Contributions to Mathematics

Geometry

When Pascal was 16 years old he wrote a paper entitled Essay pour les coniques. In it he presented Pascal's Theorem that states that the intersections of opposite sides of a hexagon inscribed in a conic section are colinear. This is a fundamental theorem in what is now called projective geometry.

Figure 1: Pascal's Hexagon Theorem

Probability Theory

It is generally accepted that probability theory had its origin in a series of letters between Blaise Pascal and Pierre de Fermat. The subject of these letters was several gambling questions posed to Pascal by Antoine Gombaud, chevalier de Méré. The chevalier de Méré had been wagering that he could roll a six with a single die in four rolls. He had won for a while, but was now losing. He switched to a different game where he bet that he could roll a double six with two dice in 24 rolls. He was losing more than he was winning. He asked Pascal if he was just unlucky or was he making a stupid bet. Pascal showed him that he had a slight advantage in the first game, but was at a slight disadvantage in the second game. In the first game the probability of not rolling a six on a single roll was $5/6$. Therefore, the probability of not rolling a six in four rolls was $(5/6)^4=0.4823$. It follows that the probability of rolling at least one six in four rolls is $1-(5/6)^4=0.5177$ — a slight advantage. Similarly, in the second game the probability of rolling a double six in 24 rolls is $1-(35/36)^{24}=0.49144$ — a slight disadvantage.

The next question posed to Pascal was more challenging and was the subject of several letters between Pascal and Fermat. This question involved the fair distribution of the pot in an unfinished game. The game involved setting a number of points for a win. The players rolled the dice and the highest roll received a point. This was repeated until one of the players had the agreed upon number of points. However, what should be done if the game was discontinued before one of the players won? How should the pot be divided? A discussion of the various solutions offered by Pascal and Fermat is contained in Appendix B: Analysis of Unfinished Game Problem.

In solving the combinatorial problems that arise in probability calculations Pascal made great use of the triangular array of numbers now known as Pascal's triangle. A discussion of this triangle and its properties is contained in Appendix A: Pascal's Triangle.

The Cycloid

A cycloid (in French la roulette) is the curve generated by a point on a circle as the circle rolls along a line (see Figure 2).

Figure 2: The Cycloid Curve

This curve had been studied by many of the great minds of the time including Galileo, Descartes, and Roberval. Pascal found a way to find the area and center of gravity of any segment under the curve. He also showed how to calculate the volume and surface area of the solid obtained by rotating the cycloid about the line it was rolling on. This work was a predecessor of the more general problem in calculus of finding the area under a curve by integration.

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