844 



SCIENCE. 



[N. S. Vol. XII. No. 309. 



All that I have hitherto sent you compares to 

 this only as a house of cards to a castle. 



" P. 8. I dare to judge absolutely and with 

 conviction of these works of my spirit before 

 you, my father ; I do not fear from you any 

 false interpretation (that certainly I would not 

 merit), which signifies that, in certain regards, 

 I consider you as a second self. ' ' 



In his autobiography J4nos says: "First 

 in the year 1823 did I completely penetrate 

 through the problem according to its essential 

 nature, though also afterward further com- 

 pletions came thereto. I communicated in the 

 year 1825 to my former teacher, Herrn Johann 

 Walter von Eckwehr (later imperial-royal gen- 

 eral), a written paper, which is still in his hands. 

 On the prompting of my father I translated my 

 paper into Latin, in which it appeared as Ap- 

 pendix to the Tentamenm 1832." 



So much for Bolyai. 



The equally complete freedom of Lobach6v- 

 ski from the slightest idea that Gauss had ever 

 meditated anything different from the rest of 

 the world on the matter of parallels I showed 

 in Science, Vol. IX., No. 232, pp. 813-817. 



Passing on to the next section, pp. 163-164, 

 in the new volume of Gauss, we find it impor- 

 tant as showing that in 1805 Gauss was still a 

 baby on this subject. It is an erroneous pseudo- 

 proof of the impossibility of what in 1733 Sac- 

 cheri had called 'hypothesis anguli obtusi.' 

 To be sure Saccheri himself thought he had 

 proved this hypothesis inadmissible, so that 

 Gauss blundered in good company ; but his 

 pupil Riemann in 1854 showed that this hy- 

 pothesis gives a beautiful non-Euclidean geom- 

 etry, a new universal space, now justly called 

 the space of Eiemann. 



Passing on, we find that in 1808, Schumacher 

 writes: " Gauss has led back the theory of par- 

 allels to this, that if the accepted theory were 

 not true, there must be a constant h priori line 

 given in length, which is absurd. Yet he him- 

 self considers this work still not conclusive." 



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 difierent 

 form." 



Thus in 1813 there is still no light. 



In April, 1816, Wachter, on a visit to Got- 

 tingen, had a conversation with Gauss whose 

 subject was what he calls the anti-Euclidean 

 geometry. On December 12, 1816, he writes 

 to Gauss a letter which shows that this anti- 

 Euclidean geometry, as he understands it, was 

 far from being the non-Euclidean geometry of 

 Lobachevski and Bolyai J&nos. 



The letter as here given by Staeckel, pp. 

 175-176, is as follows : 



* * * " Consequently the anti-Euclidean or 

 your geometry would be true. However, the 

 constant in it remains undetermined : why ? 

 may perhaps be made comprehensible by the 

 following : 



" * * * The result of the foregoing may con- 

 sequently be so expressed : 



" The Euclidean geometry is false ; but never- 

 theless the true geometry must begin with the 

 same eleventh Euclidean axiom or with the as- 

 sumption of lines and surfaces which have the 

 property presumed in that axiom. 



" Only instead of the straight line and plane 

 are to be put the great circle of that sphere de- 

 scribed with infinite radius together with its 

 surface. 



"From this comes indeed the one inconve- 

 nience, that the parts of this surface are merely 

 symmetric, not, as with the plane, congruent ; 

 or that the radius out on the one side is infinite, 

 on the other imaginary. Only it is clear how 

 that inconvenience is again overbalanced by 

 many other advantages which the construction 

 on a spherical surface offers ; so that probably 

 also then even, if the Euclidean geometry were 

 true, the necessity no longer indeed exists to 

 consider the plane as an infinite spherical sur- 

 face, though still the fruitfulness of this view 

 might recommend it. 



" Only, as I thought through all this, as I had 

 already fully satisfied myself about the result, in 

 part since I believed I had recognized the ground 

 (la metaphysique) of that indeterminateness 

 necessarily inherent in geometry — also even 

 the complete indecision in this matter, then, if 

 that proof against the Euclidean geometry, as 

 I could not expect, were not to be considered 

 as stringent ; in part, so not to consider as 

 lost all the many previous researches in plane 



