694 THE POPULAR SCIENCE MONTHLY. 



steadily advanced in public favor, and its work commands the highest 

 respect among all men competent to judge throughout the world, as 

 being not only of direct service to the nation, but as making constant 

 valuable additions to science. 



" Many monographs, bearing the marks of Peirce's individuality 

 and peculiar power, have been read by him before various academies, 

 societies, and institutions ; but only the results of most of them have 

 ever been furnished for publication. Among these may be mentioned 

 an investigation of the forms of stable equilibi'ium for a fluid in an 

 extensible sack floating in another fluid, being an a joriori embryology. 

 Also, the motions of a billiard-ball, an instance in nature of discon- 

 tinuity, when the ball leaves its curve, and goes on a tangent ; another, 

 the motion of a sling, curious from the immense variety of forms com- 

 prised under exceedingly simple uniform conditions. 



" In 1857 he published a volume, summing up the most valuable and 

 most brilliant results of analytical mechanics, interspersing them with 

 original results of his own labor. A year or two later an American 

 student in Germany asked one of the most eminent professors there, 

 what books he would recommend on analytical mechanics : the answer 

 was instantaneous, ' There is nothing fresher and nothing more valu- 

 able than your own Peirce's recent quarto.' In this volume occurs a 

 singular instance of a characteristic which I have already mentioned. 

 Peirce assumes as self-evident that a line which is wholly contained 

 upon a limited surface, but which has neither beginning nor end on 

 that surface, must be a curve reentering upon itself. By means of this 

 hyper-Euclidean axiom he reduces a demonstration, which would other- 

 wise occupy half a dozen pages, to a dozen lines. 



"In 1870, through the 'labors of love' of persons engaged on the 

 Coast Survey, an edition of a hundred lithographed copies was pub- 

 lished, of certain communications to the National Academy upon 

 ' Linear Associative Algebra.' In 1852 Hamilton, of Dublin, had pub- 

 lished his wonderful volume on quaternions ; and this had been fol- 

 lowed by various other attempts to create an algebra more useful in 

 geometrical and physical research than the coordinates of Descartes. 

 Ordinary algebra deals only with quantitative relations, and the object 

 of the arithmetic of lines and of Cartesian coordinates had been to 

 reduce distances and directions to a comparison of quantity. But 

 Hamilton introduced quality also ; and his algebra employed the di- 

 mensions of space, unchanged and essentially diverse, in computation. 

 His imitators and followers had not succeeded in improving or in 

 really adding to his methods. But Peirce, in these communications 

 to the Academy, attacks the problem, according to his wont, Avith 

 astonishing breadth of view and boldness of plan. He begins with a 

 definition of mathematics, shows the variety of processes included in 

 his definition, passes then to its symbols, shows the nature of qualita- 

 tive and of quantitative algebras, and of those which combine the two, 



