2 - Newton's methods of series and fluxions

Summary

Purpose of this chapter

Section 2.2 is concerned with a mathematical method which Newton developed from the year 1664. He labelled it the ‘analytical method of fluxions’. It can be defined as the Newtonian equivalent, or counterpart, of the Leibnizian differential and integral calculus, and briefly – but somewhat improperly – called the ‘fluxional calculus’. From 1664 to the 1690s Newton elaborated several versions of it. Furthermore, Newton distinguished between an analytical and a synthetic method of fluxions (§2.3). In this chapter I will attempt a periodization of these versions, paying attention to concepts, rather than to results. What follows cannot be taken as an exposition or introduction to Newton's mathematical discoveries. This would take us too much space, and indeed, after Whiteside's works, would also be useless. What I will do rather is to sketch the nature of the mathematical method for drawing tangents and ‘squaring curves’ that Newton had developed before the composition of the Principia.

The analytical method

Mathematical background

Newton's interest in mathematics began around 1664 when he read François Viète's works (1646), the second Latin edition of René Descartes' Geometria (1659–61), William Oughtred's Clavis mathematicae (1631), and John Wallis's Arithmetica infinitorum (1656). It was in reading this selected group of works that Newton learned about the most exciting discoveries in analytical geometry, algebra, tangents, maxima and minima, quadratures, infinite series and infinite products.