Both Blaise and Jerry are, of course, absolutely correct, and Bill's derivation is, as I said, very clever and elegant (though what *are* its implications for calculus, reformed or otherwise?). I was simply replying to the claim that the quadrature didn't involve either infinitesimals or limits.
It is not clear what future there is in a project which would attempt to do large parts of calculus without mentioning the word limit. (I am not implying that Bill advocates such a project.) That one can do small parts may be true, but to what end? Calculus is generally about limits.
Newton himself, in the Principia, seems to have had a crisis of nerve, and tried to present geometric instead of "fluxion" (calculus) arguments. This resulted in a basically unreadable treatise; furthermore, few people know or even care that he did this, and rightfully so.
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