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Blaise Pascal: Numbers
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When he was about 30, Pascal got into discussions with Fermat about the mathematics of gambling. As a result, he devised what is called Pascal's Triangle, an array of numbers which begins as shown here.
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As you may have noticed, the numbers in the chart above are actually the tip of the right-angled form of Pascal's Triangle, except the preceeding 1's in each row are missing. The circular figures are formed by simply placing a number of points on a circle and then drawing all the possible lines between them. This chart shows that for a figure with n points, all you need to do is look at the nth row of the Triangle in order to find the number of points, line segments, and polygons in the figure with ALL of their vertices on the circle.
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The mechanical calculator was devised by Pascal in 1642 and was brought to a commercial version in 1645. It was one of the earliest in the history of computing. ‘Side by side in an oblong box were places six small drums, round the upper and lower halves chich the numbers 0 to 9 were written, in decending and ascending orders respectively. According to whichever aritchmatical process was currently in use, one half of each drum was shut off from outside view by a sliding metal bar: the upper row of figures was for subtraction, the lower for addition. Below each drum was a wheel consisting of ten (or twenty of twelve) movable spokes inside a fixed rim numbered in ten (or more) equal sections from 0 to 9 etc, rather like a clockface. Wheels and rims were all visible on the box lid, and indeed the numbers to be added or subtracted were fed into the machine by means of the wheels: 4 for instance, being recorded by using a small pin to turn the stoke opposite division 4 as far as a catch positioned close to the outer edge of the box.
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Pascal employed his arithmetical triangle in 1653, but no account of his method was printed till 1665. The triangle is constructed as in the figure below, each horizontal ligne 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 ligne, namely, 20, is equal to 1 + 3 + 6 + 10.
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