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Full Idea
Quantities and the ratios of quantities, which in any finite time converge continually to equality, and, before the end of that time approach nearer to one another by any given difference become ultimately equal.
Gist of Idea
Quantities and ratios which continually converge will eventually become equal
Source
Isaac Newton (Principia Mathematica [1687], Lemma 1), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.2
Book Ref
Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.238
A Reaction
Kitcher observes that, although Newton relies on infinitesimals, this quotation expresses something close to the later idea of a 'limit'.
| 18082 | Quantities and ratios which continually converge will eventually become equal [Newton] |
| 18084 | When successive variable values approach a fixed value, that is its 'limit' [Cauchy] |
| 18092 | Weierstrass made limits central, but the existence of limits still needed to be proved [Weierstrass, by Bostock] |
| 18087 | If x changes by less and less, it must approach a limit [Dedekind] |
| 18187 | Theorems about limits could only be proved once the real numbers were understood [Maddy] |
| 22886 | The modern idea of 'limit' allows infinite quantities to have a finite sum [Bardon] |