5 22 THE POPWLAR SCIENCE MONTHLY. 



works on logic. Thenceforth was no longer applicable the latter's 

 reproach : " First, logic is the only science which has made no progress 

 since the revival of letters ; secondly, logic is the only science which 

 has produced no growth of symbols." Among De Morgan's many 

 gifts to advancement, perhaps we should select as most important his 

 founding a logic of relatives. But even he thought Boole's calculus 

 of inference the most extraordinary advance ever made in logic. At 

 a single stroke of genius, unheralded, sprang forth an algebra of logic. 

 This stroke shattered the imprisoning magic circle of Aristotle ; and 

 in the last few years four or five new algebras of logic, differing more 

 or less from Boole's, have come into being ; another is now being pub- 

 lished, and I know of two more preparing. I will not attempt here 

 any explanation or eulogy of what has been thus accomplished for 

 logic. This can already be found in English, French, and German. 

 Modern logic will date from Boole. 



But I wish to call attention to the fact that here we find the best, 

 the most satisfactory introduction to the study of modern algebras, 

 modern mathematics. When told that in these systems a product may 

 not vary with each of its factors ; that a product may vanish without 

 either of its factors vanishing ; that subtraction and division may be 

 indefinite ; that, in fact, any system, e. g., quaternions, where the 

 products and powers of the units are themselves linear functions of 

 the units, excludes the ordinary assumption that a product shall vary 

 with each of its factors ; that from q q, = o, it does not follow that 

 either q = oorq, = o; that a quadratic equation, e. g., in quaternions, 

 besides its sixteen roots proper, may have an indefinite number of 

 roots which arise from the fact that the process of division is not 

 a definite one ; when told these, and very many more such, the be- 

 ginner is only too sure to think, " This is a hard saying," and may 

 give up the subject in hopeless confusion. If, however, he will start 

 with Schroeder, " Der Operationskreis des Logikkalkuls," he will find 

 the clearest explanation and illustration of these things contained in 

 his own every-day thoughts about the commonest objects ; and, while 

 learning an elegant logic, will be mastering, perhaps, the most exquisite 

 dual algebra. 



CHEMICAL EXERCISES FOR ORDINARY SCHOOLS. 



By ELIZA A. YOU MANS. 



PROFESSOR RAINS'S little book for beginners in analytical 

 chemistry has been already noticed and commended in these 

 pages, but its method is so excellent that it needs to be more fully 

 explained. 



Chemistry is now a regular study in many if not in most schools ; 



