June 9, 1899.] 



SCIENCE. 



815- 



ski from the slightest idea that Gaviss had ever 

 meditated anything diflereut from the rest of 

 the world on the matter of parallels is demon- 

 strated most happily. 



Bartels, the teacher of Lobach6vski, never 

 saw Gauss after 1807, received at Kazan one 

 letter from him in ISOS, probably a mere 

 friendly epistle containing nothing mathemat- 

 ical, and not another word during his entire 

 stay there. 



But in November, 1808, Schumacher, in Got- 

 tingen, writes in his diary that Gauss has re- 

 duced the theory of parallels to this, that if the 

 accepted theory were not true there must be a 

 constant a priori of length, ' welches absurd 

 ist,' yet that Gauss himself considers this work 

 not yet completed. 



Thus in 1808 Gauss still vacillates. The 

 proposition about the a priori given unit for 

 length is due to Lambert, 1766, and on the 

 supposed absurdity Legendre in 1794 had 

 founded a pseudo-proof of the parallel postulate. 



Thus until after 1808 Gauss had made no 

 advance beyond the ordinary text books. 



A most fortunate piece of personal testimony 

 from the distinguished astronomer Otto Struve 

 finishes the whole matter. 



When at Dorpat in 1835 and 1836 Struve was 

 attending his lectures, Bartels repeatedly spoke 

 of Lobachevski as one of his first and most 

 gifted scholars in Kazan. 



Lobachevski had then' already sent his first 

 works on non Euclidean geometry to Bartels, 

 but, as Struve writes, Bartels looked upon these 

 works ' more as interesting, ingenious specula- 

 tions than as a work advancing science.' 



Struve adds he does not recall that Bartels 

 ever spoke of any accordant ideas of Gauss. 



Such misconception of the import of non- 

 Euclidean geometry was due iu part to that 

 lack of grit or slip in judgment which let Lo- 

 bachevski damn this child of his genius with 

 the name 'Imaginary Geometry.' 



If Lobachevski had possessed the magnificent 

 Magyar mettle of Bolyai Jiinos, and dared to 

 name his creation the Science Absolute of Space, 

 he would not have taught mathematics with 

 ability throughout his life without making a 

 single disciple. 



His 'New Elements of Geometry,' here at last 



made accessible to the world, is such a master 

 piece that it remains to-day the completest and 

 most satisfactory textbook of non-Euclidean 

 geometry. Written at the flood of hope and 

 confidence, with ardor still undampened, it is 

 in his ' New Elements ' preeminently that the 

 great Russian allows free expression to his pro- 

 found philosophic insight, which, on the one 

 hand, shatters forever Kant's doctrine of our 

 absolute a priori knowledge of all fundamental 

 spatial properties, while, on the other hand, 

 emphasizing the essential relativity of space, 

 and the element of human construction, human 

 creation in it. 



Lobachevski's position is still, after sixty 

 years, the necessary philosophy for science. No 

 one has succeeded in finding any escape from 

 its cogency. No one has gone beyond it. 



Our hereditary geometry, the Euclidean, is 

 underivable from real experience alone, and 

 can never be jiroved by experience. Not only 

 can the truth or falsity of Euclid's parallel 

 postulate never be proved a priori; not even a 

 posteriori can ever its truth be proved. There- 

 fore, Euclidean geometry, in so far as Euclid- 

 ean, must ever remain a creation of the human: 

 mind. 



The introduction to the ' New Elements ' con- 

 tains a piercing critique of Legendre's attempts 

 on the parallel-postulate. 



Here at times Lobachevski almost conde- 

 scends to be humorous. For example, he says :_ 

 "Although Legendre designates his demonstra- 

 tion as completely rigorous, he, without doubt, 

 thought otherwise, for he adds the proviso that 

 a difliculty which one would perhaps still find 

 can always be removed. For this he has re- 

 course to calculations founded on the first 

 familiar equations of rectilinear trigonometry, 

 which it would be necessary previously to 

 establish, and which just in this case are useless 

 and lead to no result." 



Here for the word trigonometry in the Rus- 

 sian of the ' Collected Works,' p. 222, Engel has 

 substituted, p. 70, by some slip, the word 

 geometry' Further on Lobachevski continues : 

 " But Legendre has not noticed here that EP 

 may possibly not- meet AC. To overcome this 

 little difficulty you have only to suppose that EF 

 is the perpendicular from F on BD ; but then. 



