November 30, 1900.] 



SGIENGR 



843 



Gauss's letter of 1779 to Bolyai Parkas the 

 father of John (Bolyai Jd.iios), which I gave 

 years ago in my Bolyai as demonstrative evi- 

 dence that in 1799 Gauss was still trying to 

 prove Euclid's the only non-contradictory sys- 

 tem of geometry, and also the system of objec- 

 tive space. 



The first is false ; the second can never be 

 proven. 



But both these friends kept right on working 

 away at this impossibility, and the more hot- 

 headed of the two, Farkas, finally thought he 

 had succeeded with it, and in 1804 sent to 

 Gauss his ' Gottingen Theory of Parallels.' 

 Gauss's judgment on this is the next thing 

 given (pp. 160-162). He shows the weak spot. 

 "Could you prove, that dkc = ckf=fkg, etc., 

 then were the thing perfect. However, this 

 theorem is indeed true, only difficult, without 

 already presupposing the theory of parallels, to 

 prove rigorously." Thus in 1804 instead of 

 having or giving any light, Gauss throws his 

 friend into despair by intimating that the link 

 missing in his labored attempt is true enough, 

 but difficult to prove without petitio principii. 



Of course we now know it is impossible to 

 prove. 



Anything is impossible to prove which is the 

 equivalent of the parallel postulate. 



Yet both the friends continue their strivings 

 after this impossibility. 



In this very letter Gauss says : "I have in- 

 deed yet ever the hope that those rooks some- 

 time, and indeed before my end, will allow a 

 through passage." 



Farkas on December 27, 1808, writes to 

 Gauss: "Oft thought I, gladly would I, as 

 Jacob for Rachel serve, in order to know the 

 parallels founded even if by another. 



"Now just as I thought it out on Christmas 

 night, while the Catholics were celebrating the 

 birth of the Saviour in the neighboring church, 

 yesterday wrote it down, I send it to you en- 

 closed herewith. 



" To-morrow must I journey out to my land, 

 have no time to revise, neglect I it now, may 

 be a year is lost, or indeed find I the fault, and 

 send it not, as has already happened with hun- 

 dreds, which I as I found them took for gen- 

 uine. Yet it did not come to writing those 



down, probably because they were too long, too 

 difficult, too artificial, but the present I wrote 

 off" at once. As soon as you can, write me your 

 real judgment. " 



This letter Gauss never answered, and never 

 wrote again until 1832, a quarter of a century 

 later, when the non-Euclidean geometry had 

 been published by both Lobach6vski and Bolyai 

 JAnos. 



This settles now forever all question of Gauss 

 having been of the slightest or remotest help 

 or aid to young Jdnos, who in 1823 announced 

 to his father Farkas in a letter still extant, 

 which I saw at the Reformed College in Maros- 

 V4sS,rhely, where Farkas was professor of 

 mathematics, his discovery of the non-Euclid- 

 ean geometry as something undreamed of in 

 the world before. 



This immortal letter, a charming and glorious 

 outpouring of pure young genius, speaks as 

 follows : 



"Temesvir 3 Nov., 1823. 

 ' ' My dear and good father, 



"I have so much to write of my new crea- 

 tions, that it is at the moment impossible for 

 me to enter into great detail, so I write you 

 only on a quarter of a sheet. I await your 

 answer to my letter of two sheets ; and perhaps 

 I would not have written you before receiving 

 it, if I had not wished to address to you the 

 letter I am writing to the Baroness, which 

 letter I pray you to send her. 



" First of all I reply to you in regard to the 

 binominal. 



" Now to something else, so far as space per- 

 mits. I intend to write, as soon as I have put 

 it into order, and when possible to publish, a 

 work on parallels. At this moment it is not 

 yet finished, but the way which I have hit 

 upon promises me with certainty the attain- 

 ment of the goal, if it in general is attainable. 

 It is not yet attained, but I have discovered 

 such magnificent things that I am myself as- 

 tonished at them. 



"It would be damage eternal if they were 

 lost. When you see them, my father, you 

 yourself will acknowledge it. Now I cannot 

 say more of them, only so much : that from 

 nothing I have created another wholly new world. 



