Colin Maclaurin, FRS
Drawing by David Steuart Erskine c. 1795, from a portrait by James Ferguson Born February 1698 Kilmodan, Cowal, Argyll, Scotland Died 14 June 1746 (aged 48) Edinburgh, Scotland Citizenship Great Britain Alma mater University of Glasgow Known for Euler–Maclaurin formula Maclaurin's inequality Maclaurin series Maclaurin spheroid Maclaurin–Cauchy test Braikenridge–Maclaurin theorem Trisectrix of Maclaurin Awards Grand Prize of the French Academy of Sciences • Scientific career Fields Mathematician Institutions Marischal College, University of Aberdeen University of Edinburgh Academic advisors Robert Simson Notable students Robert Adam
Colin Maclaurin, FRS (/məˈklɔːrən/; Scottish Gaelic: Cailean MacLabhruinn; pronunciation?) (February 1698 – 14 June 1746) was a Scottish mathematician whose profound insights reshaped our understanding of geometry and algebra. His brilliance, evident from an astonishingly young age, earned him the distinction of being a child prodigy and the holder of a record for youngest professor that stood for centuries. The Maclaurin series, a fundamental concept in calculus and a specific instance of the more general Taylor series, bears his name, a testament to his enduring legacy.
It's worth noting the evolution of his surname; originally rendered as M'Laurine, the spelling shifted to MacLaurin over time, reflecting changes in orthography. This subtle alteration, however, does nothing to diminish the impact of his mathematical contributions.
Early Life
The origins of this formidable intellect trace back to Kilmodan, in the rugged landscape of Argyll, Scotland. Maclaurin's early years were marked by profound loss; his father, John Maclaurin, a minister in Glendaruel, passed away when Colin was but an infant. His mother’s death followed before he reached the age of nine, leaving him under the guardianship of his uncle, Daniel Maclaurin, also a minister, this time in Kilfinan. It was under this care that the seeds of his extraordinary mathematical talent began to germinate.
Academic Career
At the tender age of eleven, an age when most children are still grappling with the fundamentals of education, Maclaurin entered the hallowed halls of the University of Glasgow. His intellectual prowess was so undeniable that he graduated with a Master of Arts degree a mere three years later, having delivered a compelling thesis on the Power of Gravity. He remained at Glasgow, ostensibly to pursue studies in divinity, until he reached nineteen. It was then that he achieved a feat that would echo through academic history: he was elected professor of mathematics at Marischal College and University in Aberdeen, following a fiercely competitive ten-day examination. This remarkable achievement established him as the youngest professor in the world, a record that remained unbroken until March 2008, when it was officially superseded by Alia Sabur.
During the summer recesses of 1719 and 1721, Maclaurin journeyed to London, a vibrant intellectual hub. There, he found himself in the company of luminaries such as Isaac Newton, Benjamin Hoadly, Samuel Clarke, Martin Folkes, and other distinguished philosophers. His intellectual caliber was recognized, and he was admitted as a member of the prestigious Royal Society.
In 1722, Maclaurin embarked on a significant period of travel on the Continent. He served as a tutor to George Hume, the son of Alexander Hume, 2nd Earl of Marchmont, a role that allowed him to further cultivate his understanding of the world. While residing in Lorraine, he penned his seminal essay on the percussion of bodies, Demonstration des loix du choc des corps. This work so impressed the judges that it was awarded the coveted prize by the Royal Academy of Sciences in 1724. Tragically, his pupil passed away in Montpellier, prompting Maclaurin's return to Aberdeen.
The year 1725 marked another pivotal moment in his career. Through the recommendation of the esteemed Isaac Newton, Maclaurin was appointed deputy to James Gregory, the professor of mathematics at the University of Edinburgh. Gregory himself was the brother of David Gregory and the nephew of the renowned James Gregory. By November 3rd of that year, Maclaurin had assumed the full professorship, succeeding Gregory and immeasurably elevating the university's reputation as a center for scientific inquiry. The depth of Newton's admiration for Maclaurin was such that he reportedly offered to personally cover his salary, a testament to the profound respect he held for the young mathematician's abilities.
Contributions to Mathematics
Maclaurin's intellectual output was as vast as it was impactful. In his Treatise of Fluxions, he masterfully employed Taylor series to delineate the characteristics of maxima, minima, and points of inflection for functions exhibiting infinite differentiability. While he graciously attributed the series to Brook Taylor, it is crucial to acknowledge that the fundamental concept had been explored earlier by luminaries such as Newton and Gregory, and in specific instances, by Madhava of Sangamagrama in fourteenth-century India. Nevertheless, Maclaurin's meticulous application and systematic exposition solidified his claim to recognition, and the Taylor series expanded around the origin (zero) is now widely known as the Maclaurin series.
Colin Maclaurin (1698–1746)
His investigations extended to the intricate subject of the gravitational attraction of ellipsoids, a problem that also captivated the minds of such eminent figures as d'Alembert, A.-C. Clairaut, Euler, Laplace, Legendre, Poisson, and Gauss. Maclaurin's groundbreaking work demonstrated that an oblate spheroid represented a stable equilibrium configuration within Newton's framework of gravity. This area of study continues to hold scientific relevance, with Nobel laureate Subramanyan Chandrasekhar dedicating a significant portion of his book, Ellipsoidal Figures of Equilibrium, to the exploration of Maclaurin spheroids. Maclaurin engaged in extensive correspondence with fellow scholars such as Clairaut, Maupertuis, and d'Ortous de Mairan, fostering a collaborative spirit within the scientific community.
Working independently of, and employing similar methodologies to, Euler, Maclaurin independently discovered the Euler–Maclaurin formula. He leveraged this powerful tool to sum the powers of arithmetic progressions, to derive Stirling's formula, and to establish the Newton–Cotes numerical integration formulas, which notably include Simpson's rule as a specific case. His contributions also extended to the complex domain of elliptic integrals. Maclaurin devised methods to simplify many previously intractable integrals by reformulating them as problems involving the calculation of arcs of hyperbolas. This pioneering work was subsequently built upon by d'Alembert and Euler, who offered more streamlined approaches to these complex calculations.
In his Treatise of Algebra, published posthumously in 1748, Maclaurin presented a rigorous proof for a method of solving systems of linear equations, particularly for cases involving two and three unknowns. He also delved into the complexities of the four-unknown scenario. This publication predated Gabriel Cramer's generalization of the rule for n unknowns, now universally recognized as Cramer's rule, by a mere two years.
Personal Life
In 1733, Colin Maclaurin entered into marriage with Anne Stewart, the daughter of Walter Stewart, who held the influential position of Solicitor General for Scotland. Together, they raised seven children. His eldest son, John Maclaurin, pursued a distinguished career in law, eventually becoming a Senator of the College of Justice and a Lord of Session, under the title Lord Dreghorn. He was also instrumental in co-founding the Royal Society of Edinburgh.
Maclaurin was a staunch opponent of the Jacobite rising of 1745. During this tumultuous period, he actively oversaw the defensive preparations for Edinburgh against the approaching Highland army. He meticulously documented his efforts in a diary, chronicling his exertions both within and beyond the city's walls. However, when the Highland forces ultimately entered Edinburgh, Maclaurin was compelled to flee to York, where he received the hospitality of the Archbishop of York.
The grave of Colin Maclaurin, Greyfriars Kirkyard Colin MacLaurin Road, Edinburgh
The journey south proved arduous. Maclaurin suffered a fall from his horse, and the combined effects of the fatigue, anxiety, and exposure to the cold laid the groundwork for a severe bout of dropsy. Though he managed to return to Edinburgh after the Jacobite forces had marched on, his health had been irrevocably compromised, and he succumbed to his illness shortly thereafter.
His final resting place is Greyfriars Kirkyard in Edinburgh. The tombstone, a simple slab, bears the inscription "C. M. Nat MDCXCVIII Ob MDCCXLVI," a concise epitaph marking his birth and death years. It is situated near the southwest corner of the church, but a more elaborate memorial on the outer wall offers further tribute.
Interestingly, the mathematician and former MIT President Richard Cockburn Maclaurin hailed from the same family lineage.
In recognition of his immense contributions, the Mathematics and Statistics Society at Glasgow University bears his name: The Maclaurin Society (MacSoc). Furthermore, Colin Maclaurin Road, located within the King's Buildings complex of Edinburgh University, serves as a permanent reminder of his intellectual legacy.
Notable Works
The body of work left behind by Colin Maclaurin is a cornerstone of mathematical literature. Among his most significant publications are:
- Geometria Organica – Published in 1720, this early work demonstrated his burgeoning geometric insights.
- De Linearum Geometricarum Proprietatibus – Also from 1720, this treatise further explored the properties of geometric lines.
- Treatise on Fluxions – Published in 1742, this monumental two-volume work, totaling 763 pages, stands as the first comprehensive and systematic exposition of Newton's methods of fluxions. It was a crucial text for disseminating calculus concepts.
- Treatise of Algebra – Published in 1748, two years after his death, this work contained his important contributions to the theory of linear equations.
- Account of Newton's Discoveries – Left incomplete at the time of his death, this work was published posthumously in 1748, shedding light on his understanding of Newton's profound scientific achievements.
- Account of Sir Isaac Newton's philosophical discoveries – This work, published in French in 1749, further elucidated Newton's philosophical underpinnings.
The building housing the Mathematics and Actuarial Mathematics and Statistics departments at Heriot-Watt University in Edinburgh was fittingly named Colin Maclaurin.
French edition of the Treatise of algebra (1748)
French edition of the Account of Sir Isaac Newton's philosophical discoveries (1749)