Blaise Pascal (1623 - 1662)
From `A Short Account of the History of Mathematics' (4th edition, 1908)
by W. W. Rouse Ball.
Among the contemporaries of Descartes none displayed greater natural
genius than Pascal, but his mathematical reputation rests more on what
he might have done than on what he actually effected, as during a
considerable part of his life he deemed it his duty to devote his whole
time to religious exercises.
Blaise Pascal was born at Clermont on June 19, 1623, and died
at Paris on Aug. 19, 1662. His father, a local judge at Clermont, and
himself of some scientific reputation, moved to Paris in 1631, partly
to prosecute his own scientific studies, partly to carry on the
education of his only son, who had already displayed exceptional
ability. Pascal was kept at home in order to ensure his not being
overworked, and with the same object it was directed that his education
should be at first confined to the study of languages, and should not
include any mathematics. This naturally excited the boy's curiosity,
and one day, being then twelve years old, he asked in what geometry
consisted. His tutor replied that it was the science of constructing
exact figures and of determining the proportions between their different
parts. Pascal, stimulated no doubt by the injunction against reading it,
gave up his play-time to this new study, and in a few weeks had discovered
for himself many properties of figures, and in particular the proposition
that the sum of the angles of a triangle is equal to two right angles.
I have read somewhere, but I cannot lay my hand on the authority, that
his proof merely consisted in turning the angular points of a triangular
piece of paper over so as to meet in the centre of the inscribed circle:
a similar demonstration can be got by turning the angular points over
so as to meet at the foot of the perpendicular drawn from the biggest
angle to the opposite side. His father, struck by this display of
ability, gave him a copy of Euclid's
Elements, a book which
Pascal read with avidity and soon mastered.
At the age of fourteen he was admitted to the weekly meetings of
Roberval, Mersenne, Mydorge, and other French geometricians; from
which, ultimately, the French Academy sprung. At sixteen Pascal
wrote an essay on conic sections; and in 1641, at the age of
eighteen, he constructed the first arithmetical machine, an
instrument which, eight years later, he further improved. His
correspondence with Fermat about this time shews that he was then
turning his attention to analytical geometry and physics. He
repeated Torricelli's experiments, by which the pressure of the
atmosphere could be estimated as a weight, and he confirmed his
theory of the cause of barometrical variations by obtaining at
the same instant readings at different altitudes on the hill of
Puy-de-Dôme.
In 1650, when in the midst of these researches, Pascal suddenly
abandoned his favourite pursuits to study religion, or, as he
says in his
Pensées, ``contemplate the greatness and
the misery of man''; and about the same time he persuaded the
younger of his two sisters to enter the Port Royal society.
In 1653 he had to administer his father's estate. He now took up
his old life again, and made several experiments on the pressure
exerted by gases and liquids; it was also about this period that
he invented the arithmetical triangle, and together with Fermat
created the calculus of probabilities. He was meditating marriage
when an accident again turned the current of his thoughts to a
religious life. He was driving a four-in-hand on November 23, 1654,
when the horses ran away; the two leaders dashed over the parapet
of the bridge at Neuilly, and Pascal was saved only by the traces
breaking. Always somewhat of a mystic, he considered this a special
summons to abandon the world. He wrote an account of the accident
on a small piece of parchment, which for the rest of his life he
wore next to his heart, to perpetually remind him of his covenant;
and shortly moved to Port Royal, where he continued to live until
his death in 1662. Constitutionally delicate, he had injured his
health by his incessant study; from the age of seventeen or eighteen
he suffered from insomnia and acute dyspepsia, and at the time of
his death was physically worn out.
His famous
Provincial Letters directed against the Jesuits,
and his
Pensées, were written towards the close of his
life, and are the first example of that finished form which is
characteristic of the best French literature. The only mathematical
work that he produced after retiring to Port Royal was the essay
on the cycloid in 1658. He was suffering from sleeplessness and
toothache when the idea occurred to him, and to his surprise his
teeth immediately ceased to ache. Regarding this as a divine
intimation to proceed with the problem, he worked incessantly
for eight days at it, and completed a tolerably full account
of the geometry of the cycloid.
I now proceed to consider his mathematical works in rather greater
detail.
His early essay on the
geometry of conics, written in 1639,
but not published till 1779, seems to have been founded on the
teaching of Desargues. Two of the results are important as well
as interesting. The first of these is the theorem known now as
``Pascal's Theorem,'' namely, that if a hexagon be inscribed in
a conic, the points of intersection of the opposite sides will lie
in a straight line. The second, which is really due to Desargues,
is that if a quadrilateral be inscribed in a conic, and a straight
line be drawn cutting the sides taken in order in the points
A,
B,
C, and
D, and the conic in
P and
Q, then
PA.PC : PB.PD = QA.QC : QB.QD.
Pascal employed his
arithmetical triangle in 1653, but
no account of his method was printed till 1665. The triangle
is constructed as in the figure below, each horizontal line
being formed form the one above it by making every number in it
equal to the sum of those above and to the left of it in the
row immediately above it;
ex. gr. the fourth number in the
fourth line, namely, 20, is equal to 1 + 3 + 6 + 10.
The numbers in each line are what are now called
figurate
numbers. Those in the first line are called numbers of the first
order; those in the second line, natural numbers or numbers of the
second order; those in the third line, numbers of the third order,
and so on. It is easily shewn that the
mth number in the
nth
row is (m+n-2)! / (m-1)!(n-1)!
Pascal's arithmetical triangle, to any required order, is got by
drawing a diagonal downwards from right to left as in the figure.
The numbers in any diagonal give the coefficients of the expansion
of a binomial; for example, the figures in the fifth diagonal, namely
1, 4, 6, 4, 1, are the coefficients of the expansion
.
Pascal used the triangle partly for this purpose,
and partly to find the numbers of combinations of
m things taken
n at a time, which he stated, correctly, to be
(n+1)(n+2)(n+3) ... m / (m-n)!
Perhaps as a mathematician Pascal is best known in connection with
his correspondence with Fermat in 1654 in which he laid down the
principles of the
theory of probabilities. This correspondence
arose from a problem proposed by a gamester, the Chevalier
de Méré, to Pascal, who communicated it to Fermat. The
problem was this. Two players of equal skill want to leave the
table before finishing their game. Their scores and the number
of points which constitute the game being given, it is desired
to find in what proportion they should divide the stakes. Pascal
and Fermat agreed on the answer, but gave different proofs. The
following is a translation of Pascal's solution. That of Fermat
is given later.
The following is my method for determining the share of each
player when, for example, two players play a game of three
points and each player has staked 32 pistoles.
Suppose that the first player has gained two points, and the
second player one point; they have now to play for a point on
this condition, that, if the first player gain, he takes all
the money which is at stake, namely, 64 pistoles; while, if
the second player gain, each player has two points, so that
there are on terms of equality, and, if they leave off playing,
each ought to take 32 pistoles. Thus if the first player gain,
then 64 pistoles belong to him, and if he lose, then 32 pistoles
belong to him. If therefore the players do not wish to play this
game but to separate without playing it, the first player would
say to the second, ``I am certain of 32 pistoles even if I lose
this game, and as for the other 32 pistoles perhaps I will have
them and perhaps you will have them; the chances are equal. Let
us then divide these 32 pistoles equally, and give me also the
32 pistoles of which I am certain.'' Thus the first player will
have 48 pistoles and the second 16 pistoles.
Next, suppose that the first player has gained two points and the
second player none, and that they are about to play for a point;
the condition then is that, if the first player gain this point,
he secures the game and takes the 64 pistoles, and, if the second
player gain this point, then the players will be in the situation
already examined, in which the first player is entitled to 48
pistoles and the second to 16 pistoles. Thus if they do not
wish to play, the first player would say to the second, ``If I
gain the point I gain 64 pistoles; if I lose it, I am entitled
to 48 pistoles. Give me then the 48 pistoles of which I am
certain, and divide the other 16 equally, since our chances of
gaining the point are equal.'' Thus the first player will have
56 pistoles and the second player 8 pistoles.
Finally, suppose that the first player has gained one point
and the second player none. If they proceed to play for a point,
the condition is that, if the first player gain it, the players
will be in the situation first examined, in which the first player
is entitled to 56 pistoles; if the first player lose the point,
each player has then a point, and each is entitled to 32 pistoles.
Thus, if they do not wish to play, the first player would say to
the second, ``Give me the 32 pistoles of which I am certain, and
divide the remainder of the 56 pistoles equally, that is divide
24 pistoles equally.'' Thus the first player will have the sum
of 32 and 12 pistoles, that is, 44 pistoles, and consequently the
second will have 20 pistoles.
Pascal proceeds next to consider the similar problems when the game
is won by whoever first obtains
m +
n points, and
one player has
m while the other has
n points. The
answer is obtained using the arithmetical triangle. The general
solution (in which the skill of the players is unequal) is given
in many modern text-books on algebra, and agrees with Pascal's
result, though of course the notation of the latter is different
and less convenient.
Pascal made an illegitimate use of the new theory in the seventh
chapter of his
Pensées. In effect, he puts his argument
that, as the value of eternal happiness must be infinite, then, even
if the probability of a religious life ensuring eternal happiness be
very small, still the expectation (which is measured by the product
of the two) must be of sufficient magnitude to make it worth while
to be religious. The argument, if worth anything, would apply equally
to any religion which promised eternal happiness to those who accepted
its doctrines. If any conclusion may be drawn from the statement, it
is the undersirability of applying mathematics to questions of morality
of which some of the data are necessarily outside the range of an
exact science. It is only fair to add that no one had more contempt
than Pascal for those who changes their opinions according to the
prospect of material benefit, and this isolated passage is at variance
with the spirit of his writings.
The last mathematical work of Pascal was that on the
cycloid
in 1658. The cycloid is the curve traced out by a point on the
circumference of a circular hoop which rolls along a straight line.
Galileo, in 1630, had called attention to this curve, the shape
of which is particularly graceful, and had suggested that the
arches of bridges should be built in this form.
Four years later, in 1634, Roberval found the area of the cycloid;
Descartes thought little of this solution and defied him to find
its tangents, the same challenge being also sent to Fermat who
at once solved the problem. Several questions connected with the
curve, and with the surface and volume generated by its revolution
about its axis, base, or the tangent at its vertex, were then
proposed by various mathematicians. These and some analogous question,
as well as the positions of the centres of the mass of the solids
formed, were solved by Pascal in 1658, and the results were issued
as a challenge to the world, Wallis succeeded in solving all the
questions except those connected with the centre of mass. Pascal's
own solutions were effected by the method of indivisibles, and are
similar to those which a modern mathematician would give by the aid
of the integral calculus. He obtained by summation what are equivalent
to the integrals of
,
,
and
,
one limit being either 0 or
.
He also investigated the geometry of the Archimedean spiral. These
researches, according to D'Alembert, form a connecting link between
the geometry of Archimedes and the infinitesimal calculus of Newton.
This page is included in a
collection of mathematical biographies taken from
A Short Account of the History of Mathematics
by W. W. Rouse Ball (4th Edition, 1908).
Transcribed by
D.R. Wilkins
(dwilkins@maths.tcd.ie)
School of Mathematics
Trinity College, Dublin