Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

NON-EUCLIDEAN GEOMETRIES 351
the nature of non-Euclidean geometry is at length going
to be made so plain that he who runs may read and understand? I do not; nothing is farther beyond my hope.
Much that others have said I shall omit, and most of
what I shall say has been repeatedly said, in one way or
another, by them; if I succeed in adding only a little light
to that given by the extant literature of the subject, I
shall be quite content.
In Lecture VII, as you will remember, I pointed out
that the term non-Euclidean has two meanings-one of
them specific and usual, the other one generic and less
usual; the former meaning always refers to the theory
of parallels; the latter does not. In the present discussion, the term will be used in the specific sense only.
It is customary to say that non-Euclidean geometry
is a strictly modern discovery, due to the daring genius
of a young Hungarian, John Bolyai, and independently
to that of a Russian, Lobachevski, both of whom flourished in the first half of last century. The discovery, as
I have intimated, was preceded by an immense period of
preparation in which geometricians wrestled with a very
old puzzle-the so-called problem of parallels. If you
will consult Dr. T. L. Heath's superb edition of Euclid's
Elements, you will find that controversial discussion of
that problem began in pre-Euclidean days, was but aggravated instead of terminated by Euclid's handling of
the matter, and, though culminating in the birth of the
new geometry, has continued (among the geometrically
ill informed) down to our own day, a hundred years
after the puzzle was virtually solved by the two pioneers
I have named. There is no tale more romantic, nor, in
the proper sense of the term, more human, in the whole
history of Thought. A human tale, I have said, dis