British mathematical reformers in the nineteenth century: motivations and methods Winner of the BSHM undergraduate essay prize 2022

In the eighteenth century, a rift opened between the mathematical communities in Britain and the European continent. Following the Newton-Leibniz controversy on the priority of the calculus’s discovery, most British mathematicians pledged loyalty to Newton, and by extension, to his fluxional calculus (Guicciardini 2009, 1). However, by the early nineteenth century, several British mathematicians noticed the progress made on the European continent following the work of Leibniz, to which Britain had been mostly blind. These mathematicians dedicated themselves to the reform of British mathematics by circulating differential calculus. The main difference between Newton and Leibniz’s calculus is that Newton’s ‘fluxional calculus’ rested on geometrical and physical intuition (see Stedall 2008; Guicciardini 2009; Kline 1990). His method of finding the ‘fluxion’ of a function was to consider the velocity of a particle moving along a curve. In modern terms, if a particle moves in the plane with position (x(t), y(t)), its velocity is expressed in terms of x’(t) and y’(t). Newton (1669, 178) examined the limit of the ratio of these derivatives and wrote ẋo for x’(t), where ẋ is the fluxion, or instantaneous velocity, of x, and o is an infinitely small time interval (Guicciardini 2009). By contrast, Leibniz’s ‘differential calculus’ had an algebraic basis. Leibniz directly examined the infinitely small increments in x and y (differentials) and determined their relationship without the need for physical intuition. Instead, he used infinite sums with a notation close to what we use today (Leibniz 1682; translated in Stedall 2008); in place of Newton’s ẋo, o and ẋ, Leibniz used dx, dt, and dx dt respectively (Guicciardini 2009, 3).