igoy ~\ Henderson — Foundations of Geometry. 3 
geometer H. J. S. Smith, “that the increase of mathematical 
knowledge is a necessary condition for the advancement of 
science, and, if so, a no less necessary condition for the 
improvement of mankind. I could not augur well for the 
enduring intellectual strength of any nation of men, whose 
education was not based on a solid foundation of mathemat- 
ical learning, and whose scientific conceptions, or, in other 
words, whose notions of the world and of the things in it, 
were not bound and girt together with a strong framework of 
mathematical reasoning.” 
In that charming book, cast in the dialogue form and enti- 
tled Euclid and his Modern Rirjals , by the Rev. Charles L. 
Dodgson, the brilliant “Lewis Carroll” of Alice in Wonderland 
fame, Euclid confesses with reluctance that some secret flaw 
lies at the root of the subject of parallel lines. Probabilities, 
not certainties, are all that he has in vindication of his belief. 
Here we lay our fingers on the rift in the lute; in this con- 
fession, we catch a glimpse of that ignis fatuus that mathe- 
maticians have pursued in vain for well-nigh two thousand 
years. Professor G. B. Halsted cites Sohncke* as saying that 
in mathematics there is nothing over which so much has been 
spoken, written, and striven, as over the theory of parallels, 
and all, so far (up to his time), without reaching a definite 
result and decision. It is impossible, says the great Poincare, 
to imagine the vast effort wasted in this chimeric hope, this 
evanescent dream. Indeed, it was not until the nineteenth 
century that the truth began to dawn upon the minds of 
men; and almost simultaneously from the distant frontiers of 
Europe, at Kazan on the Volga and at Maros-Vasarhely in far 
Erdely, there came the startling generalizations that have ten- 
ded to revolutionize our conceptions of geometry, and thrown 
doubts upon the very nature of the space in which we live.f 
* Encyclopcedie der Wissenchaften und Kunste; Von Ersch und Gruber, 
Leipzig, 1838, under “Parallel.” 
tOompare The Value of Non-Euclidian Geometry, by G. B. Halsted; 
Pop. Sci. Monthly, vol. 67, pp. 639-646. At the outset, I wish to acknow- 
