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When the author says infinitesimals can be used to define calculus in a perfectly rigorous way I get a bit skeptical. After reading this Wikipedia article though I see why.

http://en.m.wikipedia.org/wiki/Hyperreal_number

I still think there's a certain elegance of not needing to define a whole new set of numbers, but there's also an elegance to the intuition of infinitesimals and infinitesimals.

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Actually, Newton and Leibniz were all about this: infinitesimals. It was only in the XIX Century with D'Alembert and especially when Weisrstrass was ill-digested that we got the epsilon-delta down our throats.

Evanescent quantities are quite natural to me.

As I recall, there was a fluff-up between Bishop Berkeley[1] and the Newton camp regarding the issue[2], in the 18th century.

"Mathematical historian Judith Grabiner comments, 'Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct'."

[1] http://en.wikipedia.org/wiki/George_Berkeley

[2] http://en.wikipedia.org/wiki/The_Analyst

Oh totally. I know they were. From the wikipedia article though it seems like they were much less rigorous about their treatment and that it wasn't until the mid-20th century that there was a formal treatment of them.

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