A blink of satisfaction announces my success. I have struck home, I have
found the joint in the armor. The product of minus multiplied by minus
delivers its mysteries to us.

And thus we continued our studies: he, the passive receiver, taking in
the ideas acquired without effort; I, the fierce pioneer, blasting my
rock, the book, with the aid of much sitting up at night, to extract the
diamond, truth. Another and no less arduous task fell to my share: I had
to cut and polish the recondite gem, to strip it of its ruggedness
and present it to my companion's intelligence under a less forbidding
aspect. This diamond cutter's work, which admitted a little light into
the precious stone, was the favorite occupation of my leisure; and I owe
a great deal to it.

The ultimate result was that my pupil passed his examination. As for the
book borrowed by stealth, I restored it to the shelves and replaced it
by another, which, this time, belonged to me.

At my normal school, I had learnt a little elementary geometry under
a master. From the first few lessons onwards, I rather enjoyed the
subject. I divined in it a guide for one's reasoning faculties through
the thickets of the imagination; I caught a glimpse of a search after
truth that did not involve too much stumbling on the way, because each
step forward rests solidly upon the step already taken; I suspected
geometry to be what it preeminently is: a school of intellectual
fencing.

The truth demonstrated and its application matter little to me; what
rouses my enthusiasm is the process that sets the truth before us.
We start from a brilliantly lighted spot and gradually get deeper and
deeper in the darkness, which, in its turn, becomes self-illuminated by
kindling new lights for a higher ascent. This progressive march of
the known toward the unknown, this conscientious lantern lighting what
follows by the rays of what comes before: that was my real business.

Geometry was to teach me the logical progression of thought; it was
to tell me how the difficulties are broken up into sections which,
elucidated consecutively, together form a lever capable of moving the
block that resists any direct efforts; lastly, it showed me how order is
engendered, order, the base of clarity. If it has ever fallen to my lot
to write a page or two which the reader has run over without excessive
fatigue, I owe it, in great part, to geometry, that wonderful teacher
of the art of directing one's thoug