November 8, 1901.] 



SCIENCE. 



707 



or merely for an every way bounded piece 

 of space. In the first case we obtain, be- 

 sides the Euclidean, only the three previ- 

 ously mentioned non-Euclidean space-forms. 



" In the second case appears also a mani- 

 foldness, at present not yet dominated, of 

 different space forms. 



''The treatment of continuity and the 

 ratio-idea in Euclid gives occasion for a 

 nearer investigation of the so-called Archi- 

 medes' Axiom. 



" Finally, as the first attempt to illumin- 

 ate in conjunction all the diflerent ques- 

 tions which have grouped themselves about 

 the problem mentioned, and to collect all 

 the means, which numerous mathema- 

 ticians, and not least the author himself, 

 have made for solving the problem, this 

 work will for long retain its value. 



*' That precisely the founding of geometry 

 since the appearance of this book has been 

 advanced in a wholly unexpected way by 

 Hilbert, cannot lessen Killing's merit. His 

 work remains still by far the best means for 

 mastering the researches which have ap- 

 peared in this realm up to 1898." These 

 interesting extracts I take from the Russian 

 pamphlet just issued at Kazan and fur- 

 nished me by my friend Professor Vasiliev. 



In his paper ' Ueber Nicht-Euklidische 

 und Linien-Geometrie ' (Greifswald, 1900), 

 Professor E. Study voices a profound truth 

 when he says : " The conception of geom- 

 etry as an experimental science is only one 

 among many possible, and the standpoint 

 of the empiric is as regards geometry by 

 no means the richest in outlook. For he 

 will not, in his one-sidedness, justly ap- 

 preciate the fact that in manifold and often 

 surprising ways the mathematical sciences 

 are intertwined with one another, that in 

 truth they form an indivisible whole. 



" Although it is possible and indeed 

 highly desirable, that each separate part or 

 theory be developed independently from 

 the others and with the instrumentalities 



peculiar to it, yet whoever should disregard 

 the manifold interdependence of the differ- 

 ent parts, would deprive himself of one 

 of the most powerful instruments of re- 

 search. 



'' This truth, really self-evident yet often 

 not taken to heart, applied to Euclidean 

 and non-Euclidean geometry, leads to the 

 somewhat paradoxical result that, among 

 conditions to a more profound understand- 

 ing of even very elementary parts of the 

 Euclidean geometry, the knowledge of the 

 non-Euclidean geometry cannot be dis- 

 pensed with." 



That the world has caught one deduc- 

 tion from this deep idea, is shown by the 

 fact of the almost simultaneous appearance 

 of two text-books, manuals for class use, to 

 make universally attainable this necessary 

 condition for any thorough understanding 

 of any geometry, even the most elemen- 

 tary ; two intended, available popular treat- 

 ises on this ever more essential non Euclid- 

 ean geometry. 



One of these, just being issued by G. 

 Carre et C. Naud, 3 rue Racine, Paris, 

 is ' La geometric non Euclidienne,' by P. 

 Barbarin, professor at Bordeaux, a place 

 made sacred for non-Euclideans by the 

 memory of Hoiiel. How great and prac- 

 tical is the interest of this book can be 

 gathered from the headings of its chap- 

 ters. 



I. ' General and historical considera- 

 tions.' How the non-Euclidean doctrine 

 was born and gradually developed. 



II. ' Euclid's definitions and postulates.' 

 Study of the role that they play in the 

 principles of geometry. 



Simple and elementary expose of the 

 three geometries after the method of Sac- 

 cheri. 



III. ' Distance as fundamental notion.' 

 The definitions of the straight and the 

 plane according to Cauchy. The works of 

 M. De Tilly. 



