NOVEMBKB 30, 1900.] 



SCaJKCE. 



845 



geometry, but to be used with a few modifica- 

 tions, and that still also the theorems of solid 

 geometry and mechanics might have approx- 

 imate validity, at least to a quite wide limit, 

 which perhaps yet could be more nearly deter- 

 mined ; I found this evening, just while busied 

 with an attempt to find an entrance to your 

 transcendental trigonometry, and while I could 

 not find in the plane sufiicing determinate func- 

 tions thereto, going on to space constructions, 

 to my no small delight the following demonstra- 

 tion for the Euclidean parallel theory. * * * 



ii * * * Just in the idea to conclude I re- 

 mark still, that the above proof for the Eu- 

 clidean parallel-theory is fallacious. * * * 

 Consequently has here also the hope vanished, 

 to come to a fully decided result, and I must con- 

 tent myself again with the above cited. Withal 

 I believe I have made upon that way at least a 

 step toward your transcendental trigonometry, 

 since I, with aid of the spherical trigonometry, 

 can give the ratios of all constants, at least by 

 construction of the right-angled triangle. I yet 

 lack the actual reckoning of the base of an 

 isosceles triangle from the side, to which I will 

 seek to go from the equilateral triangle." 



As to Gauss's transcendental trigonometry, 

 nothing was ever given about it but its name. 

 Requiescat in pace. 



Yet Gauss writes, April 28, 1817 : 



" Wachter has printed a little piece on the 

 foundations of geometry. 



' ' Though Wachter has penetrated farther into 

 the essence of the matter than his predecessors, 

 yet is his proof not more valid than all others. ' ' 



We come now to an immortal epoch, that of 

 the discovery of the real non-Euclidean geom- 

 etry by Schweikart, and his publication of it 

 under the name of Astral-Geometry. 



On the 25th of January, 1819, Gerling writes 

 to Gauss : 



" Apropos of parallel-theory I must tell you 

 something, and execute a commission. I 

 learned last year that my colleague Schweikart 

 (prof, juris, now Prorector) formerly occupied 

 himself much with mathematics and particu- 

 larly also had written on parallels. 



"So I asked him to lend me his book. While 

 he promised me this, he said to me that now 

 indeed he perceived how errors were present 



in his book (1808) (he had, for example, used 

 quadrilaterals with equal angles as a primary 

 idea), however that he had not ceased to occupy 

 himself with the matter, and was now about 

 convinced that without some datum the Euclid- 

 ean postulate could not be proved, also that 

 it was not improbable to him that our geometry 

 is only a chapter of a more general geometry. 



"Then I told him how you some years ago had 

 openly said that since Euclid's time we had 

 not in this really progressed ; yes, that you 

 had often told me how you through manifold 

 occupation with this matter had not attained to 

 the proof of the absurdity of such a supposi- 

 tion. Then when he sent me the book asked 

 for, the enclosed paper accompanied it, and 

 shortly after (end of December) he asked me 

 orally, when convenient, to enclose to you this 

 paper of his, and to ask you in his name to let 

 him know, when convenient, your judgment on 

 these ideas of his. 



" The book itself has, apart from all else, the 

 advantage that it contains a copious bibli- 

 ography of the subject ; which he also, as he 

 tells me, has not ceased still further to add to." 



Now comes, pp. 180-181, the precious en- 

 closure, dated Marburg, December, 1818, which, 

 though so brief, may fairly be considered the 

 first published (not printed) treatise on non- 

 Euclidean geometry. 



It is a pleasure to give this here in English 

 for the first time. 



The non-Euclidean Geometry of 1818 : By Schwei- 

 kart. 



"There is a two-fold geometry — a geometry 

 in the narrower sense — the Euclidean, and an 

 astral science of magnitude." 



The triangles of the latter have the pecu- 

 liarity, that the sum of the three angles is not 

 equal to two right angles. 



T%is presumed, it can be most rigorously 

 proven : 



(a) That the sum of the three angles in the 

 triangle is less than two right angles ; 



(b) That this sum becomes ever smaller, the 

 more content the triangle encloses ; 



(c) That the altitude of an isosceles right- 

 angled triangle indeed ever increases, the more 

 one lengthens the side; that it, however, cannot 

 surpass a certain line, which I call the constant. 



