Junk 20, 1902.] 



SCIENCE. 



985 



is due to their logical perfection, to the ad- 

 mirable concatenation of the propositions, and 

 to the rigor of the demonstrations. 'II mit 

 dans son livre,' says Montucla, 'cat enchaine- 

 ment si admire par les amateurs de la rigueur 

 geometrique.' "In vain," he adds, "divers 

 geometers whom this arrangem.ent has dis- 

 pleased, have attempted to better it. 



"Their vain efforts have made clear how 

 difficult it is to substitute for the chain made 

 by the Greek geometer another as firm and aa 

 solid." 



'This opinion of the historian of mathemat- 

 ics,' says our author, 'retains all its value 

 even after the researches which geometers 

 have undertaken for about a century to sub- 

 mit the fundamenal principles of the science 

 to an acute and profound examination.' I 

 add that the remarkable discoveries of Dehn 

 (see Science, N. S., Vol. XIV., pp. 711-712) 

 prove an unexpected superiority for Euclid 

 over all successors down to our very day, and 

 suggest the latest advance, which, though as 

 yet unpublished, exists, for under date of 

 April 2, 1902, Hilbert writes me: 'In einer 

 andern Arbeit will ich die Lobatschefski'sche 

 Geometric in der ebene unabhangig von 

 Archimedes begrunden.' That is, Hilbert 

 will found Bolyai's geometry as he has Eu- 

 clid's, without any continuity assumption. 



To get the benefit of this brilliant achieve- 

 ment, I am holding back my own book on this 

 fascinating subject. 



Says Hilbert in his unpublished Vorlesung 

 ueber Euklidische Geometrie, "The order of 

 propositions is important. Mine differs 

 strongly from that usual in text-books of ele- 

 mentary geometry; on the other hand, it 

 greatly agrees with Euclid's order. 



"So fuehren uns diese ganz modernen Un- 

 tersuchungen dazu, den Scharfsinn dieses 

 alten Mathematikers recht zu wuerdigen und 

 aufs hoechst zu bewundem." 



Again, a propos of Euclid's renowned paral- 

 lel postulate, Hilbert says : " What sagacity, 

 what penetration the setting up of this axiom 

 required we best recognize if we look at the 

 history of the axiom of parallels. As to Eu- 

 clid hmself (circa 300 B. C.) he, e. g., proves 

 the theorem of the exterior angle before in- 



troducing the parallel axiom, a sign how 

 deeply he had penetrated in den Zusammen- 

 hang der geometrischen -Saetze." 



Professor Barbarin repeats the exploded 

 error of attributing to Gauss the discovery of 

 the non-Euclidean geometry in 1792. In the 

 introduction to my translation of Bolyai's 

 'Science Absolute of Space,' pp. viii-ix, is a 

 letter from Gauss, on which I there remark: 

 "From this letter we clearly see that in 1799 

 Gauss was still trying to prove that Euclid's 

 is the only non-contradictory system of 

 geometry, and that it is the system regnant 

 in the external space of our physical experi- 

 ence. The first is false; the second can never 

 be proven." 



In. 1804 Gauss writes that in vain he still 

 seeks the unloosing of this Gordian knot. 



Again, with the date April 27, 1813, we 

 read: "In the theory of parallels we are even 

 now not farther than Euclid was. This is 

 the 'partie honteuse' (shameful part) of 

 mathematics, which soon or late must receive 

 a wholly different form." Thus in 1813 there 

 is in Gottingen still no light. 



But in 1812 in Oharkow, the non-Euclidean 

 geometry already had been for the first time 

 consciously created by Schweikart, whose 

 summary characterization of it is given in 

 Science, N. S., Vol. XII., pp. 842-846. This 

 he conmaunicated to Bessel and sent to Gerl- 

 ing and afterward to Gauss in 1818, so that 

 it may claim to be the first published (not 

 printed) treatise on non-Euclidean geometry. 



By this time Gauss had progressed far 

 enough to be willing to signify privately his 

 acceptance of Schweikart's doctrines. 



On p. 15, Barbarin makes a brief argument 

 for Euclid's axiom, 'All right angles are equal.' 



This argument was good before Hilbert 

 and Veronese, since this axiom can never be 

 X)roved by superposition. It is already a 

 consequence of the assumptions preliminary 

 to motion. This profounder analysis Bar- 

 barin has not attained to. He still uses as a 

 postulate and supposes indispensable 'I'inde- 

 formabilite des figures en deplacement.' 

 What Jules Andrade calls 'cette malheureuse 

 et illogique definition' of Legendre, 'the 

 shortest path between two points is a straight 



