1902 Encyclopedia > Colin Maclaurin

Colin Maclaurin
Scottish mathematician

COLIN MACLAURIN (1698-1746), one of the most eminent among the mathematicians and philosophers that Great Britain has produced, was the son of a clergyman, and born at Kilmodan, Argyllshire, in 1698. At the early age of eleven years he entered the university of Glasgow, where he graduated as master of arts in his sixteenth year. While at the university he exhibited a decided genius for mathematics, more especially for geometry; and it is said that before the end of his six-teenth year he had discovered many of the theorems after-wards published in his Geometria Organica.

In 1717 he was elected professor of mathematics in Marischal College, Aberdeen, as the result of a competitive examination. Two years later he was admitted a fellow of the Royal Society, and in a visit to London made the acquaintance of Newton, whose friendship and esteem he afterwards enjoyed. In 1719 he published his Geometria Organica, sive descriptio linearum curvarum universalis. This work was inspired by the beautiful discoveries of Newton on the organic description of conic sections. In it Maclaurin introduced the well-known method of generat-ing conies which bears his name, and showed that many species of curves of the third and fourth degrees can be described by the intersection of two movable angles. In 1721 he wrote a supplement to the Geometria Organica, which he afterwards published, with extensions, in the Philosophical Transactions for 1735. This paper is principally based on the following general theorem, which is a remarkable extension of Pascal's hexagram :—" If a polygon move so that each of its sides passes through a fixed point, and if all its summits except one describe curves of the degrees m, n, p, &c, respectively, then the free summit moves on a curve of the degree 2mnp .... which reduces to mnp . . . when the fixed points all lie on a right line."

In 1722 Maclaurin travelled as tutor and companion to the eldest son of Lord Polwarth, and after a short stay in Paris resided for some time in Lorraine, where he wrote an essay on the percussion of bodies, which obtained the prize of the French Academy of Science for the year 1724. The following year he was elected professor of mathematics in the university of Edinburgh on the urgent recommenda-tion of Newton. After the death of Newton in 1728, his nephew, Mr Conduitt, applied to Maclaurin for his assist-ance in publishing an account of Newton's life and discoveries. This Maclaurin gladly undertook, but before the account was written the death of Mr Conduitt put a stop to the project. It was not until many years afterwards, and subsequently to Maclaurin's death, that this account of Newton's philosophical discoveries was published (1748).

In 1740 Maclaurin obtained the high distinction of dividing with Euler and Daniel Bernoulli the prize offered by the French Academy of Science for an essay on the flux and reflux of the sea. This important memoir was subsequently revised by him, and inserted in his Treatise on Fluxions, which was published at Edinburgh in 1742, in two volumes. In the preface he states that the work was undertaken in consequence of the attack on the method of fluxions made by Berkeley in 1734, under the title of The Analyst. Maclaurin's object was to found the doc-trine of fluxions on geometrical demonstration, after the manner of Archimedes and the ancient mathematicians, and thus to answer all objections to its method as being founded on false reasoning and full of mystery. He thus laid down the grounds of the fluxional method, regarding fluxions as velocities, after Newton. He proceeded to give an extensive application of the method to curves, surfaces, and the other subjects usually discussed in works on the differential and integral calculus, his treatment being almost exclusively geometrical; but the most valuable part of the work is that devoted to physical applications, in which he embodied his essay on the tides, as stated above.

In this he investigated the attraction of an ellipsoid of revolution, and showed that a homogeneous fluid mass revolving uniformly round an axis under the action of gravity ought to assume the form of an ellipsoid of revolu-tion. The importance of this investigation in connexion with the theory of the tides, the figure of the earth, and other kindred questions has always caused it to be regarded as one of the great problems of mathematical physics. Thus Clairaut, D'Alembert, Lagrange, Legendre, Laplace, Gauss, Ivory, Poisson, Jacobi, Chasles, and other eminent mathematicians have successively attacked the problem, and in doing so have declared their obligations to Maclaurin as the creator of the theory of the attraction of ellipsoids. Lagrange's statement as to Maclaurin's discoveries deserves to be especially cited : after observing that the attraction of a spheroid of revolution is one of the problems in which the method of the ancients has advantages over that of modern analysis, he adds that Maclaurin's investigation is "un chef d'oeuvre de géométrie qu'on peut comparer à tout ce qu' Archimède nous a laissé de plus beau et de plus ingénieux " [1] (Mém. de l'Acad. de Berlin, 1773) . It may be added that Maclaurin was the first to introduce into mechanics, in this discussion, the important conception of surfaces of level, namely, surfaces at each of whose points the total force acts in the normal direction. He also gave in his Fluxions, for the first time, the correct theory for distinguishing between maxima and minima in general, and pointed out the importance of the distinction in the theory of the multiple points of curves.

In 1745, when the rebels, having got between Edinburgh and the king's troops, were marching on that city, Maclaurin took a most prominent part in preparing trenches and barricades for its defence. This occupied him night and day, and the anxiety, fatigue, and cold to which he was thus exposed, affecting a constitution naturally weak, laid the foundation of the disease to which he afterwards succumbed. As soon as the rebel army got possession of Edinburgh, Maclaurin fled to England, to avoid making the submission to the Pretender which was demanded of all who had defended the town. He accepted the invita-tion of Dr Herring, then archbishop of York, with whom he remained until it was safe to return to Edinburgh. From that time his health was broken, and he died of dropsy on June 14, 1746, at Edinburgh, in his forty-eighth year. Maclaurin was married in 1733 to Anne, daughter of Walter Stewart, solicitor-general for Scotland. His eldest son, John, born in 1734, was distinguished as an advocate, and appointed one of the judges of the Scottish Court of Session, with the title of Lord Dreghorn. He inherited an attachment to scientific discovery, and was one of the founders of the Eoyal Society of Edinburgh, in 1782.

After Maclaurin's death his account of Newton's philosophical dis-coveries was published, and also his algebra in 1748. As an appen-dix to the latter appeared his work, De linearum geometricarum pro-prietatibus generoMbus tractatus, a treatise of remarkable elegance. Of the more immediate successors of Newton in Great Britain Maclaurin is probably the only one who can be placed in competi-tion with the great mathematicians of the Continent at the time, and his name will ever be held in remembrance in connexion with his important discoveries. Among his publications in the Philosophical Transactions the following should be noticed :—

(1) "Tractatus de curvarum constructione et mensura, ubi plurimœ series curvarum infinitse vel rectis mensurantur, vel ad simpliores curvas reducantur," May 1718. The series of curves here treated are what are now styled "pedal" curves, which hold an important place in the modern discussion of curves. Maclaurin established many geometrical properties connecting a curve with its pedal. He inves-tigated the properties of the successive pedals of a circle with respect to a point on its circumference, also those of the pedals of curves for which the perpendicular on the tangent varies as some power of the radius vector drawn to the point of contact. (2) "Nova methodus universalis curvas omnes cujus cunque ordinis mechanice describendi sola datorum angulorum et rectarum ope," January 1719. This and the preceding memoir were subsequently enlarged and incorporated by Maclaurin in his Geometria Organica. (3) "On Equations with Impossible Roots," May 1726. (4) On "Continuation of the Same," March 1729. In these papers he gave a proof of Newton's rule for the discovery of the number of imaginary roots of an equation. He added some general results on the limits to the roots, and gave the well-known method''of finding equal roots by aid of the first derived equation. (5) " Observation of the Eclipse of the Sun of February 18, 1737," January 1738. (6) " On the Bases of the Cells where Bees Deposit their Honey," November 1743.

French translations of his Treatise on Fluxions and that on Newton's philosophical discoveries were published at Paris in 1749. His algebra was also translated into French, in 1753. (B. "W.)


[1] " a masterpiece of geometry that one can compare to the most beautiful and ingenious that Archimedes has left us' -- English translation by David Paul Wagner.

The above article was written by Benjamin Williamson, F.R.S., Trinity College, Dublin.

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