Calculus rested on numbers too small to exist until 1961 rescued them
Leibniz pictured a derivative as one vanishingly small quantity divided by another. Mathematicians loved the results and distrusted the foundations, eventually swapping these infinitely small numbers for limits. Then, in 1961, Abraham Robinson showed the old idea could be made fully rigorous after all.
An infinitesimal is a quantity that is not zero yet lies nearer to zero than any nonzero real number. No such thing exists among ordinary real numbers, but extended systems such as the hyperreals and the surreals contain them alongside infinitely large numbers, each kind being the reciprocal of the other. The word itself dates from 17th-century Latin, where infinitesimus meant roughly the infinitieth item in a sequence.
The roots are Greek. Zeno of Elea's dichotomy paradox wrestled with ever-shrinking intervals, and Archimedes, in the 3rd century BC, used slicing arguments in his Method of Mechanical Theorems to find areas and volumes, while his published proofs relied on the stricter method of exhaustion. His Archimedean property gives a test: a number system passing it has neither infinite nor infinitesimal members.
Much later, Nicholas of Cusa and then Johannes Kepler treated a circle as a polygon with infinitely many sides, Bonaventura Cavalieri developed indivisibles, and John Wallis, in his 1655 Treatise on the Conic Sections, wrote one over infinity for an infinitely thin strip and imagined summing endless strips into a finite area, a forerunner of integration. The modern notion emerged around 1670 from Nicolaus Mercator or Leibniz, whose calculus rested on heuristic rules like the law of continuity: whatever works for finite numbers works for infinite ones too. Euler and Lagrange used infinitesimals routinely in the 18th century, and Cauchy employed them in defining continuity.
Rigour eventually won out through limits, computable with standard reals, and infinitesimals largely left the classroom. Yet the thread persisted: Paul du Bois-Reymond studied continua enriched by growth rates, Thoralf Skolem built the first nonstandard models of arithmetic in 1934, and Robinson, drawing on Edwin Hewitt's 1948 work and Jerzy Łoś's of 1955, created nonstandard analysis. Its transfer principle turns Leibniz's law of continuity into a theorem. Surreal numbers followed, forming the largest ordered field.
Source: Infinitesimal
