Infinitesimal - History of The Infinitesimal

History of The Infinitesimal

The notion of infinitesimally small quantities was discussed by the Eleatic School. The Greek mathematician Archimedes (c.287 BC–c.212 BC), in The Method of Mechanical Theorems, was the first to propose a logically rigorous definition of infinitesimals. His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for 1/x and the reciprocals of the positive integers. A number system is said to be Archimedean if it contains no infinite or infinitesimal members.

The Indian mathematician Bhāskara II (1114–1185) described a geometric technique for expressing the change in in terms of times a change in . Prior to the invention of calculus mathematicians were able to calculate tangent lines by the method Pierre de Fermat's method of adequality and René Descartes' method of normals. There is debate among scholars as to whether the method was infinitesimal or algebraic in nature. When Newton and Leibniz invented the calculus, they made use of infinitesimals. The use of infinitesimals was attacked as incorrect by Bishop Berkeley in his work The Analyst. Mathematicians, scientists, and engineers continued to use infinitesimals to produce correct results. In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the (ε, δ)-definition of limit and set theory. While infinitesimals eventually disappeared from the calculus, their mathematical study continued through the work of Levi-Civita and others, throughout the late nineteenth and the twentieth centuries, as documented by Philip Ehrlich (2006). In the 20th century, it was found that infinitesimals could serve as a basis for calculus and analysis.