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Blaise Pascal By Victoria Hubble Blaise Pascal was born in Clermont-Ferrand, France, on June 19 th, 1623. His mother, Antoinette Been, died when he was three; and his father, Etienne, who was a local judge with a scientific reputation, brought him up. Etienne Pascal retired and moved to Paris in 1631 to concentrate on his own scientific research and to take care of his son, Blaise, and his two daughters, Gilbert and Jacqueline. Etienne had unorthodox views of education and decided to tutor his only son himself. Etienne Pascal locked up all the mathematics texts in the house because he believed that it was too exciting for young minds to be studying mathematics before the age of 15 and he was not going to sap his gifted childs energy from all other pursuits. At age 12, Blaise was curious about geometry and deduced as far as Proposition 32 of Euclid's Elements (the sum of angles of a triangle are two right angles) by himself without any mathematics training.
When his father found out, Blaise was allowed to read his fathers mathematics books, because his father knew that he couldnt stop his genius son anymore. The young Pascal began to participate with his father in Mersennes Circle, a weekly discussion group of scientists and mathematicians, In this plantation of intellects, he learned from Girard Desargues, who had just published a projective geometry book but was not well received because of the difficult vocabulary and style. Pascal was one of the few to appreciate his work. When he was 16, he presented a projective geometry paper at the meeting about what is now called the mystic hexagon which impressed everyone. One of his sisters wrote an account of her brothers life saying the paper was so well received that young Pascal was considered to be the best mathematician since the time of Archimedes. In the next year, Blaise had to suspend his association with the geometries in the Mersennes Circle because he and his sisters had to move to Rouen where his father was appointed as a royal tax official in Upper Normandy.
Blaise Pascal was a genius with many talents. He was known in his day and ours as a mathematician and physicist, and was also a spiritual writer. In this paper, we will focus on his Projective Geometry. At the age of sixteen, he was the one of the few mathematicians who admired Desargues projective geometry at the time; others were too busy with Descartes analytic geometry. He worked on conic sections and produced important theorems in projective geometry. Pascal wrote a brief Essay on Conic Sections which contains a number of projective geometry theorems including his mystic hexagon theorem.
He presented it to one of Mersennes meetings in 1639 and published it in Rouen in February of 1640. 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 notes were taken from it and it is through these notes that a fairly complete picture of the work is now possible. Pascals Triangle. Although Pascal was not the first to study Pascals triangle (Chinese and Islamic mathematicians used it more than 500 years before), his work on the topic in Treatise on the Arithmetical Triangle in 1653 was famous for being the first extensive study of it.
Pascals work on the binomial coefficients was to lead Newton to his discovery of the general binomial theorem for fractional and negative powers. Pascaline. In 1642, after Blaise moved to Rouen with his father, he invented the Pascaline, a mechanical calculator, to help in his fathers tax work. After he successfully built a prototype, he decided to mass-produce it. However, fifty more prototypes were made, but it was too expensive and few machines were sold, so production was stopped.
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 division of the livre into 240 than he would have had if the division had The Problem of the Vacuum.
Since this paper is devoted to Pascals mathematics, I will not go into details of Pascals achievements in physics. It should be noted that because of his belief in the existence of vacuum, he had two arguments with Descartes who thought otherwise. Although Pascal was interested in problems of the vacuum from 1646, his bad health often burdened his research. 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. 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 the 23 rd of 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 Pascals experiment 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. His fathers death in September 1651 and the entry of his sister into the convent of Port-Royal in 1652 provoked a bizarre two years of parties and gambling in his life. Calculus of Probabilities. Even though Blaise was unwell, he laid down the principles for the theory of probabilities in correspondence with Fermat in 1654, the end of this period of gambling.
Their correspondence consisted of five letters, which they considered the dice problem and the problem of points. A gamester, the Chevalier De Mere, proposed the problems to Pascal, who passed them on to Fermat. The first problem concerned the probability that a player will obtain certain dice faces in a given number of throws. The second consisted of determining how to divide the stakes when a game of dice is incomplete. To solve problems, Fermat used combinatorial analysis (determination of the number of possible outcomes in ideal games of chance by computing permutation and combination numbers) and Pascal used reasoning by recursions (an iterative process which determines the result of the next case by the present case). 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 the 23 of November 1654, and he pledged his life to Christianity.
After this time Pascal made visits to the Jansenists monastery Port-Royal about 30 km southwest of Paris. He began to publish anonymous works on religious topics, eighteen Provincial Letters being published during 1656 and early 1657. Pascals most famous work in philosophy is Pensees, 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. Cycloid Work. 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 mathematics again as he lay awake at night unable to sleep because of pain.
He applied Cavalieris method of indivisible's to find the area and center of gravity of any segment of the cycloid. In addition, he also found the volume and surface area of the solid of revolution formed by rotating the cycloid around the x-axis, moreover, he investigated the geometry of the Archimedean spiral. Once he accomplished all these, he believed he had perfected the method of indivisible calculus and decided to challenge other mathematicians to solve these problems. While other leading mathematicians communicated with Pascal about the challenge, Pascal gained valuable information and further enhanced methods.
These findings formed the bridge between the geometry of Archimedes and integral calculus. Leibriz said that it was while reading Pascals Treatise on the Sines of a Quadrant of a Circle, written around 1657, that he realized the inverse relationship of tangent and area problems. Pascal published his own solution to his challenge problems in the Letters to Carcavi. After that time on he took little interest in science and spent his last years giving to the poor and going from church to church in Paris attending one religious service after another. Blaise Pascals life was fruitful and difficult. He is described as: ...
a man of slight build with a loud voice and somewhat overbearing manner... he lived most of his adult life in great pain. He had always been in delicate health, suffering even in his youth from migraines... Pascals character is described as: ... precocious, stubbornly persevering, a perfectionist, pugnacious to the point of bullying ruthlessness yet seeking to be meek and humble...
Pascal used his scientific reasoning to draw the conclusion that we must work on the theory of indivisible's. A lot of people think that Pascal might have achieved more if he did not spend so much of his time on religion. But, it was religion that saved him from the wild life after his fathers death in 1651; otherwise, he would not have laid the grounds for the probability theory and integral calculus afterwards. At the age of 39, he died in pain because a malignant growth in his abdomen spread to his brain. In terms of mathematics, Pascal did not create a piece of significant massive work; however he was unable to make major original contributions to many developing fields The following assessment is given of Pascals life: At once a physicist and a mathematician, ... Pascal was embarrassed by the very abundance of his talents.
It has been suggested that it was his too concrete turn of mind that prevented his discovering the infinitesimal calculus, and in some of the Provincial Letters the mysterious relations of human beings with God are treated as if they were a geometrical problem. But these considerations are far outweighed by the profit that he drew from the multiplicity of his gifts, his religious writings are rigorous because of his Blaise Pascal was no doubt one of the best mathematicians in the seventeenth century. Bibliography:
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