SKETCH OF PROFESSOR BENJAMIN PEIRCE. 695 



and says he will investigate the general subject of algebra. First, he 

 limits himself in this volume to algebras handling less than seven dis- 

 tinct qualities that is, not exceeding six. The notation is then dis- 

 cussed, and the necessary enlargements and modifications of the alge- 

 braic signs and symbols are clearly defined. The distributive and 

 associative principles in multiplication are adopted} but not the com- 

 mutative ; and he confines himself to linear algebras that is, to 

 those in which every expression is reducible to an algebraic sum of 

 terras 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 examina- 

 tion 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 alge- 

 bras, we have one, viz., the calculus of Leibnitz and Newton. Thirdly, 

 of over twenty quadruple algebras, only one has been used, the qua- 

 ternions of Hamilton. Such is a brief abstract of this book of mar- 

 velous 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 contin- 

 ued. In the Lowell lectures he embodies many of his views on phi- 

 losophy and religion which are peculiarly dear to him, and are always 

 listened to with profound interest, even by those of less religious na- 

 ture. In the communications to the Academy lie is discussing, with 

 all his wonted power, questions of cosraical 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 woi'ld 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 plan- 

 et ; a depth of reverence which exalts him while he abases himself." 



