Indivisibles, method of

The common name for a quite varied assortment of devices aimed at determining the relationships between areas or volumes of figures; the name dates from the end of the 16th century. The basic idea of the method of indivisibles is the comparison of "indivisible" elements (or groups of elements) that in some way make up the figures whose sizes are being compared. The actual concept of an "indivisible" was interpreted in different ways by different scholars at different times.

The method of indivisibles dates back to classical Antiquity. The Greek scientist Democritus (about 460– 380 B.C.) apparently considered solids as "sums" of a tremendous number of extremely small "indivisible" atoms; Archimedes (287–212 B.C.) found the areas and volumes of many figures by combining the principles of his theory of the lever with the idea that a plane figure consists of innumerably many parallel segments of straight lines, and that a geometrical figure consists of innumerably many parallel plane sections. However, in their time such ideas and methods were severely critized. Archimedes, for example, deemed it necessary to provide a second proof of results obtained by the method of indivisibles, based on the method of exhaustion (see Exhaustion, method of). The ideas of the method of indivisibles were revived in mathematical research at the turn of the 16th century into the 17th century by J. Kepler and, especially, by B. Cavalieri, with whose name the method is most frequently associated. Cavalieri's version of the method was later considerably transformed and served as a stage in the creation of integral calculus. See Infinitesimal calculus.

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In general, Archimedes liked to first get an idea of some relation by means of mechanics, and subsequently found it necessary to provide a proof of this relation.

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