716 



SCIENCE. 



[N. S. Vol. XIV. No. 358. 



which still disgraces the beautifully illus- 

 trated book of Phillips and Fisher of 

 Yale. 



Again, in § 12 Bonola misquotes in a very 

 important particular the title of the only 

 thing Bolyai Jauos ever published, his re- 

 nowned appendix, in which title, instead of 

 ' Johanne Bolyai de eadem,' Bonola has 

 ' Johanne Bolyai de Bolya.' Again in § 8 

 Bonola is still expressing the hope that the 

 examination of the unedited manuscripts of 

 Gauss may show s )me ground for the pre- 

 tence that Gauss had sone part, however 

 minute, in the creations of Bolyai, Loba- 

 chevski and Riemann. But these manu- 

 scripts have already been most sympathet- 

 ically edited by Professor Paul Staeckeb 

 their publication making a goodly quarto, 

 in a review of which for Science under the 

 heading ' Gauss and the non- Euclidean Ge- 

 ometry,'! find they only strengthen the al- 

 ready existing demonstration that neither 

 of the creators of the non-Euclidean geom- 

 etry owed even the minutest fraction of 

 an idea or suggestion to Gauss. 



This is reproven by the correspondence 

 of Gauss and Bolyai Farkas, so sumptu- 

 ously published in royal quarto by the 

 Hungarian Academy of Science, edited by 

 Staeckel and Franz Schmidt, chiefly valu- 

 able for its references to the immortal boy, 

 Bolyai Janos, of whom unfortunately no 

 portrait exists. 



And now a word in conclusion. 



Thinking is important for life. So much 

 so that evolution in thinking has domi- 

 nated all other evolution. In all thinking 

 enters a creative element. There is not 

 any pure receptivity. Nothing can be de- 

 scribed except in terms of a precreated 

 theory. The business of science is the 

 making of these theories, and the continual 

 remaking and bettering of these theories. 

 The higher races of mankind, and chiefly 

 the Greeks, created and elaborated a scheme 

 for dominating what a popular terminology 



calls the facts and laws presented by the 

 spatial relations of things. 



This scheme was only one of an indefinite 

 number of possible schemes, but as coordi- 

 nated and systematized by a great con- 

 structive genius, Euclid, the first professor 

 of mathematics at the University of Alex- 

 andria, it proved so efficient, so effective for 

 life, that all educated men accepted it as 

 part of their common equipment. 



Though it promises no heaven, though it 

 threatens no hell, though it mentions no 

 angels, no devils, yet Euclid's elements of 

 geometry, simply as conveying a necessary 

 instrument for the conduct of civilized life, 

 has appeared in more than one thousand 

 four hundred different editions [Professor 

 Riccardi : Saggio di una bibliografia Eu- 

 clidea (Bologna, ' Memorie ' (5), I., 1890)]. 



Euclid gave to educated mankind a com- 

 mon language for description of the spatial, 

 a common mental basis for thought about 

 extension. Euclid's geometry is a certain 

 theory for a specific natural science, a men- 

 tal construction to explain, to master, to 

 communicate or transmit, and to prophesy 

 certain physical phenomena, the spatial or 

 extensive phenomena. Therefore, the body 

 of its doctrine is a system of theorems de- 

 duced in a logical way from certain un- 

 proven and in part absolutely and finally 

 indemonstrable assumptions. Such a one 

 is the world-renowned parallel-postulate, 

 which is absolutely incapable of being 

 proved in any way whatsoever, mental 

 or physical, speculative or experimental, 

 deductive or inductive. Therefore, to sub- 

 stitute for it a contradiction of it, in 

 Euclid's scheme of fundamental assump- 

 tions, is to get with certainty another 

 equally logical theory to do all that Euclid's 

 geometry has ever done. 



Of such systems each may throw light on 

 the other, each may possess special advan- 

 tages for particular applications. 



But more than that : three such systems 



