From the examination of the first of the above tables it results that, if we adhere to the geodesic operations alone, the number 36 expresses by how much the terrestrial globe differs from the spherical form, and the numbers 6,377 and 6,358 (kilometers) give the dimensions of the greater and less or equatorial and polar radii, an approximation which might, in all probability, be qualified with a higher degree of exactness by adopting either the fundamental elements given by Bessel, the astronomer of the most enviable reputation of the current century, or those deduced somewhat later, and, of course, from more copious data by the distinguished director of the Observatory of Greenwich, Professor Airy. The

exactness, or, at least, close approximation of the number, is found, moreover, to be confirmed by another class of considerations extraneous, in a certain degree, to geodesy, and very indirectly related to those which served the two celebrated astronomers last mentioned as a basis and guide in their valuable labors of combination and analysis.

It was remarked at the close of the second part of the present article that nothing in nature is fortuitous; and it might well have been added that not only is nothing fortuitous, but there is nothing without a reason for its being as it is, nothing susceptible of being essentially modified without communicating an impression to other organic parts of the complicated mechanism of the universe. The movement by which the moon is carried around the earth does not depend exclusively on the intervening distance or the respective masses of the two bodies, but on the distribution of their masses in concentric groups or on the figure of both globes. If the earth were spherical, the movement of its satel lite would not be that which is always observed; nor if the discrepancy from that simple form had been represented by a fraction differing from 30 would this fact have failed to disclose itself in a degree more or less sensible in some of the accidents which characterize the lunar movement: theory, based upon the laws of universal attraction, laws announced by Newton and so sagaciously developed by Laplace, indicated the orbit which the moon was destined to describe on the hypothesis of the polar depression of our globe being less by than the equatorial radius, and observation promptly confirmed all the conclusions to which the theory had pointed. Few astronomical discoveries reflect more honor on the human intellect than the valuation of the earth's ellipticity based upon the principles which have been just cursorily mentioned.

But it is not necessary to withdraw our eyes from the globe we inhabit to discover other means, besides those which are strictly geodesical, not only of demonstrating the ellipticity of its form, but of verifying the limits within which the eccentricity of that new figure is comprised. Our readers will doubtless readily infer that the process alluded to consists in the use of the pendulum, whose oscillations are more or less rapid in different parts of the earth, by reason of its form being sensibly and essentially different from the spherical. When Laplace announced the relation existing between the movement of the moon and the oblateness of the earth, Clairault, in a special treatise on the subject, had already stated the law of interdependence by which the continuous depression of the globe from the equator to the poles is associated with the variations of gravitation or of the weight of bodies, and consequently with the oscillatory movement of a pendulum on the surface of that globe. By both geometers the task of verifying the truth of their theories was bequeathed to after experiment, and in both cases the previsions of mathematical analysis and the results of observations long and carefully repeated have been found to be perfectly accordant.

In the long period which elapsed from the date when the French academician Richer first noticed the retardation of the pendulum in the equatorial zone, to that when the Spanish admiral, Malespina, undertook his justly celebrated voyage of scientific exploration in 1789, the experiments made with the pendulum were numerous and interesting, in so far as they were directed to the demonstration of the ellipticity of the earth and the accidental irregularities which distinguish it; but those undertaken with a view to determine the value of that ellipticity have been neither so many nor were they so early as the former. In 1826 Bessel showed the inaccuracy or want of care in the process till then followed for deducing from the oscillations or length of a compound pendulum, moving in air and at a variable temperature, the corresponding elements of a simple pendulum, oscillating in a vacuum and in a thermal state of absolute invariability; and, even much later, Humboldt thought that experiments with the pendulum, comparable in delicacy and precision with the

multitude of other, the most common, astronomical or geodesical operations, would scarcely amount in number to sixty. So scanty a result should, in our opinion, be attributed to two quite distinct causes. In the comparative experiments made with the pendulum, there is sought, in the first place, a difference of length or of numbers so small that the least inadvertence in the operation, or a disturbing cause unworthy elsewhere of consideration, will materially influence the result and impair its exactness. And moreover, even when the observations are conducted throughout with all the requisite accuracy-a thing, we repeat, of great difficulty-still the theoretic principle of their combination for deducing the terrestrial ellipticity supposes that the density of our globe, though variable according to an arbitrary law from the surface to the centre, continues identically the same in each layer concentric with the superficial one; an hypothesis which departs in some degree from the reality of nature, and which on that account cannot lead to results of absolute certainty. After these considerations, it will not be a matter of surprise that the values of the terrestrial ellipticity, deduced from experiments made in the present era by Borda first, and afterwards by Biot, chiefly at different points of the French meridian, by Kater in England, by the navigators Freycinet, Duperry, Sabine, Foster and others, under very different and distant latitudes, should sensibly vary from one another, and likewise to some extent the final number, deduced from the examination of all of them, when compared with that which results from the sum of the principal geodesic labors. But to what at most does the difference amount? From the experiments made with the pendulum, there results as the value of the earth's polar compression the number 5, somewhat greater than the fraction 300 and less than 10; the difference of these two extreme fractions is equal to 200; so that the difference of the results obtained by help of the pendulum and by the ordinary processes of geodesy will be found to be represented by a number still less than the last. Admitting, then, that the value of the equatorial radius is in metres 6,377,397, there would remain in the length of the polar radius an uncertainty of 2.362; and this, it must not be forgotten, on the supposition, really more unfavorable than is warranted, that the doubt respecting the polar compression of the earth would present to us as equally uncertain the two fractions and But the relation of the number 2.362 to the value of the equatorial or the polar radius is lower than that of 1 to 1000: thus in the appreciation of a quantity composed of a thousand equal parts, it would be at last doubted whether we had counted one part more or less than was proper! Instead of being surprised at the existence of such an uncertainty as this, it might well cause astonishment, as Prof. Airy has remarked in reference to this subject, that man should have arrived at a knowledge so precise in a matter so difficult and obscure; while there is still room for confidence that further advances are in his power, and adequate encouragement to persist in the pursuit of the truth.

That this confidence and encouragement exist is shown by a simple reference to the projects of new geodesical operations and experiments, suggested by some of the most celebrated of cotemporary astronomers. In 1857, for example, Biot proposed to the Academy of Sciences of Paris that a new determination should be undertaken, by methods and with instruments more delicate than were before known, of the whole extent of the arc of Peru as well as of the various arcs of parallel measured in Europe; that experiments with the pendulum should be multiplied in those localities where considerable anomalies have been noted in the direction of the vertical, or where their existence is suspected, with a view to ascertain their cause or causes; in a word, that no means should be spared of discovering all the accidents of form and density which distinguish the terraqueous globe from the theoretical ellipsoid defined by Bessel and the mathematicians who, with a degree of precision difficult to surpass, have either preceded or followed him in this enterprise. The ideas of Biot, de iberately considered and digested eventually into the colossal project of measuring a new

are of meridian extending from Palermo to the parallel of Cristiania and Upsal, across seas and continents prodigiously diversified, and intermediate to the Russian arc in the cast and that stretching from Formentera to the Shetland isles in the west of Europe, have been zealously seconded by the Prussian general Baeyer, the companion of Bessel in the geodesic operations of Koenigsberg, and distinguished alike for his knowledge and experience. In the memoir relative to this matter, which he published in Berlin in 1861, Baeyer does not ask the protection of goverments, nor invoke the learned of all countries to unite their efforts, for the purpose of ascertaining whether the polar compression of the earth is a hundred-thousandth part greater or less than it is believed to be; he holds, on the contrary, that the geometrical problem is resolved; but the physi cal and geological problem, closely associated with the real figure of the globe, he regards as scarcely yet defined. The idea of Baeyer, which Biot, as we have seen, also cherished, and which equally exercises the thoughts of other savants, would doubtless be realized, if the local influences which embarrass and complicate the geodesical operations, instead of being avoided as heretofore, were purposely sought for and measured; if, wherever practicable, the net-work of triangles were extended around and over the surface of seas and of volcanic regions, and across the valleys and mountain-chains of more abnormal composition; if the instruments for measuring distances and angles were rendered comparable in some sort to the balance of the chemist and the goniometer of the mineralogist; in brief, if, after having defined the external figure of the earth, geodesy should penetrate, as it were with the eyes of induction, into the interior of the globe, in order to reveal to us the origin of that figure, the transformations it has experienced, and the stability, whether little or great, which it possesses for resisting the destructive assaults of time. Considered under this new aspect, the question presents an extraordinary interest, opens to view an indefinite and almost unexplored horizon, and affords one proof more of the close interconnexion which exists among all the natural sciences. Let the project of Baeyer or some analogous one be transferred to the field of practice, and the nineteenth century will have won yet another title to the consideration of the ages to come.