er fluid, being an a priori embryology. Also, the motions of a billiard-ball, 

 an instance in nature of discontinuity, when the ball leaves its curve, and 

 goes on a tangent ; another, the motion of a sling, curious from the immense 

 variety of forms comprised 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 Ger- 

 many 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 valuable 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 re-entering upon itself. By means 

 of this hyper-Euclidean axiom he reduces a demonstration which would 

 otherwise 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 published, of certain 

 communications to the "National Academy" upon "Linear Associative 

 Algebra." In 1852 Hamilton of Dublin had published his wonderful volume 

 on Quaternions ; and this had been followed by various other attempts to 

 create an algebra more useful in geometrical and physical research than the 

 co-ordinates of Descartes. Ordinary algebra deals only with quantitative 

 relations; and the object of the Arithmetic of Lines, and of Cartesian 

 co-ordinates, had been to reduce distances and directions to a comparison of 

 quantity. But Hamilton introduced quality also ; and his algebra employed 

 the dimensions 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 Acade- 

 my, attacks the problem, according to his wont, with 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 qualitative and of quantitative algebras, and of 

 those which combine the two, and says he will investigate the general sub- 

 ject of algebra. First, he limits himself in this volume to algebras handling 

 less than seven distinct qualities ; that is, not exceeding six. The notation 

 is then discussed, and the necessary enlargements and modifications of the 

 algebraic signs and symbols are clearly denned. The distributive and asso- 



