Brook Taylor
Brook Taylor


Brook Taylor was born into a family which was on the fringes of the nobility, and certainly they were fairly wealthy. They could afford to have private tutors for their son, and this home education was all that Taylor enjoyed before entering St John's College in Cambridge in 1703. At Cambridge, Taylor became highly involved with mathematics. He graduated in 1709, but by this time he had already written his first important mathematics paper. The paper gives a solution to the problem of the centre of oscillation of a body. It is a mechanics paper which rests heavily on Newton's approach to the differential calculus. This paper started a priority dispute with Johann Bernoulli.

In 1712, Taylor was elected to the Royal Society. He was appointed to the committee set up to adjudicate on whether Newton or Leibniz invented the calculus. In 1714, Taylor was elected Secretary to the Royal Society. The next 4 years while he was Secretary mark what must be considered his most mathematically productive time. Between 1712 and 1724, Taylor published 13 articles on topics as diverse as describing experiments in capillary action, magnetism and thermometers. He found an improved method for approximating the roots of an equation by giving a new method for computing logarithms.

Two books which appeared in 1715 are extremely important in the history of mathematics. In one, Taylor added to mathematics a new branch now called the '"calculus of finite differences", invented integration by parts, and discovered the celebrated series known as Taylor's expansion. We must not give the impression that this result was one which Taylor was the first to discover. James Gregory, Newton, Leibniz, Johann Bernoulli and de Moivre had all discovered variants of Taylor's Theorem. Gregory, for example, knew the Taylor series for arctan x. All of these mathematicians had made their discoveries independently, and Taylor's work was also independent of that of the others. The importance of Taylor's Theorem remained unrecognised until 1772 when Lagrange proclaimed it the basic principle of the differential calculus.

There are other important ideas which are contained in these books which were not recognised as important at the time. These include singular solutions to differential equations, a change of variables formula, and a way of relating the derivative of a function to the derivative of the inverse function. Also contained is a discussion on vibrating strings, an interest which almost certainly come from Taylor's early love of music.

Taylor also devised the basic principles of perspective. A book on the subject gives the first general treatment of vanishing points. There is also the interesting inverse problem which is to find the position of the eye in order to see the picture from the viewpoint that the artist intended. Taylor was not the first to discuss this inverse problem but he did make innovative contributions to the theory of such perspective problems. One could certainly consider this work as laying the foundations for the theory of descriptive and projective geometry.