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

392 MATHEMATICAL PHILOSOPHY
a thing astonishing to all, if any quantity of those
that are the smallest is not capable of being measured,
But it is necessary to draw our inquiry to a close in a
direction the contrary to this, and towards what is
better, according to the proverb. As also happens in
the case of these, when they succeed in learning those
points; for nothing would a geometrician so wonder at,
as if the diameter of a square should be commensurable
with its side.
No one who had grasped the author's thought even fairly
well could have written that. And these two specimens
are typically representative: they serve to exemplify the
comparative merits of the two translations as wholes.
The Grand Continuum.-On first encountering the sort
of contradiction or discord dealt with a moment ago, our
minds are surprised and shocked because we cannot but
believe that there must be, or ought to be, a kind of
magnitude such that a definite part or amount of it just
reaches from a corner to the diagonally opposite corner
of any given square and because we cannot but feel that
geometric lines or curves ought to be so conceived that,
if they cross, they intersect, or have a point (or points)
in common. What has the human mind done about it?
What has it done to secure release from the kind of discord in question and so to enlarge the sphere of intellectual
harmony? What it has done is this: it has assumed or
created the kind of magnitude required. This new sort
of magnitude is, of course, not sensible; it is conceptual,
but it is not of the type of the mathematical continuum
of first order, for it is in this type that the difficulties
to be overcome have their roots; the structure of the new
variety of magnitude is patterned on the structure of the
mathematical continuum of second order: this latter
continuum is the mathematical continuum proper-the