Physics of the Earth - II The Figure of the Earth Bulletin of the National Research Council (1931) / Chapter Skim
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Chapter IX. The Shape and Size of the Earth
Chapter IX. The Shape and Size of the Earth
Pages 123-150
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From page 123... ...
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.
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From page 124... ...
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.
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From page 125... ...
Being ignorant of the exact Forlorn equivalents 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.
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From page 126... ...
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.
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From page 127... ...
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 inaccurate, 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.
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From page 128... ...
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.
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From page 129... ...
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.
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From page 130... ...
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.
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From page 131... ...
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.
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From page 132... ...
Some dimensions of the International Ellipsoicl of Reference are: Equatorial radius (semi-major axis)
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From page 133... ...
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)
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From page 134... ...
134 FIGURE OF THE EARTH To fit a spheroid of different curvature it must. be bent.
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From page 135... ...
, 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)
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From page 136... ...
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.
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From page 137... ...
By using, this value ~ it is found that the corresponding spheroid is depressed in latitude 4~° some 3 meters below an exact ellipsoid of revolution having the same axes as the ellipsoid. For an exact ellipsoid t44 =0.000006.
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From page 138... ...
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 ellipsoida.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 dealing. 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~ravity.
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From page 139... ...
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.
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From page 140... ...
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.
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From page 141... ...
SHAPE AND SIZE OF THE EARTH 141 If, disre~,ardiIl~, constallts of inte~,ration we write if(~)
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From page 142... ...
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)
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From page 143... ...
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.
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From page 144... ...
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~ination of' the parallax. De Sitter concludes that the parallax calculated on the theory o:t' gravitation is more reliable than the parallax direct!
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From page 145... ...
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 uncertainty 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.
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From page 146... ...
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.
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From page 147... ...
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.
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From page 148... ...
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.
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From page 149... ...
1912. (Published nearly ~ century aft outbox de~tbJ 14.
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From page 150... ...
R Die mathematischen und physikalischen Theorieen der hoheren Geodasie.
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