448 



SCIENCE. 



[N. S. Vol. IX. No. 221. 



himself created the most important qf the 

 newer departments of mathematics. 



By the introduction of his concept of con- 

 tinuous groups of transformations he put 

 the isolated integration theories of former 

 mathematicians upon a common basis. The 

 masterly reach of Lie's genius is illustrated 

 by his encompassment of the fundamentally 

 important theory of differential invariants 

 associated with the English names Cayley, 

 Cockle, Sylvester, Forsyth. 



Thirteen years ago Sylvester announced 

 his conception of ' Reciprocants,' a body of 

 differential invariants not for a group, but 

 for a mere interchange of variables. A 

 number of Englishmen thereupon took up 

 investigations about orthogonal, linear and 

 projective groups, groups in whose trans- 

 formations interchanges of variables occur 

 as particular cases, and whose differential 

 invariants are consequently classes of re- 

 ciprocants, and of the analogues of recipro- 

 cants, when more variables than two are 

 considered. 



Now all these investigations were long 

 subsequent to Lie's consideration of the 

 groups in question as leading cases of a 

 general conception. Thus they were merely 

 secondary investigations ! 



Again, the theory of complex numbers 

 appears as a part of the great ' Theorie der 

 Transformationsgruppen.' Indeed, this con- 

 tinent of ' transformations ' opened up and 

 peneti-ated with such giant steps by Lie 

 represents the most remarkable advance 

 which mathematics in all its entirety has 

 made in this latter part of the century. 



Sophus Lie it was who made prominent 

 the importance of the notion of group, and 

 gave the present form to the theory of con- 

 tinuous groups. This idea, like a brilliant 

 dye, has now so permeated the whole fabric 

 of mathematics that Poincare actually iinds 

 that in Euclid ' the idea of the group was 

 potentially pre-existent,' and that he had 

 ' some obscure instinct for it, without reach- 



ing a distinct notion of it.' Thus the last 

 shall be first, and the first last. 



In personal character Lie was our ideal 

 of a genius, approachable, outspoken, un- 

 conventional, yet at times fierce, intractable. 

 His work is cut short; his influence, his 

 fame, will broaden, will tower from day to 

 day. 



George Bruce Halsted. 



Austin, Texas. 



SCIENTIFIC BOOKS. 

 Colour in Nature: A Study in Biology. By 



Marion J. Nbwbigin. London, John 



Murray. 1898. Pp. 344. 



Ou page 300 of this work we read: "We 

 have now completed our general survey of the 

 colours and colouring-matters of organisms. 

 ■s # « xhat the survey as a whole is halting 

 and incomplete must be obvious to all. We 

 have seen that it is as yet impossible to classify 

 pigments in a logical manner ; that most of the 

 problems connected with the subject are en- 

 tirely unsolved." These statements are indeed 

 true ; and yet the book is an interesting and 

 valuable one, and will be of real assistance to 

 the working biologist. 



The whole subject of color in animals and 

 plants has suffered from the fact that it con- 

 cerns the chemist and physicist as well as the 

 biologist, and in these days of intense special- 

 ization it is hard to find anyone competent to 

 treat the matter in all its aspects. Dr. New- 

 bigin has endeavored, with some success, to 

 take all the more important facts into consider- 

 ation ; but it is practically impossible for any 

 one individual to have that intimate acquaint- 

 ance with the vital phenomena of every group 

 of living organisms which is necessary for ,a 

 satisfactory discussion of their coloration. It 

 was Darwin's method to seek the assistance of 

 numerous specialists in different branches, who 

 supplied him with information which he brought 

 together and interpreted in a masterly manner. 

 It may be that Dr. Newbigin has not yet felt 

 justified in asking for such help, but now that 

 she has fairly won her spurs (if one may use 

 such a phrase in regard to a lady) it is not un- 

 reasonable to hope that she will adopt the 

 Darwinian system, and eventually provide us 



