Matthew DeVolder

Ellsworth High School

11/22/02

Calculus

Blaise Pascal

Although Blaise Pascal made many contributions to the mathematical world, he has a greater mathematical reputation of what he could have achieved rather than what actually achieved. This is credited to the fact that Pascal spent a sizable amount of his life devoted to his religious studies.

Pascal was born the son of Etienne Pascal, a judge with some scientific repute. In 1632, the Pascal family moved to Paris to further Etienne’s scientific studies and to educate Blaise. Etienne held many unorthodox educational views and insisted on educating the boy himself. Due to some health problems, Etienne did not want Blaise to be overworked, therefore, he forbade his son from studying mathematics. This heightened Blaise’s interests in mathematics, and at the age of twelve, he forewent his playtime to study geometry. Soon after, his father granted Blaise a copy of Euclid’s Elements, which he eagerly mastered in little time.

At the age of fourteen, Blaise was admitted to the meetings of Mersenne. At these meetings, Blaise came to admire the work of Desargues. Then, in June of 1639 at the age of sixteen Blaise presented several projective geometry theorems. In this was Pascal’s mystic hexagram. This simply states, “if a hexagon ADBFCE (not necessarily convex) is inscribed into a conic (in particular into a circle), then the points of intersections of opposite sides (AD with FC, DB with CE and BF with EA) are collinear. This line is called the Pascal line of the hexagon (Planet).”

The Pascal family left Paris in December 1639 for Rouen. Etienne was appointed tax collector for Upper Normandy. Pascal’s first paper, Essay on Conic Sections, was published in February 1640. Pascal then focused his attention on developing an arithmetical machine called the “Pascaline” to aid his father in collecting taxes. Pascal improved this original design eight years later. Pascal’s interests then ventured towards physics, and he repeated Torricelli’s experiments on atmospheric pressure. In August 1647, Pascal observed that atmospheric pressure decreases with height and inferred that a vacuum must exist outside the atmosphere. Pascal composed New Experiments Concerning Vacuums in October 1647. This paper received a considerable amount of dispute from scientists, like Descartes, who did not believe in vacuums.

In 1650, around the time of his father’s death, Pascal suddenly abandoned his mathematical studies in pursuit of religion, or as Pascal said, “’contemplate the greatness and misery of man’ (maths).” Pascal again returned to his old lifestyle when he began to manage his father’s estate. Pascal completed several experiments of the pressure exerted by gases and liquids and subsequently completed Treatise on the Equilibrium of Liquids in which he explains his law of pressure.

Pascal also developed his arithmetical triangle in 1653. “The Pascal triangle is a table of the binomial coefficients where the (n, k)th entry is . Each entry is the sum of the pair immediately above it (groups).” The numbers in each line of the triangle are called figurate numbers. The first line numbers are known as numbers of the first order. The second row numbers are called numbers of the second order or natural numbers. The third row numbers are numbers of the third order. The pattern continues from here. Pascal was not the first to study the arithmetical triangle, but his work is considered to be the most important and lead Newton to discover the general binomial theorem for fractional and negative powers.

Pascal is most famous for his work with Fermat in which the principles of probability were discovered. Pascal and Fermat worked with Cardan’s dice problem and the problem of points. The dice problem poses the question of how many times a pair of dice must be rolled before a pair of double sixes will be rolled. The problem of points involves how the winnings of an uncompleted game should be divided. Using the arithmetical triangle, Pascal was able to solve the problems for two players; however, he never figured out how to complete the problems for more than two players.

On November 23, 1654, Pascal pledged the rest of his life to Christ after a near death experience. Pascal began to write his eighteen Provincial Letters. The Provincial Letters are a group of anonymously written letters on religious topics. The letters were mainly against the Jesuits and supported the Jansenists. Pascal’s last mathematical work was with cycloid. A cycloid is the “curve traced by a point on the circumference of a rolling circle (groups).” Pascal passed away August 19, 1662 in Paris from a malignant tumor in his stomach that spread to the brain.

Works Cited

17 Nov. 2002

<http://www.sosu.edu/st/math/courses/algsci/projects/Systems/hexagon.htm>.

D. R. Wilkins. School of Mathematics, Trinity College, Dublin. 10 Oct. 2002 <http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html>.

Math Forum. Aug. 1998. Drexel University. 13 Oct. 2002 <http://mathforum.org>.

Men of Mathematics. 73-89.

O'Connor, J J., and E F. Robertson. Dec. 1996. School of Mathematics and Statistics, University of St. Andrews, Scotland. 10 Oct. 2002 <http://www-groups.dcs.st-and.ca.uk/~history/Mathematicians/Pascal.html>.

Planet Math. 21 Nov. 2002 <http://planetmath.org/encyclopedia/PascalLine.html>.