814 



SCIENCE. 



TN. S. Vol. IX. No. 232. 



rigoureuse du tli6oreme des paralleles,' " Lo- 

 batschefskij sprach es klipp und klar aiis, dass 

 das Euklidische Parallelenaxiom niemals werde 

 bewiesen werdeu kouueD, weil es nnbeweisbar 

 sei." 



At the International Mathematical Congress, 

 1893, I maintained in his presence that Felix 

 Klein was utterly in error where in his ' Nicht- 

 Euklidische Geometrie,' I., p. 174, he says of 

 the letter from Gauss to Bolyai Farkas, 1799, 

 " lu this last letter is particularly said that in 

 the hyperbolic geometry there is a maximum 

 for triangle-area;" and again where he says, 

 p. 175, " There can be uo doubt that Lobachev- 

 ski as well as Bolyai owe to Gauss's prompting 

 the initiative of their researches." 



Klein's only answer was that his position 

 would be sustained when the public got access 

 to Gauss's correspoDdeuce. 



Staeckel and Engel have now had complete 

 access to these papers, and this is what Engel 

 says, pp. 428-9 : "But at all events in Gauss's 

 letters there is nowhere a support for this tra- 

 dition ; at no point of these letters can be found 

 even the slightest intimation that Gauss con- 

 nected the discoveries of Lobachevski and J. 

 Bolyai with any direct or roundabout prompt- 

 ing from him. 



' ' On the contrary the letters show (see p. 432 

 f. and Math. Ann. 49, p. 162, Briefwechsel G. 

 B., p. 109) that Gauss throughout recognized 

 the independence of both, exactly .as he recog- 

 nized that of Schweikart, whose independence 

 of Gauss is subject to no doubt. 



"With Staeckel I am at one herein that 

 exactly this circumstance is particularly weighty 

 for the decision of the whole question." 



The whole scientific world will breathe a 

 sigh of relief that Klein's ungenerous Got- 

 tingen legend, mortally wounded in 1893, is in 

 1899 annihilated forever. 



More inexplicable is Klein's bald misinter- 

 pretation of Gauss's letter of 1799 to Bolyai 

 Farkas. I gave this letter in my Bolyai as 

 demonstrative evidence that in 1799 Gauss was 

 still trying to prove Euclid's the only non-con- 

 tradictory system of geometry, and also the 

 system regnant in the external space of our 

 physical experience. The first is false ; the 

 second can never be proven. 



Summing up this same letter, Engel, p. 379, 

 instead of finding in it the hypothetical white 

 elephant of Klein's fairy tale, gives the utmost 

 that can be attributed to it in the following 

 sentence: " Hier ist er also gauz uahe daran, 

 an der Richtigkeit der Geometrie, das heisst, 

 des Euklidischen Paralleleuaxioms zweifelhaft 

 zu werden." 



Five years later, in a letter of November 25, 

 1804, Gauss speaks of a ' group of rocks ' on 

 which his attempts had always been wrecked, 

 and adds : " I have, indeed, still ever the hope 

 that those rocks sometime, and, indeed, before 

 my death, will permit a passage. Meanwhile I 

 have now so many other affairs on hand that at 

 present I cannot think on it, and, believe me, I 

 shall heartily rejoice if you forestall me and if 

 you succeed in surmounting all obstacles." 

 " Surely," says Engel, "that does not sound as- 

 if the authority of Euclid had diminished in 

 power since the year 1799 ; on the contrary, one 

 gets the impression that Gauss in 1804 rather 

 stood more completely under its ban than 

 before." 



This was clearly the view of Bolyai JAnoSj 

 whose autobiography, after quoting Gauss's- 

 letter of 1832, says : " In a previous letter Gauss 

 writes he hopes some time to be able to circum- 

 navigate these rocks — so then he hopes ! ! " 



"These last words," say Staeckel and Engel 

 in the Maihematische Annalen, ' ' show a certain 

 suspicion on the part of John against Gauss. "" 

 But the mention of this earlier letter was highly 

 natural. 



Jauos had known of it from boyhood. The 

 joy of his triumph in solving what had baffled 

 all the world for two thousand years was inten- 

 sified by his knowing that even Gauss had tried 

 and was hoping for the impossible. 



His splendid trumpet call of glory announc- 

 ing his creation of a new universe, scientiam, 

 spatii absolute veram exhibens, is answered 

 how '? Gauss answers that method and re- 

 sults coincide with his own meditations insti- 

 tuted in part since 30-35 years. But of these 

 meditations Gauss had published never a word t 

 How natural then for J4nos to refer to his- 

 previous letter, where he still was hoping to- 

 prove Euclid's parallel postulate. 



The equally complete freedom of Lobach^v- 



