Mathematics–Geometry, Probability Theory, Mechanical Calculator, Algebra

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Blaise Pascal was a child prodigy who, before he was 16, proved an important theorem in the projective geometry of conics. The modern theory of probability is usually considered to begin with the correspondence of Blaise Pascal and Pierre de Fermat in 1654, partially in response to the gambling questions Antoine Gombaud, the chevalier de Mere, presented to Pascal.(1) Gombaud could not understand why he was losing when he bet even money that double sixes would turn up at least once on average in every twenty-four rolls of two dice. The answer is once in 24.6 rolls, which means, since dice cannot be rolled a fractional number of times, that someone could only win this bet at even money if the dice were rolled at least 25 times. Later, insurance companies adopted Pascal and Fermat’s work for actuarial tables of sickness and mortality. When he was 19, Pascal invented one of the first mechanical calculators for sale.

 

At 31, in his Traite du Triangle Arithmetique (Characteristics of Arithmetical Triangles, 1654), Pascal discovered what is now known as “Pascal’s triangle.” The technique dealt with expanding expressions of the form (a + b)n. One can simply multiply (a + b) by itself as many times as indicated by n, but this becomes tedious, say, when n is 12.

 

Pascal had discovered a way to generate the coefficients in a binomial expansion from the array now known as “Pascal’s triangle”:

1

1  2  1

1  3  3  1

1  4  6  4  1

1  5  10  10  5  1

 

For example, (a + b)4 = (1)a4 + 4a³b + 6a²b² + 4ab³ + (1)b4, which uses the fourth row of Pascal’s triangle – 1, 4, 6, 4, 1 as the coefficients in the binomial expansion of (a + b)4 . Pascal used this triangle extensively in his work on probability theory.

 

Though it is called “Pascal’s triangle” the array was known in China, India and the Islamic Empire hundreds of years before Pascal. For India, it is first mentioned in the Chandasutra by Pingala (c. 200 B.C.E.) as a method of determining the number of combinations of n syllables taking p at a time. And this was further elaborated on by the commentator Halayudha, who lived in the tenth century C.E. For Islam, the triangle appears up to the twelfth power in a work by Nasir al-Din al-Tusi in 1265. Finally, Zhu Shijie in China discusses “Pascal’s triangle” in his book Si Yuan Jian (The Precious Mirror of the Four Elements) published in 1303.(2) (see stamp image under Zhu Shijie).

 

Footnotes:

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(1) Carl B. Boyer, A History of Mathematics, 2nd Edition (New York, 1991) p. 363.

(2) George Gheverghese Joseph, The Crest of the Peacock - Non-European Roots of Mathematics (Princeton, 2011), pgs. 270-273.

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