Colin Maclaurin

Scottish mathematician

• Born: February 1, 1698
• Birthplace: Kilmodan, Argyllshire, Scotland
• Died: January 14, 1746
• Place of death: Edinburgh, Scotland

Maclaurin, the greatest British mathematician of the eighteenth century, developed and extended Sir Isaac Newton’s work in fluxions (calculus) and gravitation and made important new discoveries in geometry and mathematical analysis.

Early Life

Colin Maclaurin was born in the western Scottish county of Argyll. He was the youngest of the three sons of John Maclaurin, a learned minister of Kilmodan parish. John, the eldest son, followed in his father’s footsteps and became a minister. Daniel, the second son, manifested signs of extraordinary genius but died young. Colin, too, was a child prodigy, but he never knew his father, who died when Colin was six weeks old. Further tragedy struck when he was nine years old, with the death of his mother. His father’s brother, also a minister, became guardian of the children.

In 1709, at the early age of eleven, Colin entered the University of Glasgow, where he studied theology for a year. During this time, he became friendly with Robert Simson, a professor of mathematics, from whom he acquired a passionate interest in Euclid’s geometry and in other ancient Greek mathematics. He also became interested in Sir Isaac Newton’s work, and this led to his thesis “On the Power of Gravity,” which he publicly defended in 1715 and for which he was awarded a master of arts degree. He remained at Glasgow another year to study theology, after which he returned to live with his uncle in their Highland home beside Loch Fyne.

He enjoyed wandering over the hills as well as reading mathematics, philosophy, and the classics. Some of his notebook entries that have survived from this time reveal his sensitivity to the beauties of nature, which he deeply believed manifested God’s perfections. He abandoned this Highland life in 1717, when, following a competitive examination, he was appointed to the chair of mathematics at Marischal College, Aberdeen, even though he was only nineteen years old. This first appointment marked the start of his brilliant mathematical career.

Life’s Work

Colin Maclaurin’s accomplishments grew out of Newton’s. He first met Newton in 1719 on a visit to London. Newton was favorably impressed by the young Scottish mathematician, and they became friends. Maclaurin had already contributed papers to the Philosophical Transactions of the Royal Society, and he was soon elected a fellow of this society of which Newton was president.

During this time, Maclaurin was working on a book about geometry, which was published, with Newton’s approval, in 1720. The book, whose full title was Geometria organica: Sive, Descriptio linearum curvarum universalis (organic geometry, with the description of universal linear curves), contained new and elegant methods for generating conics (circle, ellipse, hyperbola, and parabola). Maclaurin also devised an elaborate treatment of higher plane curves that was superior to Newton’s earlier results. Maclaurin proved many important theorems that could be found, without proof, in Newton’s work. He also discovered many new theorems; for example, he showed that many curves of the second and higher degrees could be described by the intersection of two movable angles.

In 1722, Maclaurin left Scotland to serve as companion and tutor to the eldest son of Lord Polwarth, British plenipotentiary at Cambrai in northern France. Maclaurin and his young charge visited Paris for a short time and then resided for a much longer period at Lorraine, where Maclaurin wrote a paper on the impact of bodies, for which he was awarded a prize by the Academy of Sciences. The sudden death of his pupil caused him to return to Aberdeen, but, because of problems connected with his three-year absence from the university without leave, he was unable to resume his position. Newton again stepped in to help, and it was largely through his strong recommendation that Maclaurin became, in November, 1724, deputy professor for the elderly James Gregory at Edinburgh University. (This James Gregory was the nephew of the famous Scottish mathematician of the same name, who had been appointed to the first chair of mathematics at Edinburgh in 1674, a year before his tragic death at the age of thirty-six.) Newton even wrote privately to the lord provost of Edinburgh, offering to contribute œ20 a year toward Maclaurin’s salary.

A short time later, in 1725, the Edinburgh position in mathematics became available, and on the recommendation of Newton, Maclaurin took up the position that he would occupy for the rest of his life. His outstanding success at Edinburgh fully vindicated Newton’s trust in him. He lectured on a wide range of topics, including the Elements of Euclid, spherical trigonometry, astronomy, and Newton’s Philosophiae naturalis principia mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729; best known as Principia). His classes were well attended, and his lectures as well as his writings were models of lucid and logical construction.

Edinburgh provided Maclaurin with the opportunity to develop and share his many talents. He was a skilled experimenter who constructed clever and useful mechanical devices. He made valuable astronomical observations, and he advocated building an observatory in Scotland. He also made actuarial tables for the budding insurance companies of Edinburgh. He took an active part in improving the maps of the Orkney and Shetland Islands, and he was even eager to make a voyage to find a northeast polar passage by way of Greenland to the southern seas.

Maclaurin’s growing fame also gave him the opportunity to play an important role in Edinburgh society. For example, he was influential in persuading the members of the newly formed Edinburgh Society for Improving Medical Knowledge to enlarge its scope. The new organization, named the Philosophical Society, reflected this change, and Maclaurin became one of its secretaries. This society later became the Royal Society of Edinburgh.

In 1733, Maclaurin married Anne Stewart, the daughter of the solicitor general for Scotland. They had seven children, of whom two sons and three daughters survived him. An engraving of Maclaurin from the Edinburgh period depicts him as a stocky man with heavy eyebrows, a strong nose, and a weak chin. His mien is determined and serious, as befits a distinguished Scottish professor and disciple of Newton.

Throughout his Edinburgh career Maclaurin sought to silence the criticism of Newton’s work on differential calculus (which he and Maclaurin called fluxions). The most influential of these critics was George Berkeley, bishop of Cloyne. In 1734, Berkeley published The Analyst: Or, A Discourse Addressed to an Infidel Mathematician, in which he attacked Newton’s ideas on fluxions. The infidel mathematician was Edmond Halley, the great astronomer and religious skeptic, who had convinced one of Berkeley’s dying friends to refuse the last rites because of the untenability of Christian doctrines. Berkeley denied neither the utility of fluxions nor the validity of their results. He did confess, however, to confusion about the meaning of fluxions. According to their defenders, fluxions were neither finitely large nor infinitely small, and yet they were not nothing. Berkeley concluded acidly that they were the “ghosts of departed quantities.”

Berkeley’s criticism stung. Maclaurin felt that a reply was necessary. He thought that Berkeley had misrepresented the method of fluxions by depicting it as full of mysteries and based on false reasoning. Since fluxions were opaque to someone of Berkeley’s intelligence, Maclaurin believed that Newton’s method needed new and more vigorous arguments to support it.

A Treatise of Fluxions, Maclaurin’s greatest work, was published in 1742. The book was an attempt to establish fluxions on as sound a basis as Greek geometry. Maclaurin began, like Newton, with the concepts of space, time, and motion, and then he systematically elaborated Newton’s version of the calculus. Since his readers were more familiar with velocity than with strictly mathematical variables, Maclaurin approached Berkeley’s difficulties by considering motion. Maclaurin agreed with Berkeley that the infinitely small was inconceivable, but he did not see any objection to bringing into geometry the idea of an instantaneous velocity. Indeed, for Maclaurin, mathematics included both the properties of motion and the properties of figures. Using this background analysis of motion, he went on to show how fluxions were measured by the quantities they would generate if they were to continue moving uniformly.

Maclaurin’s search for a rigorous foundation for fluxions was commendable but in the end unsuccessful. Nevertheless, his analysis did leave hints that future mathematicians would fruitfully follow. Furthermore, his book was not only a defense of Newton’s methods; it was also an investigation into a variety of other problems in geometry and physics. For example, he developed a test for the convergence of an infinite series, and, for the first time, he gave the correct method for deciding between a maximum and a minimum of a function by investigating the sign of a higher derivative.

In A Treatise of Fluxions, Maclaurin also built on many of the physical principles enunciated by Newton in the Principia. For example, he analyzed the tides as a problem in applied geometry. His interest in this subject had begun in 1740 when he submitted an essay on the cause of the tides for a prize offered by the French Academy of Sciences. He shared the award with Daniel Bernoulli and Leonhard Euler, both of whom also based their work on a proposition in the Principia. In his account, Maclaurin showed that a homogeneous fluid revolving uniformly about an axis under the action of gravity assumes at equilibrium the shape of an ellipsoid of revolution.

Maclaurin’s concerns were not always centered on mathematics and physics. In 1745, when Charles Edward Stuart, the Young Pretender, landed in Scotland and proclaimed that his father James was rightful king, Maclaurin took an active role in opposing him. When the Young Pretender and his army of Highlanders marched against Edinburgh, Maclaurin helped prepare trenches and barricades for the defense of the city. Despite these efforts, the Jacobite rebels captured the city, and Maclaurin, to avoid submitting to the pretender, was forced to flee to York. Here he found refuge with Archbishop Thomas Herring. When it became clear that the Jacobites were not going to occupy Edinburgh, Maclaurin returned to Scotland, but the energy he had expended in the trench warfare and in the flight to York sapped his strength and severely undermined his health. He died soon after his return to Edinburgh, in 1746, shortly before his forty-eighth birthday. A few hours before his death, he dictated some passages of a work he had been writing on Newton’s philosophy, in which he affirmed his unwavering belief in a future life.

Significance

Colin Maclaurin was the ablest and most spirited of the defenders and developers of Newton’s methods. He was a strong advocate of Newton’s geometrical techniques, and his success in using them influenced other, less able British mathematicians to try to follow. He has been best remembered for his defense of Newton against Berkeley, but he also extended Newton’s work; for example, he applied Newton’s analysis of the gravitation of a sphere to the problem of ellipsoids.

Like Newton, Maclaurin loved geometry. One of the ironies of Maclaurin’s work is that, though he emphasized geometry over analysis, his name is commemorated for the discoveries he made in analysis. Continental mathematicians emphasized analysis over geometry, and some of them—Euler, for example—rejected geometry completely as a basis for the calculus and tried to work solely with algebraic (analytic) functions. Maclaurin, on the other hand, adopted a geometric style in his book on fluxions because of certain logical difficulties that seemed to him to be insurmountable unless one were to use geometry.

Newton and Maclaurin showed the power of the geometric approach to solve many mathematical and physical problems, but this success also had harmful consequences, for it led Britons to follow these geometric methods and to neglect the analytic methods that were being pursued so successfully in the rest of Europe. As a result, most British mathematicians came to think that many problems could be solved without using the calculus. This had the effect of retarding the more powerful analytic methods in Great Britain. Thus, after Maclaurin, British mathematics suffered an eclipse because he and Newton had inadvertently helped steer it into unproductive paths.

Despite the ultimate infertility of Maclaurin’s geometric style, he still had admirers, such as Joseph-Louis Lagrange, the great French mathematician. Lagrange was proud that his famous book on analytic mechanics contained not a single geometric diagram. Maclaurin’s work, on the other hand, dealt largely in lines and figures, and he saw the great book of the universe written in this geometric language. Lagrange, who pictured the universe in terms of numbers and equations, nevertheless appreciated the insight and integrity of Maclaurin, whose work, Lagrange once said, surpassed that of Archimedes—a compliment that would have deeply pleased Maclaurin.

Bibliography

Boyer, Carl B. A History of Mathematics. New York: John Wiley and Sons, 1968. Written by a historian of mathematics, this book is intended to be a basic textbook for students of mathematics. Also appropriate, however, for general readers.

‗‗‗‗‗‗‗. The History of the Calculus and Its Conceptual Development. Mineola, N.Y.: Dover, 1959. An unabridged reprint of the work published in 1949 under the title The Concepts of the Calculus. Boyer traces the development of calculus from antiquity to the twentieth century, and offers a good account of Maclaurin’s role as defender of Newton’s theory of fluxions.

Grabiner, Judith V. “Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions.” American Mathematical Monthly 104, no. 5 (May, 1997): 393. Explains how Maclaurin’s treatise helped to further the development of Newtonian calculus.

Guicciardini, Niccoló. The Development of Newtonian Calculus in Britain, 1700-1800. New York: Cambridge University Press, 1989. A comprehensive survey of the research and teaching of Newtonian calculus, including information on Maclaurin’s ideas about fluxions.

Hedman, Bruce. “Colin Maclaurin’s Quaint Word Problems.” College Mathematics Journal 31, no. 4 (September, 2000): 286. Discusses Maclaurin’s solutions to several algebra word problems, including the Ptolemaic riddle and wage, age, and rate problems.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. Covers major mathematical developments from ancient times through the first few decades of the twentieth century. Kline aims to present the chief ideas that have shaped the history of mathematics rather than focus on individual mathematicians. Consequently, his treatment of Maclaurin emphasizes the principal themes of his work rather than the events of his life.

Maclaurin, Colin. An Account of Sir Isaac Newton’s Philosophical Discoveries. Sources of Science 74. Introduction by L. L. Laundan. New York: Johnson Reprint, 1968. A reprint of a 1748 work on Newton by Maclaurin.

‗‗‗‗‗‗‗. The Collected Letters of Colin Maclaurin. Edited by Stella Mills. Nantwich, England: Shiva, 1982. Maclaurin’s letters provide details of his life and ideas.

Mooney, John, and Ian Stewart. “Colin Maclaurin and Glendaruel.” Mathematical Intelligencer 16, no. 1 (Winter, 1994): 48. Recounts Maclaurin’s life and career by focusing on his hometown, Glendaruel, site of a memorial in his honor.

Turnbull, Herbert Westren. The Great Mathematicians. New York: New York University Press, 1961. This brief but excellent book, a biographical history of mathematics, attempts to show how mathematicians use both imagination and reason to make discoveries.