NEWTON, ISAAC (b. Woolsthorpe, England, 25 December 1642; d. London, England, 20 March 1727), mathematics, dynamics, celestial mechanics, astronomy, optics, natural philosophy.

    Principia. Evidently Newton learned the law of centrifugal force almost a decade before Huygens, who published a similar result in 1673. One early passage of the Waste Book also contains an entry on Newton's theory of conical pendulums.109

According to Newton himself, the “notion of gravitation” came to his mind “as he sat in a contemplative mood,” and “was occasioned by the fall of an apple.”110 He postulated that, since the moon is sixty times as far away from the center of the earth as the apple, by an inverse-square relation it would accordingly have an acceleration of free fall 1/(60)2 = 1/3600 that of the apple. This “moon test” proved the inverse-square law of force which Newton said he “deduced” from combining “Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the Centers of the Orbs”—that is, by Kepler's third law, that R3/T2 = constant, combined with the law of central (centrifugal) force. Clearly if F?V2/R for a force F acting on a body moving with speed V in a circle of radius R (with period T), it follows simply and at once that

F?V2/R = 4π2R2/T2R = 4π2/R2 x (R3/T2).

Since R3/T2 is a constant, F?1/R2.

An account by Whiston states that Newton took an incorrect value for the radius of the earth and so got a poor agreement between theory and observation, “which made Sir Isaac suspect that this Power was partly that of Gravity, and partly that of Cartesius's Vortices,” whereupon “he threw aside the Paper of his Calculation, and went to other Studies.” Pemberton's narration is in agreement as to the poor value taken for the radius of the earth, but omits the reference to Cartesian vortices. Newton himself said (later) only that he made the two calculations and “found them [to] answer pretty nearly.”111 In other words, he calculated the falling of the moon and the falling of a terrestrial object, and found the two to be (only) approximately equal.

A whole tradition has grown up (originated by Adams and Glaisher, and most fully expounded by Cajori)112 that Newton was put off not so much by taking a poor value for the radius of the earth as by his inability then to prove that a sphere made up of uniform concentric shells acts gravitationally on an external point mass as if all its mass were concentrated at its center (proposition 71, book I, book III, of the Principia). No firm evidence has ever been found that would support Cajori's conclusion that the lack of this theorem was responsible for the supposed twenty-year delay in Newton's announcement of his “discovery” of the inverse-square law of gravitation. Nor is there evidence that Newton ever attempted to compute the attraction of a sphere until summer 1685, when he was actually writing the Principia.

An existing document does suggest that Newton may have made just such calculations as Whiston and Pemberton described, calculations in which Newton appears to have used a figure for the radius of the Earth that he found in Salusbury's version of Galileo's Dialogue, 3,500 Italian miles (milliaria), in which one mile equals 5,000, rather than 5,280, feet.113 Here, some time before 1669, Newton stated, to quote him in translation, “Finally, among the primary planets, since the cubes of their distances from the Sun are reciprocally as the squared numbers of their periods in a given time, their endeavours of recess from the Sun will be reciprocally as the squares of their distances from the Sun,” and he then gave numerical examples from each of the six primary planets. A. R. Hall has shown that this manuscript is the paper referred to by Newton in his letter to Halley of 20 June 1686, defending his claim to priority of discovery of the inverse-square law against Hooke's claims. It would have been this paper, too, that David Gregory saw and described in 1694, when Newton let him glance over a manuscript earlier than “the year 1669.”

This document, however important it may be in enabling us to define Newton's values for the size of the earth, does not contain an actual calculation of the moon test, nor does it refer anywhere to other than centrifugal “endeavours” from the sun. But it does show that when Newton wrote it he had not found firm and convincing grounds on which to assert what Whiteside has called a perfect “balance between (apparent) planetary centrifugal force and that of solar gravity.”114

By the end of the 1660's Newton had studied the Cartesian principles of motion and had taken a critical stand with regard to them. His comments occur in an essay of the 1670's or late 1660's, beginning “De gravitatione et aequipondio fluidorum,”115 in which he discussed extensively Descartes's Principia and also referred to a letter that formed part of the correspondence with Mersenne. Newton further set up a series of definitions and axioms, then ventured “to dispose of his [Descartes's] fictions.” A large part of the essay deals with space and extension; for example, Newton criticized Descartes's view “that extension is not infinite but rather indefinite.” In this essay Newton also defined force (“the causal principle of motion and rest”), conatus (or “endeavour”), impetus, inertia, and gravity. Then, in the traditional manner, he reckoned “the quantity of these powers” in “a double