*SKETCH OF PROFESSOR BENJAMIN PEIRCE.*

and says he will investigate the general subject 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 defined. The distributive and associative principles in multiplication are adopted but not the commutative; and he confines himself to linear algebras—that is, to those in which every expression is reducible to an algebraic sum of terms each expressive of a single quality. After a full discussion of the general results which must be found in all algebras under these conditions, he begins with single algebras, then double, then triple, and so on up to sextuple, making nearly a hundred algebras which he shows to be possible, and of which he gives the great features. There are almost no comments upon them; and it is only by a patient examination for himself that the reader discovers that, of all these numerous algebras, only three have ever been heard of before. First, of the two single algebras we have one, which is the common algebra, including its simpler form of arithmetic. Secondly, of the three double algebras, we have one, viz., the calculus of Leibnitz and Newton. Thirdly, of over twenty quadruple algebras, only one has been used, the quaternions of Hamilton. Such is a brief abstract of this book of marvelous prophecy. The most noteworthy things which he has done since its publication are a course of Lowell lectures, given about a year ago, on 'Ideality in Science,' and a series of communications to the American Academy, which, it is understood, is still to be continued. In the Lowell lectures he embodies many of his views on philosophy and religion which are peculiarly dear to him, and are always listened to with profound interest, even by those of less religious nature. In the communications to the Academy he is discussing, with all his wonted power, questions of cosmical physics, and particularly theories concerning the source and supply of the sun's heat.

"While Professor Peirce has the tenacity of grasp and power of endurance which enable him to make the most intricate and tedious numerical computations, he is still more distinguished by intensity and fervor of action in every part of his nature, an enthusiasm for whatever is noble and beautiful in the world or in art, in fiction or real life; an exalted moral strength and purity; a glowing imagination which soars into the seventh heavens; an insight and a keenness of external observation which make the atom as grand to him as a planet; a depth of reverence which exalts him while he abases himself."