Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. 

The insight with exploiting infinitesimals was that objects could still retain certain defined properties, such as angle or slope, even though 

these objects were quantitatively small.^[1] The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which 

originally referred to the "infinite-th" item in a series. It was originally introduced around 1670 by either Nicolaus Mercator or 

Gottfried Wilhelm Leibniz.^[2]

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 

Archimedes used what eventually came to be known as the Method of indivisibles in his work The Method of Mechanical Theorems to find 

areas of regions and volumes of solids.^[3] In his formal published treatises, Archimedes solved the same problem using the Method of 

Exhaustion. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular 

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.

The use of infinitesimals by Leibniz relied upon heuristic principles, such as the Law of Continuity: what succeeds for the finite numbers 

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 

infinitesimals by mathematicians such as Leonhard Euler and Joseph Lagrange. Augustin-Louis Cauchy exploited infinitesimals in 

defining continuity and an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of 

Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of

 functions. Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to 

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 

developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement 

an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function

implements Fermat's adequality.