TITLE: How did the ancients view *infinitesimals*?
QUESTION [16 upvotes]: With some category/topos theory we can now put infinitesimals on a rigorous ground, as in Bell's A Primer of Infinitesimal Analysis, where the author introduces $\epsilon$ satisfying \begin{equation}
\epsilon\ne 0, \epsilon^2=0.
\end{equation}  
However, he also points out that this version of infinitesimal is not compatible with the law of excluded middle.
Meanwhile, the author seems convinced that this $\epsilon$ is the infinitesimal in the eyes of Newton and Leibniz among many others, when they were attacking problems like instantaneous speed and area under a curve.
I wonder whether this is true. I know people like Newton and Leibniz did not use limiting argument. But this does not mean they think of infinitesimals as nilsquare elements as described by Bell, because there are still other models of infinitesimals available.
Thanks very much.

REPLY [6 votes]: Your question got me reading (googling). This paper: Leibniz's Infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russel and beyond (Katz & Sherry, 2012), in section 4, explicitly addresses your question, with a quote by Leibniz himself even. In his calculus, $\epsilon^2 \neq 0$. I'm only 1/3 in, but this is a nice paper. Here is the quote:  

"being driven to fall back on assumptions that are admitted by no one; such as that something diﬀerent is
  obtained by multiplying 2 by m and by multiplying m
  by 2; that the latter was impossible in any case in which
  the former was possible; also that the square or cube of
  a quantity is not a quantity or Zero (Leibniz translated
  by Child [28, p. 146])." (emphasis added)