What was the notion of limit that Newton used?

Question

I have read that the notion of limit became rigorous two centuries after the discover of calculus

What Newton had in his mind regarding the notion of limit?

Answer 1

Section 1 of book 1 of Principia opens with a lemma that can strike us as sounding almost modern:

"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".

But a second thought raises some doubts. First, this is a lemma, not a definition of limit. The meaning of "converge continually" and "approach" are assumed to be already understood, the lemma is meant to derive a property from that. Second, it talks of "quantities". We also talk of quantities and their limits. But Newton surely can not refer to our notion of functions which assign values to arguments that first appears in Dirichlet's work from 19th century. Or even to "analytic expressions" featured in Euler's 18th century textbooks. Finally, Newton does not even have our idea of a real line assembled from points serving as arguments and values, the arithmetical continuum of Weierstrass, Dedekind and Cantor. The 17th century line is still Euclidean/Aristotelian, with points as merely external marks on it.

It becomes clearer that Newton's limits, whatever they are, can not be modern, his primitives are different, he works in a different system of mathematical concepts. There might be a sense in which Porciau's opinion that Newton "was the first to present an epsilon argument" is justified, but it would be similar to the sense in which Eudoxus was first to work with Dedekind cuts. It only means that some of their manipulations can be closely mimicked by modern ones, and we agree to ignore the meaning of what is being manipulated. And that there is an evolutionary chain connecting one to the other.

What are Newton's "quantities" then? We find an explicit description in his Quadrature of Curves (1692):

"I don’t here consider Mathematical Quantities as composed of Parts extreamly small, but as generated by a continual motion. Lines are described, and by describing are generated, not by any apposition of Parts, but by a continual motion of Points. Surfaces are generated by the motion of Lines, Solids by the motion of Surfaces, Angles by the Rotation of their Legs, Time by a continual flux, and so in the rest. These Geneses are founded upon Nature, and are every Day seen in the motion of Bodies".

Now it becomes clear where the pre-assumed understanding of "converge continually" and "approach" comes from. Newton takes the idea of motion as intuitively given, limits with their properties are then founded on it. In the Scholium to Lemma XI of the same section Newton explicitly appeals to the intuitive idea of instantaneous velocity to justify existence of limits, for example. And in Lemma 2 of Book 2 he talks of "genita", "quantities I here consider as variable and indetermined, and increasing or decreasing, as it were, by a perpetual motion or flux".

This conception of limits, and calculus generally, relying on the given intuition of motion and its observed properties, came to be called kinematic. It has roots in some works of Archimedes, such as On Spirals, where he seems to rely on something like the parallelogram of velocities to draw tangents. Newton's teacher Barrow lectured on Archimedes, and the kinematic conception of curves is explicit in his Geometric Lectures, which Newton helped prepare for publication, see Boyer's History of Calculus, p.189. But in the early years Newton also used manipulations with infinitesimals, inherited through Barrow from Fermat, which he later found objectionable. So I would have to agree with Ferraro's assessment in Some Mathematical Aspects of Newton’s Principia:

"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him... Indeed, I think that Newton’s concept of first and ultimate ratio can be reduced to the modern concept of limits: it is true that Newton has a clear idea of what meaning “approaching a limit” [is], but this is only an intuitive and non-mathematical idea that is entirely different from the modern, mathematical concept of limit."

Indeed, the "mechanical" aspect of Newton's calculus was explicitly criticized in the 18th century, as "foreign" to pure mathematics, by D'Alambert and l'Huillier, among others. A comprehensive study of 17th century mathematical conceptions is Whiteside's Patterns of Mathematical Thought in the later Seventeenth Century (p.374ff on Newton specifically), on Principia see also his Mathematical Principles Underlying Newton's Principia Mathematica. Arthur in Leery Bedfellows: Newton and Leibniz on the Status of Infinitesimals contrasts Newton's kinematic conception of calculus to the Leibniz's one, including a detailed discussion of "quantities" and Lemma 1, and the changes from early to late works. On the later fates of the kinematic conception, developed by McLaurin and still more than visible in Cauchy (despite his common assimilation to Weierstrass, his "variables" are not unlike Newton's "quantities") see Grabiner's book Origins of Cauchy's Rigorous Calculus.

Answer 2

Newton actually did have a pretty explicit concept of limit, he set it out in section 1 of Book 1 of the Principia immediately following the definitions and axioms or laws of motion. He did not use the actual word 'limit' but the concept is clearly there in his 'first and last ratios', which by his explanations turn out to be limits of ratios of finite differences, which are approached as the relevant variable controlling the size of both numerator and denominator either declines to zero ('evanescent') or, when considered in reverse, grows from zero ('nascent'). This matter has not gone without notice in the literature. A study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates and discusses Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia, with a focus on parts of Book 1, section 1.

(When I return to my sources, I'm away from base right now, I will put in online references to the Principia in its English translation of 1729 which is a good source and is online free of copyright, and other sources cited here. For now, one may note that Book 1 in the 1729 translation is online in The Mathematical Principles of Natural Philosophy, vol.1 of 2, and Newton's discussion and explanation of limit-methods extends from page 41 to page 56.)

Newton explained among other things that he relied on limits to justify his methods because the methods of the ancients by reductio ad absurdum (or exhaustion) were too long, and the method of 'indivisibles' was too rough, although he added that 'hereby the same thing is perform'd as by the method of indivisibles'. When Newton wrote, the precursor of 'infinitesimal' methods that was perhaps best known was the much-criticised 1640s work on 'indivisibles' of Bonaventura Cavalieri. Newton clearly considered such methods as not well justified, hence his reliance on limits.

There is further material that contributes to an answer to the current question in Why is calculus missing from Newton's Principia? , (answer in a nutshell, it is not missing, and the answer also provides sources in some detail about Newton's methods and explanations), and in the descriptions of attacks on the calculus in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"? . The attacks of calculus methods in France from about 1700 onwards by Michel Rolle were defended by Pierre Varignon and then by Joseph Saurin, and the defence by Varignon is specially relevant here because he relied on Book 1 section 1 of Newton's Principia to provide the justification that did not appear to be available elsewhere. Leibniz, for his part, has been said to have been generally respectful of Newton's justification in terms of limits.

Answer 3

Newton did not have the rigorous concept of limit as in $\epsilon/N$ and $\epsilon/\delta$ formulas. Instead, he had a vague idea of limit in term of motion and used the notion of infinitesimal to calculate derivatives and integrals. For example, calculating the derivative of $y=x^2$ is like $$ \dot{y}=\frac{\Delta y}{\Delta x}=\frac{(x+\Delta x)^2-x^2}{\Delta x}=2x+\Delta x=2x $$ (Newton used $\dot{x}$ for derivative and later Leibniz improved to $\frac{dy}{dx}$). In the last step, $\Delta x=0$, but in $\frac{\Delta y}{\Delta x}$, $\Delta x$ can not be $0$ for $\frac0{0}$ makes no sense. This means that $\Delta x$ (infinitesimal) sometimes is zero and sometimes is not, a fact that Newton could not explain. Nor did Leibniz know the solution. However, this defect of infinitesimal has been largely ignored because the powerful method of Calculus has solved so many and important problems that mankind has even never dreamed before.

The rigorous explanation of infinitesimal through the notion of limit (in forms of $\epsilon/N$ and $\epsilon/\delta$ formulas), however, was not completed until two hundred years after Newton, through the work of Cauchy and Weierstrass in the 19th century. So it is an overstatement to say that Newton knew the exact notion of limit and the rigorous treatment of infinitesimal. However, Newton must be credited for his invention of Calculus through infinitesimal. Likewise, it is again an overstatement to say that someones like Cavalieri or even Archimedes had invented Calculus before Newton.