  Pascal / characteristic triangle



Although it was only one page long, Pascal(1623-1662) had published "Essay pour les coniques" when he was sixteen years old. The content of this paper was the explanation of the properties of the points of intersections and the prolonged lines of each side of the hexagon inscribed in a conic section as shown in the below diagram. Pascal designed and fabricated a computing machine and sold about 50 units.



Pascal was intrigued by hydrostatics and conducted experiments with respect to the measuring of the weight of air and water pressure.



Pascal tried to solve the problem of guaranty money in case of losing a dice game (what is called gambling today). His effort to solve this problem resulted in arithmetic triangle (Pascal's triangle) as described in the below diagram.
In order to prove his standpoint, Pascal had worked out a mathematical induction and proved the mathematical properties of his arithmetic triangle. Pascal presented a paper entitled "Traite des sinus du quart de cercle".
He created a technique that finds the dimensions of a quarter of circle by first dividing a quarter of circle into multiple strips and then adding them up. This technique is called "characteristic triangle" and it made a significant impact on Leibniz.

Draw tangent at point on the circumference. On the circumference, the same arc lengths from are to be called . For the sake of better understanding, the arcs are made longer than needed in this example. The perpendicular lines to be and the points of intersection of the perpendicular lines and tangents are . The perpendicular line from point to is to be . and are similarity.  Therefore, in the interval ,  (Caution) is not the dimensions of a trapezoid .

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