How did the ancients view *infinitesimals*?

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.

I think (but am very open to correction by those who know the history better than I do) that the situation is basically this. Newton, Leibniz and other early practitioners made a bunch of assumptions (both overt and implicit) about infinitesimals that can't coherently all be held true together. That's why from the earliest days, starting with Bishop Berkeley's mockery of the very idea, there was an unresolved anxiety about infinitesimals that lasted right into the twentieth century. They "worked" wonderfully well, but seemingly shouldn't have.

So the name of the game isn't to find the coherent account of infinitesimals that Newton and Leibniz had in view. Rather the game is one of optimal rational reconstruction. What coherent view gives us a theory which satisfies the most important features that Newton and Leibniz wanted in a calculus of infinitesimals, and does so in the neatest, most mathematically fruitful way?

But note there needn't be a fact of the matter about what are the most important features, about what's neatest, about what's most fruitful. These are going to be judgement calls, and maybe there will be different ways to go, with rather different costs and benefits. Even if we decide that the kind of theory which Bell presents is overall a "best buy", it would certainly be pushing the principle of charitable interpretation to breaking point to say that it was therefore what Newton and Leibniz "really" meant all along. Almost certainly, any coherent theory will have to deny things they would have taken for granted.

But then is Bell's smooth infinitesimal analysis the best buy? As the OP indicates, there are other and more popular modern versions of nonstandard analysis which stick closer to Robinson's original. For an extended development of one, see Nader Vakil's impressive and illuminating Real Analysis Through Modern Infinitesmials (in the Cambridge Encyclopedia of Mathematics series). But I wouldn't want to say either that such a theory rather than Bell's -- attractive though it is -- is what Newton and Leibniz really were talking about! Being the best buy in the neighbourhood is recommendation enough.

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)

The Archimedean property of the real numbers says that there is no $\varepsilon>0$ such that $$ \underbrace{\varepsilon+\cdots\cdots+\varepsilon}_{n\text{ terms}} $$ remains less than $1$ regardless of how small $\varepsilon$ is, as long as the finite cardinal number $n$ is large enough.

One explanation of why it's called "Archimedean" is that Archimedes of Syracuse (287 BC – c. 212 BC), a geometer, physicist, engineer, and inventor, said such quantities don't exist. Nonetheless, he used them brilliantly. He said his arguments using infinitesimals fall short of being complete proofs because infinitesimals don't exist. All of his arguments using infinitesimals are in one work, sometimes titled The Method.

Here is an example:

• Draw a secant line to a parabola with endpoints $A$ and $B$. (This need not be orthogonal to the axis, as one might think from some illustrations.)
• Through $B$, draw the tangent line; through $A$ draw a line parallel to the axis. These meet at $C$.

Archimedes claimed: One-third of the area of triangle $ABC$ is inside the curve.

To demonstrate this, he relied on the concept of center of gravity, which had first been introduced by him. He also relied on the concept of torque on a lever, also first introduced by him.

• Let $D$ be the midpoint between $A$ and $C$. Consider $D$ to be the fulcrum of a lever, which is the line $DB$.
• Let $E$ be on the line $DB$, just as far from the fulcrum $D$ as $B$ is, but in the opposite direction.

Archimedes showed that the center of gravity of the interior of the triangle is on this line $DB$, one-third of the way from $D$ to $B$. If one could let the whole weight of the triangle rest at that center of gravity, and a weight equal to that bounded by the curve and the secant line at $E$, then the lever is in equilibrium precisely of the proposition to be demonstrated is true.

To show that, he considered cross-sections parallel to the axis. Let the infinitesimal weight of each such cross section (proportional to its length) rest on the lever at the point where the cross-section intersects the lever. Archimedes claimed this would exert just as much torque on the lever as if the whole weight of the triangle rests at the center of gravity.

So let the infinitesimal weight one such cross-section rest on the lever $DB$ at the point where it crosses $DB$. And let a weight equal to that of the part of that cross-section that is inside the curve rest at $E$. If the lever is then in equilibrium, then we're done. But that is just what in modern language we would call the equation of the parabola. QUOD ERAT DEMONSTANDUM

Archimedes used the same method to show that the center of gravity of the interior of a hemisphere (i.e. half a sphere) is five-eights of the way from the pole to the center. And maybe more than a dozen other propositions; I don't remember exactly how many.

The reals admit to axiomatization that allows for infinitesimals. Look at http://en.wikipedia.org/wiki/Hyperreal_number

Edit: Excuse me. I misread the question. No one can say with certainty. The ancients must have thought of infinitesimals as "very small" or "infinitely small" change, as in the differential change "dx." Newton and Leibniz did not have a formal definition of the reals and were not aware of their completeness. They had superb intuition. I doubt anyone will give you a more satisfactory explanation.

p.s. This question feels a lot like the high school:

What did Dumas mean when he wrote such and such in his "blah-blah?"