Calculus has its roots in some of the oldest problems in geometry. The Egyptian Rind Papyrus (c. 1650 BCE) gives rules for finding the area of a circle and the volume of an inscribed pyramid. Ancient Greek geometers explored curves, the center of gravity of planes and solid figures, and the quantities of objects formed by revolving various curves about a fixed axis.
By 1635 the Italian mathematician Bonaventura Cavalieri had supplemented the rigorous tools of Greek geometry with heuristic methods that used the idea of lines, areas, and infinitesimal fractions of volume. In 1637, the French mathematician-philosopher René Descartes published his invention of analytic geometry to give an algebraic description of geometric figures. Descartes’ method, combined with an ancient idea of moving curves, allowed mathematicians like Newton to describe motion algebraically. Suddenly geometers could move beyond the single cases and ad hoc methods of previous periods. They could see patterns of results, and thus infer new results, that the old geometric language had obscured.
For example, the Greek geometer Archimedes (287–212/211 BC) discovered as an isolated result that the area of a segment of a parabola is equal to that of a certain triangle. But with algebraic notation, in which the parabola is written as y = x2, Cavalieri and other geometers soon noted that the area between this curve and the x-axis from 0 to a is a3/3 and that That is the same principle for the curve. y = x3—that is, that the corresponding area is a4/4. From here it was not difficult for him to deduce that the general formula for the area under the curve y = xn is a + 1/(n + 1).
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