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

    the sun, but significantly different in that the ellipse implies periodic returns of the comet—and worked out the details with Newton. In the second and third editions of the Principia, Newton gave tables for both the parabolic and elliptical orbits; he asserted unequivocally that Halley had found “a remarkable comet” appearing every seventy-five years or so, and added that Halley had “computed the motions of the comet in this elliptic orbit.” Nevertheless, Newton himself remained primarily concerned with parabolic orbits. In the conclusion to the example following proposition 41 (on the comet of 1680), Newton said that “comets are a sort of planets revolved in very eccentric orbits about the sun.” Even so, the proposition itself states (in all editions): “From three given observations to determine the orbit of a comet moving in a parabola.”

Mathematics in the “Principia.”

The Philosophiae naturalis principia mathematica is, as its title suggests, an exposition of a natural philosophy conceived in terms of new principles based on Newton's own innovations in mathematics. It is too often described as a treatise in the style of Greek geometry, since on superficial examination it appears to have been written in a synthetic geometrical style.132 But a close examination shows that this external Euclidean form masks the true and novel mathematical character of Newton's treatise, which was recognized even in his own day. (L'Hospital, for example—to Newton's delight—observed in the preface to his 1696 Analyse des infiniment petits, the first textbook on the infinitesimal calculus, that Newton's “excellent Livre intitulé Philosophiae Naturalis principia Mathematica ... est presque tout de ce calcul.”) Indeed, the most superficial reading of the Principia must show that, proposition by proposition and lemma by lemma, Newton usually proceeded by establishing geometrical conditions and their corresponding ratios and then at once introducing some carefully defined limiting process. This manner of proof or “invention,” in marked distinction to the style of the classical Greek geometers, is based on a set of general principles of limits, or of prime and ultimate ratios, posited by Newton so as to deal with nascent or evanescent quantities or ratios of such quantities.

The doctrine of limits occurs in the Principia in a set of eleven lemmas that constitute section 1 of book I. These lemmas justify Newton in dealing with areas as limits of sums of inscribed or circumscribed rectangles (whose breadth ? 0, or whose number ? ?), and in assuming the equality, in the limit, of arc, chord, and tangent (lemma 7), based on the proportionality of “homologous sides of similar figures, whether curvilinear or rectilinear” (lemma 5), whose “areas are as the squares of the homologous sides.” Newton's mathematical principles are founded on a concept of limit disclosed at the very beginning of lemma 1, “Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer to each other than by any given difference, become ultimately equal.”

Newton further devoted the concluding scholium of section 1 to his concept of limit, and his method of taking limits, stating the guiding principle thus: “These lemmas are premised to avoid the tediousness of deducing involved demonstrations ad absurdum, according to the method of the ancient geometers.” While he could have produced shorter (“more contracted”) demonstrations by the “method of indivisibles,” he judged the “hypothesis of indivisibles ”to be “somewhat harsh” and not geometrical:

I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratios of nascent and evanescent quantities, that is, to the limits of those sums and ratios; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is performed as by the method of indivisibles; and now those principles being demonstrated, we may use them with greater safety. Therefore if hereafter I should happen to consider quantities as made up of particles, or should use little curved lines for right ones, I would not be understood to mean indivisibles, but evanescent divisible quantities; not the sums and ratios of determinate parts, but always the limits of sums and ratios; and that the force of such demonstrations always depends on the method laid down in the foregoing Lemmas.

Newton was aware that his principles were open to criticism on the ground “that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none”; and he anticipated any possible unfavorable reaction by insisting that “the ultimate ratio of evanescent quantities” is to be understood to mean “the ratio of the quantities not before they vanish, nor afterwards, but [that] with which they vanish.” In a “like manner, the first ratio of nascent quantities is that with which they begin to be,” and “the first or last sum is that with which they begin and cease to be (or to be augmented or diminished).” Comparing such ratios and sums to velocities (for “it may be alleged, that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, there is none”), he imagined the existence of “a limit which the velocity