Brook Taylor

Brook Taylor is remembered today for the series that bears his name — the expansion of any smooth function around a point as a sum of derivative terms. But his contemporaries knew him equally as the man who first cracked a physical problem that had resisted everyone before him: how a plucked string moves.

In his 1713 paper to the Royal Society, Taylor argued that the restoring force on each small element of a tensioned string is proportional to the string's curvature. From this he derived the period of the fundamental mode and showed that it agreed with observation. His analysis was incomplete — he treated only the lowest mode — but it set the mathematical template that d'Alembert, Euler, and Bernoulli would finish over the next forty years.

Taylor's Methodus Incrementorum (1715) contained, buried in its pages, the series expansion now known to every calculus student. Its significance went unrecognised in his lifetime; Lagrange, decades later, called it "the main foundation of the differential calculus." Taylor died young of a fever in 1731, aged forty-six.