Men of Mathematics (16 page)

All this time Pascal was studying incessantly. Before the age of sixteen (about 1639)
^I
he had proved one of the most beautiful theorems in the whole range of geometry. Fortunately it can be described in terms comprehensible to anyone. Sylvester, a mathematician of the nineteenth century whom we shall meet later, called Pascal's great theorem a sort of “cat's cradle.” We state first a special form of the general theorem that can be constructed with the use of a ruler only.

Label two intersecting straight lines
l
and
l'
On
l
take any three distinct points
A, B, C,
and on
l
any three distinct points
A', B', C'.
Join up these points by straight lines, crisscross, as follows: A and B', A' and B, B and C', B' and C, C and A', C' and A. The two lines in each of these pairs intersect in a point. We thus get three points. The special case of Pascal's theorem which we are now describing states that these three points lie on one straight line.

Before giving the general form of the theorem we mention another result like the preceding. This is due to Desargues (1593-1662). If the three straight lines joining corresponding vertices of two triangles
XYZ
and
xyz
meet in a point, then the three intersections of pairs of corresponding sides lie on one straight line. Thus,
if
the straight lines joining
X
and
x, Y
and
y, Z
and
z
meet in a point,
then
the intersections of
XY
and
xy, YZ
and
yz,
ZX
and
zx
lie in one straight line.

In Chapter 2 we stated what a conic section is. Imagine any conic section, for definiteness say an ellipse. On it mark any six points,
A, B, C,
D,
E, F,
and join them up, in this order, by straight lines. We thus have a six-sided figure inscribed in the conic section, in which AB and DE, BC and EF, CD and FA are pairs of opposite sides. The two lines in each of these three pairs intersect in a point; the three
points of intersection lie on one straight line (see figure in Chapter
IS,
page
217).
This is Pascal's theorem; the figure which it furnishes is what he called the “mystic hexagram.” He probably first proved it true for a circle and then passed by projection to any conic section. Only a straightedge and a pair of compasses are required if the reader wishes to see what the figure looks like for a circle.

There are several amazing things about this wonderful proposition, not the least of which is that it was discovered and proved by a boy of sixteen. Again, in his
Essai pour les Coniques
(Essay on Conics), written around his great theorem by this extraordinarily gifted boy, no fewer than 400 propositions on conic sections, including the work of Apollonius and others, were systematically deduced as corollaries, by letting pairs of the six points move into coincidence, so that a chord became a tangent, and other devices. The full
Essai
itself was never published and is apparently lost irretrievably, but Leibniz saw and inspected a copy of it. Further, the
kind
of geometry which Pascal is doing here differs fundamentally from that of the Greeks; it is not
metrical,
but
descriptive,
or
projective.
Magnitudes of lines or angles cut no figure in either the statement or the proof of the theorem. This one theorem in itself suffices to abolish the stupid definition of mathematics, inherited from Aristotle and still sometimes reproduced in dictionaries, as the science of “quantity.” There are no “quantities” in Pascal's geometry.

To see what the
projectivity
of the theorem means, imagine a (circular) cone of light issuing from a point and pass a flat sheet of glass through the cone in varying positions. The boundary curve of the figure in which the sheet cuts the cone is a
conic section.
If Pascal's “mystic hexagram” be drawn on the glass for any given position, and another flat sheet of glass be passed through the cone so that the shadow of the hexagram falls on it,
the shadow will be another “mystic hexagram”
with its three points of intersection of opposite pairs of sides lying on one straight line, the shadow of the “three-point-line” in the original hexagram. That is, Pascal's theorem is
invariant
(unchanged)
under conical projection.
The metrical properties of figures studied in common elementary geometry are
not
invariant under projection; for example, the shadow of a right angle is not a right angle for all positions of the second sheet. It is obvious that this kind of
projective,
or
descriptive
geometry, is one of the geometries naturally adapted to some of the problems of perspective. The
method
of projection
was used by Pascal in proving his theorem, but had been applied previously by Desargues in deducing the result stated above concerning two triangles “in perspective.” Pascal gave Desargues full credit for his great invention.

*  *  *

All this brilliance was purchased at a price. From the age of seventeen to the end of his life at thirty nine, Pascal passed but few days without pain. Acute dyspepsia made his days a torment and chronic insomnia his nights half-waking nightmares. Yet he worked incessantly. At the age of eighteen he invented and made the first calculating machine in history—the ancestor of all the arithmetical machines that have displaced armies of clerks from their jobs in our own generation. We shall see farther on what became of this ingenious device. Five years later, in 1646, Pascal suffered his first “conversion.” It did not take deeply, possibly because Pascal was only twenty three and still absorbed in his mathematics. Up to this time the family had been decently enough devout; now they all seem to have gone mildly insane.

It is difficult for a modern to recreate the intense religious passions which inflamed the seventeenth century, disrupting families and hurling professedly Christian countries and sects at one another's throats. Among the would-be religious reformers of the age was Cornelius Jansen (1585-1638), a flamboyant Dutchman who became bishop of Ypres. A cardinal point of his dogma was the necessity for “conversion” as a means to “grace,” somewhat in the manner of certain flourishing sects today. Salvation, however, at least to an unsympathetic eye, appears to have been the lesser of Jansen's ambitions. God, he was convinced, had especially elected him to blast the Jesuits in this life and toughen them for eternal damnation in the next. This was his call, his mission. His creed was neither Catholicism nor Protestantism, although it leaned rather toward the latter. Its moving spirit was, first, last and all the time, a rabid hatred of those who disputed its dogmatic bigotries. The Pascal family now (1646) ardently—but not too ardently at first—embraced this unlovely creed of Jansenism. Thus Pascal, at the early age of twenty three, began to die off at the top. In the same year his whole digestive tract went bad and he suffered a temporary paralysis. But he was not yet dead intellectually.

His scientific greatness flared up again in 1648 in an entirely new
direction. Carrying on the work of Torricelli (1608-1647) on atmospheric pressure, Pascal surpassed him and demonstrated that he understood the scientific method which Galileo, the teacher of Torricelli, had shown the world. By experiments with the barometer, which he suggested, Pascal proved the familiar facts now known to every beginner in physics regarding the pressure of the atmosphere. Pascal's sister Gilberte had married a Mr. Périer. At Pascal's suggestion, Périer performed the experiment of carrying a barometer up the Puy de Dôme in Auvergne and noting the fall of the column of mercury as the atmospheric pressure decreased. Later Pascal, when he moved to Paris with his sister Jacqueline, repeated the experiment on his own account.

Shortly after Pascal and Jacqueline had returned to Paris they were joined by their father, now fully restored to favor as a state councillor. Presently the family received a somewhat formal visit from Descartes. He and Pascal talked over many things, including the barometer. There was little love lost between the two. For one thing, Descartes had openly refused to believe the famous
Essai pour les coniques
had been written by a boy of sixteen. For another, Descartes suspected Pascal of having filched the idea of the barometric experiments from himself, as he had discussed the possibilities in letters to Mersenne. Pascal, as has been mentioned, had been attending the weekly meetings at Father Mersenne's since he was fourteen. A third ground for dislike on both sides was furnished by their religious antipathies. Descartes, having received nothing but kindness all his life from the Jesuits, loved them; Pascal, following the devoted Jansen, hated a Jesuit worse than the devil is alleged to hate holy water. And finally, according to the candid Jacqueline, both her brother and Descartes were intensely jealous, each of the other. The visit was rather a frigid success.

The good Descartes however did give his young friend some excellent advice in a truly Christian spirit. He told Pascal to follow his own example and lie in bed every day till eleven. For poor Pascal's awful stomach he prescribed a diet of nothing but beef tea. But Pascal ignored the kindly meant advice, possibly because it came from Descartes. Among other things which Pascal totally lacked was a sense of humor.

Jacqueline now began to drag her genius of a brother down—or up; it all depends upon the point of view. In 1648, at the impressionable
age of twenty three, Jacqueline declared her intention of moving to Port Royal, near Paris, the main hangout of the Jansenists in France, to become a nun. Her father sat down heavily on the project, and the devoted Jacqueline concentrated her thwarted efforts on her erring brother. She suspected he was not yet so thoroughly converted as he might have been, and apparently she was right. The family now returned to Clermont for two years.

During these two swift years Pascal seems to have become almost half human, in spite of sister Jacqueline's fluttering admonitions that he surrender himself utterly to the Lord. Even the recalcitrant stomach submitted to rational discipline for a few blessed months.

It is said by some and hotly denied by others that Pascal during this sane interlude and later for a few years discovered the predestined uses of wine and women. He did not sing. But these rumors of a basely human humanity may, after all, be nothing more than rumors. For after his death Pascal quickly passed into the Christian hagiocracy, and any attempts to get at the facts of his life as a human being were quietly but rigidly suppressed by rival factions, one of which strove to prove that he was a devout zealot, the other, a skeptical atheist, but both of which declared that Pascal was a saint not of this earth.

During these adventurous years the morbidly holy Jacqueline continued to work on her frail brother. By a beautiful freak of irony Pascal was presently to be converted—for good, this time—and it was to be
his
lot to turn the tables on his too pious sister and drive
her
into the nunnery which now, perhaps, seemed less desirable. This, of course, is not the orthodox interpretation of what happened; but to anyone other than a blind partisan of one sect or the other—Christian or Atheist—it is a more rational account of the unhealthy relationship between Pascal and his unmarried sister than that which is sanctioned by tradition.

Any modern reader of the
Pensées
must be struck by a certain something or another which either completely escaped our more reticent ancestors or was ignored by them in their wiser charity. The letters, too, reveal a great deal which should have been decently buried. Pascal's ravings in the
Pensées
about “lust” give him away completely, as do also the well-attested facts of his unnatural frenzies at the sight of his married sister Gilberte naturally caressing her children.