984 



SCIENCE. 



[N. S. Vol. XV. No. 390. 



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SCIENTIFIC BOOKS. 

 La geometrie Non-Euclidienne. Par P. Bak- 



BAEIN. Paris, 0. Naud. Scientia, Pevrier. 



1903. Phys. Mathematique, No. 15. Pp. 



79. 



It is peculiarly appropriate that from Bor- 

 deaux, made sacred for non-Euclidean geom- 

 etry by Hoiiel, should emanate this beautiful 

 little treatise, decorated with a 'gravure' re- 

 producing part of a manuscript of Euclid, 

 also with the ofiicial portrait of Lobachevski, 

 but best of all, with a portrait of Kiemann. 



It begins from the hackneyed position: 

 'Experience therefore it is which has fur- 

 nished to the ancient geometers a certain 

 number of primitive notions, of axioms, or 

 fundamental postulates put by them at the 

 basis of the science.' But now we know there 

 never was any pure receptivity. In all think- 

 ing enters a creative element. Every bit of 

 experience is in part created by the subject 

 said to receive it, but really in great part 

 making it. 



Professor Barbarin continues: 'From the 

 epoch of Euclid, this number has been re- 

 duced to the strict minimum necessary, and 

 all the others not comprised in this list, being 

 callable of demonstration, are put in the 

 class of theorems.' Now we know that 

 Euclid omits to notice many of the assump- 

 tions he unconsciously employs, for example 

 all the 'betweenness assumptions,' while Hil- 

 bert has at last rigorously demonstrated 

 Euclid's assumption 'All right angles are 

 equal,' and in turn one of Hilbert's assump- 

 tions has just been proved (see Amer. Math. 

 Monthly, April, 1902, pp. 98-101). 



The 'Elements' of Euclid, says Professor 

 Barbarin, enjoyed throughout all the middle 

 ages and still enjoy a celebrity that no other 

 work of science has attained; this celebrity 



