Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1

Abstract

During the past 25 years there has been a controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler's area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia. It is shown here that the critics of Prop. 1 have misunderstood Newton's fundamental limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well grounded.Rigor merely sanctions the conquests of sound intuition, Jacques Hadamard

The early application of the calculus to the inverse square force problem

“…1, and it does not require the existence of vortices. In fact, such an explanation for the radial oscillations in the orbit was already given by Newton in a cryptic remark in his 1679 correspondence with Hooke, and in his 1681 correspondence with Crompton regarding a question of Flamsteed about the motion of comets near the sun (Nauenberg (1994)). The vortex interpretation had one advantage over Newton's theory, because it offered an explanation why the planets all rotated along the same direction.…”

“…A detail discussion of Prop. 1 is given in reference (Nauenberg 2003). 6 Newton claimed erroneously that Leibniz had made an error in handling second-order differentials in his equation.…”

“…4. Indeed, such a curve is essential to specify Newton's geometrical construction when "'the number of triangles be increased…indefinitely" (Nauenberg 2003). Although the necessity for assuming such a curve escaped the attention of contemporary commentators of Prop.1, it was recognized first by Leibniz and then by Herman in his diagram which represents the essential features of this proposition.…”

“…For ρ, Bernoulli wrote the expression 30 Newton had derived this relation earlier, but he did not include it in the first edition of the Principia (Nauenberg 1994). It appear in the second edition (1713) of the Principia in several new corollaries to a revised form of Prop.…”

Robert Hooke?s Seminal Contribution to Orbital Dynamics

“…6 force must be directed toward a fixed center -only after implementing Hooke's way of "compounding" velocities geometrically for a general orbit. 29 Newton had considered an approach similar to Hooke's in his earliest work on dynamics, as shown in his "Waste Book" of 1664 (figure 3), 30 but he evidently did not consider extending it to arbitrary motion until after Hooke prompted him to do so. Kepler's area law was crucial to Newton's development of orbital dynamics, because it permitted him to express the time variable in purely geometrical terms, since "The areas which bodies [planets] made to move in orbits describe by radii drawn to an unmoving center of force lie in unmoving planes and are proportional to the times."…”