Calculus

Calculus was invented, more or less simultaneously, by Isaac Newton and Gottfried Leibniz. Some of the essential ingredients, however, had their beginnings in ancient Greece. In the fourth century B.C., Eudoxus invented the so called method of exhaustion, in order to furnish proofs of certain geometric theorems without having to resort to arguments involving the infinite. Approximately a century later, Archimedes used the same method to find a formula for the area of a circle. Archimedes' method consisted of inscribing a polygon with n sides inside a circle, and circumscribing a similar polygon, again with n sides, outside the circle. Then, allowing n, in other words the number of sides, to get very large, he was able to show that the area of the circle was always greater than the area of the inscribed polygon and less than the area of the circumscribed polygon. As n grew very large, the areas of the two polygons tended to become equal, thus leading him to the area of a circle. The method of Archimedes persisted from the third century B.C. until the beginning of the seventeenth century A.D.when the work of Johannes Kepler, a German Astronomer, led to the discovery of general principles for the calculation of areas and volumes. Kepler's contribution was the notion of infinitesimals. He envisioned the inscribed polygons of Archimedes as being a collection of infinitely many, vanishingly small triangles. Thus, the area of a circle could be calculated by summing the areas of these triangles. While, Eudoxus and Archimedes had worked hard to avoid the infinite, Kepler embraced it. Simultaneous work on infinite sequences and sums of infinite sequences led the French mathematician Fermat, and others, to discover general methods for evaluating areas and volumesas the sums of infinite sequences rather than the sums of areas of common geometric figures. Finally, around the middle of the seventeenth century, Isaac Newton, in attempting to develop a universal theory of gravitation, discovered the derivative, a general method for determining the instantaneous rate of change of a function, based on the notion of infinitesimals. Though he did not explicitly define the integral at the time, Newton did recognize the need to solve differential equations. As a result, he invented methods of evaluating indefinite integrals very soon after introducing the derivative. It was Leibniz, however, whose work postdated that of Newton by some 10 years, who recognized and formulated the definite integral as an infinite sum of "lines," that is, as an area calculated by summing an infinite number of infinitely narrow rectangles.