Brook Taylor

Brook Taylor (1685–1731) was an English mathematician whose work established Taylor’s theorem and the power-series method now bearing his name.

Taylor series: A power series whose coefficients are determined by a function’s derivatives at a chosen point. The expansion method most directly associated with Taylor’s name.
Methodus Incrementorum Directa et Inversa: Brook Taylor’s 1715 book on finite differences, differential calculus, and related mathematical methods. It contains the theorem and series that became central to his legacy.
Colin Maclaurin: Scottish mathematician whose work on infinite series included a special case of Taylor expansion about zero. His name became attached to the zero-centered form of Taylor’s method.
Taylor polynomial: A finite polynomial formed from the first terms of a function’s Taylor series at a point. It is the practical finite approximation produced by Taylor’s method.
Taylor's theorem: A theorem expressing a sufficiently differentiable function as a finite Taylor polynomial plus a remainder. Taylor’s 1715 treatise gave an early general statement of this result.
Isaac Newton: English mathematician and natural philosopher who developed calculus and the theory of universal gravitation. Taylor worked within the Newtonian mathematical tradition.
Joseph-Louis Lagrange: Italian-French mathematician who developed influential formulations of calculus and mechanics. Lagrange supplied a prominent remainder formula for Taylor’s theorem.
Numerical analysis: The study of algorithms for computing numerical solutions to mathematical problems. Taylor approximations help estimate functions and analyze computational error.
Power series: An infinite sum of constant multiples of nonnegative integer powers of a variable. Taylor expansions are a central use of this broader mathematical form.
Leibniz–Newton calculus controversy: A dispute over priority for the invention of calculus, involving supporters of Newton and Gottfried Wilhelm Leibniz. The controversy shaped the intellectual climate of Taylor’s mathematical career.

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Series expansionRelated: His name became attached to the derivative-based expansion formula.