What does "ultimately vanishing" mean? (Needham) In the Prologue of the book Visual Differential Geometry and Forms by Needham the notion of two functions $A(\epsilon)$ and $B(\epsilon)$ being ultimately equal is defined: it means that
$$
\lim_{\epsilon \to 0} \frac{A(\epsilon)}{B(\epsilon)} = 1.
$$
This is all good and seems to be motivated by how Newton used limits. But then a few lines below, Needham uses the notion ultimately vanishing without giving a definition. I just don't know what precisely this means. I first thought it may mean that $A(\epsilon)$ is ultimately equal to $\epsilon$, but then $\epsilon^2$ would not be ultimately vanishing, which doesn't look right. Being ultimately equal to the zero function also doesn't work. Can someone please tell me what this means?
 A: Since on page 275 Needham refers to Newton for the notion of an "ultimately vanishing" quantity, I would interpret that in the sense of Newton, where an ultimately vanishing quantity is an infinitesimal: $\lim_{\epsilon\rightarrow 0}A(\epsilon )=0$. If "ultimately equal" is interpreted in the sense of Newton, it would mean two quantities $A$ and $B$ differ by an infinitesimal, $\lim_{\epsilon\rightarrow 0}A(\epsilon)=\lim_{\epsilon\rightarrow 0}B(\epsilon)$ so an ultimately vanishing quantity is ultimately equal to 0 in Newton's sense.
See Fleuriot's book on Newton's Principia for these two concepts.

Following the discussion in the comments, let me quote from Newton's Principia (book 1, section 1, lemma 1):

Quantities, and the ratios of quantities, which in any finite time
converge continually to equality, and before the end of that time
approach nearer the one to the other than by any given difference,
become ultimately equal.

So apparently Newton allows for both definitions of "ultimately equal", the one used by Needham (ratio tends to unity) and the alternative definition (difference tends to zero). "Ultimately vanishing" refers to "ultimately equal to zero" in the second sense, and I presume Needham follows that meaning.
I should add that the two definitions of ultimately equal ($A-B\rightarrow 0$ versus $A/B\rightarrow 1$) are equivalent for quantities which remain bounded when $\epsilon\rightarrow 0$. This may well be why Newton did not bother to distinguish the two.
