Henderson — Foundations of Geometry. 
11 
ied the problem -all his life. By the aid of the principle of 
continuity, the so-called Theorem of Archimedes, he did prove 
two well known theorems: 
1. In a triang-le, the sum of the three angles can never be 
greater than two right angles. 
2. If the sum of the three angles is equal to tvro right 
angles in one triangle, it is equal to two right 
angles in every triangle. 
But Euclid’s geometry can be built up without the contin- 
uity assumption; and only a short time ago, there was proved 
by Dehn, something that might have been inferred, viz., that 
Legendre’s first theorem does not hold, i. e. not without the 
continuity assumption.* 
In addition to Legendre, there was one other Frenchman, 
Joseph Lagrange, France’s greatest mathematician in his 
day, who attempted to prove Euclid’s parallel-postulate. 
Toward the end of his life, so the story runs, Lagrange com- 
posed a discourse on parallel lines. He began to read it in 
the Academy, but suddenly stopped, and, in confusion, stam- 
mered: “II faut que j’y songe encore” — “I’ll have to think 
about it a while longer.” He stuck his manuscript in his 
pocket, sat down, and never recurred to the subject. 
The first distinct epoch in the history of the non-Euclidian 
geometry begins with the time of the great German mathe- 
matician, Karl Friedrich Gauss. He is in no sense entitled to 
credit as a discoverer in this line, although for many years he 
occupied himself with the problem. The researches he 
claims to have made on the subject have not come down to 
us; but he was closely associated, according to abundant testi- 
mony, with Schweikart and Bolyai, two of the three indepen- 
dent discoverers of the non-Euclidian geometry. The publi- 
cation in 1900 of the eighth volume of Gauss’ Collected Works 
shows, from a letter to Bolyai, the elder, a Hungarian mathe- 
♦CJompare The Foundations of Geometry, by David Hilbert; Translation 
by E. J. Townsend, Open Court Publishing Oo., Chicago. 
