Nothing has caused philosophers so much trouble as to explain the phenomenon of gravity. Du Châtelet continues that, according to the principle of sufficient reason, everything must have a reason, or a cause, so the phenomenon of gravity, too. Heaviness, or gravity, is the effect of collision with surrounding matter. Now, this matter is not itself heavy; for if it was, it would have to have recourse to other matter with which it would collide, and so on to infinity, and therefore this objection, based as it is on the general heaviness of matter, cannot stand (InstPhy, § 340). Consequently, if matter is the cause of gravity, it is impossible that it itself is heavy. But how should a matter that is not itself heavy be measurable and empirical verifiable? The assumption that matter is dead, passive, and motionless leads to a similar dilemma. If matter is dead, passive, and motionless it cannot cause motion. If matter is animated with a living force, we avoid the assumption of a prime mover, but have to explain how a living force arises from a dead force.

Du Châtelet gives no answer to these questions, but changes the perspective by focusing on Newton’s great achievements. From observations it was obvious that there must be some form of attraction between the earth and the moon, and the sun and the planets that caused them to orbit around the Sun. Yet, it was not at all clear that the same force of attraction could be responsible for the behavior of falling bodies near the surface of the earth. Newton demonstrated that the law of universal gravitation explains both the motion of a falling body (Galileo’s law) and that of the planets (Kepler’s laws of planetary motion) (InstPhy, § 344). Further, Newton showed that the centrifugal force is an inertial force directed away from the axis of rotation, and he clarified its relation to the centripetal force. In accordance with the action-reaction principle, a body in curved motion exerts an equal and opposite force on the other body.

Du Châtelet briefly explains the planetary motion and Kepler’s laws, referring to her much more detailed explanation in her book Exposition abrégée du système du monde (InstPhy, § 351). In this book, says Du Châtelet, Newton’s explanation of the tides as a result of the earth and sun gravitational pull will be examined as well as comet trajectories.

In § 354, Du Châtelet discusses Newton’s theorem of revolving orbits. Newton derived this theorem in Propositions 43–45 of Book I of his Principia and applied it to understanding the overall rotation of orbits. Planets revolving the Sun follow elliptical (oval) orbits that rotate gradually over time (apsidal precession). Newton extended his theorem to arbitrary central forces by assuming that the particle moved in nearly circular orbit (InstPhy, § 354):

Mr. Newton, à force de sagacité & de calcul , a démontré dans le corolliitt prémier de la proposition 45. de son prémier Livre, que lorsqu’une Planète se meut autour d’un centre mobile dans un orbe fort approchant du cercle (tel que l’Orbe que décrit la Lune autour de la terre), ou peut déterminer pur le movement de ses apsides en quelle raison la puissance qui lui fait parcourir son orbite agit sur elle, & en appliquant cette proposition au cours de la Lune, il détermina que l’action de la terre sur cette Planète, décroît dans une raison un peu plus grande que la raison doublée des distances. (Amsterdam 1742)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

In Proposition 45, Newton derived the consequences of his theorem of revolving orbits in the limit of nearly circular orbits. This approximation is generally valid for planetary orbits and the orbit of the Moon about the Earth.

Newton’s method involved a comparison of the motions of the moon around the earth and an object falling towards the earth. This would not have been possible without certain measurement results which Newton used as the basis for his calculations, among them the radius of the earth, and the distance from the earth to the moon. Du Châtelet mentions Jean Picard’s commonly accepted value for the size of the Earth (Mesure de la Terre 1671), on which also Newton based his calculations.

The difference between Picard’s value for one degree of meridian arc and that determined by Willebrord van Roijen Snell led Jean-Dominique Cassini (also Giovanni Domenico Cassini) and his son, Jacques Cassini, to hypothesize that the earth is a prolate spheroid. This result, if correct, contradicted computations by Isaac Newton and Christiaan Huygens (InstPhy, § 377). Newton’s theory of gravitation combined with the rotation of the Earth predicted the Earth to be an oblate spheroid. Huygens also proceeding from a rotating earth but not from Newton’s theory but his own modification of Descartes’ vortex theory arrived at the same conclusion. Huygens formula for the centripetal force, which can be found in his posthumously published “De Vi Centrifuga,” played a central role for Newton’s theory of gravitation. In both theories, in Newton’s and Huygen’s theory, the flattening of the poles was a result of the diurnal rotation. Consequently, the force of gravity varied according to ones position on the globe and with it the length of the one-second pendulum.

Du Châtelet describes Huygen’s influence on Newton in detail, using concrete examples. She quotes from “De Vi Centrifuga,” Proposition 6, comparing it with Proposition 4, Cor. 9 of Newton’s Principia (InstPhy, § 358). In the Scholium regarding Cor. 9 of Proposition 4, Newton honored Huygens, who has compared the force of gravity with the centripetal forces of revolving bodies. Referring to that passage, Du Châtelet writes (InstPhy, § 358):

Mais Mr. Huyghens a démontré que chaque courbe dans quelqu’une de ses parties que ce soit , a la même courbure qu’un certain cercle qu’on nomme osculateur; […] cette proposition a beaucoup servi à Mr. Newton: ainsi, c’est Mr. Huyghens que l’on peur dire avoir été le précurseur de Newton, bien plus que Descartes, dont il n’a presque rien emprunté. (Amsterdam 1742)

Mr. Huyghens has shown that each curve in any of its parts has the same curvature as a certain circle called an oscillator; […] this proposition was very useful for Mr. Newton: thus, it is Mr. Huyghens who can be said to have been the precursor of Newton, much more than Descartes, from whom he has borrowed almost nothing. (my own translation)

[ Manuscript Bibliotheque nationale de France (Paris), Fonds français 12265 ]

Whereas Newton and Huygens argued that the Earth is oblate and flattened at the poles, the Cassinis maintained that the Earth is prolate or elongated at the poles. In 1734, the French Academy of Sciences dispatched two geodetic expeditions as recommended by Jacques Cassini. One expedition (1736–37) under Pierre Louis Maupertuis was sent to Torne Valley (near the Earth’s northern pole). The second mission (1735–44) under Pierre Bouguer was sent to what is modern-day Ecuador, near the equator. Their measurements demonstrated an oblate Earth, with a flattening of 1:210. This approximation to the true shape of the Earth became the new reference ellipsoid. This is often seen as a final victory of Newton’s physics over its main rival Cartesian physics. At least, Du Châtelet portrays it that way.

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