The history of calculus
Everyone knows that Newton and Leibniz are the founders of Calculus. Some may think it suffices to know just this one fact. But it is worthwhile, indeed, to go into more details and to study history of the development of Calculus, in particular, up to the time of Newton and Leibniz.
In our courses on Calculus we usually begin with differentiation and then come later to integration. This is entirely justified, since differentiation is simpler and easier than integration. On the other hand, the historical development starts with integration; computing areas, volumes, or lengths of arcs were the first problems occurring in the history of Calculus. Such problems were discussed by ancient Greek mathematicians, especially by Archimedes, whose outstanding and penetrating achievements mark the peak of all ancient mathematics and also the very beginning of the theory of integration. The method applied by Archimedes for his proofs was so-called method of exhaustion, that is, in the case of plane areas, the method of inscribed and circumscribed polygons with an increasing numbers of edges. This method was first rigorously applied, in the form of a double "reductio ad absurdum", by the great Greek mathematician Eudoxus at the beginning of the fourth century B.C. He first proved the facts, previously stated by Democritus, that the volume of a pyramid equals one third of the corresponding prism and the volume of a cone equals one third of the corresponding cylinder. The same mothod was also used by Euclid and then with the greatest success by Archimedes (third century B.C.). It is well known that Archimedes was the first to determine the area and the length of the circle, that is, to give suitable approximate values of $\pi$, and moreover to determine the volume and the area of the surface of the sphere and of cylinders and cones.
But he went far beyond this ; he found the area of ellipses, of parabolic segments, and also of sectors of a spiral, the volumes of segments of the solids of revolution of the second degree, the centroids of segments of a parabola, of a cone, of a segment of the sphere, of right segments of a paraboloid of revolution and of a spheroid. These were amazing achievements, indeed. Archimedes proved his results in the classical manner, by the method of exhaustion. Sometimes the type of approximation is just the same as we would use. For instance, in order to obtain the volume of a solid of revolution of the second degree, Archimedes approximates the volume by a sum of cylindrical slabs. But the direct evaluation of the limit of such sums was cumbersome. Hence we may ask: what was the method used by Archimedes for finding his results ?
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