Brook Taylor: Much More than a Series

Classes and research have been keeping me busy, so this post will also be on someone I have been tackling in my homework this week.  One of the main things that I have learned in graduate school thus far is that none of the equations we use are "correct."  They are all approximations of one sort or another, whether because we can't solve the real equation or because we can't take into account all of the interactions.  One of the most common tools for these approximations, when we have an equation but don't want to deal with it, is to use the Taylor expansion.  I hadn't given it or him much thought until this week, but they just keep popping up, so Taylor is this week's subject.

Modern chemistry seems to have developed in the 19th century.  That's when scientists finally agreed that atoms exist, and developed the modern concepts of energy and heat.  Mathematics, however, seems to have had a heyday in the 18th century based on the number of mathematical operators, functions, rules, etc. that have been named after the mathematicians of that century. These include Laplace, Lagrange, L'Hopital, Maclaurin, Euler, Gauss, Fourier, Legendre, and, of course (or else this interlude would be rather pointless), Brook Taylor.

Brook Taylor
(1685-1731)

In reading about Brook Taylor, I realized that, more than anyone I have discussed so far, I feel that I cannot do him justice.  This stems from two main causes: my lack of understanding of the finer points of mathematics and its history, and the number of interesting things that I discovered about Taylor.

Brook Taylor was an Englishman, born in 1685.  He went to St. John's College at Cambridge and studied mathematics, which was apparently quite popular in those days.  He began writing and publishing on mathematical subjects, but didn't publish soon enough after his discoveries to avoid trouble.  In 1708 he developed a solution to the problem of the center of oscillation.  I still haven't quite figured out what this is, but apparently it was a big deal.  He didn't publish his discovery, however, until 1713: De Inventione Centri Oscillationis.¹

Meanwhile, Johann Bernoulli had independently come to the same discovery, and argued about precidence with Taylor.  In 1715 he published Methodus Incrementorum Directa et Inversa, which first introduced to the public what became known as Taylor's Theorem.  The work was also the first discussion of what came to be known as the calculus of finite differences, for more information on which you will have to ask a mathematician.  Taylor was not the first person to use the series, but he made the most general form of it.  Specific instances had already been used by Edmond Halley, Isaac Newton, Johann Bernoulli, and Johann Kepler.  The importance of the series was overlooked for many years, until it was pointed out by Joseph Lagrange in 1772.  Other problems that he solved in this book involved oscillations of a string and a change of variables formula.  He also write papers and letters on the subjects of magnetism, the movement of fluids, and logarithms.  His writing, however, suffered from a brevity that lead to confusion about what he actually meant, which led to his being under appreciated for all of the contributions that he made to mathematics.

In 1715 Taylor also published a work on linear perspective, followed in 1719 by New Principles of Linear Perspective, in both of which he used mathematics to explain linear perspective more generally than those before him had.  Bernoulli, with whom Taylor had already had heated arguments, declared that the book was "abstruse to all," especially artists.  Bernoulli's objections were so strong that Taylor wrote a reply in the Philosophical Transactions, Apologia D. Brook Taylor, J V D. & R S. Soc. contra V. C J. Bernoullium, Math. Prof. Basileae. I think Bernoulli had a point, though, since Taylor's works on perspective contained no sketches, just written descriptions, and even when he wasn't writing about art, he had a tendency to be concise to the point of confusion.

Taylor had been elected a member of the Royal Society in 1712, and had sat on the committee which adjudicated between Newton and Leibniz on the issue of which had invented calculus (they sided with Newton).  After about 1715, Taylor began writing more philosophical papers, such as "On the Lawfulness of Eating Blood."  His final paper in the Philosophical Transactions was "An Account of an Experiment, Made to Ascertain the Proportion of the Expansion of the Liquor in the Thermometer, with Regard to the Degrees of Heat," published around 1721.  He seems to have focused more on domestic matters and his health after that time, for in 1721 he also married.  His father disapproved of his wife, which suggests that Taylor, for one, married for love.  When she died in childbirth two years later, however, he and his father became reconciled.  In 1729 (1725?) he married again, but she also died in childbirth.  Taylor died just one year later.

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1. Most articles I found said that it wasn't until 1714 that he published it, but I think this is the article in question, and according to Jstor it was published in 1713. So that is what I'm going with.

References and further information

Brook Taylor, 1911 Encyclopedia Britanica
Brook Taylor, from someone at the University of St. Andrews
Brook Taylor, by Edward Irving Carlyle, Dictionary of National Biography, 1885-1900, vol. 55.
Dr. Brook Taylor's Principles of Linear Perspective, edited by Joseph Jopling, 1835.