The heart has its reasons, which reason does not know.

— Blaise Pascal, Pensées

Pascal left few finished books. Almost everything he wrote on mathematics came out as loose sheets, in letters or after his death. Even so, few authors turn up in so many different places: in the triangle at school, in the weather forecast, in a car’s brakes, in an insurance calculation and even in the name of a programming language.

The wager

The most famous passage of the Pensées is not about geometry, but it uses the mathematics Pascal had just invented with Fermat: Pascal’s wager. It does not try to prove that God exists. Pascal says reason cannot settle the question and asks what is the better bet, comparing gains and losses weighted by the chances, as in an expected value. With an infinite gain on the side of faith, any positive chance is enough. The argument, its three versions and the objections of Diderot and Voltaire are in Pascal’s wager. It is considered the first text of decision theory, the mathematics of choosing under uncertainty.

The last letter to Fermat

In 1660 Fermat suggested they meet, each travelling half the way between Toulouse and Clermont. Pascal, ill, declined in a letter that shows how he saw mathematics at the end of his life:

To speak frankly of geometry, I find it the highest exercise of the mind; but at the same time I know it to be so useless that I make little difference between a man who is only a geometer and a skilful craftsman.

— Blaise Pascal, letter to Fermat (10 August 1660)

The two never met.

What remained

  • Probability. The 1654 letters inspired Huygens, who published the first book on the subject in 1657. Then came Jakob Bernoulli, de Moivre and Laplace. Insurance, lotteries, statistics and data science descend from those letters.
  • Combinatorics and induction. The Treatise on the arithmetic triangle organized combinations and brought the first clear statement of mathematical induction.
  • Projective geometry. The mystic hexagram became one of the founding theorems of projective geometry, taken up again in the 19th century by Poncelet and Brianchon.
  • Calculus. Leibniz said it was while reading a work of Pascal’s on the quarter circle that he had the idea of the characteristic triangle, one of the steps towards the calculus.
  • Physics. The vacuum experiments and the Puy de Dôme put an end to the horror of the vacuum and gave rise to the barometer and the altimeter. Pascal’s principle is in every hydraulic system.
  • Computing. The Pascaline was the first calculating machine to be manufactured and sold.

A name that stayed

In 1971 the International System’s unit of pressure was named the pascal (Pa). When the weather forecast talks about “1013 hectopascals”, it means $101{,}300$ Pa, the air pressure at sea level that Pascal had his brother-in-law measure.

In 1970 the Swiss computer scientist Niklaus Wirth created a programming language designed to teach programming clearly and named it Pascal, in honour of the inventor of the Pascaline. Generations of students learned to program in it.

To remember

  • Pascal lived from 1623 to 1662. At 16 he published the hexagon theorem; at 19 he started the Pascaline.
  • In the arithmetic triangle each number is the sum of the two above it, row $n$ adds up to $2^n$ and the numbers are the combinations $\binom nk$.
  • The triangle was known in India, Persia, China and Europe before Pascal; he studied it in depth.
  • With Fermat, in 1654, Pascal founded probability and the idea of expected value.
  • The area under one arch of a cycloid is $3\pi r^2$, three times the area of the wheel.
  • In a still liquid, an increase in pressure is transmitted equally to every point; the press multiplies force, but not work.
  • At the top of the Puy de Dôme the mercury fell by 8.5 cm: air has weight.
Exercise: a raffle ticket costs 5 reais and pays 100 reais with a chance of 1 in 25. Is it worth it, by expected value?

The expected gain is $100\cdot\frac{1}{25}=4$ reais. Since the ticket costs 5, the expected value of playing is $4-5=-1$ real per ticket. On average you lose 1 real per raffle.

Exercise: the forecast announces 990 hPa. How many millimetres of mercury is that?

That is $99{,}000$ Pa. The height of the mercury is $h=\dfrac{99{,}000}{13{,}600\cdot9.81}\approx0.742$ m, or $742$ mmHg. Pressure that low usually means unsettled weather.

Sources: Pascal biography, MacTutor · the letters of Pascal and Fermat, in English translation.

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