of] Henderson — Foundations of Geometry. 17 
As Poincare, perhaps the world’s greatest living- mathema- 
tician, recently said, in his review of Hilbert’s Grundlagen 
der Geometrie\ 
“Lobachevski succeeded in building a logical edifice as 
coherent as the geometry of Euclid, but in which the 
famous postulate is assumed false, and in which the 
sum of the angles of a triangle is always less than 
two right angles. Riemann devised another logical 
system, equally free from contradiction, in which the 
sum is, on the other hand, always greater than two 
right angles. These two geometries, that of Lo- 
bachevski and that of Riemann, are what are called 
the non- Euclidian Geometries. The postulate of 
Euclid then cannot be demonstrated; and this impos- 
sibility is as absolutely certain as any mathematical 
truth whatsoever.”* 
Limits of space forbid more extended treatment of the 
work of Schweikart, of Bolyai, and of Lobatchevsky. By no 
means secondary in interest to the investigations of these 
men are the researches of Riemann upon the Elliptic Geom- 
etry; Cayley’s projective theory of measurement, and the 
Absolute, leading through Klein to the non-Euclidian geom- 
etry; the hypotheses advanced by Clifford to explain the 
nature of the space in which we live; the popular expositions 
of Helmholtz; and Lie’s great group-theoretic structure 
built upon the hypothesis of Zahlenmannifaltigkeit. Nor can 
I enter, at this place, into any discussion of the recent move- 
ment toward the treatment of geometry as a whole from the 
purely synthetic standpoint, inaugurated by Pasch, carried 
on by Peano, Pieri, and Veronese, and crowned by the mas- 
terly work of Hilbert. These modern investigators in what 
has been fittingly termed abstract mathematics have exhibited 
the potency of symbolism in removing from attention the 
•Compare The Value of Non-Euclidian Geometry , by G. B. Halsted; 
Pop. Sci. Monthly, vol. 67, pp. 642-3. 
