1685: Brook Taylor is born

On 18 August 1685, the English mathematician Brook Taylor was born in Edmonton, Middlesex. Educated at St John’s College, Cambridge, Taylor became a fellow of the Royal Society and secretary of that body while still in his twenties. His most enduring contribution appeared in the 1715 treatise Methodus Incrementorum Directa et Inversa, where he stated the theorem now universally known as Taylor’s theorem: any sufficiently smooth function can be locally approximated by a polynomial whose coefficients are determined by the function’s derivatives at a single point.

The result was not immediately celebrated; it took later work by Lagrange, Cauchy and others to supply rigorous remainders and to embed the expansion firmly inside real and complex analysis. Yet once established, the Taylor series became one of the workhorses of applied mathematics. It turned the qualitative study of change into a quantitative calculus of successive approximations—an intellectual move that proved indispensable whenever scientists needed to replace an intractable function with something a machine, or a human computer, could actually evaluate.

That same habit of local polynomial approximation travels directly into modern artificial intelligence. Gradient-based optimisers that train neural networks rely on first- and second-order Taylor expansions of the loss surface. Newton and quasi-Newton methods, natural-gradient descent, and many uncertainty-estimation techniques are explicit descendants of Taylor’s idea. Even the humble backpropagation algorithm can be viewed as the systematic application of the chain rule to first-order expansions. In short, every time a deep-learning framework computes a gradient or a Hessian-vector product, it is quietly invoking a three-hundred-year-old insight born on this day.

Taylor died in 1731 at the age of forty-six, leaving behind a slender but decisive body of work. Today, when an engineer watches a loss curve descend or a researcher linearises a complicated model around an operating point, the mathematics quietly at work is still the one Brook Taylor wrote down in the early eighteenth century—an unbroken thread from the coffee-houses of Georgian London to the GPU clusters of the present.