The Crooked Made Straight: Roberval and Newton on Tangents

Summary

Introduction

In October 1665, about two years after he had first read a mathematics book, Isaac Newton began investigating a method for finding the tangents to “mechanical” curves. He can have known only vaguely that he was following a path trod previously by several outstanding mathematicians, Torricelli, Descartes, Roberval, and Barrow among them. In his ignorance of the details of their work, Newton stumbled before setting himself firmly on the way to his calculus. As he progressed, he overcame the inadequate mathematical language that had kept others from expressing—sometimes from even thinking—their ideas clearly.

Newton's method found tangents by regarding a curve as the trajectory of a moving particle, so that the velocity vector lies along the tangent. Sometimes one can easily find the velocity vector, however, by decomposing the given motion into simpler ones with known velocity vectors. This method of finding tangents to curves by decomposing the velocity vector is often called the kinematic method. Newton's first manuscript on the kinematic method included three examples of curves that had traditionally been described by the composition of motions: the spiral of Archimedes, the cycloid, and the quadratrix. In addition to these mechanical curves, described as trajectories, Newton also discussed the ellipse, a so-called geometrical curve.

Newton had not been the first to consider composition of motions in general or any of these particular examples. Of course, the general idea of composition of motions goes back to the ancient Greeks, as the examples of the Archimedean spiral and the quadratrix show.