July 28, 1905.] 



SCIENCE. 



115 



an in its aim. Just on that account, it meets 

 a need long distinctly felt in its field. It does 

 not seek to rival in their own way the logicians 

 and 'arithmetizers,' or the purists like von 

 Staudtj or the Italian geometer-ontologists like 

 Veronese, or the ' visualizers ' like Enriques. 

 It is more in the spirit of such as Poncelet, 

 Steiner and Chasle, who were less concerned 

 with foundations, which to them were obvious 

 enough, than with superstructure. But it is 

 in no sense a slavish imitator of any type or 

 school. It has a way of its own, and it is 

 refreshing in these peering microscopic days 

 to find your cruder intelligence enlisted with- 

 out ado, to be ushered at once into the midst 

 of things and in course of the first score of 

 pages to find the atmosphere charged with 

 such cardinal notions as anharmonic ratio, in- 

 vol^^tion, projective transformation, and pro- 

 jective and involutoric pencils and ranges. A 

 second chapter deals with ' Collineation,' a 

 third with ' Theory of Conies,' a fourth with 

 ' Pencils and Ranges of Conies and the Stein- 

 erian Transformation in Connection with 

 Cubics,' and a final chapter of forty-five pages 

 with ' Applications to Mechanics.' The 

 method, consistently with the emancipated 

 spirit of modern geometry, is now analytic, 

 now geometric and now a combination of 

 them. For the pure mathematician as such. 

 Professor Emch's book can not be regarded 

 as a substitute for the Cremona and Peye 

 classics, but it will serve for much more than 

 an admirable introduction to them. It differs 

 frora them in spirit, content and method. The 

 exposition of the interesting connection be- 

 tween collineations and the surprisingly 

 beautiful doctrine of linkages deserves special 

 mention, as do also the clearness, directness 

 and swiftness of style in which the book is 

 written. 



Mr. Moyer has not aimed to write for the 

 geometrician, but for the student of practical 

 engineering. Hence his book does not say 

 that, if such and such propositions be granted, 

 such and such others will follow as logical 

 consequences. It says, if you desire certain 

 specified results, you should proceed thus and 

 so. From some points of view, it seems a 

 pity that such important practice should be so 



divorced from the theory which constitutes its 

 ground and rational justification. Such de- 

 tachment, however, is not fatal. The Romans 

 were but meager mathematicians, but they 

 were excellent engineers in their day. Never- 

 theless we believe that the interests of both 

 theory and practise would be better served if 

 the instruction offered by Mr. Moyer were com- 

 bined with such a course as that afforded by 

 Professor Emch's book. The time element 

 has of course to be reckoned with. Mr. 

 Meyer's experiment of adopting the notation 

 of mechanical drawing and of introducing 

 many concrete graduated exercises has been 

 eminently successful and these features have 

 been retained in the second edition. 



The above listed books of the calculus are 

 evidently, both of them, products of con- 

 scientious workmanship. Both of them were 

 composed in the composite light of teaching 

 experience and of modern knowledge of the 

 subject. Guided by the needs of his own 

 classes and therefore excluding all topics not 

 directly bearing upon engineering. Professor 

 Campbell has been enabled, without making 

 his book large or cumbrous, to give a brief 

 introduction to mechanics and differential 

 equations and at the same time to dwell upon 

 fundamental notions and processes. In this 

 last regard, his painstaking and repeated ex- 

 planation of the way in which summations 

 and integrals are constituted deserves especial 

 mention. Though written primarily for tech- 

 nological students, the book is far from ill- 

 adapted to the uses of so-called more liberal 

 institutions. Professor Granville's work is of 

 larger range, embracing not only the subjects 

 specially adapted to the needs of the student 

 consciously destined for work in applied sci- 

 ence, but also those that particularly appeal 

 to the student looking towards specialization 

 in pure mathematics. As well in its scope as 

 in its spirit, the work is distinctly more than 

 the author modestly styles it, ' essentially a 

 drill book.' We especially like its graphic 

 treatment of limits and continuity and its con- 

 stant appeal to intuition as an indispensable 

 aid to analysis. The geometric and physical 

 illustrations of the significance of the integra- 

 tion constant are happy, to mention a single 



