Fluxions

Fluxions is the term Newton used for what we now call derivatives — the rates of change of continuously varying quantities. Newton developed the method of fluxions in the 1660s, conceiving curves as generated by the motion of a point and calling the instantaneous rate of change the 'fluxion' of the flowing quantity ('fluent'). The notation was unwieldy — dots over letters, geometric ratios of 'evanescent quantities' — and the conceptual foundations, resting on infinitesimals that were simultaneously zero and non-zero, drew criticism from Berkeley and others. Newton's geometric presentation in the Principia concealed the fluxional machinery behind classical diagrams, a strategic choice that lent authority while obscuring the radical novelty of the method.

Newton's fluxions and Leibniz's differential calculus are the same mathematics in different costumes, but the costumes mattered. Leibniz's notation won because it could be taught; Newton's notation lost because it required geometric intuition that could not be mechanized. The history of mathematical notation is not a footnote to the history of mathematical ideas — it is the history of which ideas become thinkable at scale.