/0O7] Henderson — Foundations of Geometry. 15 
has become clear to me that our geometry is incomplete, and 
should receive a correction, which is hypothetical, and if the 
sum of the three angles is equal to one hundred and eighty 
degrees, vanishes. 
“That were the true geometry, the Euclidian, th q practical, 
at least for figures on the earth.”* 
The third name most closely associated in the popular mind 
with the discovery of the non-Euclidian Geometry is that of 
Nicolai Ivanovich Lobatchevsky. This brilliant genius, 
afterwards dubbed by Hoiiel the modern Euclid, was born in 
the year 1793 near Nijni Novgorod on the Volga. He stud- 
ied under the great Bartels, was graduated with distinction, 
became professor of mathematics, and finally rector, of the 
University of Kazan. The manuscripts of certain of his 
works were lost, but fortunately there remains the world- 
famous Geometrical Researches on the Theory of Parallels .f 
While both Gauss and Lobatchevsky' were students of Bar- 
tels, there is even less reason to believe that Gauss contrib- 
uted to Lobatchevsky’s, than that he assisted in Bolyai’s, dis- 
covery of the non-Euclidian geometry. In his New Elements 
of Geometry , we find Lobatchevsky’s clear enunciation: 
“The futility of the efforts which have been made since Euclid’s 
time during the lapse of two thousand years awoke in me the sus- 
picion that the ideas employed might not contain the truth sought 
to be demonstrated. When finally I had convinced myself of the 
correctness of my supposition I wrote a paper on it (assuming the 
infinity of the straight line) . 
“It is easy to show that the straight lines making equal angles 
with a third never meet. 
“Euclid assumed inversely, that two straight lines unequally in- 
clined to a third always meet. 
“To demonstrate this latter assumption, recourse has been had to 
many different procedures. 
£ *The Philosophical Foundations of Mathematics, by Paul Carus; The 
Monist, vol. 13. p. 280. 
tCompare the English translation by G. B. Halsted, published by the 
University of Texas, Austin, 1891. 
