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What are infinitesimals?

What are infinitesimals? Here are some definitions.

Noun
  1. plural of infinitesimal
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During the nineteenth century all definitions and proofs in the Leibniz style were rewritten to talk of limits instead of infinitesimals.
In the 19th century, infinitesimals were replaced by the epsilon, delta approach to limits.
Not only did Peirce defend infinitesimals.
Since his work made its appearance just before the dawn of calculus, infinitesimals will be used in the sequel.
These two concepts, infinitesimals and infinite quantities, however, were stirring great philosophical dilemmas.
It's hard to envision neurasthenic pulling or other activity, but I don't grok Hegelian infinitesimals either.

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