Maech 11, 1904.] 



SCIENCE. 



411 



verted into the other after the manner of the 

 Chinese puzzle [by cutting it up and rearranging 

 the pieces] ) , or else can be regarded as the dif- 

 ference of polygons capable of this mode of de- 

 composition (this is really the same process, ad- 

 mitting not only positive triangles but also 

 negative triangles ) . 



But we must observe that an analogous state of 

 affairs does not seem to exist in the case of two 

 equivalent polyhedra, so that it becomes a ques- 

 tion whether we can determine the volume of the 

 pyramid, for example, without an appeal more or 

 less disguised to the infinitesimal calculus. It is, 

 then, not certain whether we could dispense with 

 the axiom of Archimedes as easily in the measure- 

 ment of volumes as in that of plane areas. More- 

 over, Professor Hilbert has not attempted it. 



Max. Dehn, a young man of twenty- 

 one, in Mathematische Annalen, Band 55, 

 proved tliat the treatment of equivalence 

 by cutting into a finite number of parts 

 congruent in pairs, can never be extended 

 from two to three dimensions. 



Poincare's review first appeared in Sep- 

 tember, 1902. But on July 1, 1902, I had 

 already presented, before this very section, 

 a complete solution of the question or 

 problem he proposes, the determination of 

 volume without any appeal to the infini- 

 tesimal calculus, without any use of the 

 axiom of Archimedes. 



19. THE TEACHING OF GEOMETRY. 



As Study has said: "Among conditions 

 to a more profound understanding of even 

 very elementary parts of the Euclidean 

 geometry, the knowledge of the non-Euclid- 

 ean geometry can not be dispensed with." 



How shall we make this new creation, 

 so fruitful already for the theory of 

 knowledge, for kenlore, bear fruit for the 

 teaching of geometry? What new ways 

 are opened by this masterful explosion of 

 pure genius, shattering the mirrors which 

 had so dazzlingly protected from percep- 

 tion both the flaws and triumphs of the 

 old Greek's marvelous, if artificial, con- 

 struction ? 



One advance has been safely won and 

 may be rested on. There should be a pre- 

 liminary course of intuitive geometry 

 which does not strive to be rigorotisly 

 demonstrative, which emphasizes the sen- 

 suous rather than the rational, giving full 

 scope for those new fads, the using of pads 

 of squared paper, and the so-called labo- 

 ratory methods so well adapted for the 

 feeble-minded. Hailmann, in his preface, 

 sums up 'the purpose throughout' in these 

 significant words: 'And thus, incidentally, 

 to stimulate genuine vital interest in the 

 study of geometry.' 



I remember Sylvester's smile when he 

 told me he had never owned a mathematical 

 or drawing instrument in his life. 



His great twin brother, Cayley, speaks 

 of space as 'the representation [creation] 

 lying at the foundation of all external ex- 

 perience. ' 'And these objects, points, lines, 

 circles, etc., in the mathematical sense of 

 the terms, have a likeness to, and are rep- 

 resented more or less imperfectly, and 

 from a geometer's point of view, no matter 

 how imperfectly, by corresponding phys- 

 ical points, lines, circles, etc' 



But geometry, always relied upon for 

 training in the logic of science, for teach- 

 ing what demonstration really is, must be 

 made more worthy the world's faith. 

 There is need of a text-book of rational 

 geometry really rigorous, a book to give 

 every clear-headed youth the benefit of his 

 living after Bolyai and Hilbert. 



20. THE NEW RATIONAL GEOMETRY. 



The new system will begin with still 

 simpler ideas than did the great Alex- 

 andrian, for example, the 'betweenness' 

 assumptions; but can confound objectors 

 by avoiding the old matters and methods 

 which have been the chief points of objec- 

 tion and contest. For example, says ]\Ir. 

 Perry, 'I wasted much precious time of 

 my life on the fifth book of Euclid. ' Says 



