do we pass half the night, in
delightful hours. We cease when sleep begins to weigh too heavily on our
eyelids.

When my comrade returns to his room, does he sleep, careless for the
moment of the shifting scene which we have conjured up? He confesses to
me that he sleeps soundly. This advantage I do not possess. It is not
in my power to pass the sponge over my poor brain even as I pass it
over the blackboard. The network of ideas remains and forms as it were a
moving cobweb in which repose wriggles and tosses, incapable of finding
a stable equilibrium. When sleep does come at last, it is often but a
state of somnolence which, far from suspending the activity of the mind,
actually maintains and quickens it more than waking would. During this
torpor, in which night has not yet closed upon the brain, I sometimes
solve mathematical difficulties with which I struggled unsuccessfully
the day before. A brilliant beacon, of which I am hardly conscious,
flares in my brain. Then I jump out of bed, light my lamp again and
hasten to jot down my solutions, the recollection of which I should
have lost on awakening. Like lightning flashes, those gleams vanish as
suddenly as they appear.

Whence do they come? Probably from a habit which I acquired very early
in life: to have food always there for my mind, to pour the never
failing oil constantly into the lamp of thought. Would you succeed in
the things of the mind? The infallible method is to be always thinking
of them. This method I practiced more sedulously than my comrade; and
hence, no doubt, arose the interchange of positions, the disciple turned
into the master. It was not, however, an overwhelming infatuation, a
painful obsession; it was rather a recreation, almost a poetic feast. As
our great lyric writer put it in the preface to his volume, Les Rayons
et les ombres: 'Mathematics play their part in art as well as in
science. There is algebra in astronomy: astronomy is akin to poetry;
there is algebra in music: music is akin to poetry.'

Is this poetic exaggeration? Surely not: Victor Hugo spoke truly.
Algebra, the poem of order, has magnificent flights. I look upon its
formulae, its strophes as superb, without feeling at all astonished when
others do not agree. My colleague's satirical look came back when I was
imprudent enough to confide my extrageometrical raptures to his ears:
'Nonsense,' said he, 'pure stuff and nonsense! Let's get on with our
tangents.'

The quarterma