A history of the calculus
Analysis index History Topics Index
The main ideas which underpin the calculus developed over a very long period of time indeed. The first steps were taken by Greek mathematicians. To the Greeks numbers were ratios of integers so the number line had "holes" in it. They got round this difficulty by using lengths, areas and volumes in addition tonumbers for, to the Greeks, not all lengths were numbers. Zeno of Elea, about 450 BC, gave a number of problems which were based on the infinite. For example he argued that motion is impossible:If a body moves from A to B then before it reaches B it passes through the mid-point, say B1 of AB. Now to move to B1 it must first reach the mid-point B2 of AB1. Continue this argument to see that A mustmove through an infinite number of distances and so cannot move. Leucippus, Democritus and Antiphon all made contributions to the Greek method of exhaustion which was put on a scientific basis by Eudoxus about 370 BC. The method of exhaustion is so called because one thinks of the areas measured expanding so that they account for more and more of the required area. However Archimedes, around 225 BC,made one of the most significant of the Greek contributions. His first important advance was to show that the area of a segment of a parabola is 4/3 the area of a triangle with the same base and vertex and 2/3 of the area of the circumscribed parallelogram. Archimedes constructed an infinite sequence of triangles starting with one of area A and continually adding further triangles between theexisting ones and the parabola to get areas A, A + A/4 , A + A/4 + A/16 , A + A/4 + A/16 + A/64 , ... The area of the segment of the parabola is therefore A(1 + 1/4 + 1/42 + 1/43 + ....) = (4/3)A. This is the first known example of the summation of an infinite series.
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Archimedes used the method of exhaustion to find an approximation to the area of a circle. This, of course, is an early example of integration which led to approximate values of .
Here is Archimedes' diagram
Among other 'integrations' by Archimedes were the volume and surface area of a sphere, the volume and area of a cone, the surface area of an ellipse, the volume ofany segment of a paraboloid of revolution and a segment of an hyperboloid of revolution. No further progress was made until the 16th Century when mechanics began to drive mathematicians to examine problems such as centres of gravity. Luca Valerio (1552-1618) published De quadratura parabolae in Rome (1606) which continued the Greek methods of attacking these type of area problems. Kepler, in hiswork on planetary motion, had to find the area of sectors of an ellipse. His method consisted of thinking of areas as sums of lines, another crude form of integration, but Kepler had little time for Greek rigour and was rather lucky to obtain the correct answer after making two cancelling errors in this work. Three mathematicians, born within three years of each other, were the next to make majorcontributions. They were Fermat, Roberval and Cavalieri. Cavalieri was led to his 'method of indivisibles' by Kepler's attempts at integration. He was not rigorous in his approach and it is hard to see clearly how he thought about his method. It appears that Cavalieri thought of an area as being made up of components which were lines and then summed his infinite number of 'indivisibles'. He showed,using these methods, that
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the integral of x from 0 to a was a inferring the general result.
/(n + 1) by showing the result for a number of values of n and
Roberval considered problems of the same type but was much more...