January 20, 1899.] 



SCIENCE. 



107 



so to speak, of the mathematical sciences. Orig- 

 inally appearing under the special guise of the 

 theory of substitutions and developed iu this 

 form by the labors of Galois, Cauchy, Serret 

 and Jordan with reference chiefly to its applica- 

 tion to the theory of equations, it has of more 

 recent years overleaped at once its soientiflc 

 and its national limitations and, receiving new 

 impulse at the hands of Kronecker and Caylej', 

 has been developed largely by Klein and Lie 

 into one of the chief general instruments of 

 mathematical research. In every branch of 

 mathematics the point of view of the theory of 

 operations is now predominant ; it is employed 

 in almost every form of mathematical investiga- 

 tion, and by the reaction the science is in turn 

 constantly enriched. Conspicuous instances are 

 Klein's theory of the modular equations and 

 Lie's theory of differential equations. 



The number of separate works devoted wholly 

 or in part to the theory of operations is com- 

 pai-atively very small. Serret's Algebra held 

 the field alone down to the appearance in 1S70 

 of Jordan's classical Traite. Netto's Theory of 

 Substitutions, published in 1882, was the first 

 German book on the subject and represents, as 

 regards its special subject, the German (Kro- 

 necker) standpoint down to that date. The 

 American translation (1892) of Netto's book 

 was the first separate work in English to touch 

 the field ; in fact, it was almost the first presenta- 

 tion of the subject in any form in English. In 

 1895-96 appeared the: two volumes of Weber's 

 Algebra, a work the value of which as a sys- 

 tematic and modern treatment of the various 

 branches of algebraic science cannot be over- 

 stated. To this work, rich in other treasures, 

 belongs the distinction of being the first treatise 

 to present the theory of operations in general 

 form independent of the particular content to 

 which the operation might be applied. Closely 

 following the work of Weber, comes now the 

 second English book on the algebra of opera- 

 tions, Burnside's Theory of Groups of Finite 

 Order. Professor Burnside's work is a doubly 

 welcome contribution to the literature of the 

 subject. It not only opens up to the English 

 reader a great and hitherto almost foreign 

 field, but it presents iu a form often original 

 and always valuable the most recent develop- 



ments in that field, to which the author him- 

 self has, in fact, made no insignificant additions. 

 Many portions of the subject, otherwise only to 

 be gathered piecemeal from the journals, are 

 here brought together for the fii'St time in 

 orderly sequence. Proofs have been recast and 

 simplified or extended, and the book contains 

 an abundance of those special 'details and ex- 

 amples, perhaps too familiar in English mathe- 

 matical works, but very acceptable here in the 

 midst of a highly abstract theory. 



To the reader whose vocation or avocations 

 have not lead him into this remote region of 

 serene thought a short excursion among the 

 groups may be instructive and more or less 

 agreeable. Let him, then, first become familiar 

 with the idea of the ' product ' of two opera- 

 tions. This is simply the single operation which 

 alone produces the same effect as the successive 

 performance of the two given operations. If it 

 be asked: "What sort of operations do you 

 mean?" I reply with unction: "Any kind 

 you please, and the more general the concep- 

 tion the better." Algebraic, geometric, phys- 

 ical, chemical, even metaphysical or ' socio- 

 logical ' operations, if nothing better offers, all 

 are taken in one net. But to condescend from 

 this lofty altitude, let us take for an example 

 the rotations of a sphere about its diameters. 

 Choosing any two of them, and applying them 

 successively to the sphere, regarded as a rigid 

 body, the resulting, or resultant, displacement 

 of the sphere is equivalent to a third rotation 

 about a proper diameter. This third rotation 

 is, then, the product of the two given ones. 

 The rotations of the sphere, taken all together 

 as a system, serve also to exemplify the next 

 important notion, that of a 'group.' When a 

 system of operations is so constituted that the 

 product of any two of them is itself an opera- 

 tion of the system, so that the system is a closed 

 one with respect to the process of forming prod- 

 ucts, then if a couple of minor conditions are 

 also satisfied, the system forms a group. And 

 now the theory of operations in its present 

 form concerns itself not with all kinds of opera- 

 tions, but with these groups. Examples of groups 

 are not far to seek, after the idea is grasped. 

 No science is exempt from them ; in mathematics 

 they simply tumble over each other. Transfer- 



