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 equilibrium for a fluid in an extensible sack floating in another 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 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 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 reentering 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 coördinates of Descartes. Ordinary algebra deals only with quantitative relations, and the object of the arithmetic of lines and of Cartesian coördinates 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 Academy, 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,