Pascal's Triangle
Pascal did not invent the triangle named after him — it appears in Chinese and Indian mathematics centuries earlier — but he systematized its properties in his 1653 Traité du triangle arithmétique. Pascal's triangle encodes binomial coefficients, combinations, Fibonacci numbers, and properties of prime numbers, and is foundational to combinatorics and probability.
Probability Theory
In 1654, Pascal and Fermat exchanged a series of letters about gambling problems posed by the Chevalier de Méré. These letters founded the mathematical theory of probability. The "problem of points" — how to fairly divide stakes when a game is interrupted — was their central question, and their solution introduced the concept of expected value that underlies all of modern statistics, insurance, and finance.
The Pascaline: First Mechanical Calculator
At age 19, Pascal designed and built the Pascaline — a mechanical calculator capable of addition and subtraction, to help his father with tax calculations. About 50 were eventually built. It was one of the first functioning mechanical calculators, predating Leibniz's more advanced stepped reckoner by decades.
Fluid Mechanics and Barometry
Pascal verified Torricelli's work on atmospheric pressure, showed that atmospheric pressure decreases with altitude, and formulated Pascal's principle: pressure applied to a confined fluid is transmitted equally in all directions. This principle underlies hydraulic presses, car brakes, and hydraulic machinery.
Frequently Asked Questions
What is Pascal most famous for?
Pascal is famous for co-founding probability theory (with Fermat), building one of the first mechanical calculators (the Pascaline), formulating Pascal's principle in fluid mechanics, and his philosophical Pensées — particularly Pascal's Wager.
What is Pascal's Wager?
Pascal's Wager argues that it is rational to believe in God because if God exists and you believe, you gain infinite reward; if God exists and you don't, you lose infinitely. The expected value of belief is therefore greater regardless of the probability of God's existence.
Did Pascal actually invent probability theory?
Pascal and Fermat jointly founded modern probability theory through their 1654 correspondence. They were not the first to think about odds and games, but they were the first to develop a rigorous mathematical framework for calculating probabilities and expected values.
