On Newton's fluxional proof of the vector addition of motive forces

Abstract
This paper consists in an exposition of a proof Newton gave in 1666 of the parallelogram law for compounding velocities, and an examination of its implications for understanding his treatment of motion resulting from a continuously acting force in the Principia. I argue that the “moments” invoked in the fluxional proof of the vector resolution and composition of velocities are “virtual times”, a device allowing Newton to represent motions by the linear displacements produced in such a time; the ratio of velocities at an instant can then be represented by assuming the velocities continue for such a virtual time. By the Method of First and Ultimate Ratios, the first ratio of the velocities is then given by the ratio of such lines, under the presupposition that in the limit they will shrink to zero magnitude. I then argue that the same device is implicit in Newton’s appeal to “moments” or “particles of time” in his proof of Kepler’s Area Law in the “Locke Paper” and in Proposition 1 of Book 1 of the Principia, and that the limiting process involved there is therefore the same as that implicit in the Method of First and Ultimate Ratios.
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