240 and find that the coefficient of cos L = (A .Z?), 
 of cos 3- = %(B C), and of cos t>L = \C\ also, of sin L 
 A -\-B, of sin 3^ = #JB +Q, and of sin *>L \C. By making 
 these substitutions, we have 
 
 x- (A - B) cos L + #B <7)cos3+Kcos5Z; (25) 
 
 ) sin 3^ + \C sin 5^. (26) 
 
 But on page 231, 
 
 x = fa sin L dL, and y =fbcos L dL.
 
 FIGURE OF THE EARTH. 245 
 
 So (25) and (26) are the values of these integrals, which if in- 
 tegrated between the limits L = o and L = 90, will give 
 the semi-axes, 
 
 (27) 
 (28) 
 
 V SJBf , 
 
 =i- = - i + , approximately. 
 
 If these values be substituted in (25) and (26), we would have 
 
 (30) 
 
 (29) and (30) are the values of the co-ordinates of a point in 
 an elliptic curve whose axes are a and b, while (25) and (26) 
 are the co-ordinates of a point in the actual curve. The dif- 
 ference between the two will be the deviation of the actual 
 from the elliptic curve at any point. 
 
 I I 
 
 x x l = { C ~ } ( cos L - cos 3^ -f - cos
 
 246 
 
 GEODETIC OPERATIONS. 
 
 Si- (32) 
 
 Suppose P be the point on the 
 elliptic curve in latitude Z, and Q 
 the point on the actual curve in the 
 same latitude. P and Q will coin- 
 
 t? 
 cide when C g-^- = o, for this will 
 
 reduce (31) and (32) to x x' = o, 
 y 1 o, and will differ from one 
 
 $1? 
 
 another as C ^-v- changes from 
 
 a zero value. 
 
 If we take PS an infinitesimal 
 distance on the elliptic curve, and QS a corresponding length 
 along the normal, , we wilt have 
 
 FIG. 31. 
 
 = y-y 
 
 = x-x\ 
 
 QS= QU+SU= QU+PV 
 
 (x x 1 ) cos L -f- (y j/ 1 ) sin L, 
 
 or dR (x x 1 ) cos L -f L(y / ) sin L. 
 
 (33) 
 
 PS= VU= TU- TV 
 
 (x x 1 } sin L -|- (y j 1 ) cos L, 
 
 $ = (x x 1 } sin L -\- (y j/ 1 ) cos 
 
 (34)
 
 FIGURE OF THE EARTH. 247 
 
 Substituting in these equations the values of (x x l ~] and 
 y) from (3 1 ) an d (32), we find 
 
 $& 
 
 Clarke's values of a and b of 1866 would give 
 
 showing but a slight deviation of a meridian section from an 
 ellipse. 
 
 The Anglo-French arc places the actual curve 3.6 feet under 
 the ellipse in latitude 58, and 18.9 feet above in latitude 44 ; 
 while the Indian arc places it 19.6 feet under, in latitude 14, 
 and 9.3 feet above, in latitude 26. 
 
 The amplitude of an arc depending upon the latitude deter- 
 minations of its extremities is subject to an error from local 
 deflection. In some cases, at least a portion of these errors 
 can be corrected by computing the effects of attraction upon a 
 physical hypothesis; but in the main they are best treated 
 as accidental, and the figure of the earth determined by the 
 principle of least squares, in which the sum of the squares 
 of all errors shall be a minimum. 
 
 This was suggested by Walbeck in 1819, continued by 
 Schmidt in 1829, and perfected by Bessel in 1837. 
 
 Laplace in 1822, published the second volume of Mtcanique 
 Celeste, in which he discussed the figure of the earth, using 
 seven arcs: the Peruvian, Lacaille's Cape of Good Hope arc, 
 Mason and Dixon's, Boscovich's Italian, Delambre and Me- 
 chain's, Maupertuis' Lapland arc, and Liesganig's Austrian arc.
 
 248 GEODETIC OPERATIONS. 
 
 The second is unreliable, from an erroneously assumed cor- 
 rection for local attraction which shortened the arc by 9" too 
 much. The third was a measured arc, and not comparable 
 with a trigonometric one. And no confidence is now placed 
 in either the fourth or the last. 
 
 Bowditch, in his translation of the above-named work, con- 
 siders only the Peru and France arcs, and adds those of Eng- 
 land and India as completed in 1832. His conclusion is: 
 
 " It appears that this strictly elliptical form of the meridian 
 is more conformable to these observations than the irregular 
 figure obtained by Mr. Airy's calculation." 
 
 Sir George Airy published in the Encyclopedia Metropoli- 
 tana, under the heading " Figure of the Earth," in 1830, a dis- 
 cussion of fourteen meridian arcs and four arcs of parallel. 
 In 1841, Bessel gave to the public the results of his laborious 
 investigation of ten meridian arcs, having a total amplitude of 
 5o-5, and embracing thirty-eight latitude stations. The re- 
 sult gave an elliptic meridian, and the elements then published 
 are still known as those of Bessel's spheroid. 
 
 In 1858, in the "Account of the Principal Triangulation of 
 Great Britain and Ireland," Captain Clarke gives a most elabo- 
 rate discussion of eight arcs, having a total amplitude of 78 
 36', and embracing sixty-six latitude stations. 
 
 Again, in 1880, he revised his previous computations, using 
 corrected positions from which slightly different results were 
 obtained. 
 
 Mr. Schott discussed the combination of three American 
 arcs of meridian for determining the figure of the earth con- 
 sidered as a spheroid. He used the Pamlico-Chesapeake, Nan- 
 tucket, and Peruvian, having a total amplitude of noi' 12", 
 and embracing twenty-three latitudes. The conclusion de- 
 duced by Mr. Schott is : " The result from the combination of 
 the three American arcs is the preference it gives to Clarke's 
 spheroid over that of Bessel."
 
 FIGURE OF THE EARTH. 
 
 249 
 
 TABLE GIVING THE ELLIPTICITY AND LENGTH OF A QUAD- 
 RANT ON THE SPHEROIDAL HYPOTHESIS. 
 
 Date. 
 
 Authority. 
 
 Ellipticity. 
 
 Quadrant in Metres. 
 
 1819 
 
 Walbeck 
 
 go2 8 
 
 68 
 O OOO 2OO 
 
 1810 
 
 Schmidt 
 
 
 
 18^0 
 
 Airy 
 
 
 
 1841 
 
 Bessel 
 
 
 10 ooo 976 
 
 1856 
 
 Clarke 
 
 
 10 ooo 050 
 
 1863 
 
 Pratt 
 
 
 10 ooi 515 
 
 1866 
 
 Clarke 
 
 
 
 1868 
 
 Fischer 
 
 288 5 
 
 
 1872 
 
 
 Sa 
 
 a 
 
 1877 
 
 
 
 10 OOO 218 
 
 1878 
 
 
 286 5 
 
 10 002 232 
 
 1880 
 
 Clarke 
 
 
 10 000 081 
 
 
 
 
 9 
 
 Data for the Figure of 
 the Earth. 
 
 Bessel, 1841. 
 
 Clarke. 1866. 
 
 Coast Survey, Clark , 88o 
 1877. 
 
 Equatorial radius, a. . 
 Polar semi-axis, b. . . . 
 a b 
 Compression, .. 
 
 Mean length of a deg. 
 
 6377397-2M 
 6356079 
 
 i : 299.15 
 in I2O.6M 
 
 6378 206. 4M 
 6356583-8 
 
 i : 294.98 
 in 132.1 
 
 6378054.3X1 
 6357175 
 
 I : 305-4S 
 i" 135-9 
 
 6 378 248. 5 M 
 6 356 514. ?M 
 
 i : 293.5 
 in 131.8 
 
 The value of the ellipticity as deduced by pendulum-obser- 
 vations in accordance with Clairaut's theorem is I : 292.2, be- 
 ing almost the same as that obtained from geodetic measure- 
 ments. 
 
 Clarke's length of the quadrant would give for the metre 
 39.377786 inches, whereas the legal length is 39.370432 inches, 
 or .0073 inch too short. 
 
 LITERATURE OF THE FIGURE OF THE EARTH. 
 
 Pratt, A Treatise on Attractions, Laplace's Functions, and 
 the Figure of the Earth. London, 1861. 
 
 Roberts, Figure of the Earth. Van Nostrand 's Engineering 
 Magazine, vol. xxxii.,.pp. 228-242.
 
 2$O GEODETIC OPERATIONS. 
 
 Merriman, Figure of the Earth. New York, 1881. 
 
 U. S. Coast Survey Report for 1868, pp. 147-153. 
 
 U. S. Coast and Geodetic Survey Report for 18/7, pp. 84-95. 
 
 Clarke, Geodesy, pp. 302-322. London, 1880. 
 
 Laplace, Mecanique Celeste. Bowditch's Translation, vol. 
 ii., pp. 358-485. Boston, 1830. 
 
 Ordnance Survey, Account of Principal Triangulation, pp. 
 733-782. London, 1858. 
 
 Bruns, Die Figur der Erde. Berlin, 1878. 
 
 Baeyer, Grosse und Figur der Erde. Berlin, 1861. 
 
 Jordan, Handbuch der Vermessungskunde, vol. ii., pp. 377- 
 463. Stuttgart, 1878.
 
 FORMUL/E AND FACTORS.
 
 FORMULA AND FACTORS. 253 
 
 TRIGONOMETRIC EXPRESSIONS. 
 sin 2 a -f- cos 3 a = i ; 
 
 sin a = i/i cos 2 a 
 _ cos # 
 ~ cot a 
 
 Vi + cot* a 
 = cos a tan a 
 = 2 sin \a cos | 
 
 I 
 ~~ cosec a 
 
 sec a 
 
 sin a 
 tan a 
 
 cos a 
 
 cot a
 
 254 GEODETIC OPERATIONS. 
 
 sin a 
 
 r i sin a 
 i cos 2a 
 
 sin 2a 
 
 i cos 2a 
 
 cot * = 
 
 i -|- cos 2a 
 I 
 
 cosec a = 
 
 versin a i cos 2 sin 9 \a 
 chord <z = 2 sin \a. 
 sin ( V) si" cos b cos sin b ; 
 
 cos ( #) = cos a cos * ^ sin * sin ^' 
 
 tan ^ tan b 
 tan ( 0) = Y^te^'a tan b' 
 
 cot # cot 
 cot ( *) = cot ^ 
 
 sin 2^ = 2 sin # cos d!. 
 
 cos 2a cos a # sin 2 a 
 
 = 2 cos* a I = I 2 sin"
 
 FORMULA AND FACTORS. 255 
 
 2 cos 4 %a = i -}- cos 0. 
 
 2 sin 2 ^ = i cos a. 
 i cos a 
 
 tan 2 * = 
 
 I -f- COS tf 
 
 sm a sm b = 2 sin $(a b) cos ( 
 cos # -j- cos b 2 cos i(tf -j- ) cos%(a b]. 
 cos a cos # = 2 sin (# -f- b] sin ( a). 
 sin 2 a sin 2 b = sin (a -\- b) sin (a b). 
 cos 2 a sin 2 b = cos (a -J- ) cos (# #). 
 
 sin 2x = 2 sin JT cos jr. 
 
 sin 3jr = 3 sin jr 4 sin 3 jr. 
 
 sin 4jr = (4 sin JT 8 sin 3 jr) Vi sin 2 jr. 
 
 cos 2x = cos 2 JT sin 2 x = 2 cos 2 JT i. 
 
 cos 3jr = 4 cos 3 JT 3 cos jr. 
 
 cos 4jr = 8 cos 4 JT 8 cos 2 x -{- i. 
 
 2 tan jr 
 tan 2x = -r . 
 
 i tan JT 
 
 3 tan JT tan 3 jr 
 
 tan 3jr = 5 . 
 
 i 3 tan jr 
 
 4 tan jr 4 tan 3 x 
 tan 4x = -- . --T-.
 
 256 GEODETIC OPERATIONS. 
 
 TRIGONOMETRIC SERIES. 
 
 x' 
 
 X" 2X" \TX 
 
 f +-+ + 7 - 
 
 I X X* 2X* 
 
 i . 
 
 cosec , = _ + _+_,_+__ 
 
 , sin 8 x 3 sin 5 ^,3.5 sin' 
 
 arc " = sm " + ^T + ^:T + ^^6 
 
 = tan x \ tan 3 x -\- ^ tan 5 ^ . . . . 
 For very small angles Maskelyne's series is best. 
 
 / - , ( ** x '\ 
 
 = x V cos ^r -]-...= ^rl i ->- --- J. 
 
 BINOMIAL, EXPONENTIAL, AND LOGARITHMIC SERIES. 
 
 ^" ^ 
 
 ' a Y72 + log ' *273
 
 FORMULAE AND FACTORS. 
 log (X + I) : 
 
 M = nodulus = 0.4342945. 
 log M = 9-6377843. 
 
 CONVERSION OF METRES TO FEET. 
 
 Metres X 3.280869 = feet, or to log of metres add 0.5159889 
 X 1.093623 yards, " 0.038867 
 
 X 0.000621377 = mile, " " 6.7933550-10. 
 
 i toise = 76.734402 inches = 864 lines. 
 
 i Prussian foot = 139.13 lines. 
 i klafter = 840.76134 lines. 
 
 The toise is that of Peru, which is a standard at 13 R. 
 
 CONVERSION TABLES. 
 
 METRES INTO YARDS. 
 
 i metre = 1.093623 yards. 
 
 Metres. 
 
 Yards. 
 
 Metres. 
 
 Yards. 
 
 Metres. 
 
 Yards. 
 
 IOO OOO 
 
 109 362.3 
 
 3 000 
 
 3 280.87 
 
 60 
 
 65-617 
 
 90 ooo 
 
 98 426.1 
 
 2 OOO 
 
 2 187.25 
 
 50 
 
 54.681 
 
 80 ooo 
 
 87 489.8 
 
 I OOO 
 
 I 093.62 
 
 40 
 
 43 745 
 
 70 ooo 
 
 76 553.6 
 
 900 
 
 984.26 
 
 30 
 
 32.809 
 
 60 ooo 
 
 65 617.4 
 
 800 
 
 874.90 
 
 2O 
 
 21.872 
 
 50 ooo 
 
 54 681.2 
 
 700 
 
 765.54 
 
 10 
 
 10.936 
 
 40 ooo 
 
 43 744-9 
 
 600 
 
 656.17 
 
 9 
 
 9-843 
 
 30 ooo 
 
 32 808.7 
 
 500 
 
 546.81 
 
 8 
 
 8.749 
 
 20 000 
 
 21 872.5 
 
 400 
 
 437-45 
 
 7 
 
 7-655 
 
 10 000 
 
 10 036.2 
 
 300 
 
 328.09 
 
 6 
 
 6.562 
 
 9 ooo 
 
 9 842.61 
 
 2OO 
 
 218.72 
 
 5 
 
 5.468 
 
 8 ooo 
 
 8 748.98 
 
 IOO 
 
 109.36 
 
 4 
 
 4-374 
 
 7 ooo 
 
 7 655.36 
 
 90 
 
 98.426 
 
 3 
 
 3.281 
 
 6 ooo 
 
 6 561.74 
 
 80 
 
 87.490 
 
 2 
 
 2.187 
 
 5 ooo 
 
 5 468.12 
 
 70 
 
 76.554 
 
 I 
 
 1.094 
 
 4 ooo 
 
 4 374-49 
 
 
 

 
 258 
 
 GEODETIC OPERATIONS. 
 
 CONVERSION TABLES Continued. 
 
 YARDS INTO METRES. 
 
 i yard = 0.914392 metre. 
 
 Yards. 
 
 Metres. 
 
 Yards. 
 
 Metres. 
 
 Yards. 
 
 Metres. 
 
 IOO 000 
 
 91 439.2 
 
 3 000 
 
 2 743-18 
 
 60 
 
 54.864 
 
 90 ooo 
 
 82 295.3 
 
 2 OOO 
 
 I 828.78 
 
 50 
 
 45-720 
 
 80 ooo 
 
 73 I5I-3 
 
 I OOO 
 
 9I4-39 
 
 40 
 
 36.576 
 
 70 ooo 
 
 64 007.4 
 
 900 
 
 822.95 
 
 30 
 
 27.432 
 
 60 ooo 
 
 54 863.5 
 
 800 
 
 73L5I 
 
 2O 
 
 18.288 
 
 50 ooo 
 
 45 7I9-6 
 
 700 
 
 640 . 07 
 
 IO 
 
 9.144 
 
 40 ooo 
 
 36 575-7 
 
 600 
 
 548.6 4 
 
 9 
 
 8.230 
 
 30 ooo 
 
 27 431-8 
 
 500 
 
 457.20 
 
 8 
 
 7.315 
 
 20 ooo 
 
 18 287.8 
 
 400 
 
 365.76 
 
 7 
 
 6.401 
 
 10 000 
 
 9 143-9 
 
 300 
 
 274.32 
 
 6 
 
 5-486 
 
 9 ooo 
 
 8 229.53 
 
 2OO 
 
 182.88 
 
 5 
 
 4-572 
 
 8 ooo 
 
 7 3I5-I3 
 
 IOO 
 
 91.44 
 
 4 
 
 3.658 
 
 7 ooo 
 
 6 400.74 
 
 90 
 
 82.295 
 
 3 
 
 2-743 
 
 6 ooo 
 
 5 486.35 
 
 80 
 
 73.I5I 
 
 2 
 
 1.829 
 
 5 ooo 
 
 4 57I-96 
 
 70 
 
 64.007 
 
 I 
 
 0.914 
 
 4 ooo 
 
 3 657.57 
 
 
 
 
 
 METRES INTO STATUTE AND NAUTICAL MILES. 
 
 I metre = 0.00062138 statute mile. 
 i metre = o 00053959 nautical mile. 
 
 Metres. 
 
 Statute Miles. 
 
 Nautical Miles. 
 
 Metres. 
 
 Statute Miles. 
 
 Nautical Miles. 
 
 IOO 000 | 62.138 
 
 53-959 
 
 9 00 
 
 0-559 
 
 0.486 
 
 90 ooo , 55.9 2 4 
 
 48.563 
 
 800 
 
 0.497 
 
 0.432 
 
 80 ooo | 49-710 
 
 43.167 
 
 700 
 
 0-435 
 
 0.378 
 
 70 ooo . 43-496 
 
 37-772 
 
 600 
 
 0-373 
 
 0.324 
 
 60 ooo ! 37.283 
 
 32.376 
 
 500 
 
 0.311 
 
 0.270 
 
 50 ooo 
 
 31-069 
 
 26.980 
 
 400 
 
 0.249 
 
 0.216 
 
 40 ooo 
 
 24.855 
 
 21.584 
 
 300 
 
 0.186 
 
 0.162 
 
 30 ooo 
 
 18.641 
 
 16.188 
 
 200 
 
 0.124 
 
 0.108 
 
 20 OOO 
 
 12.428 
 
 10.792 
 
 IOO 
 
 0.062 
 
 0.054 
 
 IO OOO 
 
 6.214 
 
 5-396 
 
 90 
 
 0.056 
 
 0.049 
 
 9 ooo 
 
 5-592 
 
 4-856 
 
 80 
 
 0.050 
 
 0.043 
 
 8 ooo 
 
 4-971 
 
 4-3I7 
 
 70 
 
 0.043 
 
 0.038 
 
 7 ooo 
 
 4-350 
 
 3-777 
 
 60 
 
 0.037 
 
 0.032 
 
 6 ooo 
 
 3.728 
 
 3-238 
 
 50 
 
 0.031 
 
 0.027 
 
 5 ooo 
 
 3.107 
 
 2.698 
 
 40 
 
 0.025 
 
 0.022 
 
 4 ooo 
 
 2.486 
 
 2.158 
 
 30 
 
 0.019 
 
 0.016 
 
 3 ooo 
 
 1.864 
 
 1.619 
 
 2O 
 
 O.OI2 
 
 O.OII 
 
 2 000 
 
 1-243 
 
 1.079 
 
 IO 
 
 O.OO6 
 
 0.005 
 
 I OOO 
 
 O.62I 
 
 o 540 
 
 

 
 FORMULA AND FACTORS. 
 
 259 
 
 CONVERSION TABLES Continued. 
 
 STATUTE AND NAUTICAL MILES INTO METRES. 
 
 i statute mile = 1609.330 metres, 
 i nautical mile = 1853.248 metres. 
 
 Miles. 
 
 Metres in 
 Statute Miles. 
 
 Metres in 
 
 Nautical .Miles. 
 
 Miles. 
 
 Metres in Metres in 
 Statute Miles. Nautical Miles. 
 
 100 
 
 160 933-0 
 
 185 324-8 
 
 9 
 
 I 448.40 I 667.92 
 
 9 
 
 144 839.7 
 
 166 792.3 
 
 .8 
 
 I 287.46 I 482.60 
 
 80 
 
 128 746.4 
 
 I 4 8 259.8 
 
 7 
 
 I 126.53 I 297.27 
 
 70 
 
 112 653. I 
 
 129 727.4 
 
 .6 
 
 965.60 I 111.95 
 
 60 
 
 96 559-8 
 
 III 194.9 
 
 -5 
 
 804.67 926.62 
 
 50 
 
 80 466.5 
 
 92 662.4 
 
 -4 
 
 643-73 
 
 741 30 
 
 40 
 
 64 373-2 
 
 74 129-9 
 
 -3 
 
 482.80 
 
 555-97 
 
 30 
 
 48 279.9 
 
 55 597-4 
 
 .2 
 
 321.87 
 
 370.65 
 
 20 
 
 32 186.6 
 
 37 065.0 
 
 .1 
 
 160.93 
 
 185-32 
 
 10 
 
 16 093.3 18 532.5 
 
 .09 
 
 144.84 
 
 166.79 
 
 9 
 
 14 483-97 l6 679.23 
 
 .08 
 
 128.75 
 
 148.26 
 
 8 
 
 12 874.64 14 825.98 
 
 .07 
 
 112.65 
 
 129.73 
 
 7 
 
 ii 265.31 12 972.74 
 
 .06 
 
 96.56 
 
 111.19 
 
 6 
 
 g 655.98 ii 119.49 
 
 05 
 
 80.47 
 
 92.66 
 
 5 
 
 8 046.65 9 266.24 
 
 .04 
 
 64-37 
 
 74-13 
 
 4 
 
 6 437-32 7 412.99 
 
 03 
 
 48.28 
 
 55.60 
 
 3 
 
 4 827.99 ; 5 559-74 
 
 .02 
 
 32.19 
 
 37.06 
 
 2 
 
 3 218.66 
 
 3 706.50 
 
 .OI 
 
 16.09 
 
 18.53 
 
 I 
 
 i 609.33 
 
 i 853.25 
 
 
 
 
 Major semi axis = a, minor semi-axis = b, ellipticity = e 
 a-b 
 
 Bessel, a 6377397 . i$M, log = 7.8046434637 
 b = 6356078 . g6M, log = 6.8031892839 
 
 Clarke, a 6378206 . 4^f, 
 b = 6356583 . SM, 
 e = m- 
 
 log = 6.8046985352 
 log = 6.8032237974
 
 260 
 
 GEODETIC OPERATIONS. 
 
 CONSTANTS AND THEIR LOGARITHMS. 
 
 
 Number. 
 
 Log. 
 
 Ratio of circum. to diameter, 
 
 Tt 3.1415926 
 
 0.4971499 
 
 
 27T 6.2831853 
 
 0.7981799 
 
 
 TT" 9,-8696o44 
 
 0.9942997 
 
 
 Vn 1.7724538 
 
 0.2485749 
 
 Number of degrees in circum. 
 
 360 
 
 2.5563025 
 
 Number of minutes in circum 
 
 ., 21600 
 
 4-3344538 
 
 Number of seconds in circum, 
 
 ., 1296000 
 
 6.1 126050 
 
 Degrees in arc equal radius, 57- 2 95779 
 Minutes in arc equal radius, 3437 -7467 
 Seconds in arc equal radius, 206264 .806 
 
 1.7581226 
 
 Length of arc of I degree, 
 Length of arc of I minute, 
 Length of arc of i second, 
 
 Naperian base, 
 
 sin i" 
 * sin i" 
 
 0174533 
 .0002909 
 .00000485 
 
 5.3144251 
 
 8.2418774 10 
 6.4637261 10 
 4.6855749 - 10 
 
 2.7182818 0.4342945 
 
 4.6855749 
 4.3845449 
 
 N is the normal produced to the minor axis. R is the radius 
 of curvature in the meridian. Radius of curvature of the 
 parallel is equal to N cos /,. 
 
 The following tables are based upon Clarke's spheroid of 
 1866, and were computed in 1882. Since then similar tables 
 have been published by the Geodetic Survey, with which the 
 appended have been compared.
 
 FORMULAE AND FACTORS. 
 
 26l 
 
 Lat. 
 
 V ~(i - ,sin)f 
 Log JV. 
 
 d - *") 
 
 Log(i+^cos'Z,). 
 
 - (i-,*sin)l- 
 Logtf. 
 
 2400' 
 
 6.8049418 
 
 6.8024790 
 
 0.0024628 
 
 10 
 
 9450 
 
 4884 
 
 4566 
 
 2O 
 
 9481 
 
 4981 
 
 4500 
 
 30 
 
 9512 
 
 5076 
 
 4436 
 
 40 
 
 9545 
 
 5174 
 
 4371 
 
 50 
 
 9577 
 
 5270 
 
 4307 
 
 25 oo 
 
 9612 
 
 5370 
 
 4242 
 
 IO 
 
 9645 
 
 5470 
 
 4175 
 
 20 
 
 96/7 
 
 5569 
 
 4108 
 
 30 
 
 9711 
 
 5667 
 
 4044 
 
 40 
 
 9744 
 
 5768 
 
 3976 
 
 50 
 
 9777 
 
 5869 
 
 3908 
 
 26 00 
 
 9812 
 
 5968 
 
 3841 
 
 10 
 
 9846 
 
 6070 
 
 3774 
 
 20 
 
 9880 
 
 6173 
 
 3706 
 
 30 
 
 9915 
 
 62/6 
 
 3639 
 
 40 
 
 9948 
 
 6379 
 
 3569 
 
 50 
 
 9981 
 
 6482 
 
 3499 
 
 27 oo 
 
 6.8050017 
 
 6585 
 
 3432 
 
 IO 
 
 0051 
 
 6688 
 
 3363 
 
 20 
 
 0086 
 
 6794 
 
 3292 
 
 30 
 
 OI2O 
 
 6899 
 
 3221 
 
 40 
 
 0156 
 
 7006 
 
 3150 
 
 50 
 
 Oigl 
 
 7111 
 
 3080 
 
 28 oo 
 
 O227 
 
 7216 
 
 3011 
 
 IO 
 
 0263 
 
 7322 
 
 2941 
 
 20 
 
 0299 
 
 7429 
 
 2870 
 
 30 
 
 0334 
 
 7537 
 
 2/97 
 
 40 
 
 0371 
 
 7644 
 
 2727 
 
 5 
 
 0407 
 
 7752 
 
 2655 
 
 29 oo 
 
 0444 
 
 7862 
 
 2582 
 
 10 
 
 0480 
 
 797i 
 
 2509 
 
 20 
 
 0517 
 
 8081 
 
 2436 
 
 3 
 
 0555 
 
 8187 
 
 2368 
 
 4 
 
 0591 
 
 8296 
 
 2295 
 
 5 
 
 0628 
 
 8408 
 
 222O 
 
 30 oo 
 
 0664 
 
 8524 
 
 2I4O 
 
 10 
 
 O7OO 
 
 8636 
 
 2064 
 
 20 
 
 0738 
 
 8747 
 
 iggi 
 
 3 
 
 0776 
 
 8858 
 
 igiS 
 
 40 
 
 0813 
 
 8972 
 
 I8 4 I 
 
 50 
 
 0849 
 
 9084 
 
 1765 
 
 31 oo 
 
 089! 
 
 9198 
 
 1693 
 
 10 
 
 0928 
 
 9310 
 
 1618 
 
 20 
 
 0976 
 
 9426 
 
 1550 
 
 30 
 
 IOI4 
 
 9540 
 
 1474 
 
 4 
 
 1054 
 
 9654 
 
 1400 
 
 50 
 
 1089 
 
 9769 
 
 1320
 
 262 
 
 GEODETIC OPERATIONS. 
 
 Lat. 
 
 jV a 
 
 ? "(i-**) 
 
 Logd+^cos'Z,)- 
 
 (i -*sin* A)f 
 Log N. 
 
 (i - sin" L)f 
 
 Log;?. 
 
 32oo' 
 
 6.8051128 
 
 6.8029885 
 
 0.0021243 
 
 10 
 
 1166 
 
 6 . 8030002 
 
 1164 
 
 20 
 
 1205 
 
 0117 
 
 1088 
 
 30 
 
 1244 
 
 0232 
 
 1012 
 
 40 
 
 1283 
 
 0349 
 
 0934 
 
 50 
 
 1322 
 
 0466 
 
 0856 
 
 33 oo 
 
 1351 
 
 0583 
 
 0768 
 
 10 
 
 1390 
 
 0700 
 
 0690 
 
 20 
 
 1429 
 
 0818 
 
 0611 
 
 3 
 
 1469 
 
 0937 
 
 0532 
 
 40 
 
 1508 
 
 1055 
 
 0453 
 
 50 
 
 1548 
 
 1174 
 
 374 
 
 34 oo 
 
 1587 
 
 1293 
 
 0294 
 
 10 
 
 1627 
 
 1414 
 
 0213 
 
 20 
 
 1667 
 
 1532 
 
 oi35 
 
 30 
 
 1707 
 
 1652 
 
 0055 
 
 4 
 
 1746 
 
 1769 
 
 0.0019977 
 
 50 
 
 1785 
 
 1889 
 
 9896 
 
 35 oo 
 
 1828 
 
 2014 
 
 9814 
 
 10 
 
 1868 
 
 2134 
 
 9734 
 
 20 
 
 1909 
 
 2255 
 
 9654 
 
 30 
 
 1949 
 
 2376 
 
 9573 
 
 40 
 
 2989 
 
 2499 
 
 9490 
 
 50 
 
 2029 
 
 2619 
 
 9410 
 
 36 oo 
 
 2070 
 
 2743 
 
 9327 
 
 10 
 
 2III 
 
 2865 
 
 9246 
 
 20 
 
 2152 
 
 2987 
 
 9165 
 
 30 
 
 2192 
 
 3110 
 
 9082 
 
 40 
 
 2233 
 
 3234 
 
 8999 
 
 5 
 
 2274 
 
 3354 
 
 8920 
 
 37 oo 
 
 2316 
 
 3480 
 
 8836 
 
 10 
 
 2358 
 
 3602 
 
 8756 
 
 20 
 
 2398 
 
 3727 
 
 8671 
 
 30 
 
 2440 
 
 3851 
 
 8589 
 
 40 
 
 2482 
 
 3975 
 
 8507 
 
 50 
 
 2523 
 
 4098 
 
 8425 
 
 38 oo 
 
 2505 
 
 4225 
 
 S340 
 
 IO 
 
 2607 
 
 4350 
 
 8257 
 
 20 
 
 2648 
 
 4475 
 
 8i73 
 
 30 
 
 260X) 
 
 4599 
 
 8091 
 
 40 
 
 2732 
 
 4726 
 
 8006 
 
 50 
 
 2775 
 
 4846 
 
 7929 
 
 39 oo 
 
 2815 
 
 4977 
 
 7838 
 
 10 
 
 2857 
 
 5102 
 
 7755 
 
 20 
 
 2899 
 
 5228 
 
 7671 
 
 30 
 
 2941 
 
 5355 
 
 7586 
 
 40 
 
 2984 
 
 5482 
 
 7502 
 
 50 
 
 3025 
 
 5608 
 
 7417
 
 FORMULA AND FACTORS. 
 
 263 
 
 Lat. 
 
 a 
 
 (!-*) 
 
 U.C.+*-* 
 
 Log AT. 
 
 (i - * sin* L)f 
 Log R. 
 
 4OOo' 
 
 6.8053068 
 
 6-8035734 
 
 0.0017334 
 
 IO 
 
 3"i 
 
 5859 
 
 7252 
 
 20 
 
 3154 
 
 5987 
 
 7167 
 
 30 
 
 
 6115 
 
 7080 
 
 40 
 
 3237 
 
 6242 
 
 6995 
 
 50 
 
 3280 
 
 6367 
 
 6913 
 
 41 oo 
 
 3321 
 
 6497 
 
 6824 
 
 10 
 
 3365 
 
 6625 
 
 6740 
 
 20 
 
 3407 
 
 6752 
 
 6655 
 
 3 
 
 3450 
 
 6880 
 
 6570 
 
 40 
 
 3592 
 
 7008 
 
 6484 
 
 50 
 
 3535 
 
 7130 
 
 6405 
 
 42 oo 
 
 3577 
 
 7263 
 
 6294 
 
 IO 
 
 3620 
 
 7392 
 
 6228 
 
 20 
 
 3663 
 
 7519 
 
 6144 
 
 30 
 
 3706 
 
 7649 
 
 6057 
 
 40 
 
 3749 
 
 7777 
 
 597 2 
 
 50 
 
 379 2 
 
 795 
 
 5887 
 
 43 oo 
 
 3832 
 
 8032 
 
 5802 
 
 IO 
 
 3877 
 
 8160 
 
 5717 
 
 20 
 
 39*9 
 
 8288 
 
 5631 
 
 3 
 
 3962 
 
 8417 
 
 5545 
 
 4 
 
 4004 
 
 8549 
 
 5455 
 
 50 
 
 4047 
 
 8680 
 
 53^7 
 
 44 oo 
 
 4090 
 
 8803 
 
 5287 
 
 10 
 
 4134 
 
 8930 
 
 5204 
 
 20 
 
 4177 
 
 9059 
 
 5116 
 
 30 
 
 4219 
 
 9188 
 
 5031 
 
 40 
 
 4262 
 
 9317 
 
 4945 
 
 50 
 
 4306 
 
 9445 
 
 4861 
 
 45 oo 
 
 4347 
 
 9575 
 
 4772 
 
 10 
 
 439* 
 
 9704 
 
 4687 
 
 20 
 
 4434 
 
 9834 
 
 4600 
 
 30 
 40 
 
 4477 
 
 9961 
 6.8040090 
 
 429 
 
 50 
 
 4563 
 
 0218 
 
 4345 
 
 46 oo 
 
 4604 
 
 0347 
 
 4258 
 
 10 
 
 4648 
 
 0476 
 
 4172 
 
 20 
 
 4690 
 
 0605 
 
 4085 
 
 3 
 
 4734 
 
 0734 
 
 4000 
 
 40 
 
 50 
 
 4777 
 4820 
 
 0860 
 0989 
 
 3917 
 3831 
 
 47 oo 
 
 4861 
 
 1118 
 
 3744 
 
 IO 
 
 4905 
 
 1247 
 
 3658 
 
 20 
 
 4948 
 
 1376 
 
 3572 
 
 3 
 
 4991 1504 
 
 3487 
 
 4 
 
 5033 
 
 1631 
 
 3402 
 
 50 
 
 5076 
 
 1759 
 
 3317
 
 26 4 
 
 GEODETIC OPERATIONS. 
 
 Lat. 
 
 v a 
 
 R d-<') 
 
 Log (i + <r' cos" L). 
 
 (i - * sin* L* 
 LogJV. 
 
 1n- + **L]K 
 
 LogK. 
 
 4Soo' 
 
 6.8055118 
 
 6.8041887 
 
 0.0013231 
 
 10 
 
 5l6o 
 
 2016 
 
 3144 
 
 2O 
 
 5202 
 
 2144 
 
 3058 
 
 30 
 
 5244 
 
 2272 
 
 2972 
 
 40 
 
 5289 
 
 2400 
 
 2889 
 
 50 
 
 5333 
 
 2528 
 
 2805 
 
 49 oo 
 
 5374 
 
 2657 
 
 2717 
 
 10 
 
 5417 
 
 2784 
 
 2633 
 
 20 
 
 5459 
 
 2909 
 
 2550 
 
 3 
 
 5501 
 
 3037 
 
 2464 
 
 40 
 
 5545 
 
 3163 
 
 2382 
 
 50 
 
 5587 
 
 3293 
 
 2294 
 
 50 oo 
 
 5629 
 
 3418 
 
 2211 
 
 10 
 
 5672 
 
 3544 
 
 2128 
 
 20 
 
 57H 
 
 3671 
 
 2043 
 
 30 
 
 5756 
 
 3798 
 
 1958 
 
 40 
 
 5798 
 
 3925 
 
 1873 
 
 50 
 
 5841 
 
 4048 
 
 1790 
 
 THE A, B, C, >, E GEODETIC FACTORS. 
 From latitude 24 to 48, inclusive. 
 
 A = 
 
 D 
 
 c= 
 
 > = 
 
 2NR arc i" 
 
 |^ 3 sin L cos L 
 (i -^sTiTZ)!' 
 
 I 4- 3 tan 3 L 
 
 Referred to Clarke's spheroid of 1866.
 
 FORMULA AND FACTORS. 
 
 265 
 
 Lat. 
 
 Log A. 
 
 Log B. Log C. 
 
 Log/?. 
 
 Log E. 
 
 2400' 
 
 8.5094834 
 
 8.5119462 
 
 1.05456 
 
 2.2629 
 
 5.8147 
 
 05 
 
 818 
 
 415 
 
 625 
 
 40 
 
 59 
 
 IO 
 
 802 
 
 368 
 
 794 
 
 52 
 
 72 
 
 15 
 
 786 
 
 320 
 
 962 
 
 64 
 
 85 
 
 20 
 
 769 
 
 271 
 
 1.06130 
 
 75 
 
 97 
 
 25 
 
 753 
 
 223 
 
 297 
 
 86 
 
 5.8210 
 
 30 
 
 738 
 
 174 
 
 464 
 
 97 
 
 23 
 
 35 
 
 720 
 
 127 
 
 631 
 
 2.2708 
 
 36 
 
 40 
 
 704 
 
 7 8 
 
 797 
 
 19 
 
 49 
 
 45 688 
 
 028 
 
 962 
 
 30 
 
 62 
 
 50 672 
 
 8.5118979 
 
 1.07128 
 
 40 
 
 74 
 
 55 6 59 
 
 030 
 
 -93 
 
 5i 
 
 87 
 
 25 oo 640 
 
 882 
 
 457 
 
 62 
 
 5-8300 
 
 05 623 
 
 833 
 
 621 
 
 72 
 
 13 
 
 10 607 
 
 782 
 
 785 
 
 83 
 
 26 
 
 15 
 
 59 1 
 
 733 
 
 948 
 
 93 
 
 39 
 
 20 
 
 573 
 
 684 
 
 i. 08111 
 
 2.2804 
 
 52 
 
 25 
 
 556 
 
 634 
 
 274 
 
 15 
 
 66 
 
 30 
 
 541 
 
 585 
 
 435 
 
 25 
 
 79 
 
 35 
 
 524 
 
 535 
 
 597 
 
 35 
 
 92 
 
 40 
 
 508 
 
 484 
 
 759 
 
 45 
 
 5-8405 
 
 45 
 
 491 
 
 437 
 
 920 
 
 55 
 
 18 
 
 5 
 
 473 
 
 383 
 
 1.09080 
 
 65 
 
 31 
 
 55 45 6 
 
 337 
 
 241 
 
 75 
 
 45 
 
 26 oo 440 
 
 283 
 
 400 
 
 85 
 
 58 
 
 05 423 
 
 232 560 
 
 95 
 
 7i 
 
 10 406 
 
 181 
 
 719 
 
 2.2905 
 
 85 
 
 15 ^88 
 
 130 
 
 878 
 
 15 
 
 98 
 
 20 
 
 372 
 
 078 
 
 i . 10036 
 
 24 
 
 5-8512 
 
 25 
 
 354 
 
 027 
 
 194 
 
 34 
 
 25 
 
 3 
 
 337 
 
 8.5"7977 
 
 352 
 
 44 
 
 39 
 
 35 
 
 320 
 
 924 
 
 509 
 
 53 
 
 52 
 
 40 
 
 33 
 
 874 
 
 666 
 
 63 
 
 66 
 
 45 
 
 287 
 
 811 
 
 854 
 
 72 
 
 79 
 
 50 
 
 55 
 
 270 
 252 
 
 770 
 718 
 
 979 
 
 i. 11135 
 
 81 
 91 
 
 93 
 5.8606 
 
 27 oo 
 
 235 
 
 667 
 
 290 
 
 2.3000 
 
 20 
 
 05 
 
 218 
 
 616 
 
 445 
 
 09 
 
 34 
 
 10 
 
 201 
 
 564 
 
 600 
 
 18 
 
 47 
 
 15 
 
 182 
 
 5" 
 
 755 
 
 27 
 
 6l 
 
 2O 
 
 1 66 
 
 458 
 
 909 
 
 36 
 
 75 
 
 25 
 30 
 35 
 
 148 
 132 
 H3 
 
 405 
 353 
 310 
 
 1.12063 
 217 
 370 
 
 45 
 54 
 63 
 
 89 
 5 .8 7 0, 
 
 ^10 
 
 095 
 
 248 
 
 523 
 
 72 
 
 30 
 
 4 U 
 
 45 
 50 
 55 
 
 077 
 059 
 041 
 
 195 
 Mi 
 089 
 
 676 
 828 
 
 980 
 
 8r 
 89 
 i 98 
 
 44 
 58 
 69
 
 266 
 
 GEODETIC OPERATIONS. 
 
 Lat. 
 
 Log A. 
 
 Log B. 
 
 LogC. 
 
 Log D. Log E. 
 
 2800' 
 
 8.5094025 
 
 8.511 7036 
 
 1.13132 
 
 2.3107 i 5-8786 
 
 05 
 
 006 
 
 8.5116983 
 
 284 
 
 15 99 
 
 IO 
 
 .8.5093989 
 
 930 
 
 435 
 
 24 
 
 5-8813 
 
 15 
 
 970 
 
 8 7 6 
 
 586 
 
 32 27 
 
 20 
 
 952 
 
 823 
 
 737 
 
 4i 
 
 4i 
 
 25 
 
 936 
 
 768 
 
 887 
 
 49 56 
 
 30 
 
 918 
 
 715 
 
 i . 14037 
 
 57 i 70 
 
 35 
 
 899 
 
 66 1 
 
 187 
 
 65 i 84 
 
 40 
 
 881 
 
 608 
 
 336 
 
 74 
 
 98 
 
 45 
 
 863 
 
 552 
 
 485 
 
 82 
 
 5-8912 
 
 50 
 
 845 
 
 498 
 
 634 
 
 90 26 
 
 55 
 
 827 
 
 444 
 
 783 
 
 98 40 
 
 29 oo 
 
 808 
 
 390 
 
 932 
 
 2.3206 55 
 
 05 
 
 790 
 
 335 
 
 1.15080 
 
 14 69 
 
 10 
 
 772 
 
 281 
 
 227 
 
 22 83 
 
 15 
 
 753 
 
 226 
 
 375 
 
 29 98 
 
 20 
 
 735 
 
 171 
 
 522 
 
 37 5-9012 
 
 25 
 
 716 
 
 116 
 
 669 
 
 45 26 
 
 30 
 
 698 
 
 061 
 
 816 
 
 53 i 4i 
 
 35 
 
 679 
 
 007 
 
 963 
 
 60 55 
 
 40 
 
 66 1 
 
 8.5"5950 
 
 1.16109 
 
 68 
 
 70 
 
 45 
 
 644 
 
 896 
 
 255 
 
 75 i 84 
 
 50 
 
 624 
 
 841 
 
 401 
 
 83 98 
 
 55 
 
 605 
 
 737 
 
 546 9 5.9"3 
 
 30 oo 
 
 588 
 
 728 
 
 691 
 
 98 
 
 27 
 
 05 
 
 570 
 
 672 
 
 835 
 
 2.3305 
 
 42 
 
 IO 
 
 552 
 
 616 
 
 981 
 
 12 57 
 
 15 
 
 533 
 
 561 
 
 1.17126 
 
 19 
 
 7i 
 
 20 
 
 5U 
 
 505 
 
 270 
 
 27 
 
 86 
 
 25 
 
 494 
 
 449 
 
 414 
 
 34 5-9201 
 
 30 
 
 476 
 
 394 
 
 558 
 
 4i 15 
 
 35 
 
 458 
 
 337 
 
 701 
 
 48 30 
 
 40 
 
 439 
 
 280 
 
 845 
 
 55 45 
 
 45 
 
 420 
 
 225 
 
 988 
 
 62 60 
 
 50 
 
 401 
 
 168 
 
 1.18131 
 
 69 74 
 
 55 
 
 376 
 
 112 
 
 274 
 
 75 j 89 
 
 31 oo 
 
 361 
 
 054 
 
 416 
 
 83 5-9304 
 
 05 
 
 339 
 
 8.5114998 
 
 578 
 
 89 19 
 
 IO 
 
 324 
 
 942 
 
 700 
 
 96 34 
 
 15 
 
 35 
 
 884 
 
 842 
 
 2.3402 S 49 
 
 20 
 
 286 
 
 826 
 
 984 
 
 09 
 
 64 
 
 25 
 
 267 
 
 769 
 
 1.19125 
 
 16 
 
 78 
 
 30 
 
 248 
 
 712 
 
 266 
 
 22 
 
 93 
 
 35 
 
 229 
 
 655 
 
 407 
 
 2 9 
 
 5.9408 
 
 40 
 
 211 
 
 598 
 
 548 
 
 35 
 
 23 
 
 45 
 
 192 
 
 539 
 
 688 
 
 4i 
 
 39 
 
 50 
 
 173 
 
 483 
 
 829 
 
 48 
 
 54 
 
 55 
 
 153 
 
 424 
 
 969 54 
 
 69
 
 FORMULA AND FACTORS. 
 
 26; 
 
 Lat. 
 
 Log A. 
 
 Log B. 
 
 LogC. 
 
 LogZ). 
 
 Log*. 
 
 32oo' 
 
 8.5093134 
 
 8.5114367 
 
 1 . 20109 
 
 2.3460 
 
 5-9484 
 
 05 
 
 "5 
 
 309 
 
 2 4 8 
 
 66 
 
 99 
 
 10 096 
 
 251 
 
 388 
 
 73 
 
 5-95I4 
 
 15 077 
 
 193 
 
 527 
 
 79 
 
 29 
 
 20 057 
 
 135 
 
 666 
 
 85 
 
 44 
 
 25 037 
 
 077 
 
 805 
 
 
 60 
 
 30 oi 8 
 
 02O 
 
 944 
 
 97 
 
 75 
 
 35 8.5092998 
 
 8.5113963 
 
 1.21082 
 
 2 3503 
 
 90 
 
 40 979 
 
 903 
 
 221 
 
 09 
 
 5.9606 
 
 45 960 
 
 844 
 
 359 
 
 14 
 
 21 
 
 50 940 
 
 786 
 
 497 
 
 20 
 
 36 
 
 55 921. 
 
 727 
 
 635 
 
 26 
 
 51 
 
 33 oo 
 
 901 
 
 669 
 
 772 
 
 32 
 
 67 
 
 05 
 
 881 
 
 611 
 
 910 
 
 37 
 
 82 
 
 10 
 
 862 
 
 552 
 
 1.22047 
 
 43 
 
 98 
 
 15 
 
 842 
 
 492 
 
 184 
 
 48 
 
 5-9713 
 
 20 
 
 823 
 
 434 
 
 321 
 
 54 
 
 2 9 
 
 25 
 
 803 
 
 374 
 
 458 
 
 59 
 
 44 
 
 30 
 
 783 
 
 315 
 
 594 
 
 65 
 
 60 
 
 35 
 
 764 
 
 257 
 
 730 
 
 70 
 
 75 
 
 40 
 
 744 
 
 197 
 
 867 
 
 76 
 
 91 
 
 45 
 
 724 
 
 137 
 
 1.23003 
 
 81 
 
 5.9807 
 
 5 
 
 704 
 
 078 
 
 139 
 
 86 
 
 22 
 
 55 
 
 684 
 
 018 
 
 274 
 
 91 
 
 38 
 
 34 oo 
 05 
 
 665 
 645 
 
 8.5112959 
 898 
 
 409 
 545 
 
 97 
 
 2 . 3602 
 
 54 
 69 
 
 10 
 
 625 
 
 839 
 
 680 
 
 07 
 
 85 
 
 15 
 
 605 
 
 779 
 
 8i5 
 
 12 
 
 5.9901 
 
 20 
 
 585 
 
 720 
 
 950 
 
 17 
 
 17 
 
 25 
 
 565 
 
 660 
 
 1.24085 
 
 22 
 
 32 
 
 3 
 
 545 
 
 600 
 
 220 
 
 27 
 
 48 
 
 35 
 40 
 
 525 
 505 
 
 540 
 481 
 
 353 
 489 
 
 32 
 
 37 
 
 64 
 
 So 
 
 45 
 5 
 
 485 
 465 
 
 420 
 363 
 
 623 
 
 757 
 
 46 
 
 96 
 
 6.0012 
 
 55 
 
 445 
 
 299 
 
 891 
 
 51 
 
 27 
 
 35 oo 
 05 
 
 424 
 404 
 
 238 
 178 
 
 1-25023 
 157 
 
 56 
 60 
 
 44 
 60 
 
 10 
 
 383 
 
 118 
 
 290 
 
 65 
 
 76 
 
 15 
 
 20 
 25 
 30 
 
 35 
 
 363 
 344 
 320 
 303 
 283 
 
 058 
 8.5111997 
 936 
 875 
 814 
 
 424 
 
 557 
 690 
 823 
 955 
 
 69 
 74 
 
 78 
 83 
 87 
 
 92 
 6.0108 
 23 
 40 
 56 
 
 40 
 45 
 50 
 
 55 
 
 263 
 24-3 
 223 
 203 
 
 753 
 6 9 3 
 633 
 571 
 
 i . 26088 
 220 
 353 
 485 
 
 92 
 96 
 2.3700 
 04 
 
 72 
 88 
 6 . 0204 
 21
 
 268 
 
 GEODETIC OPERATIONS. 
 
 Lat. 
 
 Log A. 
 
 Log. 
 
 Log C. 
 
 LogD. 
 
 Lo gJ e. 
 
 36oo' 
 
 8.5092182 
 
 8:5111509 
 
 1.26617 
 
 2.3709 
 
 6.0237 
 
 05 
 
 161 
 
 448 
 
 749 
 
 13 
 
 53 
 
 10 
 
 141 
 
 387 
 
 881 
 
 17 
 
 69 
 
 15 
 
 121 
 
 326 
 
 1.27013 
 
 21 
 
 86 
 
 20 
 
 100 
 
 265 
 
 145 
 
 25 
 
 6.0302 
 
 .25 
 
 080 
 
 20 3 
 
 276 
 
 2 9 
 
 18 
 
 30 
 
 060 
 
 142 
 
 407 
 
 33 
 
 35 
 
 35 
 
 039 
 
 080 
 
 539 
 
 37 
 
 5i 
 
 40 
 
 018 
 
 018 
 
 670 
 
 4i 
 
 67 
 
 45 
 
 8.5091998 
 
 8.5110957 
 
 801 
 
 44 
 
 84 
 
 50 
 
 978 
 
 895 
 
 93i 
 
 48 
 
 6.0400 
 
 55 
 
 956 
 
 834 
 
 I . 28062 
 
 52 
 
 17 
 
 37 oo 
 
 936 
 
 772 
 
 193 
 
 56 
 
 33 
 
 05 
 
 915 
 
 710 
 
 323 
 
 60 
 
 50 
 
 10 
 
 894 
 
 648 
 
 454 
 
 63 
 
 66 
 
 15 
 
 874 
 
 587 
 
 584 
 
 66 
 
 83 
 
 20 
 
 854 
 
 525 
 
 714 
 
 70 
 
 6.0500 
 
 25 
 
 833 
 
 462 
 
 845 
 
 74 
 
 16 
 
 30 
 
 812 
 
 401 
 
 975 
 
 77 
 
 33 
 
 35 
 
 791 
 
 339 
 
 1.29104 
 
 81 
 
 50 
 
 40 
 
 771 
 
 276 
 
 234 
 
 84 
 
 66 
 
 45 
 
 750 
 
 215 
 
 364 
 
 87 
 
 83 
 
 50 
 
 729 
 
 151 
 
 494 
 
 9i 
 
 6 . 0600 
 
 55 
 
 708 
 
 090 
 
 623 
 
 94 
 
 17 
 
 38 oo 
 
 687 
 
 027 
 
 753 
 
 97 
 
 33 
 
 05 
 
 667 
 
 8.5109964 
 
 882 
 
 2.3800 
 
 50 
 
 10 
 
 646 
 
 902 
 
 1.30011 
 
 03 
 
 67 
 
 15 
 
 625 
 
 840 
 
 140 
 
 07 
 
 84 
 
 20 
 
 604 
 
 777 
 
 
 09 
 
 6.0701 
 
 25 
 
 533 
 
 7*5 
 
 398 
 
 13 
 
 18 
 
 30 
 
 562 
 
 652 
 
 527 
 
 16 
 
 35 
 
 35 
 
 541 
 
 59 
 
 656 
 
 18 
 
 51 
 
 40 
 
 521 
 
 526 
 
 785 
 
 22 
 
 68 
 
 45 
 
 499 
 
 463 
 
 913 
 
 24 
 
 85 
 
 50 
 
 479 
 
 401 
 
 1.31042 
 
 27 
 
 6.0802 
 
 55 
 
 45? 
 
 338 
 
 170 
 
 30 
 
 19 
 
 39 oo 
 
 437 
 
 275 
 
 299 
 
 33 
 
 37 
 
 05 
 
 416 
 
 212 
 
 427 
 
 35 
 
 54 
 
 10 
 
 395 
 
 150 
 
 555 
 
 38 
 
 7i 
 
 15 
 
 374 
 
 099 
 
 683 
 
 4i 
 
 88 
 
 20 
 
 353 
 
 023 
 
 811 
 
 43 
 
 6.0905 
 
 25 
 
 S3 2 
 
 8.5108960 
 
 939 
 
 46 
 
 22 
 
 30 
 
 3n 
 
 8 97 
 
 1.32067 
 
 48 
 
 40 
 
 35 
 
 290 
 
 843 
 
 195 
 
 51 
 
 57 
 
 40 
 
 269 
 
 770 
 
 323 
 
 53 
 
 74 
 
 45 
 
 248 
 
 707 
 
 450 
 
 56 
 
 9i 
 
 5 
 
 227 
 
 644 
 
 578 
 
 58 
 
 6.1009 
 
 55 
 
 206 
 
 581 
 
 706 
 
 61 
 
 26
 
 FORMULAE AND FACTORS. 
 
 269 
 
 Lat. 
 
 Log A. 
 
 LogS. 
 
 LogC. 
 
 LogD. 
 
 LogE. 
 
 4ooo' 
 
 8.5091184 
 
 8.5108518 
 
 1.32833 
 
 2.3863 
 
 6.1043 
 
 05 
 
 I6 3 
 
 455 
 
 960 
 
 65 
 
 61 
 
 10 
 
 142 
 
 393 
 
 1.33088 
 
 67 
 
 78 
 
 15 
 
 125 
 
 327 
 
 215 
 
 69 
 
 96 
 
 20 
 
 099 
 
 264 
 
 342 
 
 72 
 
 6.1113 
 
 25 
 
 079 
 
 2OI 
 
 470 
 
 74 
 
 3 
 
 3 
 
 057 
 
 137 
 
 596 
 
 76 
 
 48 
 
 35 
 
 036 
 
 073 
 
 723 
 
 78 
 
 65 
 
 40 
 
 015 
 
 OIO 
 
 850 
 
 80 
 
 83 
 
 45 
 
 8.5090998 
 
 8.5107946 
 
 977 
 
 82 
 
 6.I2OI 
 
 50 
 
 972 
 
 883 
 
 1.34104 
 
 84 
 
 18 
 
 55 
 
 952 
 
 820 
 
 231 
 
 86 
 
 36 
 
 41 oo 
 
 930 
 
 755 
 
 358 
 
 88 
 
 54 
 
 05 
 
 909 
 
 6 9I 
 
 485 
 
 90 
 
 71 
 
 10 
 
 888 
 
 628 
 
 611 
 
 gi 
 
 89 
 
 15 
 
 867 
 
 574 
 
 738 
 
 93 
 
 6.1307 
 
 20 
 
 845 
 
 500 
 
 864 
 
 95 
 
 24 
 
 25 
 
 824 
 
 437 
 
 991 
 
 96 
 
 42 
 
 30 
 
 803 
 
 373 
 
 1. 35"7 
 
 98 
 
 60 
 
 35 
 
 781 
 
 308 
 
 244 
 
 2.3900 
 
 78 
 
 40 
 
 760 
 
 244 
 
 370 
 
 OI 
 
 96 
 
 45 
 
 739 
 
 181 
 
 497 
 
 03 
 
 6.1413 
 
 50 
 
 7i8 
 
 117 
 
 623 
 
 04 
 
 3 1 
 
 55 
 
 696 
 
 053 
 
 749 
 
 06 
 
 49 
 
 42 oo 
 05 
 
 675 
 653 
 
 8.5106989 
 925 
 
 874 
 1.36001 
 
 07 
 08 
 
 67 
 
 85 
 
 10 
 
 632 
 
 861 
 
 127 
 
 IO 
 
 6.1503 
 
 15 
 
 610 
 
 797 
 
 253 
 
 II 
 
 21 
 
 20 
 
 590 
 
 733 
 
 379 
 
 12 
 
 39 
 
 25 
 
 568 
 
 668 
 
 55 
 
 14 
 
 57 
 
 30 
 
 547 
 
 604 
 
 631 
 
 15 
 
 75 
 
 35 
 40 
 
 524 
 504 
 
 541 
 
 476 
 
 757 
 883 
 
 16 
 
 17 
 
 94 
 6.1612 
 
 45 
 
 483 
 
 413 
 
 1.37009 
 
 18 
 
 3 
 
 5 
 
 460 
 
 348 
 
 135 
 
 19 
 
 48 
 
 55 
 43 oo 
 05 
 
 439 
 419 
 396 
 
 284 
 
 220 
 156 
 
 261 
 386 
 
 5" 
 
 20 
 21 
 22 
 
 66 
 85 
 6.1703 
 
 10 
 
 376 
 
 092 
 
 638 
 
 23 
 
 21 
 
 15 
 
 354 
 
 028 
 
 764 
 
 24 
 
 4 
 
 20 
 
 333 
 
 8.5105963 
 
 889 
 
 25 
 
 58 
 
 25 
 
 312 
 
 899 
 
 1.38015 
 
 25 
 
 76 
 
 30 
 
 35 
 
 290 
 
 269 
 
 835 
 771 
 
 141 
 266 
 
 26 
 27 
 
 95 
 6.1813 
 
 40 
 45 
 
 247 
 226 . 
 
 7 06 
 6 4 2 
 
 392 
 
 27 
 28 
 
 32 
 50 
 
 50 
 
 55 
 
 204 
 
 183 
 
 578 
 513 
 
 643 
 769 
 
 29 
 29 
 
 69 
 
 87
 
 2/0 
 
 GEODETIC OPERATIONS. 
 
 Lat. 
 
 Log .4. 
 
 Log,?. 
 
 LogC. 
 
 LogZ>. 
 
 Log. 
 
 4400' 
 
 8.5090162 
 
 8.5105449 
 
 1.38894 2.3930 
 
 6.1906 
 
 05 
 
 140 
 
 375 
 
 1 . 3902.0 30 
 
 24 
 
 IO 
 
 118 
 
 3" 
 
 145 31 
 
 43 
 
 15 
 
 097 
 
 256 
 
 271 
 
 31 
 
 62 
 
 20 
 
 076 
 
 193 
 
 396 
 
 32 
 
 80 
 
 25 
 
 054 
 
 128 
 
 522 
 
 32 
 
 99 
 
 30 
 
 033 
 
 063 
 
 647 
 
 32 
 
 6.2017 
 
 35 
 
 on 
 
 8.5104999 
 
 773 
 
 32 
 
 36 
 
 40 
 
 8.5089990 
 
 935 
 
 998 
 
 33 
 
 55 
 
 45 
 
 969 
 
 870 
 
 1.40024 
 
 33 
 
 74 
 
 50 
 
 947 
 
 806 
 
 149 
 
 33 
 
 93 
 
 55 
 
 925 
 
 741 
 
 275 
 
 33 
 
 6.2II2 
 
 45 oo 
 
 904 
 
 677 
 
 400 
 
 33 
 
 3i 
 
 05 
 
 883 
 
 612 
 
 526 
 
 33 
 
 50 
 
 IO 
 
 861 
 
 548 
 
 651 
 
 33 
 
 69 
 
 15 
 
 840 
 
 484 
 
 777 
 
 33 
 
 88 
 
 20 
 
 8x8 
 
 419 
 
 002 
 
 33 
 
 6.2207 
 
 25 
 
 797 
 
 356 
 
 I.4IO28 
 
 33 
 
 26 
 
 30 
 
 776 
 
 291 
 
 153 
 
 33 
 
 45 
 
 35 
 
 754 
 
 226 
 
 279 
 
 33 
 
 64 
 
 40 
 
 733 
 
 162 
 
 404 
 
 33 
 
 83 
 
 45 
 
 711 
 
 098 
 
 530 
 
 32 
 
 6 . 2302 
 
 50 
 
 690 
 
 034 
 
 655 
 
 32 
 
 21 
 
 55 
 
 668 
 
 8.5103969 
 
 78l 
 
 32 
 
 40 
 
 46 oo 
 
 647 
 
 905 
 
 006 
 
 3i 
 
 60 
 
 05 
 
 625 
 
 841 
 
 1.42032 
 
 3i 
 
 79 
 
 IO 
 
 604 
 
 776 
 
 157 
 
 30 
 
 98 
 
 15 
 
 583 
 
 712 
 
 283 
 
 30 
 
 6.2417 
 
 20 
 
 56i 
 
 648 
 
 409 
 
 29 
 
 37 
 
 25 
 
 539 
 
 584 
 
 534 
 
 29 
 
 56 
 
 3 
 
 518 
 
 5i8 
 
 660 
 
 28 
 
 76 
 
 35 
 
 497 
 
 457 
 
 786 
 
 28 
 
 95 
 
 40 
 
 475 
 
 392 
 
 911 
 
 27 
 
 6.2514 
 
 45 
 
 454 
 
 326 
 
 1.43037 
 
 26 
 
 34 
 
 5 
 
 431 
 
 262 
 
 163 
 
 26 
 
 53 
 
 55 
 
 410 
 
 199 
 
 289 
 
 25 
 
 73 
 
 47 oo 
 
 390 
 
 134 
 
 414 
 
 24 
 
 93 
 
 05 
 
 368 
 
 070 
 
 539 
 
 23 
 
 6.2612 
 
 IO 
 
 347 
 
 005 
 
 666 
 
 22 
 
 32 
 
 15 
 
 326 
 
 8.5102941 
 
 792 
 
 21 
 
 52 
 
 20 
 
 304 
 
 876 
 
 917 
 
 21 
 
 71 
 
 25 
 
 283 
 
 813 
 
 1.44043 
 
 20 
 
 9i 
 
 30 
 
 261 
 
 749 
 
 169 
 
 19 
 
 6.2711 
 
 35 
 
 240 
 
 685 
 
 295 
 
 17 
 
 30 
 
 40 
 
 219 
 
 621 
 
 421 
 
 16 
 
 50 
 
 45 
 
 197 
 
 557 
 
 547 
 
 15 
 
 70 
 
 50 
 
 176 
 
 493 
 
 673 
 
 14 
 
 90 
 
 55 
 
 155 
 
 428 
 
 799 
 
 13 
 
 6.2810
 
 FORMULA. AND FACTORS. 
 
 271 
 
 Lat. 
 
 Log A. 
 
 Log B. 
 
 LogC. 
 
 Log/). 
 
 Log.E. 
 
 48oo' 
 
 8.5089133 
 
 8.5102364 
 
 1.44926 
 
 2.39" 
 
 6.2830 
 
 05 
 
 112 
 
 301 
 
 1.45052 
 
 IO 
 
 50 
 
 10 
 
 Ogi 
 
 2 3 6 
 
 I 7 8 
 
 09 
 
 70 
 
 15 
 
 O7O 
 
 172 
 
 304 
 
 08 
 
 90 
 
 20 
 
 4 8 
 
 108 
 
 431 
 
 06 
 
 6.2910 
 
 25 
 
 027 
 
 045 
 
 557 
 
 05 
 
 30 
 
 30 
 
 005 
 
 8.5101981 
 
 683 
 
 03 
 
 50 
 
 35 
 
 8.5088984 
 
 917 
 
 809 
 
 02 
 
 70 
 
 40 
 
 963 
 
 853 
 
 937 
 
 OO 
 
 91 
 
 45 
 
 941 
 
 789 
 
 1.46063 
 
 2.3899 
 
 6.3011 
 
 50 
 
 92O 
 
 725 
 
 189 
 
 97 
 
 31 
 
 55 
 
 899 
 
 662 
 
 316 
 
 95 
 
 51 
 
 49 oo 
 
 878 
 
 598 
 
 442 
 
 94 
 
 72
 
 2/2 
 
 GEODETIC OPERATIONS. 
 
 FROM UNITED STATES COAST SURVEY REPORT. 
 
 AUXILIARY TABLES FOR CONVERTING ARCS OF THE CLARKE ELLIPSOID INTO 
 
 ARCS OF THE BESSEL ELLIPSOID. 
 
 [All corrections are positive.] 
 
 
 Corrections to dM. 
 
 Arguments L' and dM. 
 
 dM 
 
 60' 
 
 50' 
 
 4 0' 
 
 3' 
 
 20' 
 
 10' 
 
 
 
 
 
 
 
 
 60" 
 
 So" 
 
 40" 
 
 30" 
 
 
 10 
 
 5 ' 
 
 Lat. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 23 
 
 0.481 
 
 0.401 
 
 0.320 
 
 0.240 
 
 0.160 
 
 0.080 
 
 0.008 
 
 0.006 
 
 0.005 
 
 0.004 
 
 0.003 
 
 0.00 
 
 o 0006 
 
 24 
 
 .484 
 
 43 
 
 .322 
 
 .242 
 
 .161 
 
 .080 
 
 .008 
 
 .006 
 
 .005 
 
 .004 
 
 .003 
 
 .OO 
 
 .0006 
 
 25 
 
 .486 
 
 .405 
 
 324 
 
 243 
 
 .162 
 
 .081 
 
 .008 
 
 .006 
 
 .005 
 
 .004 
 
 .003 
 
 .00 
 
 .0006 
 
 26 
 27 
 
 .489 
 .491 
 
 .407 
 .409 
 
 -326 
 327 
 
 IS! 
 
 .163 
 .164 
 
 .081 
 .082 
 
 .008 
 .008 
 
 .006 
 .006 
 
 .005 
 .005 
 
 .004 
 .004 
 
 .003 
 .003 
 
 00 
 
 .0006 
 .0006 
 
 28 
 
 494 
 
 .411 
 
 329 
 
 .247 
 
 165 
 
 .082 
 
 .008 
 
 .007 
 
 .005 
 
 .004 
 
 .003 
 
 .00 
 
 .0006 
 
 29 
 
 .496 
 
 .413 
 
 33 
 
 .248 
 
 .166 
 
 .083 
 
 .008 
 
 .007 
 
 
 .004 
 
 .003 
 
 .00 
 
 .0006 
 
 3 
 
 497 
 
 .416 
 
 332 
 
 .250 
 
 .167 
 
 .083 
 
 .008 
 
 
 s 
 
 .004 
 
 .003 
 
 .00 
 
 .0006 
 
 3 1 
 
 .502 
 
 .418 
 
 334 
 
 251 
 
 .168 
 
 .084 
 
 .008 
 
 .007 
 
 .006 
 
 .004 
 
 .003 
 
 .00 
 
 .0006 
 
 32 
 
 505 
 
 .420 
 
 336 
 
 253 
 
 .169 
 
 OS^ 
 
 .008 
 
 .007 
 
 .006 
 
 .004 
 
 .003 
 
 .00 
 
 .0006 
 
 33 
 
 57 
 
 .422 
 
 338 
 
 254 
 
 .169 
 
 .085 
 
 .008 
 
 .007 
 
 .006 
 
 .004 
 
 
 .001 
 
 .0006 
 
 34 
 
 .51 
 
 425 
 
 34 
 
 255 
 
 .170 
 
 .085 
 
 .008 
 
 .007 
 
 .006 
 
 .004 
 
 .003 
 
 .OOI 
 
 .0006 
 
 P 
 
 3 
 
 .427 
 43 
 
 342 
 342 
 
 -256 
 .258 
 
 .171 
 .172 
 
 .086 
 .086 
 
 .008 
 .009 
 
 .007 
 .007 
 
 .006 
 .006 
 
 .004 
 .004 
 
 .003 
 .003 
 
 .00 
 
 .0006 
 .0006 
 
 37 
 
 .518 
 
 432 
 
 345 
 
 259 
 
 '73 
 
 .087 
 
 .009 
 
 .007 
 
 .006 
 
 .004 
 
 .003 
 
 .OOI 
 
 .0007 
 
 38 
 39 
 
 .521 
 524 
 
 434 
 436 
 
 347 
 349 
 
 :26i 
 .262 
 
 .174 
 '75 
 
 .087 
 .088 
 
 .009 
 .009 
 
 .007 
 .007 
 
 .006 
 .006 
 
 .004 
 .004 
 
 .003 
 .003 
 
 .00 
 
 .0007 
 .0007 
 
 4 
 
 527 
 
 439 
 
 35' 
 
 .264 
 
 .176 
 
 .088 
 
 .009 
 
 .007 
 
 .006 
 
 .004 
 
 .003 
 
 .00 
 
 .0007 
 
 
 530 
 
 
 353 
 
 .265 
 
 .177 
 
 .089 
 
 .009 
 
 .007 
 
 .006 
 
 .004 
 
 .003 
 
 .00 
 
 .0007 
 
 42 
 43. 
 
 is 
 
 444 
 .446 
 
 355 
 357 
 
 .267 
 .268 
 
 .178 
 .179 
 
 .089 
 .090 
 
 .009 
 .009 
 
 .007 
 
 .006 
 .006 
 
 .004 
 .004 
 
 .003 
 
 .00 
 
 .0007 
 
 .0007 
 
 44 
 
 539 
 
 449 
 
 359 
 
 .270 
 
 .180 
 
 .090 
 
 .009 
 
 .008 
 
 .006 
 
 .005 
 
 .003 
 
 .001 
 
 .0007 
 
 45 
 
 0.542 
 
 o 45' 
 
 0.^61 
 
 0.271 
 
 o, 18 
 
 0.091 
 
 o.ooq 
 
 0.008 
 
 0.006 
 
 0.005 
 
 0.003 
 
 O.OOI 
 
 o 0007 
 
 
 Corrections to dL. 
 
 Arguments and dL. 
 
 dL. 
 
 60' 
 
 50' 
 
 40' 
 
 3 o' 
 
 20' 
 
 10' 
 
 60" 
 
 50" 
 
 40" 
 
 30" 
 
 20" 
 
 10" 
 
 5" 
 
 Lat. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 23 
 
 0.193 
 
 0.160 
 
 0.129 
 
 0.096 
 
 0064 
 
 0.032 
 
 0.003 
 
 0.00 
 
 0.002 
 
 0.002 
 
 O.OOI 
 
 O.OOI 
 
 0.0003 
 
 24 
 
 .200 
 
 .165 
 
 133 
 
 .099 
 
 .066 
 
 033 
 
 .003 
 
 .00; 
 
 .002 
 
 .002 
 
 .001 
 
 .001 
 
 .0003 
 
 11 
 
 .206 
 .213 
 
 .171 
 .177 
 
 .138 
 .142 
 
 :a 
 
 .068 .034 
 .070 .035 
 
 .003 
 
 oo- 
 
 !o02 
 
 .002 
 
 'ooi 
 
 .001 
 .001 
 
 .0003 
 
 27 
 
 .220 
 
 .183 
 
 .147 
 
 .110 
 
 073 -037 
 
 .004 
 
 .00; 
 
 .002 
 
 .002 
 
 .001 
 
 .001 
 
 .0003 
 
 28 
 
 27 
 
 .189 
 
 .151 
 
 tt3 
 
 .075 .038 
 
 .004 
 
 .00; 
 
 .002 
 
 .002 
 
 .001 
 
 .001 
 
 .0003 
 
 29 
 
 34 
 
 .196 
 
 .156 
 
 .117 
 
 .o 7 8| .039 
 
 .004 
 
 .00' 
 
 .OO2 
 
 .OO2 
 
 .OOI 
 
 .OOI 
 
 .0003 
 
 30 
 
 42 
 
 .202 
 
 .161 
 
 
 .080 
 
 .040 
 
 .004 
 
 .00; 
 
 .002 
 
 .002 
 
 .001 
 
 .001 
 
 .0003 
 
 3 1 
 
 5 
 
 .209 
 
 .167 
 
 125 
 
 .083 
 
 .042 
 
 .004 
 
 .00; 
 
 .003 
 
 .002 
 
 .001 
 
 .001 
 
 .0004 
 
 32 
 
 58 
 
 .2l6 
 
 .172 
 
 .129 
 
 .086 
 
 .043 
 
 .004 
 
 .OO' 
 
 
 .OO2 
 
 .OOI 
 
 .001 
 
 .0004 
 
 33 
 34 
 
 275 
 
 .223 
 230 
 
 .178 
 .184 
 
 133 
 137 
 
 .089 
 .091 
 
 31 
 
 .005 
 .005 
 
 .003 
 
 .00; 
 
 .002 
 
 .002 
 
 .001 
 .00 
 
 .0004 
 .0004 
 
 35 
 
 .283 
 
 .237 
 
 .190 
 
 .141 
 
 .094 
 
 .047 
 
 .005 
 
 
 .OO 
 
 .002 
 
 .002 
 
 .001 
 
 .0004 
 
 36 
 
 .291 
 
 .243 
 
 .195 
 
 145 
 
 .097 
 
 .o 4 8 
 
 
 .004 
 
 .OO 
 
 .OO2 
 
 .002 
 
 .001 
 
 .0004 
 
 3 l 
 
 .300 
 
 .250 
 
 
 .150 
 
 
 .050 
 
 .005 
 
 .004 
 
 .OO' 
 
 .OO2 
 
 .002 
 
 .00 
 
 .0004 
 
 38 
 
 .308 
 
 257 
 
 .206 
 
 154 
 
 .103 
 
 .051 
 
 .005 
 
 .004 
 
 .003 
 
 .002 
 
 .002 
 
 .001 
 
 .0004 
 
 39 
 
 3*7 
 
 .264 
 
 .212 
 
 .158 
 
 .106 
 
 .053 
 
 .005 
 
 .004 
 
 .004 
 
 .OO 
 
 .002 
 
 .00 
 
 .0004 
 
 40 
 
 325 
 
 271 
 
 .217 
 
 .162 
 
 .108 
 
 .054 
 
 .005 
 
 .004 
 
 .004 
 
 .OO; 
 
 .OO2 
 
 .00 
 
 .0005 
 
 41 
 
 334 
 
 .278 
 
 
 .167 
 
 . ii 
 
 .056 
 
 .006 
 
 .004 
 
 .004 
 
 .OO' 
 
 .002 
 
 .00 
 
 .0005 
 
 42 
 
 343 
 
 .286 
 
 .229 
 
 
 i4 
 
 .057 
 
 .006 
 
 .004 
 
 .004 
 
 .00' 
 
 .002 
 
 .00 
 
 .0005 
 
 43 
 
 352 
 
 .294 
 
 .236 
 
 .176 
 
 i? 
 
 
 .006 
 
 
 .004 
 
 .OO' 
 
 .OO2 
 
 .00 
 
 .0005 
 
 44 
 
 362 
 
 .302 
 
 .242 
 
 .181 
 
 . 20 .060 
 
 .006 
 
 .005 
 
 .004 
 
 .003 
 
 .OO2 
 
 .00 
 
 .0005 
 
 45 
 
 0.372 
 
 0-310 
 
 0.249 
 
 0.186 o. 24 0.062 
 
 0.006 
 
 0.005 
 
 0.004 
 
 0.003 
 
 0.002 
 
 O.OOI 
 
 0.0005
 
 FORMULA AND FACTORS. 
 
 273 
 
 TAKEN FROM U. S. COAST AND GEODETIC SURVEY REPORT. 
 
 SUBSIDIARY TABLE FOR REFERRING VALUES OF COEFFICIENTS A, B, C, D, E, 
 
 FROM CLARKE'S TO BESSEL'S ELLIPSOID. 
 
 Lat. 
 
 To log A add. 
 
 To log B add. 
 
 To log C add. 
 
 From log D 
 subtract. 
 
 To log E add. 
 
 23 
 
 0.0000582 
 
 0.0000233 
 
 0.00008 
 
 0.0061 
 
 O.OOOI 
 
 24 
 
 584 
 
 N 241 
 
 08 
 
 6l 
 
 
 25 
 
 587 
 
 249 
 
 08 
 
 6l 
 
 
 26 
 
 590 
 
 2 5 8 
 
 08 
 
 61 
 
 
 27 
 
 593 
 
 266 
 
 09 
 
 61 
 
 
 28 
 
 596 
 
 274 
 
 09 
 
 61 
 
 
 29 
 
 599 
 
 283 
 
 09 
 
 61 
 
 
 30 
 
 602 
 
 293 
 
 09 
 
 61 
 
 
 31 
 
 605 
 
 302 
 
 09 
 
 61 
 
 
 32 
 
 609 
 
 312 
 
 09 
 
 61 
 
 
 33 
 
 612 
 
 321 
 
 09 
 
 61 
 
 
 34 
 
 615 
 
 331 
 
 09 
 
 61 
 
 
 35 
 
 619 
 
 342 
 
 IO 
 
 61 
 
 
 36 
 
 622 
 
 352 
 
 10 
 
 61 
 
 
 37 
 
 625 
 
 362 
 
 IO 
 
 61 
 
 
 38 
 
 629 
 
 372 
 
 IO 
 
 61 
 
 
 39 
 
 632 
 
 383 
 
 IO 
 
 61 
 
 
 40 
 
 636 
 
 393 
 
 10 
 
 61 
 
 
 4i 
 
 639 
 
 404 
 
 IO 
 
 61 
 
 
 42 
 
 643 
 
 
 II 
 
 61 
 
 
 43 
 
 647 
 
 425 
 
 II 
 
 61 
 
 
 44 
 
 650 
 
 436 
 
 II 
 
 61 
 
 
 
 654 
 
 447 
 
 II 
 
 61 
 
 
 46 
 
 657 
 
 458 
 
 II 
 
 61 
 
 
 47 
 
 661 
 
 468 . 
 
 II 
 
 61 
 
 
 48 
 
 664 
 
 479 
 
 II 
 
 61 
 
 
 49 
 
 668 
 
 490 
 
 12 
 
 61 
 
 
 50 
 
 672 
 
 501 
 
 12 
 
 61 
 
 
 TABLE OF LOG F. 
 
 Lat. 
 
 Log F. 
 
 Lat. 
 
 Log-F. 
 
 Lat. 
 
 Log .P. 
 
 Lat. 
 
 Log^ 1 . 
 
 23 
 
 7.812 
 
 30 
 
 7.866 
 
 37 
 
 7.876 
 
 44 
 
 7.8 4 8 
 
 24 
 
 23 
 
 31 
 
 70 
 
 38 
 
 74 
 
 45 
 
 40 
 
 25 
 
 32 
 
 32 
 
 73 
 
 39 
 
 72 
 
 46 
 
 32 
 
 26 
 
 41 
 
 33 
 
 75 
 
 40 
 
 69 
 
 47 
 
 2 4 
 
 27 
 
 49 
 
 34 
 
 77 
 
 4i 
 
 64 
 
 48 
 
 *4 
 
 28 
 2 9 
 
 55 
 61 
 
 11 
 
 77 
 77 
 
 42 
 43 
 
 60 
 54 
 
 49 
 50 
 
 04 
 7.792 
 
 IS
 
 274 
 
 GEODETIC OPERATIONS. 
 
 TABLE OF CORRECTIONS TO LONGITUDE FOR DIFFERENCE 
 IN ARC AND SINE. 
 
 Log K(-\ 
 
 Log difference. 
 
 Log dM(+\ 
 
 Log**-). 
 
 Log difference. 
 
 Log dM(+). 
 
 3.871 
 
 0.0000001 
 
 .330 
 
 4-913 
 
 O.OOOOirg 
 
 3.422 
 
 3-970 
 
 OO2 
 
 -479 
 
 4.922 
 
 124 
 
 3-431 
 
 4-"5 
 
 003 
 
 .624 
 
 4-932 
 
 130 
 
 3-441 
 
 4.171 
 
 004 
 
 .680 
 
 4.941 
 
 136 
 
 3-450 
 
 4.221 
 
 005 
 
 730 
 
 4.950 
 
 142 
 
 3-459 
 
 4.268 
 
 006 
 
 777 
 
 4-959 
 
 147 
 
 3-468 
 
 4.292 
 
 007 
 
 .801 
 
 4.968 
 
 153 
 
 3-477 
 
 4.309 
 
 008 
 
 .818 
 
 4.976 
 
 1 60 
 
 3.485 
 
 4.320 
 
 009 
 
 839 
 
 4-985 
 
 1 66 
 
 3-494 
 
 4.361 
 
 OIO 
 
 .870 
 
 4-993 
 
 172 
 
 3.502 
 
 4-383 
 
 on 
 
 .892 
 
 5.002 
 
 i?9 
 
 3-5II 
 
 4-415 
 
 012 
 
 .924 
 
 5.010 
 
 1 86 
 
 3-519 
 
 4-430 
 
 013 
 
 939 
 
 5-017 
 
 192 
 
 3-526 
 
 4-445 
 
 014 
 
 954 
 
 5-025 
 
 199 
 
 3-534 
 
 4-459 
 
 015 
 
 .968 
 
 5-033 
 
 206 
 
 3-542 
 
 4-473 
 
 016 
 
 .982 
 
 5 040 
 
 213 
 
 3-549 
 
 4.487 
 
 017 
 
 .996 
 
 5-047 
 
 221 
 
 3-556 
 
 4-500 
 
 018 
 
 3-009 
 
 5-054 
 
 228 
 
 3-563 
 
 4-524 
 
 020 
 
 3-033 
 
 5.062 
 
 236 
 
 3-571 
 
 4.548 
 
 023 
 
 3-057 
 
 5.068 
 
 243 
 
 3-577 
 
 4-570 
 
 025 
 
 3-079 
 
 5-075 
 
 251 
 
 3-584 
 
 591 
 
 027 
 
 3.100 
 
 5.082 
 
 259 
 
 3-591 
 
 t .612 
 
 030 
 
 3-I2I 
 
 5.088 
 
 26 7 
 
 3-597 
 
 .631 
 
 033 
 
 3-140 
 
 5-095 
 
 275 
 
 3-604 
 
 .649 
 
 036 
 
 3-158 
 
 5.102 
 
 284 
 
 3.611 
 
 .667 
 
 039 
 
 3.176 
 
 5.108 
 
 292 
 
 3-617 
 
 4.684 
 
 042 
 
 3-193 
 
 5-"4 
 
 300 
 
 3-623 
 
 4.701 
 
 045 
 
 3.210 
 
 5.120 
 
 309 
 
 3 629 
 
 4.716 
 
 048 
 
 3-225 
 
 5.126 
 
 318 
 
 3-635 
 
 4-732 
 
 052 
 
 3-241 
 
 5-132 
 
 327 
 
 3-641 
 
 4.746 
 
 056 
 
 3-255 
 
 5.138 
 
 336 
 
 3.647 
 
 4.761 
 
 059 
 
 3-270 
 
 5-144 
 
 345 
 
 3.653 
 
 4-774 
 
 063 
 
 3-283 
 
 5-150 
 
 354 
 
 3-659 
 
 4.788 
 
 067 
 
 3 297 
 
 5-156 
 
 364 
 
 3-665 
 
 4.801 
 
 071 
 
 3-3io 
 
 5.161 
 
 373 
 
 3.670 
 
 4.813 
 
 075 
 
 3-322 
 
 5-167 
 
 383 
 
 3.676 
 
 4-825 
 
 080 
 
 3-334 
 
 5-172 
 
 392 
 
 3.681 
 
 4-834 
 
 084 
 
 3-343 
 
 5.178 
 
 402 
 
 3.687 
 
 4.849 
 
 089 
 
 3-358 
 
 5-183 
 
 412 
 
 3-692 
 
 4.860 
 
 094 
 
 3-369 
 
 5.188 
 
 422 
 
 3-697 
 
 4.871 
 
 098 
 
 3-380 
 
 5-193 
 
 433 
 
 3.702 
 
 4.882 
 
 103 
 
 3-391 
 
 5-199 
 
 443 
 
 3.708 
 
 4.892 
 
 108 
 
 3.401 
 
 5-204 
 
 453 
 
 3-7I3 
 
 4-903 
 
 114 
 
 3-412 
 
 

 
 INDEX. 
 
 ABULFEDA, description of Arabian arc-measurement, 3 
 
 Adjusting the azimuth 200 
 
 Adjustment, figure 168 
 
 station 146 
 
 when directions have been observed 180 
 
 Airy 248 
 
 Alexandria 2 
 
 Anaximander i 
 
 Angles, method of measuring 97 
 
 Arabian arc-determination 2 
 
 unit of measure 28 
 
 Arago 15 
 
 Argelander 18 
 
 Auzout 29 
 
 Axes of the earth, Clarke's and Bessel's values of 259 
 
 Azimuth, affected by adjustment 200 
 
 formula for computing 218 
 
 BAEYER 20 
 
 Barrow, Indian arc-measurement 13 
 
 Base apparatus, first form of 50 
 
 requisites of 50 
 
 Bache-Wurdeman 52 
 
 Baumann. 60 
 
 Bessel 58 
 
 Borda 51 
 
 Colby 59. 64 
 
 Ibafiez 60 
 
 Lapland So 
 
 Peru 50 
 
 Porro 59 
 
 Repsold 60 
 
 Struve 59
 
 2/6 INDEX. 
 
 PAGE 
 
 Base-line, probable error of 74 
 
 reduction to sea-level 78 
 
 Base-measurements 49 
 
 aligning 64 
 
 comparison of results 61 
 
 computation of results 70 
 
 erection of terminal marks 66 
 
 general precautions 69 
 
 inclination 68 
 
 instructions 64 
 
 sector error 68 
 
 selecting site 64 
 
 record, form of 66 
 
 references 79 
 
 transference of end to the ground 67 
 
 Beccaria, arc measurement to 
 
 Bessel 247 
 
 base apparatus 20 
 
 review of the French arc 15 
 
 Biot 15 
 
 Bonne 19 
 
 Borda, metallic thermometer 14 
 
 Borden, survey of Massachusetts 23 
 
 Boscovich, arc-measurement 10 
 
 Bouguer 7 
 
 Boutelle 81, 86 
 
 Brah6, Tycho 29 
 
 Briggs 29 
 
 CALDEWOOD, glass base apparatus n 
 
 Camus 8 
 
 Cape of Good Hope arc 10 
 
 Cassini 4, II 
 
 revision of the French arc 9 
 
 Celsius 8 
 
 Centre, reduction to 196 
 
 Chaldean unit of measure 28 
 
 Chauvenet 160 
 
 Circle, entire, first used 29 
 
 Clairault 8 
 
 theory of the figure of the earth 9
 
 INDEX. 
 
 Clarke, solution for the figure of the earth ........................ 
 
 reference to the great English theodolite ....... 
 
 Coast Survey, U. S., organized ................................ '"" ' 2J 
 
 form of base apparatus ......................... 52, 61 
 
 heliotrope ................................. ' 46 
 
 signals .................................... g 4 
 
 theodolite ............................. ____ 32 
 
 Colonna ..................................... 
 
 Commission for European degree-measurements ........................ 26 
 
 Comparison of base-bars with a standard .............................. j! 
 
 Condamine, De la ........................................... _ 
 
 Connection of France and England by triangulation ..................... n 
 
 Constants and their logarithms, table of .............................. 260 
 
 Correction for inclination. ............................................ -4 
 
 Correlatives, equations of ............................................. jjg 
 
 Cutts ................................................................ 86 
 
 DAVIDSON ................ . ....................................... 47, 102 
 
 Delambre ............................................................ !3 
 
 revision of the Peruvian arc .................................. 8 
 
 Des Hays, pendulum-investigations ............................ ........ 6 
 
 Directions, adjustment of .......... . ................................. 180 
 
 horizontal, copy of record .................................. 101 
 
 Dividing-engine first used ........................... .................. 30 
 
 Dixon. See Mason. 
 
 Doolittle ................................ , ............................ 195 
 
 ECCENTRIC signal .................................................... 146 
 
 instrument .................................................. 196 
 
 Ellipticity of the earth, table of ........................................ 249 
 
 Errors, mean of ...................................................... 146 
 
 probable. See Probable Errors. 
 
 Equations, conditional, number of ..................................... 179 
 
 side ...................................................... 171 
 
 solution of, by logarithms ................................... 194 
 
 Eratosthenes ......................................................... I 
 
 Everest, Indian arc ................................................... 14 
 
 Expansion coefficient, determination of ................................. 72 
 
 FERNEL ............................................................. 3 
 
 Figure-adjustment .................................................... 168 
 
 Figure of the earth .................................................... 234 
 
 literature of ....... ................................ 249
 
 2/8 INDEX. 
 
 PACK 
 
 French Academy arc-measurements in Lapland and Peru 6 
 
 Froriep I 
 
 GASCOIGNE, first to use spider-lines 4 
 
 Gauss 19, 159 
 
 Geodesic, Ecole speciale de 17 
 
 Geodetic factors, tables of 264 
 
 Godin 7 
 
 Greek unit of measure 28 
 
 HANSTEEN 18 
 
 Heights determined by barometer 94 
 
 triangulation 101 
 
 Heliotrope, description of 45 
 
 first used 45 
 
 illustration 46 
 
 use and adjustments 46 
 
 Hilgard 41 
 
 Hipparchus 29 
 
 Hounslovv Heath base . 12 
 
 Humboldt 29 
 
 Huygens's theory of centrifugal motion 6 
 
 INGENIEUR Corps, organization of 16 
 
 Instruments 28 
 
 Invention of the vernier 29 
 
 Isle, De 1' 17 
 
 Italian commission organized 25 
 
 Italy, co operation with Switzerland 17 
 
 James, Sir Henry, reference to the great English theodolite II 
 
 LACAILLE, revision of Picard's arc 4 
 
 arc-measurement at the Cape of Good Hope 10 
 
 Lahire 4 
 
 Lambton, Indian arc 14 
 
 Lapiace 13, 247 
 
 Lapland arc-measurement 8 
 
 Latitude, formula for computing 203 
 
 illustration 220, 222 
 
 Least squares, theory of 104
 
 INDEX. 279 
 
 Letronne, review of Posidonius's arc 
 
 Level, table giving difference between the true and apparent 92 
 
 Liesganig, arc-measurement lo 
 
 Littrow 
 
 Longitude determined by powder-flashes IQ 
 
 Longitude, formula for computing 217 
 
 illustrations 22O) 22 ~ 
 
 MACLEAR, continuance of the Cape of Good Hope arc 23 
 
 Maraldi "" 
 
 Mason, Maryland-Pennsylvania boundary-line lo 
 
 Maupertuis, Lapland arc 3 
 
 Mayer, repeating-theodolites I2 
 
 Mechain n, 15 
 
 Metre, determination of its length !3 
 
 legal and recent values of 249 
 
 Metres to feet, table for converting 257 
 
 miles 258 
 
 yards 257 
 
 Micrometer first used 29 
 
 determination of run 35 
 
 Miles to metres, table for converting 259 
 
 Monnier 8 
 
 Mosman 97 
 
 Mudge 15 
 
 Muffling, Von , 19 
 
 Musschenbroeck 19 
 
 NAPIER : 29 
 
 Newton, theory of universal gravitation demonstrated by Picard's arc 4 
 
 Normal equations 155 
 
 Normals, table of 261 
 
 Norwood, measurement of the distance from London to York 4 
 
 Nunez 29 
 
 OUTHIER 
 
 PALANDER 14 
 
 Pamlico-Chesapeake arc 24 
 
 Phase, correction for *44
 
 28O INDEX. 
 
 PAGB 
 
 Picard's triangulation 4 
 
 Pcle, selection of, in figure-adjustment 179 
 
 Posch, view of Ptolemy's length of a degree 2 
 
 Posidonius, arc-measurement 2 
 
 Probable error of the arithmetical mean 124 
 
 of a single determination 120 
 
 illustration 122 
 
 of a base-line 125 
 
 direction 125 
 
 in the computation of unknown quantities in triangles 127, 133, 
 
 136, 137, 138 
 
 Prussia, first geodetic work 16 
 
 Prussian-Russian connection 25 
 
 Ptolemy, value of earth's circumference 2 
 
 Puissant, review of the French arc 15 
 
 Pythagoras I 
 
 QUADRANT of the earth, length of 249 
 
 RADIUS of curvature, table of 261 
 
 Reduction to centre 196 
 
 Reichenbach 26, 31 
 
 Repetition of angles, principle first announced 12 
 
 abandoned 31 
 
 Repsold 26, 31 
 
 Riccioli 3 
 
 Richer, expedition to Cayenne 6 
 
 Roemer 29 
 
 Roy ii, 15 
 
 Russian arc, accuracy of 18 
 
 Russia, first geodetic work 17 
 
 SAEGMCLLER, principal of bisection 31 
 
 Schott 24, 179, 248 
 
 Schmidt 247 
 
 Schumacher 10, yi 
 
 Schwerd ig 
 
 Series, binomial 256 
 
 exponential 256 
 
 logarithmic 257
 
 INDEX 28l 
 
 PAGB 
 
 Signals, form used on coast-survey 84 
 
 night cost of 82 
 
 method of erection 87 
 
 size and lengths of timbers 88 
 
 Snellius' triangulation 3 
 
 Spain, first geodetic work 25 
 
 Spherical excess, computation of 168 
 
 Speyer base 10, 
 
 Spider-lines in telescope, first use of 4 
 
 Stations, description of 05 
 
 permanent markings 96 
 
 intervisibility of go 
 
 Station-adjustment ... 146 
 
 Struve 17 
 
 S van berg 14 
 
 Sweden, coast triangulation 26 
 
 Swedish arc-measurement in Lapland 14 
 
 Switzerland, first geodetic work 17 
 
 Syene i 
 
 TENNER 17 
 
 Thales i 
 
 Theodolites, adjustment of 33 
 
 construction of 32 
 
 errors of eccentricity 36 
 
 graduation 39 
 
 illustration 34 
 
 size of 32 
 
 Toise of Peru 7 
 
 Transferring underground mark to the top of a signal 96 
 
 Triangles, best composition of 86 
 
 Triangulation, calculation of 143 
 
 conditions to be fulfilled 143 
 
 Trigonometric expressions 253 
 
 series 256 
 
 ULLOA 7 
 
 VARIN, pendulum-investigations 6 
 
 Vernerius 2 9
 
 282 INDEX. 
 
 fAGK 
 
 Vernier, invention of 29 
 
 distance apart 37 
 
 WALBECK 247 
 
 Walker 22 
 
 Waugh 22 
 
 Weights, application of, in adjustments 165 
 
 YARDS to metres, table for converting 258 
 
 Yollond 184 
 
 ZACH, VON, revision of Beccaria's arc 10 
 
 of the Peruvian arc. . . 8