many arguments are based on a “Newtonian form of geometric reasoning” [p. 33] making liberal use of ultimate equality: two quantities A and B depending on a parameter are said to be ultimately equal
| A | |||
| denoted A≍B | lim | =1 | |
| ε→0 | B |
ref: Geometria differenziale. Il libro "Visual Differential Geometry and Forms - Tristan Needham" usa questo modus.
Translated and Annotated by Ian Bruce. Page 76-86
book1s1.pdf
ref:
https://www.17centurymaths.com/contents/newtoncontents.html
The mathematical foundations of the work are set out here in a series of Scholia, which are of considerable interest, as they are given in terms of what Newton calls the first and last ratios of sums and ratios, being a geometric approach to the limiting processes involved in integration and differentiation.
https://www.jstor.org/stable/2973484?seq=1
Whether certain variables can reach their limits or not is the vital issue in
the "Achilles." For that reason Newton's statements on this point are of
interest:
". . . those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits toward which the ratios of quantities decreasing without limit do always converge; and to which they approach nearer than by any given difference, but never go beyond, nor in effect attain to, till the quantities are diminished in infinitum."
Newton's Principia, Book I, Section I, last scholium.
That Newton let his variables reach their limits appears even more clearly in
the following passage:
"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."
Newton's Principia, Book I, Section I, Lemma I.
Other passages in the first book of the Principia allow variables to reach
their limits. While Newton's exposition is not as explicit as one might wish,
nor free from objection, he deserves the credit of perceiving that variables may
reach their limits and that variables arising in mechanics are usually of such a
nature that they do reach their limits.
As is well known, the foundations of the calculus were severely attacked by
Bishop Berkeley. His first published statement on this subject appears in his
Alciphron, or the Minute Philosopher (1732), penned while he and his wife were
sojourning at Newport, Rhode Island. He says that mathematical science falls
short of "those clear and distinct ideas" which many "expect and insist upon in
the mysteries of religion. . . . Such are those which have sprung up in geometry
about the nature of the angle of contact, the doctrine of proportions, of
indivisibles, infinitesimals, and divers other points."
Geometria differenziale. Il libro "Visual Differential Geometry and Forms - Tristan Needham" usa questo modus.