3, son, who had already displayed exceptional
3, and died in Paris on Aug. 19, 1662. His father, a local judge at Clermont, and himself of some scientific reputation, moved to Paris in 1631, for two main reasons, to prosecute his own scientific studies, and to carry on the education of his only son, who had already displayed exceptional ability. Pascal was kept at home in order to ensure his not being overworked. Surprisingly, Pascals family directed his education to foreign languages and did not teach him mathematics.
Naturally, this excited the boy’s curiosity, and one day, when he was twelve years old; he asked what geometry consisted of. His tutor told him that it was the science of constructing exact figures and determining the proportions between their different parts. Pascal, gave up his play-time to this new study, and in a few weeks had discovered for himself many properties of figures, and in particular the proposition that the sum of the angles of a triangle is equal to two right angles. His father, struck by this display of ability, gave him a copy of Euclid’s Elements, a book that Pascal read constantly and soon mastered. When Pascal was fourteen he was admitted to the weekly meetings of Roberval, Mersenne, Mydorge, and other French geometricians; from which, later the French Academy sprung. At the age of sixteen Pascal wrote an essay on conic sections, and in 1641, at the age of eighteen, he constructed the first arithmetical machine, an instrument which, eight years later, he further improved.
His correspondence with Fermat about this time shows that he was then turning his attention to analytical geometry and physics. In 1653 he had to administer his father’s estate. He now took up his old life again, and made several experiments on the pressure exerted by gases and liquids.
It was also about this time that he invented the arithmetical triangle, and together with Fermat created the calculus of probabilities. Pascal is also famous for his Provincial Letters directed against the Jesuits, and his Penses, which were written towards the close of his life. They are the first examples of that finished form which is characteristic of the best French literature. The only mathematical work that he produced after retiring to Port Royal was the essay on the cycloid 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, then PA.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 Pascals 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!