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

Section 12 of book I contains Newton's results on the attractions of spheres, or of spherical shells. He dealt first with homogeneous, then nonhomogeneous spheres, the latter being composed of uniform and concentric spherical shells so that the density is the same at any single given distance from the center. In proposition 71 he proved that a “corpuscle” situated outside such a nonhomogeneous sphere is “attracted towards the centre of the sphere with a force inversely proportional to the square of its distance from the centre.” In proposition 75, he reached the general conclusion that any two such spheres will gravitationally attract one another as if their masses were concentrated at their respective centers--or, in other words, that the distance required for the inverse-square law is measured from their centers. A series of elegant and purely mathematical theorems follow, including one designed to find the force with which a corpuscle placed inside a sphere may be “attracted toward any segment of that sphere whatsoever.” In section 13, Newton, with a brilliant display of mathematics (which he did not fully reveal for the benefit of the reader) discussed the “attractive forces” of nonspherical solids of revolution, concluding with a solution in the form of an infinite series for the attraction of a body “towards a given plane.”155

Book I concludes with section 14, on the “motion of very small bodies” acted on by “centripetal forces tending to the several parts of any very great body.” Here Newton used the concept of “centripetal forces” that act under very special conditions to produce motions of corpuscles that simulate the phenomena of light--including reflection and refraction (according to the laws of Snell and Descartes), the inflection of light (as discovered by Grimaldi), and even the action of lenses. In a scholium, Newton noted that these “attractions bear a great resemblance to the reflections and refractions of light,” and so

. . . because of the analogy there is between the propagation of the rays of light and the motion of bodies, I thought it not amiss to add the following Propositions for optical uses; not at all considering the nature of the rays of light, or inquiring whether they are bodies or not; but only determining the curves of [the paths of] bodies which are extremely like the curves of the rays.

A similar viewpoint with respect to mathematical analyses (or models and analogies) and physical phenomena is generally sustained throughout books I and II of the Principia.

Newton's general plan in book I may thus be seen as one in which he began with the simplest conditions and added complexities step by step. In sections 2 and 3, for example, he dealt with a mass-point moving under the action of a centripetal force directed toward a stationary or moving point, by which the dynamical significance of each of Kepler's three laws of planetary motion is demonstrated. In section 6, Newton developed methods to compute Keplerian motion (along an ellipse, according to the law of areas), which leads to “regular ascent and descent” of bodies when the force is not uniform (as in Galilean free fall) but varies, primarily as the inverse square of the distance, as in Keplerian orbital motion. In section 8 Newton considered the general case of “orbits in which bodies will revolve, being acted upon by any sort of centripetal force.” From stationary orbits he went on, in section 9, to “movable orbits; and the motion of the apsides” and to a mathematical treatment of two (and then three) mutually attractive bodies. In section 10 he dealt with motion along surfaces of bodies acted upon by centripetal force; in section 12, the problems of bodies that are not mere points or point-masses and the question of the “attractive forces of spherical bodies”; and in section 13, “the attractive forces of bodies that are not spherical.”

Book II of the “Principia.”

Book II, on the motion of bodies in resisting mediums, is very different from book I. It was an afterthought to the original treatise, which was conceived as consisting of only two books, of which one underwent more or less serious modifications to become book I as it exists today, while the other, a more popular version of the “system of the world,” was wholly transformed so as to become what is now book III. At first the question of motion in resisting mediums had been relegated to some theorems at the end of the original book I; Newton had also dealt with this topic in a somewhat similar manner at the end of his earlier tract De motu. The latter parts of the published book II were added only at the final redaction of the Principia.

Book II is perhaps of greater mathematical than physical interest. To the extent that Newton proceeded by setting up a sequence of mathematical conditions and then exploring their consequences, book II resembles book I. But there is a world of difference between the style of the two books. In book I Newton made it plain that the gravitational force exists in the universe, varying inversely as the square of the distance, and that this force accordingly merits our particular attention. In book II, however, the reader is never certain as to which of the many conditions of resistance that Newton considers may actually occur in nature.156

Book II enabled Newton to display his mathematical ingenuity and some of his new discoveries. Occasionally, as in the static model that he proposed to