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

    way: that is, according to intension or extension.” He defined bodies, in the later medieval language of the intension and remission of forms, as “denser when their inertia is more intense, and rarer when it is more remiss.”

In a final set of “Propositions on Non-Elastic Fluids” (in which there are two axioms and two propositions), axiom 2, “Bodies in contact press each other equally,” suggests that the eventual third law of motion (Principia, axiom 3: “To every action is always opposed an equal and opposite reaction”) may have arisen in application to fluids as well as to the impact of bodies. The latter topic occurs in another early manuscript, “The Lawes of Motion,” written about 1666 and almost certainly antedating the essay on Descartes and his Principia.116 Here Newton developed some rules for the impact of “bodyes which are absolutely hard,” and then tempered them for application to “bodyes here amongst us,” characterized by “a relenting softnesse & springynesse,” which “makes their contact be for some time in more points than one.”

Newton's attention to the problems of elastic and inelastic impact is manifest throughout his early writings on dynamics. In the Principia it is demonstrated by the emphasis he there gave the concept of force as an “impulse,” and by a second law of motion (Lex II, in all editions of the Principia) in which he set forth the proportionality of such an impulse (acting instantaneously) to the change in momentum it produces.117 In the scholium to the laws of motion Newton further discussed elastic and inelastic impact, referring to papers of the late 1660's by Wallis, Wren, and Huygens. He meanwhile developed his concept of a continuously acting force as the limit of a series of impulses occurring at briefer and briefer intervals in infinitum.118

Indeed, it was not until 1679, or some time between 1680 and 1684, following an exchange with Hooke, that Newton achieved his mature grasp of dynamical principles, recognizing the significance of Kepler's area law, which he had apparently just encountered. Only during the years 1684-1686, when, stimulated by Halley, he wrote out the various versions of the tract De motu and its successors and went on to compose the Principia, did Newton achieve full command of his insight into mathematical dynamics and celestial mechanics. At that time he clarified the distinction between mass and weight, and saw how these two quantities were related under a variety of circumstances.

Newton's exchange with Hooke occurred when the latter, newly appointed secretary of the Royal Society, wrote to Newton to suggest a private philosophical correspondence. In particular, Hooke asked Newton for his “objections against any hypothesis or opinion of mine,” particularly “that of compounding the celestiall motions of the planetts of a direct motion by the tangent & an attractive motion towards the centrall body....” Newton received the letter in November, some months after the death of his mother, and evidently did not wish to take up the problem. He introduced, instead, “a fancy of my own about discovering the Earth's diurnal motion, a spiral path that a freely falling body would follow as it supposedly fell to Earth, moved through the Earth's surface into the interior without material resistance, and eventually spiralled to (or very near to) the Earth's centre, after a few revolutions.”119

Hooke responded that such a path would not be a spiral. He said that, according to “my theory of circular motion,” in the absence of resistance, the body would not move in a spiral but in “a kind [of] Elleptueid,” and its path would “resemble an Ellipse.” This conclusion was based, said Hooke, on “my Theory of Circular Motions [being] compounded by a Direct [that is, tangential] motion and an attractive one to a Centre.” Newton could not ignore this direct contradiction of his own expressed opinion. Accordingly, on 13 December 1679, he wrote Hooke that “I agree with you that ... if its gravity be supposed uniform [the body would] not descend in a spiral to the very centre but circulate with an alternate descent & ascent.” The cause was “its vis centrifuga & gravity alternately overballancing one another.” This conception was very like Borelli's, and Newton imagined that “the body will not describe an Ellipsoeid,” but a quite different figure. Newton here refused to accept the notion of an ellipse produced by gravitation decreasing as some power of the distance—although he had long before proved that for circular motion a combination of Kepler's third law and the rule for centrifugal force would yield a law of centrifugal force in the inverse square of the distance. There is no record of whether his reluctance was due to the poor agreement of the earlier moon test or to some other cause.

Fortunately for the advancement of science, Hooke kept pressing Newton. In a letter of 6 January 1680 he wrote “... But my supposition is that the Attraction always is in a duplicate proportion to the Distance from the Centre Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction, and Consequently as Kepler Supposes Reciprocall to the Distance.” We shall see below that this statement, often cited to support Hooke's claim to priority over Newton in the discovery of the inverse-square law, actually shows that Hooke was not