652 



SCIENTIFIC THOUGHT. 



or philosophical one viz., the clearer definition of the 

 assumptions or principles which underlie arithmetical 

 and algebraical reasoning. And if algebraical, then also 

 geometrical reasoning. Both problems seem to have 

 presented themselves to the youthful mind of Gauss, 

 as is evident from his correspondence with Bessel 1 and 

 Schumacher, and from his direct influence on Bolyai, 2 

 Mobius, and Von Staudt, perhaps also indirectly on 

 Lobatchevsky. 3 It does not, however, appear as if he 



1 See especially the letters of 

 Gauss to Bessel, dated November 

 and December 1811 and May 1812 

 ( ' Briefwechsel,' Leipzig, 1880, p. 

 151 sqq.) 



2 Bolyai, the elder (1775-1856), 

 was a student friend of Gauss in 

 the} years 1797 to 1799, and kept 

 up a correspondence with him dur- 

 ing half a century. This correspon - 

 dence has now been published by 

 F. Schmidt and P. Stackel, Leipzig, 

 1899, with a supplement containing 

 some information about this extra- 

 ordinary man. His son, Johann 

 Bolyai (1802-60), is the author of 

 the celebrated " Appendix, scien- 

 tiam spatii absolute veram ex- 

 hibens," which was attached to 

 his father's 'Tentamen, juven- 

 tutem ... in elementa matheseos 

 pune . . . introducendi,' 1832. 

 The tract seems to have been 

 written hi 1823. A translation, 

 with introduction, has been pub- 

 lished by Dr G. Bruce Halsted 

 ('Neomonic Series,' vol. iii. 4th ed., 

 Austin, Texas, 1896). When the 

 elder Bolyai sent to Gauss in the 

 year 1831 to 1832 a copy of his 

 son's tract and of his own work on 

 Geometry, Gauss expressed great 

 surprise at the contents of the 

 former. (See his letter of March 

 6, 1832.) His remarks that the 

 younger Bolyai had anticipated 

 some of his own ideas on the 



subject, remind one of a similar 

 remark which he made, May 30, 

 1828, to Schumacher with refer- 

 ence to Abel's " Memoir on Elliptic 

 Functions " in vol. ii. of Crelle's 

 'Journal* (see Gauss, 'Werke,' vol. 

 iii. p. 495). In both cases he felt 

 himself relieved from the necessity 

 of publishing his own results, 

 though, so far as those referring 

 to the foundations of geometry are 

 concerned, it does not appear that 

 his ideas had arrived at that state 

 of maturity which the publication 

 of his posthumous papers has 

 proved to have been attained in 

 his treatment of the higher func- 

 tions. Indeed little or nothing of 

 prime importance has been found 

 among his papers referring to the 

 principles of geometry ; and he 

 stated to Bolyai that though he 

 had intended to commit his views 

 to paper, so that they should not 

 be lost, he had not intended to 

 publish anything during his life- 

 time. 



3 It is doubtful whether Gauss's 

 speculations had any influence on 

 the younger Bolyai's theory, and 

 still more so as regards Lobat- 

 chevsky, whose first tract ap- 

 peared in the ' Kazan Messenger,' 

 1829 to 1830, but dates back 

 probably to 1826. Inasmuch, 

 however, as the younger Bolyai 

 must have become acquainted 



