Indivisibles, Method of

in mathematics, a variety of significantly different methods introduced near the end of the 16th century for determining ratios of areas or volumes of figures. The method of indivisibles was based on a comparison of the “indivisible” elements (or collections of elements) that in some manner formed the figures. The very concept of indivisibles has been understood in different ways by scientists of various periods.

The method of indivisibles was first used in ancient Greek science. Democritus seems to have considered solids as “sums” of extremely large numbers of extremely small indivisible atoms. Archimedes found areas and volumes of many figures by combining the principle of the lever and the concept that a plane figure consists of an uncountable number of parallel line segments and a geometric body consists of an uncountable number of parallel plane sections. Even in antiquity, however, these concepts and methods were subjected to criticism. Archimedes, for example, believed it necessary to use the method of exhaustion to re-prove results proved by means of the method of indivisibles. Disputes regarding the structure of the continuum were revived in medieval science and have continued up to the present. The concepts of the method of indivisibles were revived in mathematical investigations at the turn of the 17th century by J. Kepler and, particularly, by F. B. Cavalieri, with whose name the method of indivisibles is most often associated. The method of indivisibles developed by Cavalieri was later substantially changed by E. Torricelli, J. Wallis, B. Pascal, and other prominent scientists and became one stage in the creation of the integral calculus.