**Colin Maclaurin** was a Scottish mathematician who published the first systematic exposition of Newton's methods, written as a reply to Berkeley's attack on the calculus for its lack of rigorous foundations.

- John Maclaurin was more of a scholar than one would expect of a parish minister, for he had translated the Psalms into Gaelic.
- Colin was the youngest.
- Colin became a student at the University of Glasgow in 1709 at the age of eleven years.
- Certainly Maclaurin's abilities soon began to show at Glasgow University.
- This was the standard text for mathematical study at this time, but Maclaurin studied it on his own, quickly mastering the first six of the thirteen books of the Elements.
- At Glasgow Maclaurin came into contact with Robert Simson who was the Professor of Mathematics there.
- Simson was particularly interested in the geometry of ancient Greece and his enthusiasm for the topic was to influence the young student Maclaurin.
- At the age of 14 Maclaurin was awarded the degree of M.A. after studying Latin, Greek, Logic, Moral Philosophy, Natural Philosophy and Mathematics.
- However, Maclaurin had to defend a thesis in a public examination for the award of this degree (which is not the case today), and he chose On the power of gravity as his topic.
- After graduating with the degree of M.A., Maclaurin remained at the University of Glasgow for a further year to study divinity.
- These were happy years for Maclaurin who studied hard and walked in the nearby hills and mountains for recreation.
- The appointment followed ten days of examinations to find the best candidate and it is clear that, despite there being another outstanding candidate, Maclaurin had the most knowledge of advanced topics.
- Maclaurin was to make two journeys to London, and the first of these he made in 1719.
- Maclaurin had already shown himself a very strong advocate of the mathematical and physical ideas of Newton, so it was natural that they should meet during Maclaurin's visit to London.
- Maclaurin received more than civility from the Royal Society, for he was elected a Fellow of the Royal Society during this visit to London.
- A rather strange event in Maclaurin's career took place during the time he held the chair of mathematics at Aberdeen.
- Maclaurin returned to Aberdeen to discover that the University was most certainly highly displeased that he had not been undertaking his duties for two years.
- It was certainly not the case that Maclaurin had been idle during his time away, for, while in France, he had been awarded a Grand Prize by the Académie des Sciences for his work on the impact of bodies.
- Despite being reinstated to his chair by the University of Aberdeen, Maclaurin sought a position in the University of Edinburgh.
- There is no evidence to suggest that Edinburgh took Newton up on his offer to contribute to Maclaurin's salary.
- Maclaurin began his appointment to the University of Edinburgh on 3 November 1725.
- Maclaurin was to spend the rest of his career in Edinburgh.
- Not long after his marriage, Maclaurin worked to expand the Medical Society of Edinburgh into a wider society to include other branches of learning.
- Maclaurin himself acted as one of the two secretaries of this expanded Society and at the monthly meetings he often read a paper of his own or a letter from a foreign scientist on the latest developments in some topic of current interest.
- This Society would, after Maclaurin's death, become the Royal Society of Edinburgh.
- Maclaurin did notable work in geometry, particularly studying higher plane curves.
- This prize was jointly awarded to Maclaurin, Euler and Daniel Bernoulli, bracketing Maclaurin with the top two mathematicians of his day.
- In 1742 Maclaurin published his 2 volume Treatise of fluxions, the first systematic exposition of Newton's methods written as a reply to Berkeley's attack on the calculus for its lack of rigorous foundations.
- Grabiner gives five areas of influence of Maclaurin's treatise: his treatment of the fundamental theorem of the calculus; his work on maxima and minima; the attraction of ellipsoids; elliptic integrals; and the Euler-Maclaurin summation formula.
- Maclaurin appealed to the geometrical methods of the ancient Greeks and to Archimedes' method of exhaustion in attempting to put Newton's calculus on a rigorous footing.
- It is in the Treatise of fluxions that Maclaurin uses the special case of Taylor series now named after him and for which he is undoubtedly best remembered today.
- The Maclaurin series was not an idea discovered independently of the more general result of Taylor for Maclaurin acknowledges Taylor's contribution.
- Another important result given by Maclaurin, which has not been named after him or any other mathematician, is the important integral test for the convergence of an infinite series.
- The Treatise of fluxions is not simply a work designed to put the calculus on a rigorous basis, for Maclaurin gave many applications of calculus in the work.
- Other topics which Maclaurin wrote on were the annular eclipse of the sun in 1737 and the structure of bees' honeycombs.
- Maclaurin did become involved in controversy with other mathematicians over a number of results.
- Again the argument, which Maclaurin calls "a disagreeable dispute", was about priority.
- We should not only comment on Maclaurin's mathematical research, however, but also on his qualities as a teacher.
- Maclaurin played an active role in the defence of Edinburgh during the Jacobite rebellion of 1745.
- As the Jacobite army marched towards Edinburgh in September 1745, Maclaurin worked endlessly in attempting to prepare the defences of the city.
- Maclaurin fled to England and while in Newcastle he received an invitation from the Archbishop of York to be his guest in York.
- When the Jacobite army marched south from Edinburgh, Maclaurin returned to the city in November 1745.
- Many wrote of Maclaurin's outstanding kindness.
- Maclaurin's Treatise on algebra was published in 1748, two years after his death.
- As mentioned above, Maclaurin is best known for the Maclaurin Series, which is a special case of the Taylor series.
- He is also remembered for the Euler-Maclaurin Summation Formula and for the Maclaurin-Cauchy Integral Test for Convergence which Maclaurin discovered 50 years before Cauchy was born.
- Maclaurin was the first to discover Cramer's Paradox on the intersection of curves.
- In this work Maclaurin considers the geometric problem of finding the difference between the volume of the frustum of a solid of revolution which is generated by a conic section and the volume of the cylinder of the same height as the frustum having diameter equal to that of the frustum at the midpoint of its height.

Born February 1698, Kilmodan (12, km N of Tighnabruaich), Cowal, Argyllshire, Scotland. Died 14 June 1746, Edinburgh, Scotland.

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Algebra, Analysis, Ancient Indian, Astronomy, Origin Scotland, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive