Method of exhaustion

In mathematics, the method of exhaustion is an old process of calculation of surfaces, volumes and lengths of geometrical figures complex. The squaring is the research of the surface of a surface, the correction is that the length of a curve.

In the case of the calculation of the surface of a figure planes, for example, the principle is to calculate the surfaces of polygons registered and circumscribed with the figure. The framing provides an approximation of as much better than the number on sides used is large. Read by the modern ones, calculations of exhaustion make it possible to arrive at the exact value of the surface has figure by passage to the Limite. It is however about a stage which was not reached by the Old ones which used a double reasoning by the absurdity: it is supposed that the surface is larger than has and a contradiction is obtained, then one supposes that the surface is smaller than has and one obtains another contradiction, from where one concludes that the surface is worth has .

One allots to Eudoxe de Cnide the paternity of this process, but it is Archimedes which made of it a method of framing precise and systematic, by making great use of the axiom which bears its name. This one was many successes, before being made obsolete by the appearance of the Infinitesimal calculus.

Exhaustion and squaring

The method of exhaustion was used in the following problems:

Quadrature of the circle

Proposal 2 of book XII of the Eléments of Euclide proves that the surface of a disc is proportional to the square of the diameter. It rests on a similar property relating to the polygons registered in a circle and previously proven by Euclide. The principle of the method of exhaustion is the following. That is to say a disc of diameter D and surface has, and a second disc of diameter Of and A' surface. It is a question of showing that A/A' = D ² /D' ². Let us suppose that it is not the case and that A/A' is larger. That is to say B a surface such as B/A' = D ² /D' ². One thus has > B. Register in the disc of surface has a polygon of surface C such as has > C > B and in the disc of A' surface a polygon of surface It similar to the polygon of surface C. According to the proposal shown on the polygons, one has C/C' = D ² /D' ² = B/A'. However It < A'. Thus C < B, which is absurd. One cannot thus have A/A' higher than D ² /D' ². One proceeds in the same way if A/A' is lower than D ² /D' ². One thus has A/A' = D ² /D' ².

Archimedes proved then that a circle delimits a surface equal to that of a right-angled triangle of which one of the east coasts equal to the ray of this circle, and of which the other side of the right angle is equal to the circumference of this same circle. By approaching the circle by a succession of polygons, one obtains approximate values of pi. Here principal openings in this field:

Squaring of the parabola

Exhaustion and volume

Volume of the pyramid and the cone

Proposal 6 of book XII of the Eléments of Euclide proves that the pyramids which have even height and of the of the same bases surface have even volume. Euclide from of then deduced that the volume of the pyramid is one the third of the base by the height. The later demonstrations of this formula call all upon methods concerned with near or by far with an integral calculus and no simpler geometrical demonstration could be found. This difficulty led Hilbert in 1900 to make appear this question in third place in its list of problems.

In proposal 10 of book XII, the preceding result is extended to the cones, third of the of the same cylinder bases and of the same height.

Volume of the sphere

By approaching a sphere by registered polyhedrons, it is shown, in proposal 18 of book XII of the Eléments of Euclide, that the volume of a sphere is proportional to the cube of the diameter. It is Archimedes which will determine then the formula of the volume of the sphere.

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