Blaise Pascal’s Contributions in Mathematics


Blaise Pascal was a French mathematician, physicist and philosopher. He was born on 19 June 1623 in Clermont-Ferrand, Auvergne, France. His father, Etienne Pascal, was of unorthodox educational views and never allowed him to attend the school. His father decided that Blaise would not study mathematics till the age of 15 and removed all mathematical texts from their house. Out of curiosity, Blaise started working on geometry at the age of 12. He discovered that the sum of the angles of a triangle is two right angles. Looking at Blaise interests in mathematics, his father allowed him to study the work of Euclid on geometry. At the age of 15, Pascal started studying the work of Girard Desargues (a French mathematician who was a founder of projective geometry). In June 1639, he presented a paper that contained Pascal’s mystic hexagon and theorems on projective geometry. In February 1640, Blaise Pascal published his first work, Essay on Conic Sections. Also, he was the second person to invent a mechanical calculator after Schickard, who invented the one in 1624. From 1642 to 1645, Blaise worked on a device called Pascaline that resembled a mechanical calculator of 1940. Blaise founded this device to help his father with his work of collecting taxes. In the year 1646, Pascal proved that the vacuum existed, and in 1647, he published a paper “New Experiments Concerning Vacuums” that was not accepted by scientists like Descartes, who refused to believe in the existence of the vacuum. In 1653, Blaise Pascal wrote Treatise on the Equilibrium of Liquids, in which he explained Pascal’s law of pressure. His work “The Generation of Conic Sections” was completed by him in 1654, is now lost. However, some notes of his work were made by Leibniz and Tschirnhaus that helped them to have a complete picture of his work. He also proposed an arithmetical triangle which is now called Pascal’s triangle. Blaise’s work, “Treatise on the Arithmetical Triangle” was the most significant one and his work on binomial coefficients helped Newton in the discovery of binomial theorem for negative and fractional powers. In 1654, along with Fermat, he led the foundation of Probability. His last work was on cycloid, the curve traced by a point on the circumference of the rolling circle. In the year 1654, Pascal was unwell and bedridden. He spent his last years attending religious services at church. He died after 2 months of his 39th birthday of a disease of the stomach that got spread in his brain.

1. Pascal’s Theorem

In projective geometry, Pascal’s theorem states that

If six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon.”

The theorem is also true in the Euclidean plane, but the statement should be modified in the cases when some opposite sides of the hexagon are parallel.

Symmetry — Pascal's theorem. No matter how you choose the red... | Theorems, Geometry, Geometry design

2. Pascal’s Triangle

Pascal’s mathematical work, “Treatise on the Arithmetical Triangle” explained a tabular representation of binomial coefficients, where each number is the sum of the two numbers directly above it. A binomial expression is an algebraic expression, where two numbers are operated on by multiplication, subtraction, addition with a whole number exponent, represented as {(x+y)}^{n}. When this binomial expression is expanded, the coefficients produce a symmetrical triangle.


Image result for sierpinski triangle gif | Geometric

Pascal was not the first person to propose this triangle. Before him, the triangle was given by Persian mathematician Al-Karaji in the 10th century, by a Chinese mathematician in the 13th Century, and as Tartgalia’s triangle in Italy in the 16th Century. But only Pascal discovered a pattern in the rows, columns, and diagonals of the array of the numbers. For example, looking diagonally, the first diagonal 1,2,3,… are all natural numbers, the second diagonal 1, 3, 6, 10,… are triangular numbers in order. The third diagonal 1, 4, 10,…. are pyramidal triangular numbers. He was likely to find prime numbers, Fibonacci numbers, Catalan numbers, and many other series.

3. Pascaline

Pascaline is the mechanical calculator invented by Pascal in the mid 17th century. Pascaline was successful because of its carry mechanism. Pascaline adds 1 to 9 on one dial and carries 1 on the other dial when the digits change from 9 to 0 on the previous dial. Pascaline was designed to add and subtract numbers directly, whereas multiplication and division were done by repeated addition and subtraction. He designed this machine in 1642 and presented it in 1645. In the year 1649, King Louis XIV of France gave him the right to design and manufacture calculating machines in France. At present, nine Pascaline calculators exist, mostly displayed in European museums.


4. Probability

Pascal’s development of the theory of probability is considered his most significant discovery in the field of mathematics. Pascal along with his friend Pierre de Fermat solved gambling problems. Specifically, one problem was Gambler’s Ruin in which one has to determine the chances of winning of each of two men playing a particular game with specific rules. Another problem was the problem of Points, in which, how a game’s winnings should be divided between two equally skilled players if the game was ended prematurely, is to be determined. After this, the term ‘expected value’ came into existence. Although Pascal and Fermat did important contributions to probability theory, they didn’t write any book on the theory. The first book on probability theory was written by Christiaan Huygens. Pascal’s and Fermat’s work on probability helped Leibniz to lead the foundation of Calculus.

Pascal's theory of probability

5. Cycloid

A cycloid is a curve traced by a point on a circle as it rolls along a straight line. In 1658, when Pascal was suffering from a toothache he started studying the topic cycloid. After his toothache disappeared, he took it as a positive sign of God and continued his research on cycloids. He prepared his paper in eight days, and to publicize his results, he organised a contest offering 20 and 40 Spanish doubloons to the winners. He also solved the problems of the volume and surface area of the solid of revolution formed by rotating the cycloid about the x-axis.

File:Cycloid animated .gif - Wikimedia Commons

6. Famous Quotes by Pascal

  • Truth is so obscure in these times, and falsehood so established, that, unless we love the truth, we cannot know it.
  • We sail within a vast sphere, ever drifting in uncertainty, driven from end to end.
  • There are only two kinds of men: the righteous who think they are sinners and the sinners who think they are righteous.
  • We view things not only from different sides but with different eyes; we have no wish to find them alike.
  • In faith, there is enough light for those who want to believe and enough shadows to blind those who don’t.
  • Nature is an infinite sphere of which the centre is everywhere and the circumference nowhere.
  • That we must love one God only is a thing so evident that it does not require miracles to prove it.
  • Nothing is so intolerable to man as being fully at rest, without a passion, without business, without entertainment, without care.
  • Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.

Add Comment