Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them.
In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or,
so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" in the
vernacular means "extremely small". An infinitesimal object by itself is often useless and not very well defined; in order to give it a
calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin's work on decimal representation of all
numbers in the 16th century prepared the ground for the real continuum.Bonaventura Cavalieri's method of indivisibles led to an extension
of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of
codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building
blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal
denoted in area calculations.
succeeds also for the infinite numbers and vice versa; and theTranscendental Law of Homogeneity that specifies procedures for replacing
expressions involving inassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of
Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of
Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. A
mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who
implements Fermat's adequality.