ciative 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 fea- 

 tures. 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 numer 

 ous 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 sim- 

 pler 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 marvellous prophecy. The most note- 

 worthy 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 lis- 

 tened 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 compu- 

 tations, 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 in- 

 sight and a keenness of external observation which makes the atom as grand 

 to him as a planet ; a depth of reverence which exalts him while he abases 

 himself. 



