strings, first reduced to mechanical principles by him. This work contains also "Taylor's theorem," the importance of which was not recognised by analysts for over fifty years, until Lagrange pointed out its power. His proof of it does not consider the question of convergency, and is quite worthless. The first rigorous proof was given a century later by Cauchy. Taylor's work contains the first correct explanation of astronomical refraction. He wrote also a work on linear perspective, a treatise which, like his other writings, suffers for want of fulness and clearness of expression. At the age of twenty-three he gave a remarkable solution of the problem of the centre of oscillation, published in 1714. His claim to priority was unjustly disputed by John Bernoulli.

Colin Maclaurin (1698–1746) was elected professor of mathematics 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 R1, R, etc., 1 1 and the system of n tangents in 71, 72, etc., then Σ = Σ OR Or

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 mnp ......., which reduces to mnp... 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

wonderful progress in the higher analysis made on the Continent.

It remains for us to speak of Abraham de Moivre (1667-1754), who was of French descent, but was compelled to leave France at the age of eighteen, on the Revocation of the Edict of Nantes. He settled in London, where he gave lessons in mathematics. He lived to the advanced age of eighty-seven and sank into a state of almost total lethargy. His subsistence was latterly dependent on the solution of questions on games of chance and problems on probabilities, which he was in the habit of giving at a tavern in St. Martin's Lane. Shortly before his death he declared that it was necessary for him to sleep ten or twenty minutes longer every day. The day after he had reached the total of over twenty-three hours, he slept exactly twenty-four hours and then passed away in his sleep. De Moivre enjoyed the friendship of Newton and Halley. His power as a mathematician lay in analytic rather than geometric investigation. He revolutionised higher trigonometry by the discovery of the theorem known by his name and by extending the theorems on the multiplication and division of sectors from the circle to the hyperbola. His work on the theory of probability surpasses anything done by any other mathematician except Laplace. His principal contributions are his investigations respecting the Duration of Play, his Theory of Recurring Series, and his extension of the value of Bernoulli's theorem by the aid of Stirling's theorem.2 His chief works are the Doctrine of Chances, 1716, the Miscellanea Analytica, 1730, and his papers in the Philosophical Transactions.

EULER, LAGRANGE, AND LAPLACE.

During the epoch of ninety years from 1730 to 1820 the French and Swiss cultivated mathematics with most brilliant success. No previous period had shown such an array of illustrious names. At this time Switzerland had her Euler; France, her Lagrange, Laplace, Legendre, and Monge. The mediocrity of French mathematics which marked the time of Louis XIV. was now followed by one of the very brightest periods of all history. England and Germany, on the other hand, which during the unproductive period in France had their Newton and Leibniz, could now boast of no great mathematician. France now waved the mathematical sceptre. Mathematical studies among the English and German people had sunk to the lowest ebb. Among them the direction of original research was ill-chosen. The former adhered with excessive partiality to ancient geometrical methods; the latter produced the combinatorial school, which brought forth nothing of value.

The labours of Euler, Lagrange, and Laplace lay in higher analysis, and this they developed to a wonderful degree. By them analysis came to be completely severed from geometry. During the preceding period the effort of mathematicians not only in England, but, to some extent, even on the continent, had been directed toward the solution of problems clothed in geometric garb, and the results of calculation were usually reduced to geometric form. A change now took place. Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected

an elaborate structure. There are few great ideas pursued by succeeding analysts which were not suggested by Euler, or of which he did not share the honour of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, Lagrange developed the infinitesimal calculus and put analytical mechanics into the form in which we now know it. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labours of Newton, gave a full analytical discussion of the solar system. He also wrote an epoch-marking work on Probability. Among the analytical branches created during this period are the calculus of Variations by Euler and Lagrange, Spherical Harmonics by Laplace and Legendre, and Elliptic Integrals by Legendre.

Comparing the growth of analysis at this time with the growth during the time of Gauss, Cauchy, and recent mathematicians, we observe an important difference. During the former period we witness mainly a development with reference to form. Placing almost implicit confidence in results of calculation, mathematicians did not always pause to discover rigorous proofs, and were thus led to general propositions, some of which have since been found to be true in only special The Combinatorial School in Germany carried this tendency to the greatest extreme; they worshipped formalism and paid no attention to the actual contents of formulæ. But in recent times there has been added to the dexterity in the formal treatment of problems, a much-needed rigour of demonstration. A good example of this increased rigour is seen in the present use of infinite series as compared to that of Euler, and of Lagrange in his earlier works.

cases.

The ostracism of geometry, brought about by the masterminds of this period, could not last permanently. Indeed, a