infinitesimal (historical)

An infinitesimal was imagined as a number bigger than zero but smaller than every ordinary positive number — a quantity so tiny it could be discarded at the end of a calculation yet was nonzero enough to divide by in the middle. Useful, intuitive, and, taken literally, contradictory.

In the calculus of Newton and Leibniz, derivatives were computed by forming a ratio dy/dx of infinitesimal increments, then ‘neglecting’ infinitesimal leftovers. The method worked astonishingly well in practice but its foundations were attacked (Berkeley's jibe about ‘ghosts of departed quantities’) because no real number is positive yet smaller than every positive number — the Archimedean property of R forbids it.

The nineteenth-century resolution replaced the infinitesimal with the rigorous limit: instead of an actually-infinitely-small quantity, one studies what happens as a quantity becomes arbitrarily small, quantified by epsilon–delta. The honest footnote: infinitesimals were later rehabilitated rigorously in Robinson's non-standard analysis (1960s) using a larger number system, but that is a separate, optional framework — standard real analysis dispenses with them entirely.