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Full Idea
When the successive absolute values of a variable decrease indefinitely in such a way as to become less than any given quantity, that variable becomes what is called an 'infinitesimal'. Such a variable has zero as its limit.
Gist of Idea
Values that approach zero, becoming less than any quantity, are 'infinitesimals'
Source
Augustin-Louis Cauchy (Cours d'Analyse [1821], p.19), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.4
Book Ref
Kitcher,Philip: 'The Nature of Mathematical Knowledge' [OUP 1984], p.247
A Reaction
The creator of the important idea of the limit still talked in terms of infinitesimals. In the next generation the limit took over completely.
| 21382 | Things get smaller without end [Anaxagoras] |
| 18081 | Nature uses the infinite everywhere [Leibniz] |
| 18080 | A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz] |
| 18091 | Infinitesimals are ghosts of departed quantities [Berkeley] |
| 18085 | Values that approach zero, becoming less than any quantity, are 'infinitesimals' [Cauchy] |
| 18086 | Weierstrass eliminated talk of infinitesimals [Weierstrass, by Kitcher] |
| 18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
| 18083 | With infinitesimals, you divide by the time, then set the time to zero [Kitcher] |
| 18834 | Infinitesimals do not stand in a determinate order relation to zero [Rumfitt] |