QB 
 8 1 
 
 KG 
 
 UC-NRLF 
 
THE FIGURE 
 
 EARTH. 
 
 SECTION I HISTORICAL 
 
 SECTION II, THE OBLATE SPHEROIDAL 
 HYPOTHESIS. 
 
 FRANK C. ROBERTS, C.E. 
 
 REPRINTED, WITH ADDITIONS, 
 FROM TAN NOSTRAND'S ENGINEERING MAGAZINE. 
 
 NEW YOEK: 
 
 ;i). VAN NOSTRAND, PUBLISHER, 
 23 MURRAY AND 27 WARREN STREETS. 
 
 1885. 
 
COIPTEIGHT D. VAN NOSTRAND, 1886. 
 
PREFACE. 
 
 IN presenting the following pages to 
 the public our object has been twofold. 
 
 In the first place, we have endeavored 
 to place before our readers such historical 
 data, in connection with the Figure of 
 the Earth, as may prove of an interesting 
 and instructive character. 
 
 Secondly, we have sought to arrange, 
 in a compact form, the important mathe- 
 matical principles for the deduction of the 
 Figure of the Earth upon the spheroidal 
 hypothesis. Hitherto these principles 
 were to be found only among a mass of 
 purely mathematical discussions of the 
 Earth in the fluid state, the theory of 
 attractions, etc. ; and in omitting these 
 latter in the present treatise we hope 
 to have simplified the study of the sub- 
 ject, and thereby to have supplied a want 
 long felt by students of Geodesy. 
 
 770^ 
 
IV 
 
 In the preparation of this essay, we 
 have had access to works of standard au- 
 thors, and we cordially acknowledge our 
 indebtedness to many, but more espe- 
 cially to the essays of Airy, Pratt, and 
 Clarke. 
 
 PRINCETON, N. J., 
 
 March 12, 1885. 
 
SECTION I. 
 
 THE FIGURE OF THE EARTH, 
 
 HISTORICAL. 
 
THE FIGURE OFTHE EARTH 
 
 SECTION I.-HISTORICAL. 
 
 CHAPTEK I. 
 
 EARLY SUPPOSITIONS. 
 
 THE progress of the science of astron- 
 omy is the continued triumph of the 
 powers of the intellect over the first er- 
 roneous conceptions of the senses ; and 
 its history is so allied to that of the hu- 
 man mind that we cannot help feeling a 
 strong inclination to know at what time, 
 and by what people, the hypotheses upon 
 which the science is based were first ad- 
 vanced. 
 
 The value of an hypothesis is estimated 
 by the number of difficult phenomena it 
 explains. From this standpoint few sup- 
 positions can be found to have been more 
 important than that which assigned to 
 
the earth its approximate figure and mag- 
 nitude. 
 
 Little is known of the early history of 
 this hypothesis, for it is enveloped in 
 those dark ages of antiquity when the 
 revolutions of empires were imperfectly 
 recorded, not to speak of the calm specu- 
 lations of quiet and thoughtful men. 
 
 To its first inhabitants the earth must 
 have appeared as an extended fixed plane, 
 the extremities of which apparently sup- 
 ported the vast dome of the heavens. 
 Among the ancients the prevailing opin- 
 ion was that the surface of the earth was 
 flat, that the visible horizon was the 
 boundary of the earth, and the ocean the 
 boundary of the horizon ; that the earth 
 and heavens were the whole visible uni- 
 verse, and that all beneath the earth was 
 Hades. 
 
 The Chaldees had quite an opinion of 
 their own regarding the shape of the 
 earth. They believed it to have the 
 form of a boat turned upside down. 
 This theory was thoroughly believed in 
 by the Chaldean astronomers, and they 
 
9 
 
 endeavored to support it by scientific 
 argument. We should express the same 
 idea, at present, by comparing the earth 
 to an orange, the top of which had been 
 cut off, leaving the orange upright upon 
 the flat surface thus produced.* 
 
 Evidently ignorant of many important 
 physical laws, the early suppositions of the 
 ancients regarding the Figure of the Earth 
 are in a great measure ludicrous. Being 
 but imperfectly acquainted with the sci- 
 ence of astronomy, and their observa- 
 tions being rude and inaccurate, they 
 were led to base their theories upon false 
 assumptions. It might be well to except 
 from this sweeping statement the theories 
 of the early Eastern astronomers. The 
 earliest astronomical records that can. be 
 conceived of as authentic are found in 
 China, and go back as far as 800 B. C., 
 when we find eclipses observed and reg- 
 istered. This would naturally lead us to 
 conclude that the observers were ac- 
 quainted with the fact that the sun and 
 heavenly bodies were visible to people 
 * Lenormant's Chaldean Magic. Page 150. 
 
10 
 
 towards the east sooner than people to- 
 wards the west. As this could not occur 
 unless the earth were curved, we may as- 
 sume with almost a certainty that the 
 globular figure of the earth was known 
 in China at least 800 years B. C. 
 
 Again, astronomy was evidently culti- 
 vated as a science, and whatever may have 
 been their suppositions as to the Figure 
 of the Earth, we may suppose that at 
 the first they were led to conclude that 
 the heavens were spherical by observing 
 that those stars sufficiently elevated to- 
 wards the north pole, performed their en- 
 tire revolution around the pole without 
 interruption ; from which it might by an 
 easy inference be concluded that the 
 other stars, though concealed from view, 
 pursued their course in the same man- 
 ner. When once the theory of a revolv- 
 ing heaven was accepted, it would be 
 comparatively an easy matter to conclude 
 that the earth was globular. 
 
 The most authentic records of early re- 
 searches in connection with the Figure of 
 the Earth come to us from Greece. The 
 
11 
 
 discovery that the earth is not a plane is 
 ascribed to Thales, of Miletus> B. C., 640. 
 Anaximander, B.C. 570, Anaxagoras, B.C. 
 460, claimed a cylindrical shape for the 
 earth, estimating the height as three times 
 the diameter, the land and water being on 
 the upper base. Plato, B. C. 400, called 
 the earth a cube. Aristotle is commonly 
 supposed to have first advanced the 
 theory that the earth was spherical. This 
 supposition we consider unwarranted. 
 
 Egypt in the days of early Greece, 
 is known to have taken the lead in all 
 philosophical pursuits. Thales, the 
 first Grecian mathematician, studied 
 in Egypt, but long before his time, 
 geometry, astronomy, and other sciences, 
 had been held in high repute. Plato 
 ascribed the invention of geometry to 
 Thoth ; lamblicus says it was known in 
 Egypt during the reign of the gods, and 
 Manetho attributes a knowledge of 
 science and literature to the earliest 
 kings. These reputed scientific atttain- 
 ments are fully verified by the testimony 
 of the ancient monuments of Egypt. If 
 
12 
 
 we are to credit Eustathius, the Egypti- 
 ans marked their lines of march and con- 
 quest on maps. This would, of course, 
 necessitate a knowledge of mensuration 
 and surveying, and we can now explain 
 how it' was that the Isrealites were 
 enabled to divide the land of Canaan 
 when commanded to do so by Joshua.* 
 It was this known progress made in the 
 branches of scientific research, that in- 
 duced the Greeks to study in Egypt, and 
 later, actuated Pythagoras, Eudoxus, 
 Plato and Aristotle, to pay extended 
 visits to this fountain head of knowl- 
 edge.f 
 
 It can hardly be supposed that Pythag- 
 oras suggested without previous experi- 
 ence the Copernican theory, the sun 
 being the center of our system ; or the 
 obliquity of the ecliptic, or the moon's 
 " borrowed light ; " or the explanation 
 of the milky way being a collection of 
 stars. These theories, according to 
 
 * Joshua XVIII. Kitto's Daily Bible Illustrations. 
 Page 286. 
 t Kawlinson's Herodotus. 
 
13 
 
 Aristotle,* were taught by Democritus, 
 and by Anaxagoras, the former of whom 
 studied astronomy for five years in 
 Egypt. 
 
 The same may be said of the principles 
 by which the heavenly bodies were at- 
 tracted to a center and impelled in their 
 order, the theory of the eclipses, and the 
 proof of the Earth being spherical.^ 
 These and many other ideas were doubt- 
 less borrowed from the Egyptians and 
 Babylonians, from whose early discover- 
 ies so much has been derived concerning 
 the heavenly bodies. $ From these state- 
 ments it seems highly probable, that the 
 spherical hypothesis was held before the 
 time of Aristotle, and that the basis for 
 his conclusions in regard to the Figure of 
 the Earth, was derived from his knowl- 
 edge of the science of astronomy as 
 taught by the Egyptians. Moreover, 
 Cicero, on the authority of Theophrastus, 
 mentions Hycetas, of Syracuse, a Pythag- 
 
 * Arist. Met. 1.8. 
 
 t Aristotle de Coel, 11, 14. 
 
 $ Aristotle de Coel, 11, 12. 
 
14 
 
 orean, who claimed that the earth re- 
 volved in a circle around its own axis. 
 Aristotle himself also observes, that 
 though most philosophers say the earth 
 is the center of the system,* "the 
 Pythagoreans, who live in Italy, claim 
 that fire is the center, and the earth 
 being one of the planets rotates about 
 the center and makes night and day.'' 
 
 It appears also, that the celebrated 
 ancient Hindo astronomer, Aryabhatta, 
 maintained the doctrine of the earth's 
 diurnal revolution around its axis.f 
 The " sphere of the stars," he affirms, 
 " is stationary," and the " earth making 
 a revolution produces the daily rising 
 and setting of the stars and planets." 
 Now it is evident that these theories 
 could not be held without the assump- 
 tion of a spherical earth, and we there- 
 fore conclude that the spherical hypothe- 
 sis was advanced long before the time of 
 Aristotle. 
 
 The progress of knowledge concern- 
 
 * Aristotle de Coel, 11, 13. 
 
 t Researches Bengal Asiatic Society. Vol. 12. 
 
15 
 
 ing the Figure of the Earth has from the 
 earliest times been closely connected with 
 the study of astronomy. As in this latter 
 science, first impressions are abandoned, 
 and all conclusions are in striking contra- 
 diction to those of superficial observa- 
 tion ; so, as man progressed, the earth 
 became divested of its flattened shape 
 and character of fixidity, and was shown 
 to be a globular body turning swiftly 
 upon its own axis, and moving through 
 space with great rapidity. 
 
 The idea of the earth being a globe is- 
 now so familiar to us that arguments in 
 proof of it are almost unnecessary. Yet 
 familiar as this fact now is, many ages 
 must have elapsed before it was univers- 
 ally received. So difficult was it to con- 
 ceive how the inhabitants of the opposite 
 hemisphere could exist with their heads- 
 down wards, that we find St. Augustine 
 in the 5th century vehemently contend- 
 ing against the possibility of the exist- 
 ence of an antipodes. 
 
16 
 
 CHAPTER II. 
 
 THE SPHERICAL HYPOTHESIS. 
 
 It may be well to understand at this 
 point that when we speak of the Figure 
 of the Earth, we mean the figure which 
 would be assumed by the earth were it 
 covered entirely with water, and more 
 specifically water at mean sea level.* As 
 for the inequalities on the surface of the 
 arth owing to mountains and valleys, 
 they are of no moment in the estimation 
 of the general figure, being of much less 
 account with respect to relative propor- 
 tion than the asperities on the surface of 
 an orange with regard to the orange it- 
 self. Upon an artificial globe of 6f feet 
 in diameter, Mount Chimborazof would 
 be represented by a grain of sand less 
 than 1-20 of an inch in thickness. 
 
 The active curiosity of man did not 
 rest contented with having assumed that 
 
 * Scientifically speaking the figure of the earth is 
 interpreted as meaning the mean surface of the sea 
 imagined to percolate the continents by canals. 
 
 t|21,424 feet above mean tide. 
 
17 
 
 the earth was a sphere, but proceeded 
 to ascertain the exact dimensions of the 
 planet. In the course of his discussion 
 Aristotle states, as does Archimedes (B. 
 C. 250), that mathematicians estimated 
 the circumference of the earth at 300,000 
 stadia. 
 
 The first approximation to the magni- 
 tude of the earth, however inaccurate, 
 must have been at that time a most im- 
 portant addition to the stock of natu- 
 ral knowledge. Indeed, except with a 
 view to some very refined scientific in- 
 vestigation, the general idea which the 
 ancients had of the magnitude of the 
 earth differs but little from that of the 
 moderns ; for we are so incapable of the 
 appreciation of number or magnitude 
 when either exceeds a certain limit, that 
 the difference between their results and 
 ours, makes little or no difference in the 
 general idea which we hold as to the size 
 of the earth. 
 
 Eratosthenes, B. C. 230, was apparent- 
 ly the first to conceive a method for the 
 deduction of the length of the circumfer- 
 
18 
 
 ence of the earth. Although his results 
 are probably sadly inaccurate, the method 
 which he adopted is in essence identical 
 with that followed at the present time. 
 
 As it is impossible for us to occupy *a 
 position from which the earth may be 
 viewed as a whole and compared with 
 some standard of measure, we are com- 
 pelled to resort to geometrical principles 
 in the determination of its figure and 
 magnitude. The problem is rendered 
 more difficult by the fact of there being 
 no fixed landmarks or standard lines 
 upon the surface of the earth, indicating 
 aliquot parts of the earth's circumfer- 
 ence. It therefore becomes necessary 
 to refer our situation on the earth to 
 objects external to our own planet. 
 Such marks are afforded by the heavenly 
 bodies. 
 
 By observations of the meridian alti- 
 tudes of stars, and from their known 
 polar distances, we determine the altitude 
 of the pole-star, which is also the lati- 
 tude of the place at which the observa- 
 tions are made. 
 
19 
 
 Let us suppose then, that we wish to 
 determine the length, on the surface, of 
 one degree of the earth's circumference. 
 Let us suppose also that we know the 
 distance between two places on the same 
 meridian. Then having determined the 
 latitudes of the two places, their differ- 
 ence may be taken as representing the 
 angle at the center of the earth corre- 
 sponding to the measured distance on 
 the surface. Dividing the distance by 
 the angle, we find the length of a merid- 
 ian arc equivalent to one degree of lati- 
 tude, and this multiplied by 360 gives 
 us the length of the earth's circumfer- 
 ence. 
 
 Where local difficulties compel the ob- 
 servers to deviate, in the measurement of 
 the distance, from the line of the merid- 
 ian, the amount of the deviation must be 
 noted. A very simple calculation will 
 enable us to reduce the measured dis- 
 tance to the corresponding length on the 
 meridian. It seems hardly necessary to 
 add that this measurement must be 
 made with the greatest care and accuracy,. 
 
20 
 
 for an error in the measured length of 
 one degree is multiplied 360 times in the 
 circumference, and nearly 58 times in the 
 deduced radius of the earth. 
 
 Such in its simplest form is the geodetic 
 operation called the measurement of an arc 
 of a meridian, and is in essence the method 
 employee! by Eratosthenes. He knew 
 that at Syene, in S. Egypt, on the day of 
 the summer solstice at mid-day, objects 
 cast no shadows, whence he concluded 
 that the sun was in the zenith. In Alex- 
 andria, at the same periocf, he observed 
 that the sun made an angle with the ver- 
 tical of 7 12', or -g* th of a circumference. 
 Assuming Alexandria to be directly north 
 of Syene,* he concluded the length of 
 the circumference to be 50 times the dis- 
 tance between these two places, or 250,000 
 stadia. Of course this determination was 
 very imperfect, for with the instruments 
 of his time he was compelled to neglect 
 the diameter of the sun in the determin- 
 ation of declination. This occasioned 
 
 * The error of this assumption was about three de- 
 grees. 
 
21 
 
 an error of J in the length of the celes- 
 tial arc at Syene. The measurement of 
 the distance between Syene and Alex- 
 andria was probably also very ina,ccu- 
 rate. 
 
 The next attempt to solve the problem 
 was made by Posidonius, B. C. 90. In- 
 stead of using the sun for the determin- 
 ation of the difference of latitude, he 
 found the celestial arc by means of the 
 star Ganopus. At Rhodes this star when 
 on the meridian is just visible above the 
 horizon, while at Alexandria its meridian 
 altitude is 7 30'. The distance between 
 the two places being known, he deduced 
 240,000 stadia as the circumference of 
 the earth. 
 
 Ptolemy, an astronomer, A. D. 160, in 
 his treatise on Geography, gives 500 
 stadia as the length of a degree. This 
 value would give for total length of the 
 circumference 180,000 stadia a result 
 widely different from any previously de- 
 duced. 
 
 Unfortunately the degree of approxi- 
 mation attained in these results cannot 
 
be known, as we have no value for the 
 length of the stadium. It probably had 
 different values dependent upon time and 
 place. 
 
 In the year 819 the Caliph Almamoun 
 caused the astronomers of Bagdad to 
 measure an arc of the meridian on the 
 plains of Mesopotamia by means of 
 wooden rods. Authorities differ as to 
 the resulting deduction of the length of 
 a degree ; some claiming that, failing in 
 their own observations owing to insur- 
 mountable obstacles, the Arabians adopt- 
 ed the result of the Grecian astronomer 
 Ptolemy. Others hold that they found 
 for the length of a degree 56f Arabian 
 miles, or approximately 71 English miles. 
 
 From this time until the revival of let- 
 ters, interest in this subject seems to have 
 disappeared. Speculation was at a stand- 
 still for 700 years. Former theories and 
 suppositions were forgotten and lost amid 
 the social storms of the middle ages. Man 
 was again ignorant of the form and dimen- 
 sions of the planet which had been as- 
 
23 
 
 signed to him in the immensity of 
 space. 
 
 Early in the 15th century the question 
 of the form of the earth began a second 
 time to attract the attention of thought- 
 ful men. The prevalent idea was that 
 the earth was a plane. This time it was 
 not the philosophers but the navigators 
 who looked with doubt upon this sup- 
 position. Columbus fearlessly asserted 
 the earth to be globular, and after the 
 voyage of Magellan around the earth, the 
 globular hypothesis was once more ac- 
 cepfced. Immediately endeavors were 
 made to determine the size of the earth. 
 In 1525 Fernel made a determination of 
 the length of a degree by deducing the 
 difference of latitude between Paris and 
 Amiens, and measuring the distance by 
 observing the number of revolutions 
 made by his coach wheel in traveling 
 from one place to the other. From these 
 observations he deduced the length of 
 one degree to be 57,050 toises or 364,960 
 English feet. (The toise an old French 
 measure is practically equal to 1.949 
 meters, or 6.3946 English feet.) 
 
24 
 
 In 1617 Willebrord Snell conceived 
 the idea of the deduction of the length 
 of distances by means of a series of tri- 
 angles measured from a known base. This 
 was the first instance of the application 
 of the invaluable principle of trigono- 
 metrical surveying which, since that time, 
 has become general in all extensive sur- 
 veys. 
 
 Snell measured his base line upon the 
 frozen surface of the meadows between 
 Leyden and Soeterwood. The angles 
 he measured by means of a quadrant of 
 5J feet radius . His result for the length 
 of a degree was 55,020 toises, or approxi- 
 mately 66.63 miles. 
 
 In 1633 Norwood, in England, adopt- 
 ing a method similar to that of Fernel r s 
 deduced 57,424 toises for the length of a 
 degree. 
 
 We are now brought down to the time 
 of Picard, whose invaluable adaptation 
 of the telescope to circular instruments 
 for measuring angles, marks an era in the 
 progress of geodetic science. Hitherto 
 the measurement of angles was roughly 
 
25 
 
 made by the use of sights similar, only 
 much more unreliable, to those used on 
 rifles. Picard first introduced spider lines 
 in the focus of the telescope, whereby 
 a far higher degree of precision in de- 
 termination of the position of a distant 
 point may be obtained, by covering the 
 point in question with the intersection of 
 spider lines, which is so placed as to be 
 exactly in the center line of the telescope. 
 In his determination of the length of a de- 
 gree he used the trigonometrical method, 
 measuring twice a base line of nearly 
 seven miles in length. His measurement 
 is the first executed with anything like 
 scientific precision. He even calculated 
 the error produced by his instrument be- 
 ing out of the center of his station, and 
 determined his difference of latitude by 
 means of the zenith distance of a star in 
 Cassiopeia measured with a sector. At 
 this time the effect of aberration, refrac- 
 tion and nutation were unknown, never- 
 theless his result of 57,060 toises, or 
 nearly 69.76 miles is marvelously near 
 that of later determinations. 
 
26 
 
 The measurement of an arc of the 
 meridian, although the most reliable, is 
 not the only method by which the Figure 
 of the Earth upon the spherical hypoth- 
 esis may be deduced. One simple ex- 
 pedient consists in determining the dip 
 or angle of depression of the horizon. 
 Take, for instance, the case of a mountain 
 near the sea coast.. Knowing the height 
 of the mountain above the sea, and the 
 angle of depression from its top to the 
 horizon, we can, by an easy mathematical 
 formula, deduce the following equation 
 in which r is the radius of the earth, h 
 the height of the mountain, and d the 
 distance from the mountain to the hori- 
 izon: 
 
 This principle was applied more than 
 200 years ago at Mount Edgecombe, and 
 since that time at Ben Nevis. 
 
 Or, again, we may by the application 
 of the following proposition form a pro- 
 portion from which the diameter of the 
 earth may be found the earth's diameter 
 bears the same proportion to the dis- 
 
27 
 
 tance of the visible horizon from the eye 
 as that distance does to the height of 
 the eye above the sea level. 
 
 Both these methods, of course, only 
 furnish means of determining the size of 
 the earth with a rough approximation. 
 Refraction bends the visual lines out of 
 the truly rectilinear direction, and, there- 
 fore, introduces a serious error in the 
 result. 
 
28 
 
 CHAPTER HI. 
 
 SPHEROIDAL AND ELLIPSOIDAL HYPOTHESES. 
 
 Up to 1690 astronomers supposed the 
 form of the earth to be nearly that of a 
 perfect sphere, and consequently the 
 length of degrees in all latitudes pre- 
 cisely equal. 
 
 In 1690-1718, J. and D. Cassini pub- 
 lished results showing, that although the 
 measures of meridional arcs made in va- 
 rious parts of the globe agreed suffi- 
 ciently to prove that the supposition of a 
 spherical figure is not very remote from 
 the truth, yet exhibited discordances far 
 greater than could be attributable to er- 
 rors of observation, and which rendered 
 it evident that the spherical hypothesis 
 was untenable. Immediately upon this 
 discovery, new interest was awakened in 
 the subject. The works of previous sci- 
 entists upon this subject were carefully 
 examined, and, as a result, it became 
 known that Picard, as early as 1671, in 
 his work on the Figure of the Earth, 
 
29 
 
 mentions a conjecture proposed to the 
 French Academy that, supposing the di- 
 urnal motion of the earth, heavy bodies 
 should descend with less force at the 
 equator than at the poles ; and that, for 
 the same reason, there should be a varia- 
 tion in the length of the pendulum vi- 
 brating seconds in different latitudes, for 
 the time of oscillation of a pendulum of 
 constant length depends upon the intens- 
 ity of the force of gravity. 
 
 In the same year Richer was sent to 
 Cayenne, in equatorial S. A., and was 
 especially charged by the Academy to ob- 
 serve the length of the pendulum vi- 
 brating seconds. On his return he stated 
 that the difference between the seconds 
 pendulum at Paris and Cayenne was one 
 line and a quarter, that at Cayenne be- 
 ing the shorter. Moreover, the clock 
 which Richer took to Cayenne, having 
 been adjusted to beat seconds at Paris, 
 retarded two minutes a day at Cay- 
 enne, so that no doubt remained of the 
 diminution of the force of gravity at the 
 
30 
 
 equator.* This, as it was the first direct 
 proof of the diurnal motion of the earth, 
 was also what led Huygens to suspect 
 that there was a protuberance of 
 the equatorial parts of the earth, and 
 a corresponding depression of the poles. 
 Cassini had already observed this phe- 
 nomenon in the figure of Jupiter, which 
 analogy strongly favored the supposition 
 of a similar peculiarity in the shape of 
 the earth, f Since, then, it was evident 
 that the meridian section of the earth was 
 not a circle, what was the next simplest 
 supposition that could be made respect- 
 ing the nature of the meridian. In the 
 flattening of a round figure at two oppo- 
 site points, and its protuberance at points 
 rectangularly situated to the former, we 
 recognize the distinguishing feature of 
 
 * See Newton's Principia, Book III. 
 
 t The difference of the diameters of Jupiter amounts 
 almost to l-10th, and when we compare the exact 
 measure of this depression, the dimensions of Jupiter 
 and the time of his rotation, with like phenomena con- 
 nected with the earth, we find for this latter planet a 
 proportional depression of l-388th, which is very near- 
 ly identical with the value deduced from the great 
 French measurement. 
 
31 
 
 the elliptic form. Thus mathematicians, 
 after discarding the spherical hypothesis 
 assumed the meridian to be an ellipse. 
 The geometrical properties of that curve 
 enabled them to assign the proportion be- 
 tween the lengths of the axes which 
 would correspond to any proposed rate 
 of variation in its curvature, as well as 
 to fix upon the absolute lengths corre- 
 sponding to any assigned length of a de- 
 gree in a given latitude. 
 
 Spheroids are generated by the revolu- 
 tion of ellipses about one or another of 
 their axes. Every ellipse has two axes, 
 one passing through the foci is called the 
 major axis, while the other perpendicu- 
 lar to the major axis at its middle point 
 is called the minor axis. When the el- 
 lipse revolves about its minor axis it gen- 
 erates what is called an oblate spheroid, 
 and when it revolves about its major 
 axis the figure generated is named a pro- 
 late spheroid. The ellipticity is the 
 amount of variation of the form of the 
 spheroid from a sphere of like content, or 
 the amount of flattening at the poles* 
 
32 
 
 This is expressed by dividing the differ- 
 ence of the semi-major and minor axes 
 by semi-major axis. The eccentricity of 
 an ellipse is equal to the distance from 
 the center to one of its foci divided by 
 the semi-major axis. 
 
 Huygens was the first person who at- 
 tempted to determine the Figure of the 
 Earth by direct calculation, but in his in- 
 vestigation he assumes that the whole of 
 the attractive force resides in the center 
 of the earth, and that its power varies as 
 the square of the distance. This hypoth- 
 esis, since the discovery of the law of 
 universal gravitation, has been found in- 
 admissible, and therefore his results were 
 largely in error. 
 
 In the course of the discussion of Cas- 
 sini's observation of the variation in 
 length of the seconds' pendulum in differ- 
 ent latitudes, and consequent diminution 
 of the force of gravity at the equator, it 
 was claimed that this diminution might 
 be due to the counteracting effect of the 
 centrifugal force occasioned by the rota- 
 
33 
 
 tion of the earth. Newton* showed that 
 even after making allowances for this ef- 
 fect, the difference between the force of 
 gravity at Paris and Cayenne was too 
 great for the spherical hypothesis, and 
 further, upon the assumption that the 
 earth is a homogeneous fluid, and sup- 
 posing its density to be the same through- 
 out the whole mass, and assuming that 
 the constituent molecules attract one 
 another in proportion to the inverse 
 square of the distance, he demonstrated 
 that, in consequence of rotation, the 
 earth would assume the form of an oblate 
 spheroid, whose ellipticity would amount 
 to ^th. 
 
 Clairaut was the first to advance a gen- 
 eral solution of this problem adapted to, 
 the hypothesis of a variable density. He 
 proved that, if the density of the strata 
 of which the earth is composed increases 
 towards the center, the ellipticity will be 
 less than in the hypothesis of Newton, 
 and greater than in that of Huygens ; 
 and, again, that the sum of the fraction 
 
 * Pi-incipia, Book III. 
 
34 
 
 representing the ellipticity and the frac- 
 tion expressing the augmentation of grav- 
 ity at the poles will always make \ r con- 
 stant quantity, which is equal to | of the 
 fraction which expresses the proportion 
 which exists between the centrifugal 
 force and gravity at the equator. It is 
 by means of this theorem we are enabled 
 to ascertain the Figure of the Earth by 
 pendulum experiments. 
 
 These theoretical determinations of 
 Huygens, Newton, and Clairaut were, up- 
 on the completion of surveys made by 
 Cassini in France, found to be at variance 
 with his results. He found the length of 
 one degree of a meridian south of Paris 
 to be 57.092 toises, while north of the 
 city it was only 56.960 toises. This led 
 to the conclusion that the earth is a pro- 
 late spheroid. Here, of course was ma- 
 terial for a controversy ; in view of this 
 fact the French Academy sent out two ex- 
 peditions to make measurements that 
 would definitely settle the matter. These 
 expeditions set out in 1735 ; Bouguer, 
 Godin and La Condamine proceeded to 
 
35 
 
 Peru, and after ten years' work they 
 measured an arc of above 3 between the 
 parcels 2' 31" N., and 3 4' 32" S. 
 latitude. Maupertius, Clairaut, Camas 
 a nd Le Monnier, arriving in Lapland, 
 measured an arc of 57 minutes, and re- 
 turned within 16 months. The results 
 deduced from these observations con- 
 curred in proving that the degrees of the 
 meridian increase very sensibly in length 
 from the equator to the high latitudes, 
 and from this time dates the undisputed 
 conclusion that the a earth is an oblate 
 spheroid, rather than a sphere or prolate 
 spheroid. 
 
 The deviation from the spherical form 
 is evidently very slight, the difference be- 
 tween the equatorial and polar diameter 
 being only 27 miles. As an illustration, 
 on a globe 24 inches in equatorial diam_ 
 eter, and on which the thickness of a 
 sheet of writing paper would represent 
 the elevation of the lands above the 
 waters, the polar axis would be 23.928 
 inches, or in other words, the difference be- 
 tween the polar and equatorial axes would 
 
36 
 
 be but one-fourteenth of an inch. For 
 this reason the spherical hypothesis is 
 sufficiently accurate for many purposes. 
 When this hypothesis is used in geodeti- 
 cal operations, the radius of the earth as 
 a sphere is taken as the average of all 
 the radii of the spheroid. This radius is 
 equal to 6.370 kilometers, or 3.958 miles. 
 In the ^determination of the mean length 
 of an arc of 1 , -g^- of the length of an el- 
 liptical quadrant of the spheroid is taken. 
 Yarious values for this quadrant have 
 been computed by different mathemati- 
 cians. The one deduced by Bessel has 
 been in long use for geodetical computa- 
 tions, and is very nearly the mean of the 
 values found by other investigators. 
 The mean length of one degree is, ac- 
 cording to Bessel, 111.121 meters, or 
 69.043 miles. 
 
 It is thus seen that when the spherical 
 hypothesis is applied the assumed sphere 
 is one having an equal volume with that 
 of the oblate spheroid. 
 
 There is, however, in these values, a 
 serious inconsistency, for the quadrant of 
 
37 
 
 a circle corresponding to the above mean 
 radius is nearly 6 kilometers greater than 
 Bessel's value used in the above deter- 
 mination of the length of 1. For this 
 reason the value sometimes used for the 
 radius is that of a circle whose circum- 
 ference is equal to the circumference of a 
 meridian ellipse, or 3.956 miles 6.367 
 kilometers. This value is 3 kilometers 
 too small, but the error is unavoid- 
 able. 
 
 That the science of the mathematicians 
 had described in a general way the fig- 
 ure of our globe, was sufficient to satisfy 
 the curiosity of the ordinary individual, 
 but not the zeal of scientists for exact 
 knowledge ; they further endeavored i^o 
 obtain the precise amount of the depres- 
 sion at the poles, whose existence had 
 been proven by so many experiments. 
 Material was accumulated, new arcs were 
 measured, but the difficulty of an exact 
 determination only increased. The dif- 
 ferent measures of degree lengths gave 
 varying values for this depression upon 
 the oblate- spheroidal hypothesis. An 
 
38 
 
 Italian mathematician, named Frisi, 
 showed the variation of the calculated de- 
 pression very clearly by a comparison of 
 the measures then known. The follow- 
 ing is a list of the arcs used by him in his 
 computations, and also of the astrono- 
 mers to whom we are indebted for their 
 determination : 
 
39 
 
 ill 
 
 o> . 
 
 T3T3 
 
 2 2 
 
 ii 
 
 i i i i 
 
 f 
 I 
 
 o 
 
 . 
 .0 
 
 . 
 
40 
 
 Frisi, in his calculations, sought to de- 
 termine, according to Newton's theory, 
 the data for a regular curve from which 
 could be derived the above values. In 
 this he was unsuccessful. The curves 
 were either too large or too small. 
 
 The values for the ellipticity of the 
 earth's meridian deduced from the sur- 
 veys instituted by the French Academy 
 are as follows : 
 
 Lapland and French arcs, 
 Lapland and Peruvian arcs, -3^0 th ; 
 French and Peruvian arcs, 
 
 There was evidently a serious discrep- 
 ancy either in the assumption as to the 
 form of the earth, or in the accuracy of 
 the determinations, for if the earth were 
 a spheroid of revolution these results 
 should be identical. Following this dis- 
 covery numerous measurements of arcs 
 of meridian were made in different parts 
 of the world. The most important of 
 these, however, were executed under the 
 direction of the French Government in 
 the determination of the length of the 
 
41 
 
 meter taken as one ten-millionth part of 
 the quadrant of the earth's meridian. 
 These latter observations when combined 
 with the corresponding values in the 
 Peruvian arc gave for the ellipticity, 
 -g-g^th. The nearest approximation of 
 the calculated curve to an ellipse whose 
 minor axis would be to its major in the 
 ratio of 230 to 231 involved an error of 
 more than 100 toises to the degree. 
 Frisi then determined the mean value for 
 the various depressions resulting from 
 the above data and found, for the mean 
 term, a depression almost identical with 
 that furnished by the observations of the 
 pendulum and the measurements for the 
 determination of the French measures. 
 
 The evident impossibility of finding a 
 regular curve to correspond to the dif- 
 ferent degrees measured, gave rise to 
 doubts as to the possibility of measuring 
 a degree of the meridian with accuracy. 
 The instruments then employed in the 
 determinations were liable to errors of 
 three or four seconds for the celestial 
 
42 
 
 arc, or 60 toises for a terrestrial de- 
 gree.* 
 
 The attraction of mountains upon the 
 plumb line, causing a deviation of the 
 vertical, was another source of er- 
 ror. Thus, if the direction of the plumb 
 line at the extremities of the arc meas- 
 ured deviated from the normal by 15 sec., 
 it would cause an error of 500 toises or 
 533 fathoms in the final result, a quantity 
 greater than the presumed difference of 
 the two extreme degrees under the equa^ 
 tor and the pole.f Towards the end 
 of the last century various attempts 
 were made to reconcile the accumu- 
 lating data with the spheroidal hypoth- 
 esis. Among the most prominent investi- 
 gations are those of Boscovich in 1760 r 
 and Laplace in 1793 and 1799. Laplace 
 took, as the basis of his combinations, 
 nine of the measurements used by FrisL 
 The curve which he calculated gave for 
 the length of a degree a value too small 
 by 137.7 toises (= nearly 268 meters), or 
 approximately nine seconds of latitude. 
 
 * Bouguer, Fig. de la Terre, sect. 1, 4. 
 t Malt-Brun. p. 25. 
 
43 
 
 These errors, sajs Laplace, are too great 
 to be admitted, and it must be concluded 
 that the earth deviates materially from 
 the elliptical figure.* 
 
 When compared with the great size of 
 the earth, this deviation of the figure of 
 the earth from the oblate spheroidal 
 form is very slight. As previously stated, 
 for many practical problems it is suffi- 
 ciently accurate to consider the earth as 
 a sphere, but where, for the purposes of 
 science, it is necessary to apply the spher- 
 oidal hypothesis, mathematicians have 
 deemed it expedient to determine the ele- 
 ments of an ellipse agreeing as nearly as 
 possible with the actual meridian section 
 of the earth, and to base their calcula- 
 tions upon the resulting spheroid. . 
 
 In 1805 Legendre announced the meth- 
 od of least squares for the adjustment of 
 observations, and during the present cen- 
 tury numerous applications of this prin- 
 ciple in the determination of the mean el- 
 lipse of the earth's meridian have been 
 made, the principal of which are given in 
 the table on page 44. f 
 
 * Hist. Acad. Paris, 1789. t Jordan. 
 
^S?5?Qpioioo6ci' 
 
 'GJQOlOiaoOCiOCo" 
 1 V* WJ CS O^ OO OO QO Gti 
 'G^C^CiCQOiOJC^XTi 
 
 
45 
 
 Of these, the values of Bessel and 
 Clarke are considered the most reliable, 
 and the spheroids deduced from the ele- 
 ments calculated by these investigators 
 are called respectively the Bessel and 
 Clarke spheroids. The dimensions of 
 the terrestrial spheroid deduced by Bes- 
 sel are as follows : 
 
 Greater or equatorial diameter, 7925.604 
 miles. 
 
 Lesser, or polar diameter, 7899. 114 miles. 
 
 Difference of diameters, or polar com- 
 pression, 26.491 miles. 
 
 Proportion of diameters as 299.15 to 
 298.15. 
 
 Probably the value for the ellipticity 
 deduced from pendulum experiments* is 
 nearer the truth than any deduced from 
 geodetic data. The latter values have 
 been continually approaching those of the 
 former, and we have every reason to be- 
 lieve that when perfection of geodetic op- 
 erations is more nearly approached, the 
 results will be practically identical. We 
 give below the elements of the earth's 
 
46 
 
 figure deduced from pendulum observa- 
 tions : 
 
 Ellipticity = -S55-T-. Eccentricity ^^th. 
 
 ^OO.O L& 
 
 Quadrant of E's Meridian section =10001 
 kilometers, or 6214.62 statute miles. 
 
 In 1859 Gen. de Schubert, in attempt- 
 ing to find a continuous curve for the 
 meridian which would satisfy all meas- 
 ured geodetic arcs, suggested the hy- 
 pothesis of an elliptic equator and an el- 
 lipsoidal figure. The ellipsoid is not a 
 figure of revolution. The meridian sec- 
 tions, as in the spheroid, are ellipses, but 
 the equator, instead of being a circle, is 
 an ellipse. The curves of latitude, how- 
 ever, except the equator, are not plane 
 curves, and consequently not true paral- 
 lels. 
 
 Thus, we see that the ellipsoid has 
 three unequal axes at right angles to each 
 other. 
 
 Gen. de Schubert embodied his idea in 
 his "Essai d'une Determination de la 
 veritable Figure de la Terre," and de- 
 
47 
 
 duced from eight meridian arcs an ellips- 
 oid of the following elements :* 
 
 tf, =6,378,566 metres 
 
 ^ 2 =6,377,837 " 
 
 b =6,356,719 " 
 
 ^ 1= 2927l f* = lm' ^8881 
 where a } 2 are the semi-equat. axes. b = 
 semi- polar axis, /, /'.,=the ellipticities of 
 the greatest and least meridian ellipses, 
 F = the ellipticity of the equator. 
 
 In 1860 and 3,866 similar calculations 
 were made by Capt. A. E. Clarke. Sub- 
 sequent investigations led Clarke in 1878 
 to publish the results of a third discus- 
 sion, giving as the elements of the ellips- 
 oid the following : 
 
 a t = 20,926,629 feet. 
 
 a, = 20,925,105 " 
 
 =20,854,477 " 
 
 111 
 ^ = oH7v /' = 296 73' F = Wm 
 
 The present opinion in regard to the 
 
 *Mem. d TAcad. Imp. des Sciences de St. Peters 
 bourg, VII. Serie, Tome 1, No. 6. 
 
48 
 
 ellipsoidal hypothesis is that until data of 
 a more general and accurate kind have 
 been accumulated, the elements of a sat- 
 isfactory ellipsoid cannot be computed. 
 Arcs of longitude are needed, for the el- 
 lipticities of the meridians differ by such 
 small quantities, that measurements in 
 their directions alone, are insufficient to 
 determine with much precision the form 
 of the equator and parallels. 
 
 Aside from this, the physical improba- 
 bilities of an ellipsoidal figure are so 
 great that it seems more reasonable to at- 
 tribute any apparent departure from the 
 spheroidal figure to effects of local at- 
 traction. Again, there are physical rea- 
 sons for supposing a spheroidal earth, 
 but the existence of a fluid ellipsoid can 
 only be explained by supposing the exist- 
 ence of an ellipsoidal nucleus, which all 
 speculators in cosmogony agree in re- 
 garding as highly improbable. Arch. 
 Pratt remarks, concerning the ellipsoidal 
 figure, that " if a very large number of 
 arcs in all parts of the world were meas- 
 ured, and local attraction being taken in- 
 
49 
 
 to account, the result gave an ellipsoid 
 with its two equatorial axes differing by a 
 quantity, important when compared with 
 the residual errors of observation, there 
 might be some argument for an ellipsoid- 
 al figure." 
 
 During the present century the ex- 
 tensive trigonometrical surveys under- 
 taken by many countries have been the 
 means of furnishing a number of long 
 and accurately measured arcs. Of these 
 the most important are the Anglo- Gallic, 
 the Russian, the Indian, and the U. IS. 
 Coast Survey. The first three have been 
 used in most of the later determinations 
 of the mean ellipse mentioned in the 
 table on page 44. 
 
 Now, it is highly satisfactory to find 
 that the oblate spheroidal figure, thus 
 practically proved to exist, is what theo- 
 retically ought to result from the rotation 
 of the earth on its axis. The form of 
 the earth is owing to the reciprocal at- 
 tractions of its component particles. 
 When a weight is whirled around, it ac- 
 quires a tendency to recede from the cen- 
 
50 
 
 ter of its motion, as a stone whirled 
 around in a sling. This tendency is 
 called centrifugal force. Supposing the 
 rotation of the earth, a centrifugal force 
 is generated whose general tendency will 
 be to cause objects to fly off the sur- 
 face. This force diminishes the gravity 
 of particles, and hence they recede 
 from the axis of the earth until, by 
 their number and attraction, they counter- 
 balance the centrifugal force. This is con- 
 firmed by experience. There is an actual 
 difference in the force of gravity or down- 
 ward tendency of the same body when 
 conveyed successively to points in differ- 
 ent latitudes. Delicate experiments, con- 
 ducted with the greatest care, have .fully 
 demonstrated the fact of a regular and 
 progressive increase in the weight of 
 bodies, corresponding to the increase of 
 latitude. 
 
 Now, let us suppose a globe of the 
 size of the earth to be uniformly covered 
 with water. So long as the body re- 
 mained fixed, the surface of the water 
 would outline a perfect sphere. Imme- 
 
51 
 
 diately attending the introduction of ro- 
 tation on its axis, a centrifugal force 
 would be developed which would act upon 
 every particle in such a way as to tend to 
 cause it to recede from the axis of rota- 
 tion. But now every particle would be 
 subject to the action of two forces the 
 one just mentioned, and that of gravity. 
 The direction of the former would be 
 perpendicular to the axis of rotation, and 
 that of the latter perpendicular to the 
 surface of the water. Since at no 
 position but the line of the equator 
 are these forces directly opposite, 
 they combine to form a third force 
 which urges every particle not situated 
 in the equator towards it with a force de- 
 pendent upon the velocity of rotation. 
 This latter force and the figure of the re- 
 sulting surface of the water are so con- 
 nected that an increase of centrifugal 
 force is always counterbalanced by a pro- 
 portionate change in the direction of gra- 
 vity. Therefore the water would recede 
 from the poles and heap itself on the 
 equator. This would leave the polar re- 
 
52 
 
 gions, in the case of the earth, protuber- 
 ant masses of land. Now, the sea is con- 
 stantly washing and grinding away the 
 land, and carrying and depositing pebbles 
 and fragments over its bed. Thus, in 
 the case considered, the water beating 
 the polar continents would gradually 
 wear them down, and, as with the mole- 
 cules of water, so, in turn, the worn-off 
 particles and fragments of the polar land 
 would would be forced towards the equa- 
 tor, till the earth would assume by de- 
 grees the form we have shown it to ap- 
 proximate the oblate spheroid. 
 
SECTION II. 
 THE OBLATE 
 
 SPHEROIDAL HYPOTHESIS. 
 
SECTION II. 
 
 It is not our purpose to enter into the 
 mathematical discussion of the fluid the- 
 ory of the Figure of the Earth, but simply 
 to place before our readers such princi- 
 ples of the spheroidal hypothesis as will 
 lead to practical results. For a complete 
 investigation of the form assumed by a 
 revolving fluid on the principle of gravi- 
 tation, we would refer our readers to sec- 
 tion 2 of Mr. Airy's essay upon the Figure 
 of the Earth contained in the " Encyclo- 
 pedia Metropolitans" 
 
 We have stated that if the earth be 
 considered a fluid mass, the form of the 
 surface will be an oblate spheroid of 
 small ellipticity. Further, its axis will 
 coincide with the axis of revolution, and 
 the surface will everywhere be perpendic- 
 ular to the direction of gravity. It fol- 
 lows also upon the assumption that the 
 
56 
 
 density of the strata varies according to 
 a certain probable law, that the ellipticity 
 is jfa. 
 
 Let us assume, then, that the mean 
 Figure of the Earth is an oblate spheroid, 
 and endeavor to show by what methods 
 an ellipse can be found cutting the plumb 
 line at right angles and with its minor 
 axis coinciding with the axis of the earth. 
 This end may be reached by four methods, 
 and first we will consider how the Figure 
 of the Earth may be determined from 
 geodetic operations.* 
 
 * We do not purpose to deal with the theory of the 
 deduction of the Figure of the Earth from measure- 
 ments of the Arcs of Longitude, or the determinations 
 of Azimuths. The underlying principles of these 
 methods have had a very limited application ; and, 
 aside from the Indian Arc, little material is available. 
 The method of Azimuths depends entirely upon angles, 
 and, therefore, can only assist in determining the 
 ellipticity. 
 
57 
 
 CHAPTEK I. 
 
 THE FIGURE OF THE EARTH DETERMINED BY 
 GEODETIC OPERATIONS. 
 
 The first step in this method is to 
 measure as accurately as possible a base 
 line of any convenient length, not less 
 than 5 or 6 miles, and as near as possible 
 to the meridian upon which we are to 
 base our calculations. 
 
 For this purpose wooden or metal 
 rods, steel chains, or compensation bars 
 are used. The latter consist of a series 
 of compound bars, self-correcting for 
 temperature.* Two bars, one of brass, 
 the other of iron, are laid side by side 
 firmly united at their centers, while their 
 ends are free to expand or contract. 
 These bars, at the standard temperature, 
 are of the same length. The following 
 is the principle of their construction: 
 Let AB be one bar, A'B' the other. 
 Draw lines through similar extremities 
 
 * Encyc. Brit. 
 
58 
 
 AA/ BB' to P and Q. Make A'P=B'Q, 
 AA'=BB'. Now if A'P is to AP as the 
 rate of expansion of the bar A'B' is to 
 that of the bar AB, then the distance PQ 
 will be nearly invariable. In the com- 
 pleted instrument P and Q are dots 10 feet 
 apart. In the measurement, the bars when 
 aligned do not come in contact. A space 
 of six inches or thereabouts, is left be- 
 tween each bar and its neighbor, and the 
 interval is measured by an ingenious 
 micrometrical device, constructed upon 
 the same principle as the bars them- 
 selves. 
 
 The United States Coast Survey in 
 their measurements, generally use an 
 apparatus consisting of four bars, pro- 
 tected by a wooden covering. The bars 
 are placed so as to leave between them a 
 small interval, which is measured by 
 wires brought into contact and adjusted 
 by means of micrometer microscopes. 
 The bars are also provided with ther- 
 mometers. 
 
 In performing the operation of measur- 
 ing the base it is very important that 
 
59 
 
 at least two of the rods or bars should 
 be in position before a third is applied. 
 
 The base being thus measured, and 
 the reduction for inclination applied 
 together with the correction for tem- 
 perature, if necessary, there re- 
 mains but to reduce the length to 
 sea level. Let r Dearth's radius, or the 
 radius of the surface or the sea, and h 
 the elevation ; the measured lengths must 
 
 be multiplied by the fraction - - or 
 
 1 , in order to obtain the length at 
 
 the level of the sea. 
 
 Having now completed the operation 
 known as the measurement of the base, 
 we next proceed to measure the angles 
 between the base line, and visual lines 
 joining the extremities of the base to 
 distant points, taken as near .the meridian 
 as convenient. The instruments which 
 have been used for measuring these 
 angles are quadrants, theodolites and re- 
 peating circles. With the quadrant and 
 repeating circle, the angle actually sub- 
 
60 
 
 tended by the points or signals is ob- 
 served, while with theodolites the hori- 
 zontal angle is measured. Knowing the 
 length of one side, and two of the 
 angles of a triangle, we can, by trigono- 
 metrical formulae deduce the lengths of 
 the other sides. Repeating the operation 
 with the sides already calculated, and 
 selecting new points to suit the emergen- 
 cies of the case, we establish a connec- 
 tion between the original base line and a 
 second base at the termination of the 
 chain of triangles, and obtain the length 
 of this second base by calculation. It is 
 then measured, and by a comparison of 
 the calculated and measured results the 
 correctness of the operations is tested. 
 The next step is to determine the direc- 
 tion of one of fche sides of the chain of 
 triangles with regard to the meridian. 
 This is called the determination of the 
 azimuth, and in principle is as follows a 
 temporary mark is fixed as near as 
 possible to the meridian ; a transit in- 
 strument is adjusted upon it, and the 
 transits of stars at different polar dis- 
 
61 
 
 tances are observed. The deviation of 
 the transit instrument, or the azimuth of 
 the mark, is thus found with great accu- 
 racy. By means of a theodolite or re- 
 peating circle, the angle between the 
 mark and one of the signals can be ob- 
 served, and there results the azimuth of 
 the station. 
 
 This having been satisfactorily per- 
 formed, we are now able to calculate the 
 length of meridian arc contained between 
 the two parallels passing through the ex 
 treme stations of the chain of triangles. 
 The simplest means "for performing this 
 operation is known as the method of 
 parallels and perpendiculars, and consists 
 mainly in finding the projections upon the 
 meridian of the various sides of the system 
 of triangles. Only one precaution is ne- 
 cessary, and that is. that the perpen- 
 diculars to the meridian must not be of 
 such length as to make the difference be- 
 tween the spherical length of the sides 
 and their length when projected on a 
 tangent plane a sensible quantity. 
 
62 
 
63 
 
 Let the accompanying figure represent 
 a chain of triangles, in which A and G 
 are the extreme stations. Let AC be the 
 base line, its length together with the 
 azimuth CAc are known, and therefore 
 the remaining sides and their projections 
 upon the meridian AB maybe calculated. 
 The sum A# of the projections might be 
 supposed to represent the distance on 
 the meridian separating A and G; but 
 this is not the case, since there is a 
 difference between the perpendicular and 
 the arc of a small circle passing through 
 G. The true distance is A,/, found by 
 producing A# to B, the pole of the 
 earth, and with a radius BG describing 
 a small circle Gg'. The difference gg' 
 of the two distances must be subtracted 
 from the ascertained length A<7, in order 
 to obtain the true distance Kg'. The 
 quantity gg' may be found as follows : 
 
 ,_ (Gg.)* tan, lat. G. 
 . gg ~ earth's diain. 
 
 Another method considers BG^ a right- 
 angled spherical triangle, and using an 
 
64 
 
 approximate value for the arc of a great 
 circle, corresponding to the length Gg, 
 determines the side B<? by 
 
 cos. BG 
 Cos. B# = - 
 
 cos. Gg 
 
 If we join B and G by the arc of a 
 great circle, it is evident that BG is not 
 perpendicular to Gg, and therefore not 
 parallel to Bg. This is what is under- 
 stood as the convergence of the meridians. 
 The angle may be calculated from the 
 spherical triangle BG<7, as follows : 
 
 The determination of the distance - be- 
 tween the parallels completes the 
 Geodetic operations, and we have now 
 but to determine the latitudes of the ex- 
 tremities, an operation upon the accuracy 
 of which the whole result depends. We 
 will not attempt a description of the 
 various methods employed, but deem 
 it sufficient to mention the application of 
 Airy's Zenith Sector. 
 
 The observation consists in measuring 
 
65 
 
 with the telescope micrometer the differ- 
 ence of zenith distances of two stars 
 which cross the meridian, one to the 
 north, and the other to the south of the 
 observer, at zenith distances which differ 
 by not much more than 10' or 15', the 
 difference of the times of transit being 
 not less than one or more than twenty 
 minutes. The peculiar advantage of this 
 method is that refraction is well nigh 
 eliminated. 
 
 Having determined the latitudes of the 
 two extremities, the ellipse of which 
 the arc is part is found as follows :* 
 Let I and I' be the latitudes of the ex- 
 tremities of the arc. 
 
 " m be the mean of I and I', or the 
 middle latitude. 
 
 " A be the amplitude, or I I'. 
 
 " a and b be the semi- axes. 
 
 " e be the ellipticity. 
 
 " s be the length of the arc. 
 
 " r be the radius vector. 
 Let 6 be the angle r makes with the ma- 
 jor axis. 
 
 * Pratt's Figure of Earth. 
 
66 
 
 Then 
 1 cos. 2 # sin. 2 <9 
 
 ^ = ^- + -F-' ^ 
 
 L 2 cos. 2 +6 8 sin. 8 
 
 placing b = a (1 e) 
 r=a(l esin*l) neglecting 2 , 
 
 dr . , , dO 
 
 in. 2 / 
 
 = (! Je f^cos. 20 
 
 .-. s=a[(l-e)(l I'} ff(sin.2/ sin.2^')] 
 =i (a + 6) A | (ab} sin. A, cos. 2m. (1) 
 
 If A be small, not exceeding 12, we 
 may place sin. A=A, then 
 
 when the lengths, amplitudes and lati- 
 tudes of two arcs are known. 
 
67 
 
 s s' 
 
 ab 
 
 S n s ' r 
 
 , cos.zm cos.zm 
 ^^_A_ A 7 3) 
 
 2 cos. 2 m' cos. 2 m 
 
 From these formulae the semi-axes, 
 a and b, may be computed and the value 
 for e, the elliptic! ty follows. 
 
 If we were to substitute in these equa- 
 tions observed values resulting from the 
 various measurements of meridian arcs 
 indifferent parts of the earth, it would 
 be found that the calculated semi-axes 
 and ellipticities in the various cases 
 would be different. If the figure of the 
 earth were truly spheroidal, and there" 
 were no errors in the data, i. e., in the 
 observed amplitudes and measured arcs, 
 the results would come out in complete 
 accordance with each other. But this is 
 evidently not the case. 
 
 Let us inquire, then, as to the source 
 of these variations in the calculated spher- 
 oid, and in attaining this end we will first 
 
68 
 
 state the assumptions we have made in 
 deducing the above formulae. These are 
 (1) that the meridian arc is an. ellipse in 
 accordance with the fluid theory. (2) 
 That the plumb line at all stations is nor- 
 mal to this ellipse. 
 
 It is plain that the conclusions of the 
 fluid theory are not applicable to the 
 earth in its present state. The irregu- 
 larities in the external form of the earth 
 are considerable when theoretically con- 
 sidered, and, of necessity, these irregu- 
 larities of the surface must be accom- 
 panied by irregularities in the mathemati- 
 cal figure of the earth. The height of 
 mountains and depth of seas are, in some 
 places, equal to \ the difference between 
 the equatorial and semi-polar axes, and 
 we are, therefore, prepared to admit that 
 the surface of the earth is not one of rev- 
 olution. Nevertheless, there must be 
 some spheroid that agrees very closely 
 with the mathematical figure of the earth 
 and having the same axis of rotation. 
 
 The errors due to local attraction re- 
 sult from the deflection of the plumb 
 
69 
 
 line from the normal to the assumed el- 
 lipse. This is sufficient in certain parts 
 to produce material errors in the vertical, 
 and therefore in the difference of latitude 
 determined by the zenith distances of 
 stars. 
 
 Thus, suppose A and B to be two sta- 
 tions, and that at A there is a disturbing 
 force drawing the plumb line through an 
 angle tf ; then it is evident that the ap- 
 parent zenith of A will in reality be that 
 of some other place A', whose distance 
 from A is r#, when r = earth's radius. 
 Similarly, if there is a disturbance at B 
 of the amount #', the apparent zenith of 
 B will be that of some other place B', 
 whose distance from B is rd' '. 
 
 Mountain masses, oceans, and varia- 
 tions in the density of the interior of the 
 earth are the chief causes of local attrac- 
 tion. 
 
 It is then evident that our assumptions 
 in the preceding formulae are at fault, and 
 it is now necessary to discover some 
 means whereby the resulting errors may 
 be eliminated. 
 
70 
 
 Bessel was the first to devise a method 
 whereby the results of all the surveys in 
 the different parts of the earth might be 
 brought to bear simultaneously upon 
 this problem. He was followed by Capt. 
 A. R. Clarke, who discussed the problem 
 at the end of the Ordnance Survey Vol- 
 ume. 
 
 Bessel' s method is in essence as fol- 
 lows : Corrections expressed in algebraic 
 terms are applied to the latitudes of the 
 several stations, dividing the arcs into 
 their subordinate parts, such as to make 
 their measured lengths exactly fit an el- 
 lipse. The values of the axes of this el- 
 lipse are then determined so as to make 
 the sum of the squares of these correc- 
 tions a minimum, i. e., an ellipse is de- 
 duced which most nearly represents the 
 observations. 
 
 We will now endeavor to state briefly 
 how this may be done. 
 
 Let us first obtain a formula for cor- 
 recting the amplitude of an arc so as to 
 make its measured length accord with a 
 given ellipse. 
 
71 
 
 We have equation (1), 
 
 s=% (a + b) A f (a b) sin. A cos. 2 m. 
 
 Suppose now that x, x' are small cor- 
 rections which must be applied to the ob- 
 served latitudes to make the measured 
 arc fit the ellipse of which a and b are the 
 semi-axes ; A and m when substituted in 
 the above formula will not give the meas_ 
 ured value of s. Instead of them 
 
 X + x'x and 2m + x' + x 
 must be substituted. Hence omitting 
 very small quantities 
 
 2s= (a + b) A - 3 (a b) sin. A cos. 2m 
 + (x f x)[a + b 3(a b) cos.Acos. 2m\ 
 
 /. x' x=( r A + 3 7 8in.Acos.2m\ 
 
 \a + b a + b J 
 
 (l + 3^r 
 
 cos. A cos. 2m . (4) 
 
 Now the mean radius of the earth is 
 known not to differ much from 20890000 
 feet, and the ellipticity not much from 
 g-J-g-. It is therefore convenient to put 
 a + b under the form 
 
72 
 
 ab 
 
 where the squares of u and v may be neg- 
 lected. 
 
 Substituting these in formula (4) and 
 changing the form, we have 
 
 x'= 
 
 where m, a, /?, are functions of the ob- 
 served latitudes, the measured lengths 
 and other numerical quantities.* 
 
 In every measured arc, not only are 
 the extreme stations determined in lati- 
 tude, but also a number of intermediate 
 stations ; one of them should be taken 
 as a station of reference,, and each separ- 
 ate error should be written in terms of 
 the reference station error. 
 
 * The values of m, a and (3, have been calculated for 
 the principal arcs and may be found in the British 
 Ordnance Survey volume. 
 
73 
 Thus if we have five stations, 
 
 Now, since these equations are affected 
 by errors of observation, it is impossible 
 to find values of u and v which will 
 satisfy all the equations, and therefore it 
 is necessary to find their most probable 
 values. According to the method of 
 least squares, those values of u and v 
 are the most probable, which render the 
 sum of the squares of all the errors a 
 minimum. 
 
 In proceeding with this method, we 
 first deduce normal equations for u by 
 multiplying each observation equation by 
 the coefficient of u in that equation, and 
 add the results ; likewise, deduce normal 
 equations for v by multiplying each ob- 
 servation equation by the coefficient of v 
 in that equation, and add the results. 
 Thus we will have two normal equations, 
 each containing two unknown quantities, 
 
74 
 
 and the solution of these will give us the 
 most probable values for u and v. Sub- 
 stituting these values in equations (5), (6), 
 (7), we obtain values for the two semi- 
 axes and the ellipticity. 
 
 In conclusion, let us examine what ef- 
 fect any error in the amplitudes will 
 have upon the resulting axes. 
 
 If we differentiate equations (2, 3), 
 there will appear in the denominators of 
 the resulting expressions the quantity 
 
 cos 2m cos 2m'. 
 
 The errors in the axes dependent upon 
 the amplitude errors will therefore be 
 least when this quantity is a maximum 
 If an arc is located in the southern half 
 of the quadrant, cos 2 m is positive, 
 then, 
 
 2m' =180, or m'=90 
 
 will give the best result. If another 
 arc is in the northern half, cos 2m is nega- 
 tive, and 
 
 cos 2m' =0 
 
 will give the best result. 
 
 Therefore the nearer one arc is to the 
 
75 
 
 pole, and the other to the equator, the 
 less will be the effect upon the calcula- 
 tions of any errors in the observed am- 
 plitudes. 
 
76 
 
 CHAPTER II. 
 
 THE FIGURE or THE EARTH DEDUCED FROM 
 PENDULUM EXPERIMENTS. 
 
 Upon the hypothesis of the earth be- 
 ing a fluid mass, Clairaut deduced an 
 equation showing that the increase of 
 gravity in passing from the equator to 
 the poles varies as the square of the sine 
 of the latitude, and that a certain rela- 
 tion must necessarily exist between the 
 ellipticity and the force of gravity. 
 Therefore, if we desire to use this the- 
 orem in the determination of the Figure 
 of the Earth, we must first obtain some 
 practical method of measuring the force 
 of gravity in any latitude. The neces- 
 sity of using the pendulum for this pur- 
 pose will easily be seen if we consider 
 the impossibility of ascertaining the 
 magnitude of the force by an experiment 
 upon the single descent of free bodies. 
 The quantity to be measured is the veloc- 
 ity which gravity creates in any freely 
 
77 
 
 descending body by its action continued 
 during a second of time. In con- 
 ducting a series of experiments it is 
 usual to observe the number of vibra- 
 tions made by the same seconds' pendu- 
 lum in the different places at which it is 
 proposed to compare the force of grav- 
 ity, and likewise the number of vibrations 
 made at London or Paris. The com- 
 parative number of vibrations being 
 found, the comparative force of grav- 
 ity or the comparative length of the sec- 
 onds' pendulum can be deduced. Since 
 the length of the seconds' pendulum has 
 been very accurately determined at Lon- 
 don and Paris, its length in any latitude 
 may be found. 
 
 In order to understand the principles 
 of the deduction of the Figure of the 
 Earth from pendulum experiments, let us 
 examine the theory of the pendulum. 
 
 In the following investigation the 
 pendulum is taken to be a material point 
 suspended by a string without weight 
 an evident impossibility. It may be 
 shown, however, that if a compound 
 
78 
 
 pendulum be given, it is possible to calcu- 
 late the length of a simple pendulum 
 with the same vibration period; there- 
 fore, what is true of the imaginary 
 pendulum, is true of the pendulum used 
 in practice, and the latter may be used 
 for all the purposes of the former. 
 
 It is known that the time of vibration, 
 in seconds, of a simple pendulum, whose 
 length is I, in double the circular arc, 
 whose versed sign is b, will be nearly 
 expressed by ' 
 
 * 
 
 where g equals the space through which 
 a body would fall in a second of time. 
 If the arc is indefinitely small, the time 
 will not vary much from 
 
 Now let us assume the length of 
 pendulum, such that the time of vibra- 
 tion shall be one second, or 
 
79 
 
 The space g then results from multiply- 
 
 2 
 
 ing the length of the pendulum by -^ 
 
 Since gravity is not the same at all 
 places on the earth's surface, the length 
 of the pendulum vibrating seconds will 
 vary. However, when the lengths at the 
 different places are determined, the same 
 proportion holds between those lengths 
 as between the forces of gravity at the 
 various stations. 
 
 Since it is very difficult to make a 
 pendulum which shall vibrate in exactly 
 one second, it is expedient in some cases 
 to observe the time in which a pendulum 
 of known length vibrates, and to deduce 
 from this the length of the seconds 
 pendulum. In the expression 
 
 where t= the time of vibration, it is evi- 
 dent that the length of the pendulum is 
 proportional to the square of its time of 
 vibration. ' Thus it is seen that the 
 length of the seconds' pendulum bears to 
 
80 
 
 the known length the same ratio as unity 
 to the square of the number of seconds 
 in the observed time of vibration. 
 
 In this method the absolute length of 
 the pendulum is determined at every 
 station of observation, and while in some - 
 respects this is preferable, there are seri- 
 ous disadvantages. For this reason a 
 method has been used of finding the pro- 
 portion between the force of gravity at 
 each station, and at some place of refer- 
 ence where its absolute length has been 
 very accurately determined. This is 
 done by " transporting the same pendu- 
 lum, after having observed it at the 
 reference station, to all the different 
 stations, and observing the time of vibra- 
 tion at all." 
 
 Since 
 
 and I is the same at all stations, g is 
 inversely proportional to t*. Therefore, 
 the force of gravity at any new station is 
 to the force of gravity at the place of 
 reference, inversely, as the square of 
 
81 
 
 the time of vibration at the new station, 
 to the square of the time of vibration at the 
 place of reference. Or again directly as 
 the square of the number of vibrations 
 per day at the new station, to the number 
 of vibrations per day at the place of refer- 
 ence. The length of the seconds' pendu- 
 lum being in the same ratio, that at the 
 new station may be deduced. In the pre- 
 ceding investigation certain suppositions 
 are made, which are not tenable when 
 the nature of the experiments partake of 
 great accuracy. These are, namely 1st, 
 that the arcs of vibration are extremely 
 small ; 2d, that the length of the pendu- 
 lum continues invariable ; 3d, that the 
 condition of the atmosphere is the same 
 under all circumstances. We will con- 
 sider each of these in the above order. 
 
 When the arc of vibration is so large 
 that the term depending upon it cannot 
 be neglected, the time of vibration is ex- 
 pressed by 
 
If the arc of vibration includes n de- 
 grees on each side of the vertical. 
 
 07 . 2 n n 3 sin. a l 
 0=1 versm n =21 sin. ~^-~^ - o - 
 
 2 2 
 
 nearly ; and the time of vibration 
 
 If the experiments continue for so 
 short a time that n remains practically 
 invariable, the observed time of vibration 
 should be divided by 
 
 , sin. 3 l 
 
 1 + n --[6- 
 
 or the number of vibrations per day 
 should be multiplied by the same 
 quantity, in order to represent the time 
 or number of vibrations in indefinitely 
 small arcs. When the observations con- 
 tinue for a long period, it is necessary to 
 know what is the law of decrease of the 
 arc. Borda and Airy found, by direct ex- 
 periment, that the decreasing arcs form 
 very nearly a geometrical series. Let 
 m= the number of vibrations observed; 
 and p= the proportion of each arc to the 
 
83 
 
 preceding arc, where 1 p isjvery small. 
 Let m', the number of degrees in the last 
 
 m-l 
 
 arc, =np. Now the lengths of the suc- 
 cessive arcs are n, np, 
 
 m-l 
 np* np. 
 
 The sum of all the times of vibration 
 are: 
 
 16 
 
 The sum of the degree series is : 
 
 l_2m l_ p 2ro-2 
 
 -~ r = * a nearly; 
 1 j9 2 1 p* 
 
 therefore the sum of all the times 
 
 -l 
 
 approx., 
 
 o 
 
 .-. log. /j 2 = (log. n' log. n) 
 
 m 
 
84 
 
 Now log. p*= modulus (p 2 1), nearly, 
 since p* 1 is very small. 
 2 
 
 .'. 1 # a = =, .(log.W-log.7l'). 
 
 modulus X m v 
 Consequently the sum of all the times 
 
 " 9 \ 
 
 modulus X (ri* n'*) sin. 2 l ) 
 
 log. n log. n' ' 32 ) ; 
 and the mean of the observed times 
 
 modulus X(^ 8 -^ /8 ) sin. 2 1 ) 
 log. n log. n' 32 ) 
 
 The quantity within the brackets is 
 that by which the observed mean time of 
 vibration should be divided, or the 
 number of vibrations per day should be 
 multiplied, in order to reduce the obser- 
 vations to vibrations in indefinitely small 
 arcs. 
 
 In the second place, we have supposed 
 the length of the pendulum to continue 
 invariable. It is well known that a me- 
 
85 
 
 tallic wire or bar undergoes considerable 
 changes in its length with changes of 
 temperature, and it is therefore necessary 
 to reduce the number of vibrations to 
 the number which would have been 
 made if the pendulum had remained of 
 constant length. Suppose that the 
 pendulum's length is increased above 
 that at the standard temperature in the 
 ratio of l:l + y (y being very small.) 
 Then the time of vibration is increased 
 
 in the ratio of 1: VTTy, or 1: 1+| 
 
 nearly, and the number of vibrations per 
 day is diminished in that ratio. There- 
 fore the observed number of vibrations 
 should be increased in this ratio in order 
 to find the number which would have 
 been made if the pendulum had been 
 of constant length. The same law holds 
 if the temperature were decreased in- 
 stead of increased. 
 
 Thirdly, we have supposed that the 
 condition of the atmosphere remains un- 
 changed. The vibrations are observed 
 in air, and since the state of the air 
 
86 
 
 is variable it is desirable to calculate the 
 number of oscillations which would have 
 been made in a vacuum. Now the effect 
 of the air is to diminish the weight of 
 the pendulum by a quantity equal to the 
 weight of air displaced. If this diminu- 
 tion be in the ratio of 1:12, then the 
 time of vibration (as will appear by 
 changing g into g (1 2) in the preceding 
 expression), will be increased in the ratio 
 
 of 1 : -==: and .'. the number of vibra- 
 tions per day diminished in the ratio of 
 1 : V 1 2, or 1:1 jr nearly. Therefore 
 
 to get the number of vibrations in a 
 vacuum, we must increase the observed 
 
 number in the ratio of 1:1 + ^. 
 
 In calculating the above correction, it 
 is necessary to observe the height of the 
 barometer, since the height of the latter 
 is nearly proportional to the weight of 
 the atmosphere. 
 
 This correction, in the early part of this 
 century, was applied for the reduction of 
 
87 
 
 the observations to the sea level. Dr, 
 Young, in the Phil. Trans, for 1819, re- 
 marked that since this correction neg- 
 lected the attraction of the elevated mass 
 it was too great. He adds that, assum- 
 ing the density of the elevated mass to be 
 
 o r 
 
 the mean density of the earth, the cor- 
 5.5 
 
 rection would be .66, that of the usual 
 reduction. 
 
 Having shown how refinement of ex- 
 periment is obtained, we will now en- 
 deavor to explain by what means we 
 deduce the Figure of the Earth from 
 Pendulum Experiments. 
 
 We have for the attraction at any point 
 of a spheroid the following expression : 
 
 
 m, in the earth, = -^^ 
 
 In which m= ratio of centrifugal force to> 
 gravity, / and g the co-ordinates of the 
 
88 
 
 point, in the directions of CD and CE re- 
 spectively, b= polar axis, =ellipticity and 
 
 *?. ?-^= gravity at the equator. 
 3 o 
 
 For the force at the pole we must make 
 f=o, go, and the expression therefore 
 
 For the force at the equator, we must 
 2 + ^=a a =:5 2 , nearly and there- 
 fore equatorial gravity. 
 
 The excess of the former above this is 
 
 and the ratio of this excess to equatorial 
 
 gravity is s m f. Let this = w, then 
 Jj 
 
 n-f- f=-m, or we have deduced Clairaut's 
 theorem. 
 
IFlg. 3. 
 
 Let EF represent the earth's surface, and 
 let PQ be the normal at P. 
 
 Then P Q N is the latitude of P and 
 
 QN=PQ, cos. I. 
 
 Now as we shall have to substitute only 
 in the small terms of the equation, PQ=& 
 nearly, and QN=CN nearly, 
 
 nearly. 
 
 Substituting this in equation (8), we 
 have for the force of gravity 
 
 . e4 . 
 
 + 
 
90 
 
 Gravity, therefore, may be generally ex- 
 pressed by the formula 
 
 , ... (9) 
 where E = equatorial gravity and 
 
 5m 
 
 n + e=-. 
 
 Now let p and p' be the lengths of the 
 seconds pendulum in latitudes I and V, 
 P that at the equator, then from equation 
 (9) we have 
 
 p=P(l + n8m*l) . . (10) 
 
 p'=P(l + nsm.'l') 
 where 
 
 ,.n= 
 
 " * 2 Tt 
 
 sin.Y sin.V 
 P 
 
 , 5m 
 
 and e= n 
 
 In applying the preceding equations to 
 a series of pendulum observations for the 
 determination of the Figure of the Earth, 
 the principle of Least Squares is applied 
 as in the last chapter to a number of ob- 
 
91 
 
 servation equations of the form (10), there- 
 by obtaining the most probable values for 
 P and 71. 
 
92 
 
 CHAPTER III. 
 
 1. THE ELLIPTICITY OF THE EARTH DEDUCED 
 
 FROM OBSERVED INEQUALITIES OF THE 
 
 MOON'S MOTION. 
 
 In the expression for the tangent of th 
 moon's latitude there is this term* 
 
 / m\ 4.(60) s . s 
 
 I -~n I \ -sm. parallax. 
 \ 2 / n 3 TT // 
 
 sin. obliquity, cos. obliquity, sin. 6 
 
 Earth's Mass 70 
 .Now = - - - =r-= - =rr, nearly, 
 fA Earth + Moon 71 
 
 27.25 
 
 The mean horizontal parallax =57', 
 
 obliquity = 23 28' nearly, 
 
 .-. term=-/f-^) .4891" sin. 8 
 
 Now this inequality has been found to 
 exist and its magnitude has been inferred 
 from observation. It has the effect of 
 fncreasing the apparent inclination of the 
 
 * Airy's Math. Tracts. 
 
93 
 
 moon's orbit in one position of her nodes 
 and diminishing it as much in the oppo- 
 site position. 
 
 It is found by observation that the co- 
 efficient = -8" 
 
 ^=4frl= 001635 
 
 and = .001730 
 2 
 
 2. THE EARTH'S ELLIPTICITY DEDUCED FROM 
 THE PRECESSION OF THE EQUINOXES. 
 
 The formula expressing the annual Pre- 
 cession is the following : 
 
 in which 1= the obliquity of the eclip- 
 tic = 23 28' 18", i = inclination of 
 moon's orbit to ecliptic =5 8' 50", 
 n and n' are the mean motions of the 
 earth around its axis and around the sun 
 and their ratio =365.26, n" the mean 
 motion of the moon around the earth 
 
= 27.32 days, v= ratio of the masses of 
 earth and moon =75. 
 
 Substituting these quantities 
 
 C A 
 
 Annual Precession = 16225''.6 ^ 
 
 where A and C are the principal moments 
 of inertia of the mass, the latter about 
 the axis of revolution. 
 
 Now ^^=1.98177 (*-iro)'* 
 
 .-. Annual Precession =32155" (e %m) 
 But the Precession by observation = 50". 1 
 
 .-. e J m = 50.1-=-32155 = 0.0015581 
 /. =0.0015581 + 00017271=^ 
 
 In estimating the reliance to be placed 
 on these results it must be observed that 
 unless the observations extend over a 
 period greater than 20 years they are in- 
 sufficient. This renders the resulting de- 
 terminations less reliable than they other- 
 wise would be, as in all probability the 
 observations which are compared have 
 been made by different persons and in 
 different manners. The small lunar in- 
 
 * See Pratt's Fig. of the Earth. Page 151. 
 
95 
 
 equalities, besides, are involved among a 
 mass of terms greater than themselves. 
 Airy remarks concerning this fact that an 
 error in their determination has less in- 
 fluence on the value of than an equal 
 error in the determination of nutation. 
 These facts show clearly that the deduc- 
 tion of the two preceding divisions cannot 
 be compared with those of geodetic meas- 
 ures and pendulum observations. How- 
 ever, the close agreement of the results 
 with those deduced from geodetic meas- 
 ures and pendulum experiments is sig- 
 nificant. It shows that in the main the 
 spheroidal hypothesis is correct. 
 
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