114 



SCIENCE. 



[N. S. Vol. XXII. No. 552. 



theories, the motivity and principles of classi- 

 fication are presented in fresh and simple 

 fashion, and the three grand types are made to 

 appear vividly in their true light as together 

 constituting a single exhaustive geometric 

 unit or whole. To this end it is shown, in an 

 introductory chapter, by easy considerations 

 connected with symmetry, reversibility (re- 

 tourndbilite) and curvature, that the concept 

 of super-dimensional spaces is logically indis- 

 pensable. The second chapter accordingly 

 deals with the concept and certain illumi- 

 nating properties of Euclidean space of four 

 dimensions. The essay closes with a third 

 chapter devoted to analogous features of the 

 geometry of spaces of negative curvature. 

 From the simplest and most familiar consider- 

 ations to the most remote and recondite, the 

 reader is conducted swiftly and along geodesic 

 paths, but the scenery along the way is rich 

 and stimulating. We take pleasure in record- 

 ing our judgment, independently formed and 

 concordant with the aiithor's, that the geom- 

 etry of hyperspace is, at least for certain 

 temperaments, genuinely geometric and not 

 merely analytic as some have claimed. The 

 notion of hyperspace, indeed, rests upon 

 spatial intuition, and the higher spaces that 

 reason has created, the imagination may very 

 well yet learn to picture and illuminate. To 

 those who may aspire to a reasonably com- 

 petent acquaintance with the elements of gen- 

 eral geometry, we commend this little book 

 along with Jouffret's ' Traite elementaire de 

 Geometrie a quatre dimensions,' from which 

 the former draws to some extent, Schubert's 

 essay on fourfold space in his ' Mathematical 

 Essays and Eecreations,' and ' Mehrdimen- 

 sionale Geometrie ' by Paul Schoute. 



The late Judge Cooley was accustomed to 

 advise his students never to buy a law book 

 not provided with an index. It wovild be 

 harsh to apply that maxim to Professor 

 Candy's book, for, though it lacks the fea- 

 ture mentioned, it contains others which make 

 it one of the very best among the works of its 

 class. We refer particularly to its emphasis 

 upon the general analytic method as distin- 

 guished from limited methods especially avail- 

 able for the conies, and to its fusion and 



simultaneous treatment of algebra, geometry, 

 equation theory, analytics and calculus, as con- 

 trasted with the usual practice of presenting 

 these subjects separately in succession and of 

 so giving the injurious impression that they 

 are as so many separate instruments, or in- 

 sulated departments, of thought. In the re- 

 spects indicated we believe the book will prove 

 to be something of a pioneer. 



Doctors Smith and Gale ' have endeavored 

 to write a drill book for beginners which pre- 

 sents the elements in a manner conforming 

 with modern ideas.' But these words are ob- 

 viously not to be taken quite literally. For, 

 on the one hand, a book that is relieved by 

 such live and life-giving topics as invariants, 

 parametric equations, conic systems, homo- 

 thetic, similitude and symmetry transforma- 

 tions, inversion, systems of orthogonality, 

 poles, polars and polar reciprocation, can 

 hardly be adequately described as a mere drill 

 book ; and, on the other hand, its modernity, 

 for it is modern in many respects, would have 

 been, in our judgment, not^a little improved 

 by a bold introduction and use of the infinite 

 elements and by allowing imaginaries to play 

 conspicuously their familiar enlightening role. 

 It is true that imaginary elements are not 

 properly intuitable; they lack the property of 

 definitely localizable exteriority, and on that 

 account they tend to confuse at first, but only 

 at first, and afterward they give a light which, 

 if it be invisible to sensuous vision, reason at 

 all events demands. We do not believe in 

 the subordination of the intelligence to sight 

 and imagination. It is to the rational spirit, 

 which beholds many things inaccessible to the 

 ordinary imagination, that analytic geometry, 

 strictly and properly conceived, addresses it- 

 self. And in this subject, the languages of 

 analysis and geometry should be coextensive. 

 We are glad to note that in this work, as in 

 Candy's, the accent falls upon the general 

 analytic raethod rather than on specific curves, 

 as those of second order. The book abounds 

 in concepts, and these after all are the con- 

 stituents of intelligence. These being well 

 formed, ratiocination is easy, while without 

 the former the latter is empty and vain. 



Professor Emch's book is avowedly utilitari- 



