BLAISE PASCAL

Blaise Pascal is known as one of the co-founders of present day probability theory. He is known by many titles such as geometer, philosopher, mathematician, probabilist, physicist, and inventor.

Pascal was born in Clermont-Ferrand on June 19, 1623. He was the third of Etienne Pascal's children and his only son. Etienne was a scientist and a government official. His mother died when he was three years old. In 1632 the Pascal family, Etienne and his four children, left Clermont and settled in Paris. Blaise Pascal's father had unorthodox educational views and decided to teach his son himself. Etienne had associations with the likes of Descartes, Mersenne, and Fermat and decided that Blaise was not to study mathematics before the age of 15 and all mathematics texts were removed from their house. Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12. He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he relented and allowed Blaise a copy of Euclid. Pascal soon proved himself a mathematical prodigy, mastering Euclid's Elements by the age of 12.

At the age of 14 Blaise Pascal started to accompany his father to Mersenne's meetings. Mersenne belonged to the religious order of the Minims, and his cell in Paris was a frequent meeting place for Gassendi, Roberval, Carcavi, Auzout, Mydorge, Mylon, Desargues and others. Soon, certainly by the time he was 15, Blaise came to admire the work of Desargues. At the age of sixteen, Pascal presented a single piece of paper to one of Mersenne's meetings in June 1639. It contained a number of projective geometry theorems, including Pascal's mystic hexagon.

In December 1639 the Pascal family left Paris to live in Rouen where Etienne had been appointed as a tax collector for Upper Normandy. Shortly after settling in Rouen, he formulated one of the basic theorems of projective geometry, known as Pascal's theorem and described in his Essai pour les coniques (Essay on Conics, 1639)

Pascal invented the first digital calculator to help his father with his work collecting taxes. He worked on it for three years between 1642 and 1645. The device, called the Pascaline, resembled a mechanical calculator of the 1940s. This, almost certainly, makes Pascal the second person to invent a mechanical calculator for Schickard had manufactured one in 1624.

There were problems faced by Pascal in the design of the calculator which were due to the design of the French currency at that time. There were 20 sols in a livre and 12 deniers in a sol. The system remained in France until 1799 but in Britain a system with similar multiples lasted until 1971. Pascal had to solve much harder technical problems to work with this division of the livre into 240 than he would have had if the division had been 100. However production of the machines started in 1642 but, as Adamson writes in [3],

By 1652 fifty prototypes had been produced, but few machines were sold, and manufacture of Pascal's arithmetical calculator ceased in that year.

Events of 1646 were very significant for the young Pascal. In that year his father injured his leg and had to recuperate in his house. He was looked after by two young brothers from a religious movement just outside Rouen. They had a profound effect on the young Pascal and he became deeply religious. Pascal converted to Jansenism, the Catholic sect rivaling with Jesuits, who had the support of the King, Louis XIV. Pascal's sister, Jaqueline, entered the Jansenist convent of Port-Royal in south-west Paris and became one of the most passionate advocates of the sect. The Jansenists, who were never officially accepted by the Catholic Church, were named after Cornelius Jansenius (1587-1638), a Flemish theologian. The Jansenists argued that since the Fall in the Garden of Eden, all humankind has been corrupted by sin. Their objection to the Jesuits stemmed from what they saw as the over-reliance of the Jesuits on human fee will, to the detriment of divine grace. Jansenius, in his book Augustinus, stated that free-will of the natural effort and ability of the individual man and also supernatural grace, are both required, in co-operation, for salvation.

From about this time Pascal began a series of experiments on atmospheric pressure. By 1647 he had proved to his satisfaction that a vacuum existed. Descartes visited Pascal on 23 September. His visit only lasted two days and the two argued about the vacuum which Descartes did not believe in. Descartes wrote, rather cruelly, in a letter to Huygens after this visit that Pascal

...has too much vacuum in his head.

In August of 1648 Pascal observed that the pressure of the atmosphere decreases with height and deduced that a vacuum existed above the atmosphere. Descartes wrote to Carcavi in June 1647 about Pascal's experiments saying:-

It was I who two years ago advised him to do it, for although I have not performed it myself, I did not doubt of its success ...

In October 1647 Pascal wrote New Experiments Concerning Vacuums which led to disputes with a number of scientists who, like Descartes, did not believe in a vacuum.

Pascal returned to Paris in 1647 on his father's second retirement. In 1647, a few years after publishing Essai pour les coniques he suddenly abandoned the study of mathematics. Because of his chronically poor health, he had been advised to seek diversions from study and attempted for a time to live in Paris in a deliberately frivolous manner. His interest in probability theory has been attributed to his interest in calculating the odds involved in the various gambling games he played during this period.

Pascal proved by experimentation in 1648 that the level of the mercury column in a barometer is determined by an increase or decrease in the surrounding atmospheric pressure rather than by a vacuum, as previously believed. This discovery verified the hypothesis of the Italian physicist Evangelista Torricelli concerning the effect of atmospheric pressure on the equilibrium of liquids. Six years later, in conjunction with the French mathematician Pierre de Fermat, Pascal formulated the mathematical theory of probability, which has become important in such fields as actuarial, mathematical, and social statistics and as a fundamental element in the calculations of modern theoretical physics. Pascal's other important scientific contributions include the derivation of Pascal's law or principle, which states that fluids transmit pressures equally in all directions, and his investigations in the geometry of infinitesimals.Pascal also made major contributions to physics, including a treatise on hydrostatics and experiments with his barometer to determine the cause of the mercury's suspension.

From May 1653 Pascal worked on mathematics and physics writing Treatise on the Equilibrium of Liquids (1653) in which he explains Pascal's law of pressure. Adamson writes in [3]:-

This treatise is a complete outline of a system of hydrostatics, the first in the history of science, it embodies his most distinctive and important contribution to physical theory.

He worked on conic sections and produced important theorems in projective geometry. In The Generation of Conic Sections (mostly completed by March 1648 but worked on again in 1653 and 1654) Pascal considered conics generated by central projection of a circle. This was meant to be the first part of a treatise on conics which Pascal never completed. The work is now lost but Leibniz and Tschirnhaus made notes from it and it is through these notes that a fairly complete picture of the work is now possible.

Although Pascal was not the first to study the Pascal triangle, his work on the topic in Treatise on the Arithmetical Triangle was the most important on this topic and, through the work of Wallis, Pascal's work on the binomial coefficients was to lead Newton to his discovery of the general binomial theorem for fractional and negative powers.

In correspondence with Fermat he laid the foundation for the theory of probability. This correspondence consisted of five letters and occurred in the summer of 1654. They considered the dice problem, already studied by Cardan, and the problem of points also considered by Cardan and, around the same time, Pacioli and Tartaglia. The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete. They solved the problem of points for a two player game but did not develop powerful enough mathematical methods to solve it for three or more players.

Through the period of this correspondence Pascal was unwell. In one of the letters to Fermat written in July 1654 he writes

... though I am still bedridden, I must tell you that yesterday evening I was given your letter.

However, despite his health problems, he worked intensely on scientific and mathematical questions until October 1654. Sometime around then he nearly lost his life in an accident. The horses pulling his carriage bolted and the carriage was left hanging over a bridge above the river Seine. Although he was rescued without any physical injury, it does appear that he was much affected psychologically. Not long after he underwent another religious experience, on 23 November 1654, and he pledged his life to Christianity.

After this time Pascal made visits to the Jansenist monastery Port-Royal des Champs about 30 km south west of Paris. He began to publish anonymous works on religious topics, eighteen Provincial Letters being published during 1656 and early 1657. These were written in defence of his friend Antoine Arnauld, an opponent of the Jesuits and a defender of Jansenism, who was on trial before the faculty of theology in Paris for his controversial religious works. Pascal's most famous work in philosophy is Pensées, a collection of personal thoughts on human suffering and faith in God which he began in late 1656 and continued to work on during 1657 and 1658. This work contains 'Pascal's wager' which claims to prove that belief in God is rational with the following argument.

If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.

With 'Pascal's wager' he uses probabilistic and mathematical arguments but his main conclusion is that

...we are compelled to gamble...

His last work was on the cycloid, the curve traced by a point on the circumference of a rolling circle. In 1658 Pascal started to think about mathematical problems again as he lay awake at night unable to sleep for pain. He applied Cavalieri's calculus of indivisibles to the problem of the area of any segment of the cycloid and the centre of gravity of any segment. 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.

Until 1654 Pascal devoted himself to mathematics and scientific studies. After a mystical experience on November 23-24, 1654, he had a second conversion, and defended Jansenism against the Jesuits in LETTRES PROVINCIALES (Provincial Letters). At the end of 1654, after several months of intense depression, Pascal had a religious experience that altered his life. He entered the Jansenist monastery at Port-Royal, although he did not take orders, and led a rigorously ascetic life until his death eight years later. He never published in his own name again. The Jansenists encouraged him in his mathematical studies, which he resumed. To assist them in their struggles against the Jesuits, he wrote, under a pseudonym, a defense of the famous Jansenist Antoine Arnauld, the famous Lettres provinciales (Provincial Letters), in which he attacked the Jesuits for their attempts to reconcile 16th-century naturalism with orthodox Roman Catholicism. His most positive religious statement appeared posthumously (he died August 19, 1662); it was published in fragmentary form in 1670 as Apologie de la religion Chrienne (Apology of the Christian Religion). In these fragments, which later were incorporated into his major work, he posed the alternatives of potential salvation and eternal damnation, with the implication that only by conversion to Jansenism could salvation be achieved. Pascal asserted that whether or not salvation was achieved, humanity's ultimate destiny is an afterlife belonging to a supernatural realm that can only be known intuitively.
Pascal's most famous work is the Pensees (published 1670), a set of deeply personal meditations in somewhat fragmented form on human suffering and faith in God.

Pascal lived in the time, when Copernicus' discovery - that the earth moves round the sun - had made human beings insignificant factors in the new disenchanted world. Facing the immensity of the universe, Pascal felt horror - "The eternal silence of these infinite spaces terrifies me." For him the world seemed empty of ultimate meaning or significance without Christianity, which he defended against the assaults of freethinkers. When Montaigne lived at ease with skepticism, Pascal was tormented by religious doubt, and took the question Why are we here? with the utmost seriousness, revealing his thoughts in his most famous book, the posthumous PENSÉES.

"Pascal's disillusioned analysis of human bondage is sometimes interpreted to mean that Pascal was really and finally an unbeliever, who, in his despair, was incapable of enduring reality and enjoying the heroic satisfaction of the free man's worship of nothing. His despair, his disillusion, are, however, no illustration of personal weakness; they are perfectly objective, because they are essential moments in the progress of the intellectual soul; and for the type of Pascal they are the analogue of the drought, the dark night, which is an essential stage in the progress of the Christian mystic."

(T.S. Eliot in Selected Essays, 1960)

In the Pensees he attempted to explain and justify the difficulties of human life by the doctrine of original sin, and he contended that revelation can be comprehended only by faith, which in turn is justified by revelation. "Pascal's wager" expresses the conviction that belief in God is reasonable on the ground that there are no rational grounds either for belief or disbelief, so belief is not less reasonable than disbelief; but this being so it is wiser to gamble on the truth of religion since this policy involves success if religion is true and no significant loss if it is false. He had admirers both Roman Catholic and Protestant, including John Wesley, the founder of Methodism, who praised an essay he wrote on the psychology of conversion. Pascal died at the age of 39 from a combination of tuberculosis and stomach cancer.

Reference:

1.http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html

2.http://www.island-of-freedom.com/PASCAL.HTM

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