846 



SCIENCE. 



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



Squares have consequently the following 

 form : 



Is this constant /or us half the earth's axis (as 

 a consequence of which each line drawn in the 

 universe from one fixed star to another, which 

 are ninety degrees apart from one another, 

 would be a tangent of the earth-sphere), so is it 

 in relation to the spaces occuri-ing in daily life 

 infinitely great. 



The Euclidean geometry holds good only 

 under the presupposition that the constant is 

 infinitely great. Only then is it true that the 

 three angles of every triangle are equal to two 

 right angles ; also this can be easily proved if 

 one takes as given the proposition that the con- 

 stant is infinitely great. 



Such is the brief declaration of independence 

 of this hero. 



Nor were Schweikart's courage and inde- 

 pendence without farther issue. Under his 

 direct influence his own nephew, Taurinus, 

 developed the real non-Euclidean trigonometry 

 and published it in 1825, with successful ap- 

 plications to a number of problems. 



Moreover, this teaching of Schweikart's made 

 converts in high places. 



In the letter of Bessel to Gauss of 10 Feb. , 

 1829 (p. 201), he says : " Through that which 

 Lambert said, and what Schweikart disclosed 

 orally, it has become clear to me that our 

 geometry is incomplete, and should receive a 

 correction, which is hypothetical and, if the 

 sum of the angles of the plane triangle is equal 

 to a hundred and eighty degrees, vanishes. 



" That were the true geometry, the Euclidean 



ihei practical, at least for figures on the earth." 



The complete originality and independence of 

 Schweikart and of Lobachevski are recognized 

 as a matter of course in the correspondence be- 

 tween Gauss and Gerling, who writes, p. 238 : 

 " The Russian steppes seem, therefore, indeed 

 a proper soil for these speculations, for Schwei- 

 kart (now in Konigsberg) invented his ' Astral- 

 Geometry ' while he was in Charkow. ' ' 



This fixes the date of the first conscious crea- 

 tion and naming of the non-Euclidean geom- 

 etry as between 1812 and 1816. 



Gauss adopts and uses for himself this first 

 name, Astral-Geometry (1832, p. 226 ; 1841, p. 

 232). 



At length the true prince comes. On Feb- 

 ruary 14, 1832, Gauss receives the profound 

 treatise of the young Bolyai J&nos, the most 

 marvellous two dozen pages in the history of 

 thought. Under the first impression Gauss 

 writes privately to his pupil and friend Gerling 

 of the ideas and results as ' mit grosser Eleganz 

 entwickelt.' He even says 'I hold this young 

 geometer von Bolyai to be a genius of the first 

 magnitude. ' 



Now was Gauss's chance to connect himself 

 honorably with the non-Euclidean geometry, 

 already independently discovered by Schwei- 

 kart, by Lobach6vski, by Bolyai J4nos. 



Of two utterly worthless theories of parallels 

 Gauss had already given extended notices in 

 in the Gottingische gelehrte Anzeigen (this vol- 

 ume, pp. 170-174 and 183-185). 



To this marvel of J4nos, Gauss vouchsafed 

 never one printed word. 



As Staeckel gently remarks, this certainly 

 contributed thereto, that the worth of this 

 mathematical gem was first recognized when 

 John had long since finished his earthly career. 



The 15th of December, 1902, will be the cen- 

 tenary of the birth of Bolyai J4nos. 



Should not the learned world endeavor to 

 arouse the Magyars to honor Hungary by hon- 

 oring then this truest genius her son ? 



Geoege Bruce Halsted. 



Austin, Texas. 



SCIENTIFIC JOURNALS AND ARTICLES. 

 In the July number of the American Journal 

 of Insanity, Dr. J. G. Eogers, of Indiana, pre- 



