Brook Taylor

Brook Taylor. He was an influential English mathematician of the early 18th century, best known for the foundational result that bears his name, Taylor's theorem. His work in calculus and the study of infinite series provided crucial tools for the development of mathematical analysis and mathematical physics. Though his life was marked by personal tragedy and ill health, his contributions left a lasting legacy on the work of major figures like Leonhard Euler and Joseph-Louis Lagrange.

Early life and education

Born in Edmonton to a family of comfortable means, he was educated at St Paul's School, London before entering St John's College, Cambridge in 1701. At the university, he studied mathematics under the guidance of John Keill, a prominent early advocate for the Newtonian methods developed by Isaac Newton. He earned his Bachelor of Laws degree in 1709 and his Doctor of Laws in 1714, though his true passion remained in the mathematical sciences. During this period, he began corresponding with other leading intellectuals of the Royal Society, to which he was elected a fellow in 1712.

Mathematical work

His most enduring contribution is the formal introduction of what is now called the Taylor series in his 1715 work *Methodus Incrementorum Directa et Inversa*. This theorem provides a method to approximate smooth functions using polynomials derived from their derivatives at a single point, a cornerstone of calculus. He also made significant advances in the mathematics of perspective, publishing *Linear Perspective* in 1715 and *New Principles of Linear Perspective* in 1719, which influenced later artists and architects. His investigations extended to the physics of vibrating strings, the mathematical theory of musical temperament, and the calculation of the centers of mass of several bodies, engaging in scholarly disputes with contemporaries like Johann Bernoulli.

Later life and legacy

His later years were troubled by family disputes and deteriorating health following the death of his wife in 1730. He served as Secretary to the Royal Society from 1714 to 1718 but was often absent from London due to illness. Despite the relative obscurity of his work during his lifetime, its importance was championed and generalized by later mathematicians, most notably Colin Maclaurin, whose Maclaurin series is a special case of his expansion. The full power of his theorem was realized through the efforts of the Bernoulli family, Leonhard Euler, and Joseph-Louis Lagrange, who placed it at the heart of analytical mechanics and differential calculus. Today, his name is immortalized in fundamental concepts across engineering, physics, and applied mathematics.

Selected publications

* *Methodus Incrementorum Directa et Inversa* (1715) – containing the first published statement of his series. * *Linear Perspective* (1715) – on the geometrical principles of drawing. * *New Principles of Linear Perspective* (1719) – a revised and expanded work on the same subject. * Several important papers in the *Philosophical Transactions of the Royal Society* on topics ranging from the motion of projectiles to the calculus of finite differences.

Category:1685 births Category:1731 deaths Category:English mathematicians Category:Fellows of the Royal Society Category:Alumni of St John's College, Cambridge