SKETCH OF CARL FBIEDRICH GAUSS. 697 



lytic metliod, 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 im- 

 portant 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 Legen- 

 dre, 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 181G. 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 Arithmetica?," 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 mathemati- 

 cians." The author discovered it by induction before he was 

 eighteen years old, and worked out the first proof which he pub- 

 lished 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 



