lytic method, were exceedingly clear expositions; in them lie liked to discuss the methods and the roads by which he had arrived at his great results. He required the closest attention, and objected to the taking of notes, lest his hearers should lose the thread of his argument. The students seated round the lecture-table listened with delight to the lucid and animated addresses of their master; addresses more resembling conversations than set lectures."
Gauss's writings are upon subjects of arithmetic, algebra, and astronomy. The fullest list, that given in the Royal Society's catalogue of scientific papers, contains one hundred and twenty-four titles, but does not include his largest works. The most important papers are on arithmetic, while only a very few of the number are algebraic, and they all relating to a single theorem. Prof. Cayley remarks that of the memoirs in pure mathematics "it may be safely said that there is not one which has not signally contributed to the progress of the branch of mathematics to which it belongs, or which would not require to be carefully analyzed in a history of the subject." One of his earliest discoveries was "the method of least squares," which, though first published by Legendre, he applied as early as 1795. His first published paper—a thesis for the Doctorate of Philosophy, in 1799—was devoted to the demonstration that every equation has a root; and of this theorem he made two other distinct demonstrations in 1815 and 1816. But these works, though he was the first in the field on the subject, gave him no fame. Lagrange seems not to have heard of the first one; and Cauchy, whose subsequent demonstrations have been preferred, received in France all the praise due to a first discoverer. The "Disquisitiones Arithmeticæ," which is perhaps his principal work, contains many important researches, one of which, known as the celebrated Fundamental Theorem of Gauss, or the law of Quadratic Reciprocity of Legendre, of itself alone. Prof. Tucker says, "would have placed Gauss in the first rank of mathematicians." The author discovered it by induction before he was eighteen years old, and worked out the first proof which he published of it in the following year. He was not satisfied with this, but published other demonstrations resting on different principles, till the number reached six. He had, however, been anticipated in enunciating the theorem, but in a more complex form, by Euler, and Legendre had unsuccessfully attempted to prove it. "The question of priority of enunciation or of demonstrating by induction," says Prof. Tucker, "in this case is a trifling one; any rigorous demonstration of it involved apparently insuperable difficulties." Another discussion involves the theory of describing within a circle the polygon of seventeen sides; another, the theory of the congruence of numbers, or the relation that exists between
