In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, (...) as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83). (shrink)

Newton characterizes the reasoning of Principia Mathematica as geometrical. He emulates classical geometry by displaying, in diagrams, the objects of his reasoning and comparisons between them. Examination of Newton’s unpublished texts shows that Newton conceives geometry as the science of measurement. On this view, all measurement ultimately involves the literal juxtaposition—the putting-together in space—of the item to be measured with a measure, whose dimensions serve as the standard of reference, so that all quantity is ultimately related to spatial extension. I (...) use this conception of Newton’s project to explain the organization and proofs of the first theorems of mechanics to appear in the Principia. The placementof Kepler’s rule of areas as the first proposition, and the manner in which Newton proves it, appear natural on the supposition that Newton seeks a measure, in the sense of a moveable spatial quantity, of time. I argue that Newton proceeds in this way so that his reasoning can have the ostensive certainty of geometry. (shrink)

By a simple revision of Newton's diagram for Proposition 6 of the third edition of the Principia, one can see directly how the mathematics of uniform circular motion have been employed to solve the Kepler problem of elliptical planetary motion in Proposition 11. Newton strove initially to build his dynamics on the linear kinematics of Galileo; and, in this utilization of uniformly accelerated linear motion to solve more complicated problems, he can be seen as revolutionary. But he could not escape (...) completely from the coils of celestial circularity, and in his utilization of uniform circular motion to solve problems of elliptical motion, he can be seen as reactionary. The key to understanding Newton's mature dynamics resides in the discussion of the alternate dynamics ratio, as presented here in section six. (shrink)

For about the last thirty years Newton scholars have carried on a discussion on the meaning of Newton’s second law and its place in the stucture of his physics. E. J. Dijksterhuis, Brian D. Ellis, R. G. A. Dolby, I. Bernard Cohen, and R. S. Westfall in their treatments of these matters all quote a passage that Newton added to the third edition of the Principia. This passage, beginning “Corpore cadente” (“when a body is falling”), was inserted into the Scholium (...) following the Laws. The present article is an analysis of the literature that treats this passage and its relation to the second law. The conclusion is that they are right who see a deep duality in Newton’s mind regarding this law. (shrink)