Dt2 r3 tr2

and, hence, that gravitation must vary inversely with the square of the distance.

The solar vortex provided the basic mechanism for the system, but Leibniz did not manage to develop a theory free of inconsistencies. For example, the fluid matter that made up the vortex was supposed to carry the planet in the transradial direction but not to interfere at all with the radial motion. In order to explain the radial motion, Leibniz introduced the concept of an ether that was distinct from the coarser matter that made up the vortex; this ether moved about in all directions, much as in Huygens' account of terrestrial gravity. Huygens, however, did not see the need for the harmonic fluid vortex that Leibniz had introduced; he realized that all that was needed to explain Kepler's laws was a satisfactory mechanism that led to inverse square attraction. Thus, he wrote to Leibniz:

It is clear that the gravity ofthe planets being supposed in the reciprocal proportion ofthe squares of their distances from the Sun, this, together with the centrifugal force gives the eccentric ellipses of Kepler. But how, in substituting your harmonic circulation, and retaining the same proportion ofthe gravities, you deduce the same ellipses, it is this that I have never understood... since the said proportion ofthe gravities, with the centrifugal force, alone produce the Keplerian ellipses according to the demonstration of Mr Newton.

Of course, the harmonic vortex was superfluous, but Leibniz could not see that the circulation according to his harmonic law (i.e. the component of velocity perpendicular to the radius decreasing in proportion to distance from the Sun) followed from the law of attraction.

Quoted from Aiton (1972).

There were other problems. Comets did not fit easily into Leibniz's framework and, perhaps more significantly, the fact that different planets obeyed Kepler's third law did not fit in with the harmonic circulation. Leibniz got around the latter problem by hypothesizing that the harmonic law existed only in narrow bands around each planet, but that the fluid in the large spaces between the planets did not rotate according to the same law. While this saved the phenomena, it was hardly convincing, and Newton found it easy to find fault with Leibniz's approach. Newton also accused Leibniz of making errors in his mathematical analysis, but in most cases these criticisms were misplaced.

Another factor that increased French opposition to Newtonian ideas was the measurement of arcs of the surface of the Earth by Jacques Cassini (son of G. D. Cassini). If the Earth were an oblate spheroid, the length of 1° of latitude (corresponding to a change of 1° in the altitude of the celestial pole) would increase as one moved from the equator to the pole, but Cassini's measurements suggested that the Earth bulged at the poles rather than at the equator. Many people were happy to believe this result, since it cast doubt over Newton's theory, but it was by then an observed fact that gravity decreased toward the equator, and this did not appear to fit in with Cassini's data. As a result, the staunch defender of Cartesianism, J. J. d'Ortous de Mairan, invented an ad hoc law of attraction that (at least qualitatively) reproduced this behaviour.97

In a treatise on the shape of celestial bodies in 1732, Pierre-Louis-Moreau de Maupertius, one of the few supporters of Newton in France, compared the theories of Descartes and Newton and found in favour of the latter. However, his calculations on the shape of the Earth produced a very different value from that of Newton, and he concluded that in order to settle the question, more accurate measurements were needed. Accordingly, two groups set out to measure degrees of latitude and longitude; one travelled to Peru (the northern part, now Ecuador) in 1735, and another, headed by Maupertius himself, went north to Lapland in 1736. Maupertius' group returned in 1737 with data that showed that the degree

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