# What was the notion of limit that Newton used?

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?

• @MathWizard title has been changed. – KCd May 9 at 21:21
• He used what is called kinematic conception, relying on the intuition of converging motion:"Those ultimate ratios... limits towards 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". See Ferraro's discussion:"In effect, Newton does not define the terms “limit” and “ultimate ratio”: these terms have a clear intuitive meaning to him." – Conifold May 9 at 21:41
• @Conifold Thanks for citing Ferraro (2011). He disagrees with Pourciau (2001), but the foundation of his disagreement is left as bare assertion:- "differently from what Pourciau stated, Newton does not define the word "limit" by referring to quantities that approach a certain value becoming less than any fixed quantity epsilon". Ferraro acknowledges that Newton wrote of quantities to which the ratios "approach nearer than by any given difference". But he doesn't explain the difference if any between that and "less than any fixed quantity epsilon", his preferred formulation. – terry-s May 9 at 22:15
• @terry-s I think "less than any fixed quantity epsilon" is an allusion to Weierstrass's technique, which Newton certainly does not use. A more plausible reference for "approach nearer than by any given difference" is the Greek style double reductio, but even that is mostly rhetorical. A closer match to how Newton actually handles limits is Archimedes's kinematic conception in On Spirals, for example. – Conifold May 9 at 22:31
• @terry-s You mean "nearer than by any given difference" counts as "quantified idea" and "remarkable closeness"? Shouldn't we look at what Newton actually does to see how close it is, and whether it amounts to any Weierstrass-style "quantification"? On its face, it is just as close to Proclus saying that horn angle is smaller than any rectilinear angle, for example, or generic colloquial descriptions of how something gets "infinitely small" used in a hand-waivy manner in calculus classes. To me, assimilating a turn of phrase to a developed technique two centuries later is very suspect. – Conifold May 9 at 23:17

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.

• "But Newton surely can not refer to ... "analytic expressions" featured in Euler's 18th century textbooks." Why are you so sure about that? – Michael Bächtold May 11 at 11:47
• It seems to me that in the quoted lemma, Newton is not defining the notion of a limit, but proving a property of continuous functions. Here's how I interpret it: If two quantities $u$ and $v$ are continuous functions of time $t$ and $\lim_{t\to t_0}u=\lim_{t\to t_0}v$ (converge to equality) then $u|_{t=t_0}=v|_{t=t_0}$ (ultimately equal). – Michael Bächtold May 11 at 12:05
• @Conifold I fear you're on shaky ground relying on Ferraro's paper. E.g. he stated (p.7-8) "Newton does not distinguish between the limit process lim A(t) (for x->c), and the ultimate value of this process |A(t)| (at x=c)". But Newton insisted on the distinction: "Those ultimate ratios with which quantities vanish, are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities, decreasing without limit, do always converge; and to which they approach nearer than by any given difference..." Ferraro overlooks the significance of that and much more. – terry-s May 11 at 15:43
• @MichaelBächtold Arthur (I added a reference) interprets the lemma as "a synthetic version of the Archimedean Axiom" (pp.2,12): less than "any given difference" is only zero, there are no non-zero infinitesimals. Of course, Newton derives it from the kinematic "definition" of his quantities rather than takes it as an axiom, as Greeks did for their magnitudes. – Conifold May 13 at 0:22
• Of Newton's calculus-related methods, DTW 1961 considers fluxions, but I find no mention of first/prime and last/ultimate ratios as in Principia Bk.1 sec.1. On the other point, "less than 'any given difference' " is in general neither zero nor infinitesimal because for a 'given difference', all its fractions e.g. 1/2 are clearly both less than it and also non-zero finite, also Newton excludes the zero cases if any by restricting the considered point to 'before the end of that time', i.e. before the considered difference reaches zero. I will try to address this in an edit/amendment. – terry-s May 14 at 12:10

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.

• @math-wizard : I question your interpretation of the meaning of 'first and last ratios', and refer you both to the prime source, the Principia, and to Bruce Pourciau's discussion (cited in the answer above). If you think it can be concluded that there are infinitesimals in Newton's cited treatment, please would you explain with sources the reason and justification for that conclusion? – terry-s May 9 at 19:54
• To attribute something to an individual, major part of the problem has to be solve and major breakthrough follows after it. It is not enough just working out slight clue or hint. In this sense, Calculus based on infinitesimal should be credited to Newton, not Cavalieri or Archimedes because differentiation and integration were known only after Newton, and not before. However, it is well known that Newton did not know why infinitesimal sometimes is zero and sometimes not. Nor did Leibniz and others then. This was only clear after work of Cauchy and Weierstrass in the 19 century – Math Wizard May 9 at 22:39
• @Math-wizard : I believe you are changing the subject and not answering the question. Where is the evidential support for the assertions you made before and the new ones you are making now? – terry-s May 9 at 22:42
• Overstatement often happens in mathematics or science. Another example is to attribute arithmetic (place value system) to Babylonians (base 60). This is not correct because the most important part of arithmetic involves 2 tables for addition and multiplication, which are possible only for decimal and not base 60. So arithmetic as a whole should not be credited to Babylonians, even though it has something similar. – Math Wizard May 9 at 22:52
• @math-wizard Again the unsupported bare assertions, I regret to have to say, and moving away from the question. If you look at this answer hsm.stackexchange.com/questions/7704/… you will find references to 18th-century work and conclusion that calculus was effectively defended with rigour by then already. I'm aware that it's repeatedly said to be a 'well-known fact' that the job was not effectively done until later, Cauchy and Weierstrass, but where do you find evidential support for that? – terry-s May 9 at 22:53

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.

• It would be helpful to offer references in support of your assertions. The question, just to remind, is about limits, and Newton's conception of them. His treatment of limits in the Principia is offered in answer, with online reference/s and text/s supplied. The subject is his justificatory work. No-one here denied that in other writings he used the equivalent of infinitesimals, no-one suggested there was any rigorous account of infinitesimals apart from limit-arguments. No-one suggested that Cavalieri or Archimedes invented calculus. And so on. It would be helpful to read before arguing! – terry-s May 9 at 23:26
• I think the books on history of mathematics by Morris Kline or Alexandrov contain the references of this view. – Math Wizard May 10 at 0:19
• "Newton could not explain why" --- By phrasing your comment this way, you appear to be suggesting that Newton tried and failed to explain something along these lines, and I rather doubt this accurately reflects what actually took place. I suspect you're talking about George Berkeley's essay The Analyst; or a Discourse Addressed to an Infidel Mathematician and the subsequent controversy it generated. The essay appeared in 1734, and Newton died in 1727. – Dave L Renfro May 10 at 16:39
• I realize that this view could depend on countries. British mathematicians and those from English speaking countries may believe that Newton already knew the notion of limit, including George Berkeley. But (most) continental European mathematicians and others think differently. Remember there had been a huge debate on who invented Calculus (Newton or Leibniz) between British and continental European mathematicians. Since I learned Calculus in non-English environment, I knew only that Newton could not explain fact about infinitesimal which is true. – Math Wizard May 10 at 19:06
• Shouldn't it read $\dot{x^2}$ if anything at all? And I doubt that Newton would have written the calculation you did, since, as far as I know he did not use differences/differentials. Or could you point to a place where he wrote $\Delta x$? – Michael Bächtold May 11 at 11:42