This experiment, the importance of which was foreseen by the Academy of Sciences,* perfectly coincided with the rcasoning of mathematicians, who began to consider the earth as depressed towards the poles; and the cause of the augmentation of gravity, or the attracting force, was explained by the depression of the surface, which therefore approaches nearer to the centre. BOOK II. theory. Huyghens, a Dutch mathematician, had the glory of Hayghens's divining that truth, even before the experiment of the pendulum was known. Considering that bodies which revolve round a centre, or an axis, acquire a centrifugal force, which tends constantly to make them fly off from this centre or axis, as we observe in a stone whirled about in a sling, he concluded, that the fluid diffused over a considerable part of the surface of the earth, could not assume a form perfectly spherical, as it must be affected at the same time by the force of gravity impelling it towards the centre. He supposed, therefore, that the earth must be depressed towards the poles, and that the axis of rotation was shorter than the equatorial diameters by which is equal to about four sea leagues. This consequence, deduced from the centrifugal force by Huyghens, may be made sensible to the eye, by turning rapidly a wet bladder round an axis, which then assumes the form of a spheroid flattened towards the extremities, contiguous to the axis. theory. The immortal Newton, who, by profound reflection on Newton's the laws of the planetary motions ascertained by Kepler, discovered the principle of universal gravitation, no longer considered gravity at the surface of the earth as a constant force, everywhere directed towards the centre of our globe, but as the result of the mutual attractions of all the particles of the earth to each other; he found that this force varied a little in intensity and direction, from the earth not being perfectly spherical. If the figure of the earth depended upon gravity, gravity would regulate itself according to the * Lalande, Abrégé d'Astronomie, art. 742 and 805, II. BOOK figure which the earth had; this accelerating force, as to terrestrial bodies, ought to be perpendicular to the surface, and proportional to the distance; the earth having once assumed the oblate figure, this figure alone, independently of the centrifugal force, ought to render gravity weaker under the equator than under the poles. Newton, calculating on this principle, and supposing the earth homogeneous in all its parts, found, for the quantity of depression, rз, or 10 sea leagues.* Investigations of Maclau raut, &c. Those conclusions, differing as to the quantity of the result, but agreeing with respect to the alteration which the rin, Clai- figure of the earth ought to undergo in consequence of the centrifugal force, have been developed by the most delicate and profound calculations. Of these we can only here point out the results. It has been demonstrated, that the earth could not be a homogeneous mass, but that its density ought to increase in descending to the centre; and that, in all cases, an elliptical figure satisfies the laws of the equilibrium of fluids. Inequality of the degree or the oblate plane. At the same time, the theory of the diminution of gravity towards the equator, was confirmed by a great number of observations on the pendulum, from Lapland to the Cape of Good Hope; and from their general agreement it has been concluded, that the depression of the globe is equal to the 332nd or 336th part of its axis. The theory of the depression of the earth might also be verified by measures taken on the terrestrial globe; for it results from this theory, that the degrees of latitude cannot be equal throughout the whole extent of the meridian, * Newton Principia, b. iii. prop. 19. + Clairaut, Theorie de la Figure de la Terre. Maclaurin, Memoir on the Flux and Reflux of the Sea. D'Alembert, Recherches sur le Système du Monde, &c. Dubourguet, Traité de Navigation, note i. page 290, 291. Histoire de l'Acad. des Sciences, passim. Laplace, Système du Monde, p. 250. Svanberg, Exposition de la Mésure d'un Degré. II. but that they ought to be augmented, in the flattened part BOOK of the meridian, that is, towards the poles, and diminished in the convex part of the same meridian, or near the equator. Those consequences, which flow from the fundamental notions of elementary geometry, were however for some time mistaken by men of very great merit, such as Cassini, and d'Anville. It may not therefore be improper to insist a little on the demonstration. A degree of the meridian is the portion Aa, Fig. 14, of that curve, when the radii CA, Ca, which intercept that part of the arc, from an angle ACa, equal to the 360th of the circle. In consequence of this definition, it is easy to perceive that the radii CA, Ca perpendicular to the tangents, will meet at the same distance in the curve only when this curve is a circle; that to the same arc will correspond the same angle, and that in this case the degrees throughout the whole extent of the curve will be of the same length. But this does not apply to curves whose curvature is not uniform. In the ellipse, for example, if we take two arcs of the same length, as Mm, and Nn, Fig. 15, one in the most concave part, and the other in the flattest part of the figure, the perpendicular Mc, mc, drawn to the extremities of the former arc, will meet nearer in this arc, than the perpendiculars NC, ne drawn to the extremities of the other arc. The angle Ncn is therefore visibly less than the angle MCm, and consequently if the latter be equal to one degree, the arc Nn, equal in length to Mm, does not correspond to a degree. To obtain this angle in the part NP of the curve, it is necessary to take in a space greater than Mm. Therefore the terrestrial degrees must be greater in the flat part of the globe, if we would have them correspond to the celestial degrees which are all equal, not being real arcs, but only angular distances. We may also reason in the following manner:-The point where two verticals meet, is the centre of the terrestrial arc contained between them; if this arc were a right line, the verticals would be parallel, or would meet only BOOK II. Paralogism on this sub ject. at an infinite distance. The greater the curvature of the arc, the more the verticals converge, therefore they meet at a less distance. Thus the part of an ellipse near its great arc being the most curved, the verticals which are perpendicular to it, will meet at a smaller distance. The radius of the arc intercepted between them will be shorter, and consequently, the absolute length of the arc itself will be less. On the contrary, near the small arc, the verticals meet at a greater distance, showing that the radius of the intercepted arc, and the arc itself, are longer. By not tracing those notions to their source, the contrary was concluded at the commencement of the last century, because it was supposed that the degrees were determined by the angles M om, N om, formed by the lines drawn to the centre of the ellipse EPQ p; but this hypothesis was not conformable to the principles of the operation, for the lines OM and O m, ON and On, not being perpendicular to the curve, differ entirely, both in magnitude and direction, from the verticals to which are referred the points of the celestial arc. The measures of Cassini having seemed at first to indicate a diminution of the degrees from south to north, several learned Frenchmen maintained, by means of the paralogism above cited, that this diminution was a proof of the depression at the poles; the mathematicians demonstrated that it was rather proof of the contrary. The error of the principle was at length discovered, and it has not been since revived but by entire strangers to geometry. But Cassini and D'Anville, in deducing from the pretended diminution of the degrees towards the north, the natural conclusion, affirmed that the earth swelled out in its polar direction; or, in other words, that the terrestrial ellipsoid performed its revolution round its major axis; which was contrary to the theory of gravity and the equilibrium of fluids. Bernardin de St. Pierre, Etudes de la Nature, &c. II. In France, the notion prevailed for forty years, that the BOOK earth is a spheroid protracted towards the poles.* At length, the Academy of Sciences resolved to ascertain the truth of Measurethe theoretical conclusions on the subject, and selected from ment cartheir own body two companies of mathematicians, who were ried on in Lapland dispatched, the one in 1736 to Peru, and the other in 1737 and Peru, to the polar circle, to measure a degree of the meridian in the regions bordering on the equator and the pole. The results obtained by these companies, compared with each other, and with the degree measured in France by Picard, though they did not entirely agree with respect to the quantity of the depression of the earth at the poles, yet they completely dissipated all doubts of the fact. The degree measured at the polar circle exceeded that of the equator by 669 toises, or 703 fathoms; and the French degree, though smaller than that of the polar circle, still surpassed that of the equator by 307 toises, or 327 fathoms. The Cassinis themselves, after having verified their measures, had the candour to avow that they had fallen into some slight errors, and that the degrees measured by them in France concurred to prove the depression of the globe towards the poles.‡ other mea sures. It was not enough that the science of mathematicians Different had described, in a general way, the figure of our globe, they further endeavoured to discover the exact quantity of that depression, the existence of which had been proved by so many experiments. But, in this investigation, the accumulation of the materials only increased the difficulty of the question. The degrees successively measured in different parts of the world gave very different quantities for the depression. This was demonstrated with great perspicuity by an Italian mathematician, by comparing the twelve best measures known for half a century back. We shall first Bossut, Hist. des Mathématiques, p. 273. + Bouguer, Figure de la Terre. Maupertuis, Elémens de Geographie, &c. Cassini et de Thury, Meridienne de l'Observatoire royale verifié, 1744. Pauli Frisi, Cosmographia, tom. ii. Chap. de Figura Planet. comp. íd. opera omnia, t. iii. p. 123, 599. VOL. I. 7 |