Blaise Pascal was born in Clermont, now called Clermont-Ferrand, in Auvergne, in 1623. His mother died when he was only three years old. A few years later the family moved to Paris, where Blaise Pascal's father, Etienne, decided that his son should be home-schooled. No mathematics texts were allowed in the house: this prohibition arousing the young Pascal's curiosity, he began working on geometry himself at the age of twelve. His father gave in, and allowed the boy to accompany him to meetings of mathematicians. The family moved to Rouen when Pascal's father was named tax collector for the region. From 1642 to 1645, Pascal worked on a calculator to help his father in his work. The device, which came to be known as the "pascaline," is considered to be among the first mechanical calculators ever invented. With the illness of his father and a life-threatening accident he endured himself, Pascal became increasingly religious. By the end of his life, in 1662, he was as well known for his religious publications as for his mathematical papers. In "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.
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." Here are three images of Pascal: |