GAUSS AND HIS AMERICAN DESCENDANTS 109 



theory of equations, the calculus of variations, the theory of proba- 

 bility, the geometry of surfaces, and the subject of infinite series. Like 

 Sir Isaac Newton, he at times displayed a disinclination to enter upon 

 a prompt publication of his scientific deductions. As a consequence 

 of this, others rediscovered and published results which Gauss might 

 have claimed for himself. Thus it is now known that some of the dis- 

 coveries on elliptic functions made by Abel and Jacobi had been worked 

 out by Gauss thirty years earlier but not published. According to Pro- 

 fessor Felix Klein, some Gaussian manuscripts reveal a knowledge of 

 the fundamental ideas of quaternions, a subject fully elaborated later 

 by the genius of the Irish astronomer, Sir William Eowan Hamilton. 1 

 Perhaps the most striking case of loss of priority of discovery due to 

 failure to place his results at the disposal of the general scientific public, 

 is that of non-euclidean geometry. For many years Gauss permitted 

 his mind to dwell upon the subtle subject of parallel lines, and he 

 reached some exceedingly original results. But he did not write down 

 in full what he had worked out in his mind, and nothing was published 

 by him on this topic. Off and on he would touch upon this subject in 

 letters to scientific friends. He expressed to them his intention not to 

 allow any part of this research to reach the general public during his 

 lifetime. On January 27, 1829, he wrote to Bessel : "Probably I shall 

 not be ready for a long time yet, to prepare for publication my very 

 extensive researches on this subject and perhaps this will not happen 

 during my lifetime, for I would dread the clamor of the Boeotians, were 

 I to speak out in full." Imagine his surprise when the Hungarian 

 Wolfgang Bolyai, a close friend of his during their student days at the 

 university, sent a printed document of twenty-six pages written by 

 Wolfgang's son, John Bolyai, in which the young Bolyai had worked 

 out with wonderful clearness and originality the fundamental proposi- 

 tions of non-euclidean geometry. Gauss saw at once that he had been 

 anticipated. How did the world-renowned mathematician of Gottingen 

 behave toward the young and unknown Hungarian ? Students of scien- 

 tific history know that on questions of priority of discovery many a 

 bitter battle has been fought. Scientific men are only human, and they 

 frequently fail to see the full merits of rival claimants. But Gauss 

 showed himself as generous as a man as he was great as a scienti>t. 

 After reading John Bolyai's published dissertation, he wrote to his 

 friend Gerling as follows (February 1-4, 1832) : "I consider this young 

 geometer v. Bolyai a genius of the first rank." To his old friend 

 Wolfgang Bolyai, Gauss wrote (March 6, 1832) in this manner: 



1 Professor P. G. Tait declared that Klein is mistaken and that the Gaussian 

 restricted forms of linear and vector operators do not constitute an invention of 

 quaternions. Klein's article is in Math. Annalen, LI., 1898. A note by Tait 

 appeared in Proc. of the Royal Soc. of Edinburgh, December 18, 1899. 



