Untitled Essay, Research Paper

Blaise Pascal was born at Clermont on June 19, 1623, and died in Paris on Aug. 19, 1662. His male parent, a local justice at Clermont, and himself of some scientific repute, moved to Paris in 1631, for two chief grounds, to prosecute his ain scientific surveies, and to transport on the instruction of his lone boy, who had already displayed exceeding ability. Pascal was kept at place in order to guarantee his non being overworked. Surprisingly, Pascal? s household directed his instruction to foreign linguistic communications and did non learn him mathematics. Naturally, this excited the male child & # 8217 ; s wonder, and one twenty-four hours, when he was 12 old ages old ; he asked what geometry consisted of. His coach told him that it was the scientific discipline of building exact figures and finding the proportions between their different parts. Pascal, gave up his play-time to this new survey, and in a few hebdomads had discovered for himself many belongingss of figures, and in peculiar the proposition that the amount of the angles of a trigon is equal to two right angles. His male parent, struck by this show of ability, gave him a transcript of Euclid & # 8217 ; s Elementss, a book that Pascal read invariably and shortly mastered.When Pascal was 14 he was admitted to the hebdomadal meetings of Roberval, Mersenne, Mydorge, and other Gallic geometers ; from which, subsequently the Gallic Academy sprung. At the age of 16 Pascal wrote an essay on conelike subdivisions, and in 1641, at the age of 18, he constructed the first arithmetical machine, an instrument which, eight old ages subsequently, he farther improved. His correspondence with Fermat about this clip shows that he was so turning his attending to analytical geometry and physics.In 1653 he had to administrate his male parent & # 8217 ; s estate. He now took up his old life once more, and made several experiments on the force per unit area exerted by gases and liquids. It was besides about this clip that he invented the arithmetical trigon, and together with Fermat created the concretion of probabilities.Pascal is besides celebrated for his Provincial Letters directed against the Jesuits, and his Pens? Es, which were written towards the stopping point of his life. They are the first illustrations of that finished signifier which is characteristic of the best Gallic literature. The lone mathematical work that he produced after retiring to Port Royal was the essay on the cyc

loid in 1658. He was suffering from sleeplessness and toothache when the idea occurred to him, and to his surprise his teeth immediately ceased to ache. Regarding this as a divine intimation to proceed with the problem, he worked nonstop for eight days at it, and completed a full account of the geometry of the cycloid.His early essay on the geometry of conics, written in 1639, but not published until 1779, seems to have been based on the teaching of Desargues. Two of the results are important as well as interesting. The first of these is the theorem known now as “Pascal’s Theorem,” namely, that if a hexagon were inscribed in a conic, the points of intersection of the opposite sides will lie in a straight line. The second, which is really due to Desargues, is that if a quadrilateral be inscribed in a conic, and a straight line be drawn cutting the sides taken in order in the points A, B, C, and D, and the conic in P and Q, thenPA.PC : PB.PD = QA.QC : QB.QD.Pascal employed his arithmetical triangle in 1653, but no account of his method was printed until 1665. The triangle is shown on the title page, each horizontal line is being formed form the one above it by making every number in it equal to the sum of those above and to the left of it in the row immediately above it; ex. gr. the fourth number in the fourth line, namely, 20, is equal to 1 + 3 + 6 + 10. The numbers in each line are what are now called figurate numbers. Those in the first line are called numbers of the first order; those in the second line, natural numbers or numbers of the second order; those in the third line, numbers of the third order, and so on. It is easily shown that the mth number in the nth row is (m+n-2)! / (m-1)!(n-1)!Drawing a diagonal downward from right to left as in the figure gets Pascal?s arithmetical triangle, to any required order. The numbers in any diagonal give the coefficients of the expansion of a binomial; for example, the figures in the fifth diagonal, namely 1, 4, 6, 4, 1, are the coefficients of the expansion (a+b) to the 4th power . Pascal used the triangle partly for this purpose, and partly to find the numbers of combinations of m things taken n at a time, which he stated, correctly, to be (n+1)(n+2)(n+3) … m / (m-n)! This figure is one of the greatest mathematical discoveries of all time, and will live on forever!