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Life | Accomplishments | Pascal's Theorem | Bibliography | Back to the front page

Accomplishments
Julia Chew

Pascal's life, aside from religion, was centered around math and science. He managed to work these into other projects such as his calculating machine. Pascal's machine, also called la Pascaline, was inspired by his desire to help his father whose many duties as tax commissioner included the tedious work of calculating many numbers. The idea came to him in 1640, but the actual completion of the machine took him more that four years and many models.

Among some of his mathematical accomplishments are his works on the arithmetic triangle, the cycloid and conic sections. We know the arithmetic triangle by the name Pascal's Triangle. Although the triangle has been around long before Pascal, it is named for him because he studied the triangle and published the Traite du Triangle Arithmetique. The triangle is constructed by starting with a 1 in the upper left corner and each number in the triangle is a result of adding the number to its left to the number above it.
1 1 1 1 1 1 1 ...
1 2 3 4 5 6 ...
1 3 6 10 15 ...
1 4 10 20 ...
1 5 15 ...
1 6 ...
1 ...
Pascal used this triangle to conclude many results. One of these is his solution of the division problem. This involves the division of stakes in a fair game between two people when the game in ended early. Other results from Pascal's triangle is the fact that the numbers in the diagonals of the triangle are the binomial coefficients. For example, corresponds to the fourth diagonal row and 1 3 3 1 are the coefficients for .

The cycloid problem had already been studied by many others before Pascal. This problem involves finding the area of the curve constructed by fixing a point on the circumference of a circle and turning it like a wheel. Pascal proposed a contest to solve various problems involving the cycloid. Many submitted solutions but Pascal submitted a solution of his own under a pseudonym. He then announced himself as the winner.