*186. Recent developments.* It would be
only confusing to the student to attempt to trace
here the later developments of the science of protective
geometry. It is concerned for the most part with
curves and surfaces of a higher degree than the second.
Purely synthetic methods have been used with marked
success in the study of the straight line in space.
The struggle between analysis and pure geometry has
long since come to an end. Each has its distinct
advantages, and the mathematician who cultivates one
at the expense of the other will never attain the results
that he would attain if both methods were equally
ready to his hand. Pure geometry has to its credit
some of the finest discoveries in mathematics, and
need not apologize for having been born. The
day of its usefulness has not passed with the invention
of abridged notation and of short methods in analysis.
While we may be certain that any geometrical problem
may always be stated in analytic form, it does not
follow that that statement will be simple or easily
interpreted. For many mathematicians the geometric
intuitions are weak, and for such the method will
have little attraction. On the other hand, there
will always be those for whom the subject will have
a peculiar glamor—who will follow with
delight the curious and unexpected relations between
the forms of space. There is a corresponding pleasure,
doubtless, for the analyst in tracing the marvelous
connections between the various fields in which he
wanders, and it is as absurd to shut one’s eyes
to the beauties in one as it is to ignore those in
the other. “Let us cultivate geometry,
then,” says Darboux,(23) “without wishing
in all points to equal it to its rival. Besides,
if we were tempted to neglect it, it would not be
long in finding in the applications of mathematics,
as once it has already done, the means of renewing
its life and of developing itself anew. It is
like the Giant Antaeus, who renewed, his strength by
touching the earth.”

(The numbers refer to the paragraphs)

Abel (1802-1829), 179

Analogy, 24

Analytic geometry, 21, 118, 119, 120, 146, 176, 180

Anharmonic ratio, 46, 161, 184, 185

Apollonius (second half of third century B.C.), 70

Archimedes (287-212 B.C.), 176

Aristotle (384-322 B.C.), 169