Mr. Thomson was elected in 1815 to the Professorship of Mathematics in the Royal Academical Institution of Belfast, and held the post for seventeen years, building up for himself an excellent reputation as a teacher, and as a clear and accurate writer. Just then analytical methods were beginning to supersede the processes of geometrical demonstration which the form adopted by Newton for the Principia had tended to perpetuate in this country. Laplace was at the height of his fame in France, and was writing the great analytical Principia, his _Mécanique Céleste_, applying the whole force of his genius, and all the resources of the differential and integral calculus invented by Newton and improved by the mathematicians of the intervening century, to the elucidation and extension of the "system of the world," which had been so boldly sketched by the founder of modern physical science. In that period Fourier wrote his memoirs on the conduction of heat, and gave to the world his immortal book to be an inspiration to the physical philosophers of succeeding generations. Legendre had written memoirs which were to lead, in the hands of Jacobi and his successors, to a new province of mathematics, while, in Germany, Gauss had begun his stately march of discovery. The methods and results of this period of mathematical activity were at first hardly known in this country: the slavish devotion of Cambridge to the geometrical processes and the fluxional notation of Newton, an exclusive partiality which Newton himself would have been the first to condemn, led analytical methods, equally Newtonian, to be stigmatised as innovations, because clothed in the unfamiliar garb of the continental notation. A revolt against this was led by Sir John Herschel, Woodhouse, Peacock, and some others at Cambridge, who wrote books which had a great effect in bringing about a change of methods. Sir John thus described the effect of the new movements:--"Students at our universities, fettered by no prejudices, entangled by no habits, and excited by the ardour and emulation of youth, had heard of the existence of masses of knowledge from which they were debarred by the mere accident of position. They required no more. The prestige which magnifies what is unknown, and the attractions inherent in what is forbidden, coincided in their impulse. The books were procured and read, and produced their natural effects. The brows of many a Cambridge examiner were elevated, half in ire, half in admiration, at the unusual answers which began to appear in examination papers. Even moderators are not made of impenetrable stuff, though fenced with sevenfold Jacquier, and tough bull-hide of Vince and Wood."