Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
CHAPTER IX THE SHAPE ANT) ASIDE OF THE 1SARTI:T N)T D . LA M BERT U.- S. CYO~'Si and Geodetic Surgery The idea that, apart I'ron~ the hills and valleys that d;versi1';y its surface, the earth is essentially flat is of course ogle that natty ally presents itself to all primitive peoples. The notion of a flat earth would, however, present ciifficulties to ~ though.t-t'ul observer of the stars who had not.icecl that as he traveled no~thwarcl or southward the aspect of' the heavens changecl; it would present difficulties also to a thoughtful seaman who had noticed the gradual disappearance of' a receding vessel, hull first and then the masts alla jigging. In China, Egypt, and Babylonia astronomy was cultivated by priests and astrologers centuries and millenniums be-fore tile beginning, of our era, but it does not appear that their speculations on the subject led there to formulate the idea of the rotundity of' the earth. At any rate no such speculations have come down to us. These phenomena previously me tioned, being but roughly obser~:ecl, would, of course, prove merely that the surface of' the earth is rounded and convex, not that it is exactly spherical.* The Greeks were both a seafaring and speculative people, and it is id Greek thought that we first find the idea of' a spherical earth. It is not likely that these earliest advocates of' the idea of' a spherical earth made measurements to find out how closely the earth approximates to a sphere. The exact sphericity was considered as proved, or as made plausible, by metaphysical considerations. The first clear and unequivocal statement of the sphericity of the earth is due to the Greek philosopher Pythagoras (about 340 B. C.) or to his followers. No doubt the evidence in favor of the sphericity of' the earth found increasing favor among, thou`,hti'ul men, but even too centuries later Aristotle (384-322 B. C.) thought it worth while to refute the arguments against' the sphericity of' the earth and to set forth those in favor of its (See numbered references at end of chapter). Some of his ar~,u~r~er~.ts are very :t'oreign to our ways of thinli- ing; others sound very modern. * Theoretically at least, the difference in curvature of the sea. surface in different directions could be measured by measuring the varying dip of the horizon in different azimuths. 123
12! F7~ OF r~ ~6 r~ Evidently the astronomers Off Aristotle time h~4 some idea of the size of the earth. The concluding passage of the Depone took of his truth se; Ma Ago, aeons: ~ Forever those m~thPm~ici~ns ago try to romputc the cir~nmference of the each say that it is i00~000 strain, which initiates got only Ant the earthy mass is sphrric~1 in shape hut Vito that ~ is of no grant size as compare] gab the be~venly todTes/, four hundred thong Athc static Could he 74~000 kilometers. This is rert~i~T~ Lot ~ p~rticul~ly good ~pproxim~Hon to the rectum hauler 40~000 kilometers; but it is at least of the right order of magnitude. Hog this (aura ~ reached does lTOt appears but it eras prol~hl~: by some such p] in] 11S a\' t1.S 111.tOl! 11SP] by Lr~toatbenes. About ~ bUndre] Bears later ~ better dete~in~tio~ of the aide of tar each gas made by Fr~tosthe~s Plied about I93 B. C.~, libr~ri~ at Alert and ~ Max of gibe attainments TO Various herds of learning. Fr~tos~enes assume] Act Spend; Me modern Assuage on the Upper Age lay OU the Myopic of cancer ~D] that therefore the sun at the summer solstice Could cast ~ vertic~1 shadow. He assumed furler that Syene ~] Alexandria mere OU the same meridian: tbo~gh their dT~erence of longitude is some two degrees. The distance between them be supposed to Be S;000 static. Some commentators bade thought that this figure derive] from the computations of the o~ci~1 surveyors who kept track of property hag, ~bieb were likely to become distinct owing to the ~nDu~1 inundations of the gild tUt more pivotally it represents merely an cstim~te Based on the accounts of trsvele~s. Abed Er~tostbenes observe] at Alexandria the zenith distance of the sun at the summer solstice and found it to he one-hitietb of ~ ci~cumfer- once ~° IS ~bicb is therefore [~fn~ffAnA~, vn]~P fan 1~ ~1]~r ~ ^ ~ ~ v ~ ~ ~ v ~ v ~ . ~ _ ~ _ ~ _ _ ~ _ ~ meridion~1 distance between suede and Alexandria. Since one-hItietb of the Circumference corresponds to J:000 studio, obvionsl~ the Whole rircU~Terence is 30 x 5~000 strata Or 230~000 static. To make the length of ~ degree come out in round numbers the result gas taken as 252~000 stat or 700 attain to He degree. It is not certain that those mere Attic studio of 183 meters each. However, with any re~son~tle value of the stadiums Fr~tostbenes, result is closer to the truth tan the one cited LY Eli stotle Rnd slthoUgb methods of oBserv~tioD mere improve] in ., course of fume, the simple priDriple Use] ty Er~tostbenes regained the Stogie] nntM the eipUteentb century of OUI elan Ruben the necessity arose for disLblgUishing between ~ spberic~1 ~] ~ s~beroida1 ebb. Abe Greek geographer; Str~bo; ago wrote at the Beginning of OUP elan and the Gr~ero-Egypti~ astronomer Cladding Ptolemy; hobo live] in the second census> of one elan both give the circumference of the ebb
SBrAPE AND SIZE OF THE EARTH 125 as 180,000 staclia. Willis figure cl.oes not appear in. either case to represent an independent deter~nina.tion; its chic-i' interest for us is the use made of it by Columbus. The Arabs of the ~.~;n.th. century are reported to have ~llarl.e a deter~ni.na- t.ion of the size of' the earth. by direct measurement manacle ;~ level country over two degrees of latit.uLle. Being ignorant of the exact Forlorn equiva- lents of the units userl, we cannot judge the accuracy of their work, but it was probably greater than the accuracy attained by lTra.tosthenes. They had fairly accurate astrolabes and the meridional distances are spill to have been Pleasured with cords. Before col~siderin~, ally modern determi.natiolls it may be well to say sol~lethi~l`, concerning, the cloctri~le of the sphericit;y of' the earth. It is sonnetizes said that the Glen ol' the Middle Ages believed the earth to be flat and that Columbus proved it to be rouncl. No doubt ignorant need of' the Middle Ages, and of other times as well, believed the earth to be flat, but the idea of' the sphericitsr of' the earth never died out amoral the learned. It its true that during the early centuries of the Christian cra there were those among, the fathers o-E the Church who, basing their objections on a lit.era.1 interpretation of scriptural la.n~,ua`~,e, denier1 the sphericity of the earth. Put there were always some among them who put the matter aside as not involving, any vital Christian doctrine and still others who franl~ly accepted the view, handed down from classical times, that the earth is spherica.1. At least the doctrine of a spherical earth was latent alive by the attacks upon it, and gradually it gained ground. In the thirteenth and l'ourteenth centuries it may be considered as the generally accepted doctrine among, the lear~ecl.3 For example, the idea of a spherical earth is a fundamental part of the scheme of' Dante's Divine Comedy. Columbus, nearly two centuries later than Dante, tool: the idea of a spherical earth for granted. Columbus' chief' point was that, given the sphericitv of the earth, and assuming as its most probable circum-i'erence the small figures of' Ptolemy and Strabo, and assuming, further the great eastward extent of' Nsia--whi.ch, from the accounts of land travelers, it was easy to exaggerate, so that Asia would wrap itself pretty well around this small world of' ours then it was clear that the distance westward from western Europe to eastern Asia could not be so very great and that Asia could readily be reached by sailing westward. After the time of Columbus we next note the world of Snell, who is also letdown for his discovery of the sine law of refraction. Calling himself' the " Batavian ~,ra.tosthenes," he published in 1617 at Leyden the results of' the length of' a cle~,ree obtained by a then novel methocl. Instead of obtaining his distance along a meridian by estimate or by direct measure
126 FIGURE OF THE EARTH ment as his preclecesso~s hacl done, he laid out a chairs of triangles between Alkma.ar in North Holland and Bergen-op-%oom near the present Belgian frontier, and measured their angles; he also measured a comparatively short base from which all lengths in the triangulation could be determined. The principles usecl. were in no way different from those of modern geo- cletic operations. The length of a cle¢,ree obtained by him was, however, too small by mole than three pet cent. The work of Picard in France, begun in 166O, was also ba.secl ore the method of triangulation and was notable for two other things: it was the first geodetic work done with telescope and reticule; and its results were used by Newton' in his publish.erl calculation to prove that terrestrial gravity controls the motions of: the moon.* Picard was followed by the Cassinis who gradually extended his arc and improved upon his methods. With the acceptance of the Copernican theory and the development of mechanics at the hands of Newton, Euyghens and their contemporaries, there appeared reasons for thinking that the earth is cot spherical but approximately an oblate spheroid, i. e., flattened at the poles. This would make the length of a degree Increase from the equator to the poles,! but from the work of Picard and Jacques fin.~sini the length of a degree having its mid-point in latitude 45° 4l' ~ ~ . . . was b?,097 toises and that of a degree with its mid-point in latitude 49° 56' was only 56,960 toises,; a decrease instead of: ax increase. To us there is nothing surprising in this, as the combined deflections of the vertical necessary at the four astronomical stations two for each arc- even after allowing for the normal increase in length, is only some 10", quantity that we now know to be often found in practice. More- ove~, the errors both in triangulation and in the astronomical determina- tions were by no means negligible in comparison with this discrepancy. These fact were realized to a certain extent even then, but not so strongly as today, and of course the physical reasons given by Newton and others * Newton probably used Picard's de-termination merely as the latest and. best. It has been said that Newton was not satisfied of the validity of his conclusion until Picard's work was available. This seems to misrepresent the development of Newton's thought. See reference :No. 9 at end of chapter. tIt should be noted that the length of a, degree of latitude on the elliptical meridian of a spheroidal earth means the length of a degree of astronomical lati- tude and is therefore governed, not by the direction of the radius vector, but by the direction of the normal. The normal changes direction most rapidly within a given distance along the meridian where the meridional curvature is greatest, that is, at the equator of an ablate spheroid, and it is here that the length of one degree is shortest. ~ A toise was composed of six French feet, or a little less than 6 feet 5 inches of English or American measure.
SHAPE AND SIZE OF THE EARTH 127 for the flattening did not then carry nearly so much weight as they do now, so that for a time there was a lively' controversy between the earth- flatteners and the earth-elon~ators. The major axis and the ellipticity of the earth, supposed to be an ellip- soid of revolution, may be determined from the lengths of a degree of latitude in two different latitudes and it is fairly obvious that the greater the difference in latitude of the mid-points of the two degrees the less will the values of the major axis and the ellipticity be a.ffectecl by the inevitable errors of observation and by the errors of the underlying assumption that the earth is an ellipsoid of revolution, these last-named errors being the deflections of the vertical under another namers With this in view and in order to settle the dispute between the earth-flatteners and the earth-elon~,ators, the Royal Academy of Sciences of Paris or~,~.n- ized two geodetic expeclitions, one to Lapla.nd arid the other to " Peru.": The n~riv in TJ~nl~nd ~ndn.r the o,nn~rn1 Blare Of 1\/~n~rh~i.~ T.o ~ Ar · ~ ~ · c~ ~ ~ ~ · r. r. ~ lVlonuler, anti as representing ~weclen, cels.lus, professor ot ast.ronom: at Upsala, ~neasurecl in the years 17'3(~-~737 an are of a? mi-r~utes in 1~id- latitude 66° 20' along the river Tornea, which empties. into the Gulf' of Bothnia. From this they concluded that the length of one degree at this point is b7,437.9 tosses, a value which we nowl~now to be rather inac- curate, but since this ler~',th is greater than the lengths previously found by Picarcl and Cassini in Glance, it was co~.~clucled that the earth is flattened at the poles. The party bound :t'or " Peru " set out ahead of the Laplan~l party in 37la.y, 1735; it was composed of three members of' the Paris Academy, Bouguer, Godin and de la Conclamine and their assistants. Two Spanish officers representing the Spanish government joined-it in Colombia. On the way to the scene of their future operations the party made some studies of' refraction and determined the position of the Equator, chiseling in the rocks an appropriate inscription. It was not until June of 173G that the entire party was together in Quito. Base measurements, trian~u- lation and astro~:~on~ical world were not completed until 1743. The total amplitude of the are was 3° 0?' with its micl-point in latitude 1° 3l' S. The length of' a degree as derived by different members of the party varied :t'rom bC,7'46 toises to 56,768 toises. Even the greater :figure is less than the length of a degree in France and still less than the 1 ~1 1 .. . .. .. . . .. . * The arc is always called the Peruvian arc and the bar with which bases were measured is called the " toise of Peru," but the territory within which operations were conducted is part of the present Ecuador, then-a portion of the Spanish province of Peru The " t.oise of Peru," being a more definite standard than other units then in current use. was used as the standard for much of the geodetic worl; down to the time of the introduction of the meter. and even later. 9
128 FIGURE OF THE EARTH length of' a de'~,ree in La.pland, so that the question of whether the earth is flattened or elongated at the poles could be considered as definitely settled by observation in favor of' the flattening,, but the amount of the flattening was far from being accurately determined and so remained i'or a long time thereafter.* Other meridional arcs were measured; among, them may be mentioned that of La.caille in South Africa in 1762, the seco~cl arc in ~no~lern times outside of Europe, and that of Mason and Dixon in America in 176S, the third arc outside of' Europe. The line used for the latter was not the east-and-west line now forming, the boundary between ~IarvlanUl and Pennsylvania and ordinarily thought of as Mason and Dixon's lines, but a line containing as its principal part the line running nearly north arid south and forming the boundary between Maryland and Delawarc. Although this line was chosen by the Royal Society at the instance of' Mason and Dixon because of its supposed freedom from deflections o-t' the vertical, it was evidently the immediate surroundings that were ;I1 the Society's mind. It seems to have been assumed that the effect of' more clista.nt topography such as the Alleghenies and the Atlantic could be neglected. But when the question was examinecl by Ca.vendish it was found that these geo~,raphic features might have a very perceptible effect There was no question of isostatic compensation The name was not vet invented and the thintr itself' was hardly suspected. To be sure ., , ~ ~ 1.__ ~ 1 ~ ~ ._ 1 ~ _ ~ 1 __.; ~ 1~ ~ '.; ~ ~ .- ~ 4~ an; ~ A 1~ A, 1 ~ou~uel~ in comparing, ~ravlty at sea-leve1 wlln gravely al ~UIIo Ll~t(l found that the 10,000 feet of matter between sea-level and Quito seemed to have strangely little edect on ~,ravity there, but there had been loo systematic study of the effect of topography, although the subject of local attraction was much in the minds of' ¢,eodesists and the mathematical knowledge of the day was quite adequate for the necessary computations. It might be supposed that the explanation of this was a lack of knowledge of' the topography and a lacl; of scientific institutions capable of the necessary organized effort to malice the computation; yet nearly a. huncl.recl years later Pratt, with only a rather vague l~nowledt,-e o-t' the topo<,raph:' of the Himalayas and central Asia and working, sint,-le-handed, was able to bring I'orward convincing, evidence that t.opo~raphic features should not be talker at their face Value, that is, that there is compensation, encl. probably some sort of isostatic con~pensation. The state of' mind of' geodesists of the time in question, and for a generation later, may be illustrated by a passage -i'rom Delambre's Grancleur et figure de la Terre. The book is a mine of' information on the history of ei~,hteenth-century * By combination of the Lapland arc with talc arc between Paris and Amiens, the flattening comes out as 1/178;; by talking the Peruvian arc with the Lapland arc, the flattening collies out 1/213. The French and PeIUN ion arcs giN e 1/304.
SHAPE AND SIZE OF THE EARTH 129 <,eodesy. Delambre died in 1827 but the latest geodetic world passed in review was published in 1808. On page 199 Delambre says: It was at first believed that this degree (Mason and Dixon's) could not have been changed by local attractions. But Cavendish after examining the question more carefully found that the excess attractions of the Allegheny mountains on one side and the deficient attraction of the Atlantic Ocean on the other might hare shortened this degree by 60 or 100 toises. He likewise found that; these two causes might have appreciably affected the Italian degree and that of the Cape of Good Hope. In our :F'ranco-Spa,nish arc the greater attraction of the continent must have attracted the plumb line to the northward at D!unl~irk as well as in Spain; these two effects must have to a great extent canceled one another. All this is quite possible but rather vague; and the most certain conclusion is that it, is not well to rely on anything. There are few degrees that are not liable to greater uncertain- ties than has hitherto been believed. We may assume the average degree to be 57.000 toises in round n~mbeis arid change the others in aGeordance with a, flat- tenint, of 1/308, Lehigh today seethes; to havoc the support of sure eyors and ~, astronomers. Cavendish~s work has not been published in full. Some account of it is given in The Scientific Papers of the Honourable Henry Cavendish, Cambridge University Press, vol. I, p. 401-407 (1921~. The number of meridional arcs measured now began to increase rather rapidly; geodetic work was begun in England in 1787 and in India in :1790, but it seems hardly worth while to go into details. Methods were of course gradually improvecl. In geodesy, as in astronomy, the work of Bessel in Germany introduced standards of accuracy comparable with those of the present time. The introduction of the method of least squares marked a great advance; it substituted a rational method for the arbi- trary procedures hitherto adopted in reconciling inevitable discordances and it increased to some extent the real accuracy of the final ~ esults. It provided a procedure for determining, the dimensions of the spheroid best adapted to represent as nearly as might be all the observations under dis- cussion, whether they included many arcs or few. Bessel's ellipsoicl published in lS41 and the Clarke ellipsoids of 1806 and 1880, all of which have been widely used in geodetic world, were derived bar the method of least squares. Latitude determinations combined with measurements of meridional arcs give only the meridional component of the deflection of the vertical. Theoretically the prime-vertical component might be determined from a comparison of the astronomic azimuth with the geodetic azimuth, and the resulting information used to improve our knowledge of the figure of the earth. In practice, however, the geodetic azimuth is liable to such an accumulation of error as the length of a chain of triangulation in- crec~ses, Plot little confidence call be placed in the pritne-:-ertical con~-
130 FI GURE OF THE EAR TH ponent of the deflection obtained from a comparison of astronomic and geodetic azimuths unless some control for the latter is provided. This control comes from the possibility of the astronomic determination of differences of longitude. Before the invention of the electric tele- graph, longitude determinations by the transportation of chronometers or by lunar methods were so laborious and inaccurate as to be unsatis- factory for geodetic purposes. The invention of the electric telegraph introduced the possibility of obtaining deflections in the prime-vertical comparable in accuracy with those in the meridian. There are two ways in which the advantages of the telegraph may be utilized to give the prime-vertical component. First, by means of Laplace stations ~ see Chapter X)T) the ~,eocletic azimuths may be controlled and the undue accumulation of error prevented; the prime-vertical component may then, as previously stated, be obtained from a comparison of <,eocletic and astronomic azimuths, the formula being prime-vertical component of cleflection= (GAG-ally cot ¢. Here a denotes azimuth, the subscripts A and G serving to distinguish the astronomic from the geodetic azimuth, and ~ is the la.titucle. The prime-vertical component may also be determined directly from a comparison of the longitudes, the formula bein`, prime-vertical component of' cleflection= (AA-AG) COS ¢, A denoting west longitude, and the subscripts A and G and the symbol having the same meanings as before. In mountainous regions clift'erence in longitude may be determined by recording the local times of the same event, such as the flash of' a prearranged explosion on a high peak visible from both points. This was done in a few cases, and differences of longitude were also determined by the transport of chronometers. Eclipses of the moon and the phe- nomena of Jupiter's satellites provide signals visible over a much greater extent of the earth's surface than an explosion of gunpowder on a mountain peal<, hut the phenomena in question are so ill-clefined as to be nearly useless for marking a definite instant of time. The method involving the flash o-i' an explosion is available only ill the neighborhood of mountains, where local deflections are likely to be large. There is another method for determining differences of longitude, which requires for its effective use the existence of two intervisible points with a considerable difference of longitude between them, a condition again usually realized only in mountainous regions. The astronomical azimuths are determined at both places and the astronomical difference in longitude is determined frown the cli-tt'erence in azimuth by Dalty's theorem, on which is based the ordinary routine computation of geodetic
SHAPE AND SIZE OF THE EARTH 131 ~.~osit.ions from triangulation. The deflection in the prime vertical is cletermined by a comparison of the astronomic difference in longitude with the geodetic differences, nibs IS The method gives poor results near the Equator. E;~,hteen geodetic ares were discussed by Airy ~ but the four arcs of parallel were not used in his final determination, and some of the meridional arcs were rejected for one reason or another. Bessel ~ nude a calculation of the figure of the earth that is still the standard for many countries and with about the same material at his disposal as Airy he again rejected the available arcs of parallel. In 1858, in the Account of: the Principal Triangulation of Cleat Britain a.ncl Ireland, Clarke lab uses longitudes, though not telegraphic ones, and azimuths in deriving a spheroid fitting as closely as possible the region covered by the triangu- lation-he expressly repudiates the idea that he is here determining the figure of the earth and also derives a figure of the earth as a whole from meridional arcs only, but later in his Geodesy * ii~ he mentions tele- ~,raphic longitudes and uses them in deriving a figure of the earth-the Clarke spheroid of 1880. Under the auspices of the International Geocletic Association, Ilelmert, Borsch and Krtiger 5 computed the deflections of the vertical in Europe and the partial derivatives of these deflections with regard to the major axis and the eccentricity of the ellipsoid. These partial derivatives could be used in determining the dimensions of the ellipsoid that would reduce the deflections in the region under discussion to a minimum ~ and hence this ellipsoid might be said to be the one that best represents the figure of the earth for the particular region under discussion. The deflections of this or that arc or this or that region have been discussed, but no ellipsoid derived from the deflections of the whole of Europe has become either well ltnown or generally accepted, much less an ellipsoid derived from European deflections and deflections outside Europe. The only ellipsoid since Clarke's time based on deflections both in the meridian and in the prime vertical that has found any general ac- ceptance is Hayford's. The deflections on which lIayford based his work were confined to the continental United States but since the actually observed deflections were corrected for topography and isostatic compen- sation (see; page 108), the spheroid that best fits these corrected deflec.- tions in the United States probably approximates closely to the spheroid that best fits. the earth as a whole. * P. 3~4. ~ The actual criterion is that the sum of the squares of the deflections-or per- ha.ps the sum of the squares multiplied by certain preassigned weights-shall be . . ;1 mlmmum. .
132 FI CURE OF THE EAR TH With this idea in mind the Section of Geodesy of the International Geodetic and Geophysical Union in 1924 adopted the Hayford ellipsoid of 1909 as the basis of the International Ellipsoid of Reference. Heis- kanen ~ b has shown that European observations of the deflection of the vertical, when isost.atically reduced are in excellent agreement with IJayford's dimensions. Some dimensions of the International Ellipsoicl of Reference are: Equatorial radius (semi-major axis).. Polar radius (semi-minor axis)....... Quadrant of meridian... Quadrant of equator.... Ellipticit',r (flattening) . ...... 6,378,388' meters ...... 6~356,912 meters ...... 10,002,288 meters ..... 10,019,148 meters . . 1/297.00 = 0.00336700 ~.. ... Square of eccentricity of me~i~li:~n ellipse 0.00672267 Deflections are found by assuming, the latitude and longitude of some point, usually a point near the center o:t the t.ria.n~ulation where the deflections are to be clet.ermined, and the azimuth of ~ line of the triangu- lation radiating, from the point. These quantities together with the dimensions of the ellipsoid used constitute a Geodetic dating. For instance, the latitude and longitude of the triangulation point Meades Ranch in Kansas, the azimuth of the line from Wades Ranch to the triangulation point Waldo, and the dimensions of Clarke's spheroid of 1866 taken. to- ~ether constitute the North American Datum, which is the basis of the geodetic positions in the United States, Canada and Mexico. In theory it would be necessary to go back to the fundamental geodetic observations in the field and express each observation by means of an observation equation with correction terms depending, on the still undetermined elements of the new geodetic datum. Such a fundamental process would, however, entail considerable labor. In actual practice the best. ellipsoid -for a giver region, or the so-called -fi¢,ure of the earth, has beets determ:inerl by a less rigorous method. A change in the dimensions of the assumed spheroid changes the compu- tation of the triangulation ab Ritz, and hence the length and azimuth of a geodesic line connectin, two points of the triangulation. In the less rigorous method, however, these changes are ignored. A central point is taken (as O. see Figure 1 i and geodesic lines OA, OR, OC, etc., are drawn to points of the triangulation at which astronomic observations have been made and deflections of the vertical are therefore available. The length and azimuth of OA, OB, etc., are computed and it is assumed that these lengths and azimuths will remain invariable when the lines are transferred to a new ellipsoid. On this assumption the change in the deflections due to Mien changes in the ellipsoid may readily be computecl. By a least-squares adjustment it is possible to determine the changes in the ellipsoid that will reduce the suns of the squares of the deflections
SHAPE AND SIZE OF THE EARTH 133 to a minimum. This is by definition the best ellipsoid of reference for a given region or for the earth as a whole (that is, for all the regions considerecl), as the case may be. The difficulty with this method of adjustment, is that the deflection obtained at a given point depends on the route followed to reach that B / I l l F ~ H \ \ A/ /1 \ \ \ E FIG. 1. - - point, that is, the deflection at ~ -i'ound by going clirectly from O to A is clid'erent -'roan that IGUnCI by going, via the lines OB and BA, and diff'er- ent also front that found by goin, via OC, CB and BA, and so for each of tile other points reached by the radial lines OA, OB, OC, etc.~9 sob The comparison of the framework of' the triangulation to a network of wires may make the matter clearer. This network is necessarily curved to fit the curvature of' the spheroid :t'or which it was originally designed.
134 FIGURE OF THE EARTH To fit a spheroid of different curvature it must. be bent. If we have merely to bend the ra.dia1 wires OA, OB, OC, etc., each independently, allowing each to preserve its original angle at O with each of the others, the matter is very simple. But if there are cross wires from A to B. i'rom :S to C, etc., to say nothing of other cross wires, like BD or ATI, then it is clear that the bending to a different curvature requires that these cross ties AB, BC, BD, AH, etc., be changed in length, either by stretch- ing, contraction or crun~plin~, and that our simple method of adjusting, to the new spl~eroicl leaves sometl~i~,- to lie desired on the side of' mathe- matical rigor. The areas hitherto cl.iscusser1 by the method of lines radiating I'rom a central point have not been so large as to introduce serious errors by this method of approxima.tior~. But if' grid where the problem arises of' dis- cussing triangulation covering all ol' North America, or Eurasia with perhaps Africa adcled, it will evidently be desirable to introduce further refinements. The practical problen~ will then be to find a method simpler than Helmert's exact method and snore exact than the central point method. Perhaps progress may be made in this direction along the lines ]'ollowed in the readjustment of the western half' of' the United States.0 DETERMINATIONS OF THE FIGURE OF THE1 EARTH BY OBSERVATIONS OF GRAVITY Gravity observations have hitherto been used almost entirely to deter- mine the ellipt.icity (flattening,) of' the earth as a whole. Theoretically there is the possibility of' determining also the linear dimensions of the terrestrial ellipsoid by gravity observations alone, but the results of such a determination would be extremely unreliable, even if observations of gravity over the earth were much more numerous and more accurate than they actually are, and so this theoretical possibility may be at once dismissed frown further consideration. It appears from the theory of' the apparent attraction of a rot.atill3 spheroid of revolution that gravity on an equipotential or level surface enclosing all attracting matter of' the spheroid may be written ap- proximately g=ge(~+b sins ¢). (1) Here ~ is ~t,ra.vity in latitude +, go gravity at the Equator, an.cl b is coefficient depending on the distribution of mass within the spheroid arid its angular velocity o-t' rotation. The quantities q~ and ~ may be * To determine ~ length such as one of the linear dimensions of the earth, we must in the final analysis measure ~ length. In the present instance the length would be the length of ~ certain pendulum. From this measured length the re- quired dimensions of the earth would be found by calculation.
SHAPE AND SIZE OF THE EARTH 135 determined by observation in various latitudes. Of course on account of local anomalies in gravity the various observed values of gravity will not all fit the same 'formula but a least-squares adjustment may be made if' desired. When values of go and b have been adopted the value of the flattening, f, may be found by the formula .6 Lea f= ~ q--b (2) where ~ is the equatorial radius of the earth, assumed to be otherwise determined, (I the angular velocity of the earth's rotation, so that Lea is the ce~tril'uga1 force at. the Equator, and where q~ and ~ are the same as previously defined. This i'ormula is known as Clairaut's * is equation from its discoverer. Clairaut supposed the earth to consist of a~ ellip- soidal nucleus covered with fluid. Some later commentators supposecl. that he assumed the earth to be entirely fluid the limiting case as the nucleus shrinks down to nothing and from the approximate agreement of' the observed values of ~,ravit.y with equations (~) and (2) they drew conclusions in favor o:t' the fluidity of' the earth's interior. Their view o:t' Clairaut's work was, however, incorrect and it was found later by Stokes''5 that be could dispense entirely with hypotheses about the interior of the earth and still prove Clairaut's theorem. The coefficient b in the preceding I'ormulas is a measure of the increase in gravity from the Equator to the poles and Clairaut's theorem says in effect that I'or a given centrifugal force a greater change in <,ravity from Equator to poles means a decreased flattening. This may appear paradoxical and in fact Newton himself' seems to have been in error on a matter o-I' similar nature. The explanation of the paradox is, first, that we assume the angular velocity of' rotation to be giver and., seconcl., that we are tallying, about gravity on a surface of' equilibrium, a surface maintained by a suitable arrangement of' densities. Suppose, for example,. the surface of equilibrium to be spherical in spite of the centrifugal force due to the rotation. Then from equation (2), b will be as large as it possibly can be, unless indeed we suppose f negative, that is, the spheroid prolate instead of' oblate. To maintain such a spherical surface of' equilib- rium against the effects of' rotation there must be such an internal arrange- ment of matter as to chive an excess of purely gravitational attraction in ~ 1 * Alexis Claude Clairaut (or Clairault), 1713-1765, whose great mathematical abilities showed themselves so early that he was admitted to the French Academy when he was below the legal age. The theorem is given in his Theorie de la figure de la Terre (1743), page 249. Clairaut evidently had the idea of a potential func- tion in mind, though he gave it no special name. He uses explicitly the term level surface " Surface courbe de Niveau," and shows that the distance of two level surfaces very close together is inversely proportional to the intensity of gravity. c'
13 ~ FI GURE OF THE EAR TI-I high. latitudes, and it is this arrangement of matter that Haloes ~,ravity increase towards the poles, that is, that makes b large. Hi' course a nearly spherical spheroid of' revolution is not necessarily an exact ellipsoid, but to the degree of approximation adopted in equa- tions (~) and (2), no distinction is made between them, nor can we make any valid distinction between the different kinds of latitude, such as geographic, geocentric, and reduced latitude. If we Lisle to malice such a distinction we must add certain small terms to both equations ancl specify the kind of latitude used in the formula for gravity. If I, tl:~e Flute;, tl~e coelTicicnt 1' <~r~1 tile quantity a be considered as small quantities of the first order, the acl.~le~l terms are of the second. orcler. Sunder present limitations of' the accuracy o:l' observations, terns off tl~e third and hi~,her orders are entirely ne¢,li3ible. When seconcl-or~ler terms are includecl and the ~eo<,raphic latitude is used, the formula for gravity at the surface o1:' a spheroid of revolution may be written q=g~+bsin ~- 44 sin''2~.* (3) The quantity be its of' the second orcler. Its value clepe:nds on Better the sphero;.cl its an exact ellipsoid or not. For an exact ellipsoid and 4 S ~ ~ ~ ~ -f ~ ~ 4 19 2 96 f 11- qua f ~ ~ Equation ~ ~) is Cla.iraut's :formula extended to terns of tilde seco~1 orcler. 1i' we assign in advance a value to the small quantity b4, the elevation of the spheroid implied by this assi~ ment above an exact ellipsoid of revolution having tl-~e same polar and equatorial axes is approxi~natel:,~ 2a4 (f-+2fb-IBM. (6) The maximum radial difference between spheroid and ellipsoid is found in latitucl.e 43°. Following the example of' Hel~nert, the value of 4t is o:t'ten taken as 0.000007'. This is based on the conditions of equilibrium * The subscript 4 and the divisor 4 in the coefficient of sin 2¢ occur merely be- cause Helmert'3 notation (Hohere Geodasie, vol. II, chap. II) is followed in order that those interested In the derivation may follow Helmert's work more readily, except that Helmert uses characters in German type instead of the corresponding ones in Latin type, used here for convenience.
SHAPE AND SIZE OF THE E'ARTII 137' for a fluid sphel~o;~1 having the size, mean density and angular velocity of the earth. The exact value depends on the assumption as to the distri- bution of density but Darwin it and Wiechert 20 making quite different assumptions found very nearly the value of b4 above cited. By using, this value ~ it is found that the corresponding spheroid is depressed in lati- tude 4~° some 3 meters below an exact ellipsoid of revolution having the same axes as the ellipsoid. For an exact ellipsoid t44 =0.000006. If we start out with assumed values of by and b. Else value o-! f is ~,;ven by .' ·',~a 610 ~ ·,i2~,: ~ 17 i,,': b' f=~ 1~- ~-~-~ ---- 21 ~ - 2~ 1~.] . (I) This equation may he used to field ~ atom letdown or assumed values of I Gild be; in tile square 1~r~ GLets the value of ~ ;l1 tile small terms ~ 4 am 26~ may be tallest HIS simply 2 --f Although thee principle of isostatic equilibrium has now found general acceptance, this floes not mean that the form of the actual Void is that of a spheroid of revolution. The actual <,eoid, according to this principle, would rise above the spheroid over the continents and would be depressed. below the spheroid over the deeper parts of the ocean. The departures from the spheroid would be at most a few tens of meters. One of the objects of geodesy is to determine the actual form of the geoid. Another object, which might be said to be geological or geophysical rather than strictly geodetic, is to test the principle of isostasy both as to its general validity and as to the validity of some particular method of computing the compensation. Both these objects may he served by the discussion of gravity observations. Here we shall treat chiefly of the first, although the second is incidentally involved. Suppose we have a mass of matter, self-attracting according to the Newtonian law and rotatin, with uniform velocity about an axis, and suppose we consider the level or equipotential surfaces resulting from the combined self-attraction and centrifugal force of rotation and in particular some one of the level surfaces that is exterior to all attracting matter. If we know exactly the form of this level surface and the value of gravity at a definite point either on it or outside of it, then from these facts alone we can deduce unequivocally the value of S,rasTity at any other * The exact values of f and b do not greatly matter here, but they were taken in this computation as 1,7297 and 0.00528D, values consistent with one another and with the assumed value of b`.
138 FIGURE OF THE EARTH point either on the surface or outside of' it. This theorem is due to Stokes. Thus if we assume the earth to be an exact ellipsoid of revolution and determine the value at any point on the surface, say at the equator, then we can write down formulas for gravity not only for points on the ellip- soida.1 surface but for all points outside of it.24 We Ludlow also that, although we may determine the level surfaces due to a given distribution of matter, we cannot reverse the process and deduce from the form of the level surfaces the distribution of matter within the inmost of' the level surfaces that surrounds all the attracting matter with which we are deal- ing. We know indeed that the matter witl~in this inmost surface may be rearranged ifs an infinite number of ways each o:t' which will give precisely the same family of' enveloping, level surl'aces and the same values of~rav- ity. There may be physical objections to some of the proposed arrange- ments, but at present ale are re:l'erri~:~3 only to the mathematical I'orn~ula- tion of the problem. Is the converse of Stokes' theorem true, that is, can we deduce the form of the level surl'ace I'ron~ a knowledge of the values of gravity on it? We cannot answer for the roost general case of the problem, but if the level surface in questions is nearly spherical, then the answer is " yes." From a knowledge of' the values of:' gravity, we clan find the borne of' the surface and the methods of' confutation are again due to Stokes, who -,ave two of them. One method involves the use of spherical harmonics, which are so convenient in the problems involving the attraction of' nea.rlv spherical loonies. Suppose us, Us, qli to represent surface spherical harmonics, that its, functions of the latitude and longitude of' certain special forms, the degree of the harmonic being, implicated by the subscript, and suppose that the gravity anomaly fig at the surface is expressible in the form fig= G(u~+u3+~ . . . ~ (8) where G is ~ mean value of gravity.* Stokes shows why the harmonics of' cle,ree zero and unity ma.;- be omitted. The anomalies imply ~ standard gravity I'ormula. and therefore an ellipticitv. Let V denote the elevation of' the geoid above the spheroid of' this ellipticity; the value of 1V is given by N=a(u.~+ ~23 +73~4 (9) where the HE'S denote the shine spherical harmonics as before and a denotes mean radius of the earth. In order to apply this method the values * Just how the mean is taken is not important. The~degree of approximation given by this formula and it is amply sufficient for present needs- does not enable any valid distinction to be made between conceivable various mean values.
SHAPE AND SIZE OF THE EARTH 139 of gravity must be known in all parts of the globe at enough points to make equation (8) an adequate expression of the gravity anomaly, which in practice would be necessarily smoothed out and generalized by this method of' representation. The method of spherical harmonics, although convenient in theoretical cliscussions, is apt in practice to lead to numerical computations of in- ordinate length and complexity Stokes devised his second method to meet this difficulty; he based the proof on a transformation of the spheri- cal harmonic expressions in equations (~) and (9), but the same result has been reached by Pizetti'''2 and others without the use oft' spherical harmonics. This alternative method of proof avoids certain troublesome mathematical difficulties that Stokes more or less passed over in silence. In whatever way it may be proved, this second method of calculation would be extremely simple in application. When Stokes' Th,eo reran is referred to without any qualifying explanation, it is usually this formula and the accompanying method of calculation that are meant. The phrase " would be simple " its used because Stokes' Theorems hats never been a1~- l~lied except to sets of data that were necessarily largely hypothetical? because no gravity data were available on the high seas. Now that Veiling Meinesz hats shown how to determine gravity aboard a submarine and the various maritime nations are beginning to apply Lois methods, the prospect seems bright that within a few years I'rom now, but more than eighty years after Stokes enunciated his theorem, the data will be available :L'or a preliminary application of the method and we shall have a Hooch iclea of the departures of the geold I'rom a perfect ellipsoid, not only over the land but over the oceans also. The practical application of Stokes' Theorem resembles the process of correcting, either gravity anomalies or the deflections of the vertical :T'or topography and isostat.ic compensation. The place at which the eleva- tion of' the geoid above the spheroid (negative elevations are clepressions) is sought is the center of' a series of' concentric circles extending to the Tripodal point of' the earth. Within each of the zones bounded by these circles the Verge ~:a,lue of the gravity a~o~al,y must be determinecl,* and this average value must be multiplied b.y the average value of a certain f'unetion of the radii of the zone and the width of the zone in radians and the results for each zone be added together. The result is the elevation of the geoid above the spheroid at the pole of the zones. *The anomaly is dependent on the formula. for theoretical gravity and the latter is dependent on the spheroid implied by the formula. Hence the elevations and depressions are with respect to this spheroid. A constant error in the values of gravity, or what is practically the same thing. an error in the value of gravity at the equator, go or in gravity at latitude 45°, if the formula is written in terms of this latter quantity, is without effect on the results.
140 FIGURE OF THE EARTH Art example entirely hypothetical will make the process clearer. The formulas for the various functions are given for completeness, and in several forms in view of practical applications. No especial geomet- ri.cal or mechanical meaning can be given to the several terms, am more than a similar meaning can be given to the various terms in the formulas for the attraction of various bodies. The terms are simply the result of integration and other mathematical transformations. Let ~ denote in general the angular distance of a point from the center of the system of circles, that is, for a point where the elevation or depression of' the Void is sought. With subscripts 2 or 1 it may denote respectively the outer or inner radius of the zone under con- sideration. In its most general forte Stokes' Theorem may be written N= i~gf(~)dS DIG (~9a) where dS is an element of' solid angle, fig is the gravity anomaly for the element concerned., the integration covering the entire sphere, and feel =~4cosec lb+!-6 sin -by-~ cos -3 cos .b loge Sin :b ~ ~ + sin A) ~ t. ~10 ~ In practice the integration will be effected by mechanical quadrature. IT we specialize the element of solid angle dS as sin (dads, where is an azin~uthal angle, we have 1\~= ~ day l~(7f(~) Sill Fd§ 27rG ~0 (it) If we uIlcle-lstalld b;y A~1lt1 the mean value of' /`y around ~ circle of:' a~,ula adius §, the -l'ormula may be written 1V= G tj ~mqf(~) sin fd§. (~12) Suppose we divicle the ir~ter~-al from O to ~ into zones, of' which a typical one has the radii 4, and §,, and let /`'mg denote the mean value of the anomaly over the entire zone, and If' sin 522 the mean value between the limits H.' and §, of fact sin §. Then since id practice we substitute summation for integration, we write ~7\' = G ~ /` rug [f ( ~ ) sin H] ~ ( ~ ~-§1 ) (13) The summation covers all zones from §=0 to the antipodal point where §=7r. The quantity b'-§, is to be expressed in radians. The mean value in question is given by ~~f(~) sir1 ~ = h i ~ ~ fib) sill (cay it'
SHAPE AND SIZE OF THE EARTH 141 If, disre~,ardiIl~, constallts of inte~,ration we write if(~) sil1bd§=~(~) we find for d>(~) the expression 'A i b) = ~; 4 sir1 2 + 2 sin' ~ -(; sift ;: - 4 sill 1 ~ -Silly blow |sin ~ (~+Sill 6 ~ ~ ~ (ill) ~2 `~.~.~.`l As ~ wo~lL;n~, for no Viol Stoli~es"Theo:ren~ that. is I:articula~ly- co~ve~ie~.t Beheld tile solves are wicle, we get 1;-- G `'~'~ d)(b ~-~(~1.) The summation extends groin .§ = 0 to ~ = Or. The -following, table gives an idea o:L how fact, ft §) sin §, and ~) fiery. It is given at closer ;~te~vals near the center of the zones on account o:t the panics variations there of tl~e first two -fU~letiO~lS. V.~1 I,ES OF THE FUNCI'1ONS USED IN STOKES THEOREM ~f (I) f (I) bits ~'t, (I) ~f (I)f (I) Sit, ~'i, ~ d) 0° GO 1.000 0.0000 80° - 1.10- 1.082 +0.0138 2° 32.6 1.139: + 0.0378 90° - 0.911- 0.914 - 0 -1626 4° 17.2 1.200 + 0.0787 100° - 0.63- 0.623 - 0.2981 6° 11.7 1.227 + 0.1212 110° - 0.0291- 0.270 - 0.3714 S° S.84 1.230 +0.1641 120° + 0.09+0.077 - 0.392S 10° 6.99 1.215 + 0.2068 130° + 0.417+ 0.358 - 0.3534 20° 27.5 +0.941 +0.3989 140° +0.82+0.526 - 0.2745 30° 0.95 a- 0.474 + 0.5241 150° + 1.12+ 0.559 - 0.17~8 40° - 0.08 - 0.054 + 0.5608 160l° + 1.35+0.461 - 0.0873 50° - 0.70 - 0.537 + 0.5078 170° + 1.49+ 0.259 - 0.0230 60° - 1.03 - 0.896 + 0.3806 180l° + 1.540.0100 0.0000 70° - 1.15 - 1.082 + 0.2053 Let us world out an entirely hypothetical example, merely to illustrate the use of the tables ancl formulas. Divide the entire globe into zones <30° wicle centering, at tile place -here lie wiser to Ludlow the elevatior~ of the ¢,eoicl arid assuage the lol]ow.i~, values of of,,, for the several zones: Rail; i ot a',,, ~ zone cm/see' 0 ° _30 ° 0.000 30° 60° + O 050 60°-90° + 0.0~60 90 °-120 ° + 0.030 20°-150° - 0.010 00°-180° - 0.040 Tl~; r ty den ecs = =/(; i: Clips = 0. ~23(; = ~onstal:~ t value of §,-As, c~=(;....3,'1 x 10'; metes, G=981 em.isec.2
142 PI CURE OF THE EAR TH (~9 alla G must be in the same unit, which, however, need not be the linear unit as for a; the unit for a will be the unit for 7V). Tapir the constant value of (~-b~) outside the summation in (13) we have N= 6 3 ~ i x i0 x 0..6236~/i'~,,g if (~) sin b j1= 3400~!` myrf (~. Let us use the simple trapezoidal rule to find the mean values for fall sin b. Then we have the following, using, for uniformity only the 10° interval even in the first zone. Since fig is zero for this zone, the error is of no consequence here. In other cases a more accurate evaluation is desirable, since near the origin fall sin ~ changes rapidly. zone Mean valises of f(~) sin ~ 30 .; [ ~ (1.000) + 1.215 + 0.941 + ~ (0.474) ~ = + 0.964 30° - 60° I- _(0.474) - 0.0~04 - 0 537 - --(0.896)] = - 0.267 60 _ 90 1~ [ _ -1- (0-896) - 1-082 - 1.082 - 4- (0.914) ~ = - 1.023 !)0° - 120° 1` [ _- (0.914) - 0.623 - 0.~:'0 + -in (0.077) ] = - 0.437 120 - 1.50 -~1 +3,-(0-O:~) +iO.~08+0-,)26+ (0 S59) 1 +0-401 150° - 180° -.3 [ +- (0-~69) +'0 461 + 0259 + -1- (0.000) ] = + 0.333 +0.964X - 0.267 X 0.000 = 0.000 + 0.050 = - 0.013 - 1.023 X T 0.060 = - 0.061 - 0.437 X +10.030 = - 0.013 + 0.401 X + 0.010 = + 0.004 + 0.333 X ( - 0.040) = - 0.013 Sum= - 0.096 Sum X 3400 = N = - 326 Inetei s, or the geoid is 326 meters below the spheroid corresponding to the gravity formula on which the gravity anomalies are supposed to be based. If we use the function Ah), which leads to a little simpler :form of computation, we get: G = 64~30 .~=6490~ ~n1~) ~] = 6490 ~0.000 X ~ 0.,0241 - 0.000 ~ + 0.0,30 ~ 0.~806 - 0.,5241 ~ + 0.060; ~-0.1626 - 0.3806) + 0.030 ~-0.3928 + 0.1626) + 0.010 ~-0.~78 + 0.3928) - 0.040 (0.0()0 8- 0.1778) l = 6490 x ~-0.0~16) = - 33o meters. The slight discrepancy is due to the fact that the process of taking mean values of fib) sin d in the first method is not particularly accurate for such wide zones. The second method is here better and slightly easier.
SHAPE AND SIZE OF THE EARTH 143 The values of ~/q7',7 are not such as would be given by a properly ad- usted gravity formula; their average value, when the areas of the zones to which they apply is consiclered, is not zero, as it should be, but 0.0308. Pv subtractin, this irons the >,;ve~ anomalies Ale get for the various zones-0.0308, + 0.0192, + 0.029?, - ().0()063, - ().208 a-cl-0.0708. If we apply these values in tl~e second met]~o~l. we get Y = 6400 ~~ - 0.0308 x (). ~241 + 0.0192 x ~-0.:1435 ) + 0.0292 x ( -0.~432? -0.0008 x ( -0.2302) -().0208 x 0.2150 -0.07()8 x 0.177 8] = - :~:~.~ Dieters, as before. There is another fault to lie found with the ave~a~,e ~,r.a~ity anomalies as Mien. If we consider the anomalies as thy;. sumac leavers of positive or negative netter with thicknesses p:ro~'ortio~.~al to the anomaly, then the center of gravity of this surface shell o-t matter should coincide with the center of the sphere over which the shell is spread. This would be true of actual gravity anomalies with re:terence to the earth, treated (with suffi- cient approximation) as a sphere, but it is not true of these hypothetical anomalies. This defect could be corrected but, just as in the case of the correction to reduce the average anomaly to zero, this second correction would leave the value of N unchanged. The simplest. type of ideal distri- bution of gravity anomalies that is correct from the start is one that is s\m~netrical (as to average value) Title respect to tile equator for which the center of the zone distend and its antipode are the poles. LUNAR ~II',T'HO])S OF DETERMINING THE FLATTENING OF TH]3 EARTH * Thee e are two lunar telethons of determining the flattening of the earth: the first, a mixed geometrical arm dynamical method dependent on the parallax of the moon and going back in principle to Newton's -fan~ous calculation that identified terrestrial ¢,ra.vitatio~ with the force that retains the moon in its orbit; the second method, purely dynamical and depenclen.t on the lunar perturbations. This seco:~cl method might lie subdi.vicled. ;~:~t.o several methods according, to tile particular pertur- bation usecl. Newt.or1 calculated what the parallax of the moon would be if terrestrial ¢,ravita.tion, dim;~;shin~, inversely as the square of the distance, were the controlling, force, and on comparing this with the observed parallax he found them to a<,ree within the uncertainties clue to' errors of ob- servation.T The so-called observed parallax, lo-ever, that is, the equa *Professor E. W. Brown gave valuable assistance in the preparation of this portion of the text. He is not, however, to be considered responsible either for the statements made or for the opinions expressed. ~ There are terms involving also the mass of the moon and the effect of the sun which require no more than mention in this connection. 10
I~14 FIGURE OF THE EARTH torial horizo~ta1 parallax. r eall\ involves an assun~pti.on as to the ellipticity of the earth, acid still other. assumptions as to the local deflections of the vertical and as to the elevations of the ~eoicl above the spheroid at the observatory ies where the parallax is cletermirred. The usual and almost i.nevitahle assun~l:,tion l~;therto has been that these deflections ancl elevations are `~11 zero. The calculation bv which ter restrial gravitation is extencle`1 out to the moon and the parallax o L the latter thus louncl is also a-tTe~te~l l.'v the assumed flatte.~in~, of the earth, though not so much as the obserwecl parallax. The flattening is found by assuming, di.~.-~:ere~t \!alues for it until the calculated equatorial horizontal parallax comes out equal to the obse~vecl parallax. This happens for a flatte::~i.r~ equal to -1/~.'3~3.4, according, to do Sitter,~4;1 on the assu~-nptio~. that the ~leflectio~s o:t the vertical at Greenwich and Cape o:t Good TIope observatories are zero. To make the two parallaxes equal for a flatten;, of 1/2'37 would requite a deflect;.o~. of about 11" at both observatories. A calculation of these deflections on the theory of isostasy, with uniform compensation down to a depth of' 120.D lam., gave deflections of' 1"16 at Greenwich and 3",, at the Cape. These ha:1 the proper signs to shift the reciprocal of' the flattening -l'rom 293. L ~1 ~ A JO 1 1 ~ towarcts ;~J s IOU: are not. large enough to change it all the wa,v.~iC They could be increased somewhat by adopting, some other clist.ributio~-~ of' isostatic compensation, but the most probable explanation of much of' the discrepancy in the flattening is observational error in the deter~ina- tion of' the parallax. De Sitter concludes that the parallax calculated on the theory o:t' gravitation is more reliable than the parallax direct!:! observed and the parallax used in Brown's new lunar tables is likewise derived 'row theory. These conclusions of' de Sitter and letdown and the uncertainty due to id orance of' the actual deflections o:t' the vertical are thus unt'avorable to the accuracy of' the flattening, obtained from it. The purer,\ cl,nI~>lmical lu.n.lr methods do Mot cleterm-ine tile flattell:in<`r l.i.rectlv but ;~ste`~.l. tl~ev `'let.crm:i:~e the qu..~titV .i .. .. J C--~-( 1+]~) Eat where A, i? anal C are the pr'incipa,1 moments of' inertia of the earth Hi ascencli~lo order ol' ~na~itucle, E is the mass 0:; the earth and a, its equa ~ ~ , , ~ · ~ ~ ~ ~ · ~ · r ~ ~ 1 , 1 I L' I 1_ tor~a.1 radius. Lams quallt,]ty J occurs in the developments in series of tne earths gravitational potential. It' the earth were composed of homo ,eneous concentric spherical shells, J would evidently be zero. Groin J the flattening,, f, is deduced by a simple I'ormula. *The deflections and elevations are understood to refer to the unknown ideal ellipsoid that fits the earth as ~ whole as closely as possible.
SHAPE AND SIZE OF THE EARTH 145 The fact that the earth is not spherical, that is, that J is not zero, causes a number of perturbations in the moon's motion, both periodic and secular. From the observed values of these perturbations the value of J can be deduced and hence the flatt.enin`,. The periodic perturbations due to the fi.~,ure ~ of the earth have been used to determine the flattening by various astronomers, I'rom Laplace down to the present. These periodic perturbations are, however, rather difficult to disentangle by observation from the perturbations due to other causes and hence the flattening deduced Tom them is not entitled to much weight. The principal secular perturbations due to the earth's figure are those of the node and perigee of the lunar orbit. These, being cumulative with the lapse of time, can be determined very accurately. Uni'ortuna.tely, however, for our present purpose much the larger part of these perturba- tions is due directly or inclirectly- to the action of' the sun and this must be determined and subtracted from the observed effect in order to determine the portion due to the figure of' the earth.! Tl~ere is also a portion, minute but by no means negli,,ible in the present problem, due to the figure of' the ~noon. Fortunately the portion due to the sun depends mainly on the ratio of the month to the year, a quantity known with great precision, so that apart from possible but not very probable errors in the theory ol' the smaller terns representing the sun's effect 4; and the un- certainty due to the figure of the moon, the perturbation due solely to the figure of the earth may be considered as having almost the same observational error as the entire observed secular perturbation, whether of perigee or node. In connection with the preparation of new lunar tables the flattening was determined by Brown,7 who found 1/293..~ from the combined results for the perigee and node. His adopted result 1/294.0 comes from includ- i~g his value 1/294.4 determined from the Greenwich-Cape of Good Hope parallax observations. By changing the assumed data for the figure of' the swoon do Sitter 34a found 1/296. Tl~e data for the figure of the moon depend partly on observation o:t' the lunar librations, partly on * The word Azure is used in the extended sense of figure and constitution, more specifically its figure and constitution as they affect the value of J. t Of the total annual motion of 146,435" in the perigee, and 69,679" in the node, all but about six or seven seconds of each are due to the action of the sun. 1: The formulas for this occupy many lines of the quarto page of Delaunays " Lunar Theory." Even so, they are not sufficiently accurate for the present pur- pose and the more modern theory, such as Brown's, does not use a complete and explicit formula, but is based on a. process of numerical approximations. Delaunay's formulas, however, bring. out the fact that the solar effect depends essentially on quantities known to ~ high degree of precision.
146 FIGURE OF THE EAR TI-I assumptions as to the way in which the density of the moor varies from center to surface. The process of deducing the flattening, from lunar perturbations is thus seen to be far from simple. The possibilities for reconciling the flattening of 1/294 :lound by this method with the 1/297 of the inter- nat.ional ellipsoid lie: 1) in changes in the secular variations deduced from observation; 2) in data for the Inure of the moon; ancl 3) in possible corrections to a very complex mathematical theorem. Jones has used occultations at the Cape o:l Good I]ope to correct the motions of the perigee and node, and his corrections tend to diminish the disagreement between Brown's value of the flattening and the new international value. The older lunar observati.o~.s, however, are not so well represented and it is questionable whether the full amount of his corrections can be accepted without further consideration. The figure of the moon, as de Sitter has shown, gives one way of reconciling, partially at least, the two values of the flattening,. Because the moon is smaller and cooler than the earth, its figure might depart relatively more from hydrostatic equilibrium than the figure of the earth can, and there is evidence to show that it does. In the present state of our knowledge, or ignorance, a wide range of suppositions is admissible. In the absence of direct evidence on this point the disagreement between Brown's value of the flattening and the value obtained by other methods might be used as evidence to show what the figure of the moon actually is. The theory of these secular variations, like the lunar theory in general, is exceedin<,ly long and intricate, but it has been so much worked over that it is probably now correct and must be assumed to be so, at least until some one with the necessary ability, energy and inclination discusses the matter further and finds an error. There seems to be room, however, for some slight improvement in a much simpler theoretical matter, the calculation of the flattening,, I, from J. Astronomers, as a rule, have omitted certain second-order geodetic terms in the equation connectin, these two quantities.* The inclusion of these terms increases the re * The approximate relation often used is 3 1 ma f 2 JO 2 ge (for meaning of w", a, and go see paragraphs dealing with Clairaut's formula). This corresponds in accuracy to the simpler form of Clairaut's equation. The relation correct to terms off the second order is ~_ .T ~ -An. -as.'= 15 m.,7 ~9 '7' 2 ' ' 2 cola where m is written for- g . r) 6 2 S s
SHAPE AND SIZE OF THE EARTH 147 ciprocal of the ellipticity by 0.4 or 0.5, or perhaps more, the exact amount depending on the exact form of the approximate equation between J and f, which may be stated in more than one way when correct to the first order only. These second-order terms would thus help to diminish the discrepancy still further. The only conclusion that can safely be drawn at the present time is that the discrepancy between the value of the flattening from lunar observations and the new international value is probably not quite so large as it at first appeared to be, since various considerations all working in the same direction tend to bring the two values together. The ideal and the problem of astronomers and geodesists is, of course, complete reconciliation of the flattening as determined by geodetic methods with the values determined by all the various astronomical methods. We can also get a hold on the flattening of the earth by studying another mechanical effect also due to the moon and in lesser degree to the sun and planets also but of somewhat different character from the perturbations of the moon's orbital motion just considered. This is the effect of the moon and the sun and the planets on the direction in space of the earth's axis of rotation, in short the precession of the equinoxes. The moon, the sun and the planets,: but chiefly the moon, exert a torque on the earth's equatorial protuberance tending to bring flee terrestrial equator into coinciclence with their own orbital planes, that is, approximately with the ecliptic. But since the earth is rotating rapidly its equator never is brought into coincidence with the plane of the ecliptic. Instead, the pole of the equator describes about the pole of the ecliptic a small circle of approximately constant radius equal to the obliquity of the ecliptic.! The pole o:t the equator completes its circuit about the pole of the ecliptic, which retains a nearly constant (Erection in space, in a period of about 26,000 years. Obviously, the greater the equatorial protuberance of the earth, that is, the greater the flattening, the greater will lie the torque exerted on the earth and the faster will be the motion o1: the pole of: the equator about the pole of the ecliptic, so that by working, Awards :from the observed rate of pre- cession the flatte~-~ir~, o:t the earth may be cletermined, subject to certain assumptions. * The pull of the planets on the earth's equatorial bulge is very minute. The so-called planetary precession is the effect of the planets on the position of the plane of the earth's orbit. ~ The obliquity of the ecliptic is not constant but its variations are slow and relatively small and a consideration of their causes and periods would lead us far afield from our subject.
118 FIO~ 0F T~ ~ In the lunar perturbations the quantity determined is-0 <~+ ) (for notation see page 144~; to pass from ~bicb to Battening no ~ssu~p- tion Shout the interior of the e~rtb is necessary. From the precesaion the quantity determined ie 1~+ ), from Ibid we cannot pass to the Battening without making ~ssnmptions. Let US assume abut the density distribution vitbin the earth is con- sistent with bydrost~tir equilibrium, any since me know isost~Uc ~djust- ment to he almost complete, this supposition seems Unlikely to lead to serious error. Theoretically; in order to derive ~ flattening I~om the precession, me must assumes not merely b~rost~tic equilibrium in gen- er~l, bat ~ JeRnite 1~v of V8liRtiOD from center to surface. Ibe eject of cb~nging the 1~v is, hoverers surprisingly smelt in fact, Afloat negligible. Any reasonably plausible law of density any some laws not so pl~nsitle-will serve. It was at one hme suppose] that the agreement of the Battening de- rived Idiom the observed precession and the assumption of I;~pl~re? (otherwise Legendre\) 1~v of density Aim the Battening derived in other Alas was ~r argument in Indoor of the [~pl~Ge-Legendre 1~; then Inch in vane; hut when other laws were tang ~ was found that the agreement was just as good. Abe explanation was given by H~d~u ~ and mount to this: ~ certain m~tbem~tic~1 expression on which the Battening de- pOD4S ~D] whim in turn depends OD the 1~v of denser is in form v~ri~tle ~] in Act Jocs vary, but so Mighty that, Then the density is restricted to Abut is pllysi rally 1jossible, the Battening deduce] Irma the precession is practically independent of the 1~v of density, if hydro- st~tic equiLbrium prevails within the earth. Y6ronnet derive] Irom the precession ~ Battening of 1/29~.12~0.38. Here We ~ indicates; not probable erroT in the ~di~ry senses but iDe estimated range of UDce~taint~ fine to -r ignorance of the 1~ of Jensi#, no ~llov~Dce being made Ior the Uncertainty Of the observed value of the constant of precession nor of other quantities involved. [e Sitter 14~ 14c Kiter ~ c~reIu1 JiSCUSaion finds even narrower Watts. Ibis result is 1/296.92~0.IS6 or in ~ later revision 1/29~.96. Most of this small provable error he finds to be dUe to the Uncertainty in the mass of the moon, We Uncertainty due to oar ignorance of the law of density Being hut ~ comp~r~t~ely small parts Aid Ho U~rert~inty JUe to the probable error in the precession~1 coust~Dt being negligible. The elUpticit~ OF ~HtteniDg of the ebb seems then to be Wont 1/297, whatever method of determination may be nseJ. .. . . . .
S~ ]\~ ~Z~ OF T~ ~T49 I. Any, C. B. Figure of the eurtb. Article in Encyclopedia ~e~opol~n~, [on- don, 18~, vol. 3. (An Decedent account of the Bole subject, including its m~tbem~tio~1 aspects, with much bistoric~1 info~ion.) 2. Aristotle. He cello. Eng. trains. by J. L. Bomb Oxford, 1~. (See especially Book II.) 3. Be~zley, C. R. The din of modern geography. 3 vole London, 18g?-1~5. (An interesting add v~uablo Coopt of the prowess of ieogr~bic kooky edge, including ~ dis~s~on of speculation about the figure of the fib.) 4. Beset F. W. B~sEmmun~ tier Axed des elEpti~cben Botudo~bEroids, Cubes den vorbundenen ~essungen am fisted entspricht. Astron. Cache vOl. 14, no. 333: 333 (1887); or, Abh~ndl. (ea. by R. En~el~unn) Leipzig, 3: 41 (1876) . . l}eber einen Fehler in tier fr~zO~iscb~ Or~dme~ng and seinen FinOuss Of die Beshmmuni der Hour der Erde. Ahb~ndL, 3: SS. (Con- t~ins the dimensions of the ~pberoid gener~Uy known us BesseLs) S. BGrscb, As and KrUger, L. (also F. R. Helmert for Heft 1). Lot~eichungen Repts. to Internal Oeod. Assoc. Reran, 1 (1~6); 2 (1002j. On Genie, Ha. Coodetic o~er~ion~ in Abe United Stoutest Washington, 1027 (Ce S Coat Ceod. Rev Spec. PuhL ~^ 134). 7. Brown, F. W. The determination of the oon~nts of the node, Me iicEn~tion, the e~rtb's eUiptic~y Ad the Inequity of the eoLptic from the Oreen- wicb meridian observations of the moon, 1847-1901. Roy Astron. Boas Ho. Notices, 74: S~ (1914). 8e Wilds A De ~ bigotry of the determine of the figure of the eurIh from arc measurements. Worcester (amp, 19~. 9. Cujort P. Charter in On Sir Isaac Kenton, 1727-192~ bicenten~^ ev~ln~- tion of his Folk, ad. ~ F. F. Br~scb. Baltimore, 1928. lOe Club~nt, Ae ThEprie de 1~ figure de 1~ Beige. Maria 1743. lie Clarke Ae R. Ceodesy. Oxford, 1380. (A good genera treatise, ~1 standard in spite of its dam.) - . Ordnance tri~onometricu1 survey of Ct. Britain and Irelund ~ccou~ of observ~i~ and cation of the princip~I tri~gul~on ~nJ of the fire, dim~<ons, and mean ~>eciEc Amity derive] Ceremony. London, 1~8. (Compiled under the dkechon of Lt. Colonel H. James) 12. Darwin, C. H. Abe theory of Me hgurc of the earn curried to the second order of small quantities. Roy. Astron. Soc., Ho. Notices, 50: ~ (1900) or, ScL Furlers C~m~d~e (Fng.), 3: 78 (1910). 13. Lelumbre, J. B. J. Histoire de l'ustronomie uDcienne. 3 vols Paris, 1S17. (Lseful for the early history of determinuti~s of the azure of the eurtb.) . Fd. by Bigourd~, C. Orundeur et hgure de 1~ Terry. Perk. 1912. (Published nearly ~ century aft outbox de~tbJ 14. de Sitter, W. On the Aeon ladies of the earth, the intensity of gravity, and the ~oon's Purdue. G. Akin. Wetted Amsterdam, Proo. 17:12~ (1013). . On the Fattening and the ConshtutioD of the earth. G. Eked. Weten- scbu~pen Amsterdam, Proc. 27: 244 ( 1924) . . For ~ rig r~76 ~6 8~6~y ~r~6, ~ Astron. Inst. ~ether- lunds, BulL v. 4, no. 1~: 60 (1927).
130 FIGURE OF THE EARTH 15. Fiske, John The discovery of America. Boston, 1892, vol. 1. (Contains an account of the clevelopr~ent of geographical knowledge of the prevailing ideas about the figure of the earth.) 16. Hayford, J. F. Supplementary investigation in 1909 of the figure of the earth and isostasy. Washington, U. S. Coast Geod. Surv., 1910, 80 p. (Not to be confused with work published in 1909, entitled The figure of the earth and isostasy from measurements in the United States.) 17. Heiskanen, W. Untersuchungen uber Schwerkraft und Isostasie. Veroffent- lich. des finnischen Geodatischen Institutes, no. 4 (19;24)- To be sum plemented by . irber die E,rddimensionen. Vierteljahrsschr. Astron. Gesellschaft, 61. Jahrgang, p. 215, and . Ist die Erde ein dreiachsiges Ellipsoid? Gerlands Beitr. Geophysil<., 19:356 (lD28). 18. Helmert, F. R. Die mathematischen und physikalischen Theorieen der hoheren Geodasie. 2 vols. Leipzig, Teubuer, 1880-84. (Still a standard work.) For deflections of the vertical see 9. Hunter, J. G. The earth's axes and triangulation. Survey of Indi<~, Prof. Paper No. 18 (19~18). 20. Lambert, W. D. The figure of the earth and-the new International Ellipsoid of Reference. Science, (i3: 242 (1926). -. The effect of variations in the assumed figure of the earth on the mapping of a large area. Washington, 1924. (U. S. Coast Geod. Surv., Spec. Publ. No. 100.) . The figure of the earth and the parallax of the moon. Astron. J., ~rol. 38, no. 908 :181 ( 1928) . 21. Muller, Q. Ges;chichte der Gradmessungen bis zur peruanische Gradmessung. (Doctor's thesis) Rostock, 1871. (Convenient historical summary.) 22. Pizetti, P. Sopra il calcolo teorico delle devia%ione del Geoide dall' Ellissoide. Atti Acc~d. sci. Torino, 46: 331 (1911). 23. Radau, R. Sur les lois des densites ;~ l'interieur de la Terre. Compt. rend., 100: 972 (1885); o' ir~ ~absta~ce in Tisserand's Traite de mecanique celeste, Paris, 2: 221 ( 1891 ) . 24. Rudzki, M. P. Physik der Erde. Leipzig, 1911. (A very convenient book. Contains the forn~ulas (p. 46) for gravity outside an exact ellipsoid of revolution. On p. 62, in the calculation of the rate of change of ~'r~vity with elev:~tion, the factor 1.0006821 has been used instead of 1.00682;1. By using tl~e correct value, the forrn;ul;1 for l becomes -0.000003088 dr~ ~ 0.000000004 sin B. which is consistent with the formulas commonly used.) 25. Stokes, G. G. On the variation of gravity at the surface of the earth. Cam- bridge Phil. Soc. Trans., 8:672 (1849); or, (Stokes') Mathematical and Physical Papers, Cambridge (En<,r.), 1883, p. 13. 26. Wiechert, E. Ueber die Massenvertheilung im Innern der Erde. Nachrichten kg. Ges. Wiss. Gottingen. Math. Phys. K1., 1897, p. 221.
