ABC of Mathematicians

ABC of MathematiciansAuthor(s): Michael HoltSource: Mathematics in School, Vol. 3, No. 6 (Nov., 1974), pp. 10-12Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211273 .

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Blaise Pascal is most widely known as a philosopher and, by some, as a religious bigot. He could have been a mathematician in the top flight if he hadn't devoted so much time to a self- imposed religious martyrdom. So tormented was he by a sense of sin-sin he often saw more clearly in others than himself!- that he even wondered if doing mathematics was not wicked. He was also tormented by physical suffering: rarely a day passed free from the agonies of acute stomach ache or tooth ache. Despite all this, or perhaps because of it, he wrote two imperish- able masterpieces of French literature, his Pensdes (Thoughts) and his Provincial Letters.

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Blaise Pascal was born on 19 June 1623 at Clermont Ferrand, France. Descartes was 27 years old; Newton had died the year before. His father, Etienne Pascal, was president of the court of aids at Clermont; his mother, Antoinette Bhgone, died when Blaise was only four. He had two beautiful and talented sisters, Jacqueline and Gilberte. They were to play no small part in his life ... and he in theirs, as we shall see.

When he was seven, the Pascals moved from Clermont to Paris. His father at first taught Blaise at home. He was a sickly boy. For fear of risking his health, Pascal pere made mathematics taboo. This not unnaturally excited Blaise's curiosity.



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One day when he was twelve, Blaise demanded of his father to know what geometry was. His father told him. He set to with a will and discovered many theorems for himself. In particular he found the famous fact that the sum of the angles of a triangle is 1800. And he is supposed to have used a paper triangle to do so: he bent over the points so that they all met at a point on the base. Can the reader do this? (Answer next issue.) This, of course, did not prove the fact; it merely showed it. Still, for a twelve-year-old it was a remarkable discovery to make. So remarkable that his father gave him a copy of Euclid's Elements. Pascal devoured the book and quickly mastered it.

A recognized mathematical prodigy, young Pascal attended weekly meetings with a Professor Roberval and later Mersenne (see Descartes); out of these meetings was to grow the French Academy of Sciences which was formed six years after the Royal Society in England. By the age of sixteen Pascal had written a book on conic sections. In it he proved 400 proposi- tions on conics, including those of Apollonius. To do so he used his 'Mystic Hexagon' (Fig. 1): Pascal proved that if any hexagon is inscribed in any conic section-parabola, ellipse, circle, or hyperbola-then the points of intersection of pairs of opposite sides will always... We leave this as a second puzzle for the reader to solve. To fix ideas, draw a hexagon (not regular!) inside a circle. Prolong pairs of opposite sides. Look at the three points of intersection. What do you see?

When young Pascal was 17 his father was forced to go into hiding for questioning Cardinal Richelieu's reducing the interest payable on bonds held by Pascal phre! Richelieu was furious. A story goes that Richelieu happening to see the beautiful Jacqueline in a play was so captivated by her performance that, on learning who she was, forgave P~re Pascal" more, he gave him the plum job of tax-collector in Rouen. Another story attributes the Cardinal's change of heart to the special pleading of Pascal's neice, Madame d'Aiguillon.

In the event the Pascals moved once more, to Rouen. Here Blaise amused himself by making the first calculating machine. It consisted of a series of gear wheels, each with ten notches, like a modern milometer. When the first wheel on the left had turned through nine notches, the tenth tripped the next wheet. The numbers showed through little windows. (The idea for this machine had been thought of, nearly 200 years before, by Leonardo da Vinci; he never made it, only drew it in his note- books. Pascal made this machine to help his father calculate tax returns. But he never made his fortune out of it; businessmen thought it too expensive toy and difficult to repair.

Not a day was to go by without Pascal suffering acute pain. At this time the whole Pascal family came under the influence

of a religious sect known as Jansenists, after Corneluis Jansen, a Dutch reformist, who had written a vast tome attacking the Jesuits. God, he believed, had chosen him as the scrouge of the Jesuits. All the Pascals, but especially Blaise, embraced this horrid creed of hatred.

In 1647 the family returned to Paris but left the following year for Clermont again.

Something must have rekindled his passion for science. For he took up Torricelli's work on the barometer and went on to show he was the equal of Galileo in understanding the scientific method (see Galileo). His sister Gilberte had married a M. Pcrier.

At Pascal's suggestion, P~rier took a barometer with him up the Puy de Dome in the Auvergne district, central France; he noted that the atmospheric pressure fell the higher he went.

In 1650 the restless Pascals moved once more to Paris; Pascal's father was fully restored to favour and was now a state councillor. Descartes made a formal visit to the Pascals. He and Pascal talked over many things, including the barometer. But they did not get on well. Descartes flatly refused to believe a boy of sixteen could have written the famous essay of conics. Worse still, Descartes suspected Pascal of lifting Descartes' idea of experiments on the barometer from himself. To make matters even worse, Descartes' and Pascal's views on religion were totally opposed: Descartes had received nothing but kindness from the Jesuits whom Pascal as a devoted Jansenist was committed to hate. And lastly, according to the outspoken Jacqueline, both Descartes and Pascal were intensely jealous of each other's brilliance. (The reader may recall how Descartes had shunned meeting Galileo for similar reasons.) However, Descartes gave Pascal good advice: to stay in bed till 11 each morning, and for Pascal's weak stomach he prescribed beef tea. Neither piece of advice was Pascal inclined to follow-perhaps to the irreparable loss of mathematics.

Jacqueline morbidly began to work on her brother to bring him more fully within the Jansenist fold. That he was not a con- firmed Jansenist is revealed by the rumour that he had had a mild love affair with a Clermont girl. But ever since Pascal passed into the Christian hall of fame on his death every effort has been made to cover up such vile rumours! However, anyone who reads the Pensdes cannot fail to be struck by Pascal's harping on lust. The book gives him away completely-as do his well-known loathing of the sight of his married sister Gilberte caressing her children.

Figure 1

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"Mystic hexagon'

In 1650 the restless Pascals moved once more to Paris. The next year Pascal pere died which gave Pascal the opportunity to write a lengthy heartless sermon on death for Gilberte. The letter is much admired to this day. Now her father was dead, Jacqueline joined the Port Royal convent thus leaving Pascal free to take up the less morbid task of administering the family estate. The division of the estate led to an unseemly squabble between brother and sister-hardly becoming in two such devout believers!

Following in Descartes' footsteps Pascal wrote a letter to the

11



Swedish Princess Christina. In it he reveals a side of him not shown in his Pensees: he humbly begged to be allowed to lay at the feet of the "greatest princess in the world" his calculating machine. What she did with it is not recorded. She had killed off Descates but declined to invite the obsequious Pascal to her court. (see Descartes).

Meanwhile Pascal and Fermat were corresponding over the Chevalier de Mere's famous dicing problem (see Fermat). Between them they solved the problem but gave different proofs of their solution. Out of this correspondence arose the whole subject of mathematical probability. To work out such problems

Pascal made use of a famous triangle of numbers, known today as Pascal's triangle (see below). Actually he didn't invent it; it was known to the Chinese centuries earlier.

I

1 1 1 2 1

1 3 3 1 1 4 6 4 1

1 5 10 10 5 1

Look at the triangle of numbers a moment. You see two sides composed entirely of ls. To see how it builds up, look at the first 3 in the third row: you form it by adding the two numbers either side of it in the row above. All numbers are formed in this way. Pascal's remarkable feat was to discover how to work out combinations from the triangle. For example, how many ways can you pick two things out of four. Look at the number three along from the left in the 4th row: 6, which is the correct answer. How many ways can you pick from 3 things from 5 things is found 4 along in the 5th row: that is, 10 ways.

Soon after laying the foundations of the theory of probability Pascal became converted to religion. On 23 November 1654, while he was driving a four-in-hand to Neuilly, just outside Paris, his horses bolted. The first pair plunged over the parapet of the bridge at Neuilly. But the traces broke, thus saving Pascal's life. Pascal regarded this narrow escape as a personal message from God, to whose services he now devoted all his time. Ever after- wards he always carried a parchment slip bearing words of mystic devotion; it has become known as "Pascal's amulet". Now only 31, Pascal retired from the world to Port Royal where he wrote his Pensdes and his Provincial Letters, the latter a scathing attack on the Jesuits.

Then one night he almost slipped from grace: this was the famous episode of the cycloid. He had been troubled by raging toothache and could not sleep. To soothe his nerves he began to ponder on the famous cycloid (see Cycloid). To his surprise his toothache vanished; he took this as a sign from God that it was not sinful to think about mathematics instead of his cus- tomary dwelling on the misery of man. For eight days he worked on the cycloid and managed to solve many of the then unsolved problems connected with it. In 1658 he published his discoveries under the pen-name of Amas Dettonville. It was to be his last contribution to science.

His health rapidly declined; racking headaches deprived him sleep. In June 1622, as a final act of self-denial, he gave up his house to a poor family suffering from small-pox, and went to live with Gilberte and her husband. On 19 August 1662 he died of convulsions. He was only thirty-nine.

His life was a sad paradox. He could coin such phrases as "The heart has its reasons which the head knows not of". And yet he himself always tried to be ruled by his head, as he showed in the famous "wager with God". As he wrote in his Pensees the "expectation" in gambling is the value of the prize multiplied by the probability of winning it. Pascal considered the value of eternal happiness to be infinite. He reasoned as follows: the probability of winning eternal happiness by leading a virtuous life is extremely small; all the same the expectation is infinite and so it is worth anybody's efforts to lead a life of virtue. This Pascal certainly tried to do. But a note of doubt creeps into another passage in the Pensees where he says: "Is probability

probable?" Such a profound question makes one wonder what, had ne not been diverted to religion, might he hot have achieved in mathematics. g

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1

by D. B. Eperson

1. A Knight's Tour on a Chess-Board In Rouse Ball's Mathematical Recreations and Problems the author discusses the problem of "moving a Knight on a chessboard in such a manner that it moves suc- cessively to every cell once and once only". Eminent mathematicians such as Euler and Demoivre have shown how to obtain solutions on a full-size chessboard, but here are .some simpler problems on a five-by-five mini- chessboard that anyone can tackle.

Figures 1, 2, 3

12 7 2

6 1 13 8

111 1 3

5 9

10 4

The numbered chart, Fig. 1, shows how to make a short tour visiting 13 cells numbered 1 to 13, so that the black and white pattern Fig. 2 made by the visited and unvisited cells has a line of symmetry, the tour diagram being shown in Fig. 3. Note that the middle cell of the tour, No. 7, is on the line of symmetry of the pattern. A tour can obviously be numbered in reversed order.

(a) Find the numbered chart and the tour diagram of the symmetrical pattern shown in Fig. 4.

(b) Show that there are two different numbered charts and tour diagrams for the pattern in Fig. 5.

(c) The pattern in Fig. 6 has two lines of symmetry, and rotational symmetry about the central cell. How many different numbered charts and tour diagrams can you find for Fig. 6?



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ABC of Mathematicians