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

    far beyond that commonly associated with the Principia. In the ancillary proposition 40, for example, Newton (again under the most general conditions of force) had sought the velocity at a point on an orbit, finding a result that is the equivalent of an integral, which (in E. J. Aiton's words) in “modern terms ... expresses the invariance of the sum of the kinetic and gravitational potential energies in an orbit.”153

In section 11, Newton reached a level of mathematical analysis of celestial motions that fully distinguishes the Principia from any of its predecessors. Until this point, he there explained, he had been “treating of the attractions of bodies towards an immovable centre; though very probably there is no such thing existent in nature.” He then outlined a plan to deal with nature herself, although in a “purely mathematical” way, “laying aside all physical considerations”—such as the nature of the gravitating force. “Attractions” are to be treated here as originating in bodies and acting toward other bodies; in a two-body system, therefore, “neither the attracted nor the attracting body is truly at rest, but both ... being as it were mutually attracted, revolve about a common centre of gravity.” In general, for any system of bodies that mutually attract one another, “their common centre of gravity will either be at rest, or move uniformly” in a straight line. Under these conditions, both members of a pair of mutually attractive bodies will describe “similar figures about their common centre of gravity, and about each other mutually” (proposition 57).

By studying such systems, rather than a single body attracted toward a point-center of force, Newton proved that Kepler's laws (or “planetary hypotheses”) cannot be true within this context, and hence need modification when applied to the real system of the world. Thus, in proposition 59, Newton stated that Kepler's third law should not be written T12:T22 = a13:a23, as Kepler, Hooke, and everybody else had supposed, but must be modified.

A corollary that may be drawn from the proposition is that the law might be written as (M + m1)T12:(M + m2)T22:a13:a23, where m1, m2 are any two planetary masses and M is the mass of the sun. (Newton's expression of this new relation may be reduced at once to the more familiar form in which we use this law today.) Clearly, it follows from Newton's analysis and formulation that Kepler's own third law may safely be used as an approximation in most astronomical calculations only because m1, m2 are very small in relation to M. Newton's modification of Kepler's third law fails to take account of any possible interplanetary perturbations. The chief function of proposition 59 thus appears to be not to reach the utmost generalization of that law, but rather to reach a result that will be useful in the problems that follow, most notably proposition 60 (on the orbits described when each of two bodies attracts the other with a force proportional to the square of the distance, each body “revolving about the common centre of gravity”).

From proposition 59 onward, Newton almost at once advanced to various motions of mutually attractive bodies “let fall from given places” (in proposition 62), “going off from given places in given directions with given velocities” (proposition 63), or even when the attractive forces “increase in a simple ratio of their [that is, the bodies'] distances from the centres” (proposition 64). This led him to examine Kepler's first two laws for real “bodies,” those “whose forces decrease as the square of their distances from their centres.” Newton demonstrated in proposition 65 that in general it is not “possible that bodies attracting each other according to the law supposed in this proposition should move exactly in ellipses,” because of interplanetary perturbations, and discussed cases in astronomy in which “the orbits will not much differ from ellipses.” He added that the areas described will be only “very nearly proportional to the times.”

Proposition 66 presents the restricted three-body problem, developed in a series of twenty-two corollaries. Here Newton attempted to apply the law of mutual gravitational attraction to a body like the sun to determine how it might perturb the motion of a moonlike body around an earthlike body. Newton examined the motion in longitude and in latitude, the annual equation, the evection, the change of the inclination of the orbit of the body resembling the moon, and the motion on the line of apsides. He considered the tides and explained, in corollary 22, that the internal “constitution of the globe” (of the earth) can be known “from the motion of the nodes.” He further demonstrated that the shape of the globe can be derived from the precession constant (precession being caused, in the case of the earth, by the pull of the moon on the equatorial bulge of the spinning earth). He thus established, for the first time, a physical theory, elaborated in mathematical expression, from which some of the “inequalities” of the motion of the moon could be deduced; and he added some hitherto unknown “inequalities” that he had found. Previous to Newton's work, the study of the irregularities in the motion of the moon had been posited on the elaboration of geometric models, in an attempt to make predicted positions agree with actual observations.154