Newton to Euler

at Aberdeen at the age of nineteen by competitive examination, and in 1725 succeeded James Gregory at the University

of Edinburgh. He enjoyed the friendship of Newton,

and, inspired by Newton's discoveries, he published in 1719 his Geometria Organica, containing a new and remarkable mode of generating conics, known by his name. A second tract, De Linearum geometricarum Proprietatibus, 1720, is remarkable for the elegance of its demonstrations. It is based upon two theorems: the first is the theorem of Cotes; the second is

Maclaurin's: If through any point $O$ a line be drawn meeting the curve in $n$ points, and at these points tangents be drawn, and if any other line through $O$ cut the curve in $R_{1}$ , $R_{2}$ , etc., and the system of $n$ tangents in $r_{1}$ , $r_{2}$ , etc., then $\sum \frac{1}{O R} = \sum \frac{1}{O r}$ . This and Cotes' theorem are generalisations of theorems of Newton. Maclaurin uses these in his treatment of curves of

the second and third degree, culminating in the remarkable theorem that if a quadrangle has its vertices and the two points of intersection of its opposite sides upon a curve of the

third degree, then the tangents drawn at two opposite vertices cut each other on the curve. He deduced independently Pascal's theorem on the hexagram. The following is his extension of this theorem (Phil. Trans., 1735): 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 $m$ , $n$ , $p$ , etc., respectively, then the free summit moves on a curve of the degree $2 m n p \dots$ , which reduces to $m n p \dots$ when the fixed points all lie on a straight line. Maclaurin wrote on

pedal curves. He is the author of an Algebra. The object of his treatise on Fluxions was to found the doctrine of fluxions on geometric demonstrations after the manner of the ancients, and thus, by rigorous exposition, answer such attacks as Berkeley's that the doctrine rested on false reasoning. The Fluxions contained for the first time the correct way of distinguishing between maxima and minima, and explained their use in the

theory of multiple points. "Maclaurin's theorem" was previously given by James Stirling, and is but a particular case

of "Taylor's theorem." Appended to the treatise on Fluxions is the solution of a number of beautiful geometric, mechanical, and astronomical problems, in which he employs ancient methods with such consummate skill as to induce Clairaut to

abandon analytic methods and to attack the problem of the figure of the earth by pure geometry. His solutions commanded the liveliest admiration of Lagrange. Maclaurin investigated

the attraction of the ellipsoid of revolution, and showed that a homogeneous liquid mass revolving uniformly around an axis under the action of gravity must assume the form of an ellipsoid of revolution. Newton had given this

theorem without proof. Notwithstanding the genius of Maclaurin, his influence on the progress of mathematics in Great Britain was unfortunate; for, by his example, he induced his countrymen to neglect analysis and to be indifferent to the