14 
Journal of the Mitchell Society. 
book.” On the publication of volume 8 of Gauss’s Collected 
Works , in 1900, light is at last thrown upon Schweikart’s 
discovery. Here we find Gerling’s actual letter to Gauss, 
written in 1819, in which he says, among other things: 
“Apropos of the parallel-theory, I learned last year 
that my colleague Schweikart had written on paral- 
lels He said that he was now about convinced that 
without some datum the Euclidian postulate could not be 
proved, also that it was not improbable to him that our geom- 
etry is only a chapter of a more general geometry.”* En- 
closed in this letter was a paper by Schweikart, dated Mar- 
burg, December, 1818. From this we learn: 
“There is a two-fold geometry — a geometry in the nar- 
rower sense— the Euclidian, and an astral- science of 
magnitude. 
“The triangles of the latter have the peculiarity, 
that the sum of the three angles (of a triangle) is 
not equal to two right angles. 
‘ This presumed, it can be most rigorously proven: 
(a) That the sum of the three angles in the tri- 
angle is less than two right' angles; 
(b) That this sum becomes ever smaller, the more 
content the angle encloses; 
(c) That the altitude of an isoscles right angled 
triangle indeed ever increases, the more one length- 
ens the side; that it, however, cannot surpass a cer- 
tain line, which I call the constant .” 
It can be easily proved that if this constant is infinitely 
great, then, and then only, is the sum of the three angles of ; 
every triangle equal to two right angles. 
That the doctrine made converts in high places is evidenced ; 
by Bessel’s letter to Gauss, Feb. 10, 1829: “Through that 
which Lambert said, and what Schweikart disclosed orally, it 
*Gauss and the non-Euclidian Geometry ; 'by G. B. Halsted; Science, N. S. 
Vol. XII, No. 309, pp. 842-846, Nov. 80, 1900. 
