# Why did Newton want lines to be generated by continued motion of points rather than by apposition of parts?

The following passage has been extracted from the Newton's (John Stewart's English translated version) "Sir Issac Newton's two Treatises: Of the Quadrature of Curves, and Analysis by equations of an infinite number of terms":

I consider mathematical quantities in this place not as consisting of parts; but as described by a continued motion. Lines are described, and there by generated not by the apposition of parts, but by the continued motion of points; superficies's by the motion of lines; Solids by the motion of superfices's; Angles by the rotation of the sides; Portion of time by a continual flux: and so in other quantities. These geneses really take place in nature of things, and are daily seen in the motion of bodies. And after this manner the ancients, by drawing moveable right lines along immoveable right lines taught the genesis of reflection...

Here Newton doesn't provide any reason on why he wants to describe lines to be generated by the "continued" motion rather than by the appositon of parts (= points??). Is there any reason for his preference for motion view?

And I noticed that Newton doesn't define point. I don't understand whether he is following Euclid's method of having some of the terms to be undefined, or some other philosophy. I want to know Newton's view on mysterious points. I will be really happy if sources on this regard (Newton's view on points) is provided.

Meaning of Apposition from "The New Oxford American Dictionary": The positioning of things or the condition of being side by side or close together. So, I interpret apposition of parts to be positioning of points/parts side by side or close together to form a line.

References to the full latin text:

I have asked same question in PSE too.

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• It can be noticed that the words "points" and "parts" are used, where I think, both mean the same. – Immortal Player Aug 19 '15 at 18:52
• If you carry on and read the next sentence or two in Newton's text, you will see that the mathematical quantities Newton has in mind include lines, surfaces, solids, angles and other things. – Rob Arthan Aug 19 '15 at 20:35
• I suspect a better translation of Newton's "moto continuo" would be "continuous motion". – Rob Arthan Aug 19 '15 at 21:26
• You can start with some historical overview : Carl Boyer, The History of the Calculus and Its Conceptual Development (1949 - Dover reprint) and Margaret Baron, The Origins of the Infinitesimal Calculus (1969 - Dover reprint). – Mauro ALLEGRANZA Aug 21 '15 at 7:10
• A shorter overview is into Ivor Grattan-Guinness (editor), From the Calculus to Set Theory 1630-1910, Ch.2. – Mauro ALLEGRANZA Aug 21 '15 at 7:20

## Problem: classical geometry is not happy with infinitesimals

Newton is systematically trying to avoid basing calculus on infinitesimal geometric quantities. We can see this from how he emphasizes that his method is consistent with the "ancient" standard of rigor:

To institute an Analysis after this manner in finite Quantities... is consonant to the Geometry of the Ancients... in the Method of Fluxions, there is no necessity of introducing Figures infinitely small into Geometry.

(Page 4 of the English translation cited in the question, page 6 of this scanned Latin edition - note that the page numbers refer to the numbering as printed on the actual scanned dead-tree pages, not the electronic numbering.)

Recall that before the modern era, the standard for rigor in mathematics was Euclid, and Newton followed the tradition by using geometric figures and arguments whenever possible. Now sure, classical geometry had what we now call the method of exhaustion. But it did not have the notion of limits or infinitesimals the way Newton did. Especially since the foundations of calculus were controversial from the start, Newton needed his ideas to be agreeable with the standards of classical geometry.

## Solution: frame the foundation in terms of motion, not geometry

Calculus has both a geometric interpretation and a dynamic interpretation - think tangent lines vs. rates of change. You learn this early on. But arguments directly in terms of infinitesimal geometric figures don't fit into the classical geometry. If you appeal to motion and change, though, you can use perfectly reasonable and familiar concepts. For example, a secant line that moves and becomes a tangent line sounds a lot more believable than supposing you can just take two "infinitely close" points and take the secant line through them. There's some hand-waving going on either way, but the latter flies directly in the face of classical geometry, which, remember, was the standard of rigor. Framing calculus in terms of motion shifts the controversial foundation further from geometry, where it would face stricter scrutiny.

With this in mind, we can interpret the quote as establishing the emphasis on the dynamic viewpoint. Essentially, Newton is saying: Don't worry, we're not gonna do infinitesimal geometry. We're not gonna break a line or anything down into absurd infinitesimal parts which may not really be possible. Just view a line as the path traced by a moving point. With this viewpoint, we can appeal to concepts like velocity to justify our arguments. We can usefully apply calculus to geometry, without having overtly infinitesimal geometry.

And of course the perspective helps teach the important interplay between the geometric and dynamic interpretations of calculus.

## P.S. Notes on the quote

For completeness, here is the quote in Latin:

Quantitates Mathematicas non ut ex partibus quam minimis constantes, sed ut motu continuo descriptas hic considero. Lineae describantur ac describendo generantur non per appositionem partium, sed per motum continuum punctorum; ...

(page 3)

You had some hypotheses about some particular words and phrases in the quote, so here are my comments on them specifically.

• You guessed that "mathematical quantities" ("Quantitates Mathematicas") means curves. It looks it just means quantities. For example, compare with page 5 of the Latin, page 4 of the English: "Fluat quantitas $x$ uniformiter & invenienda sit fluxio quantitatis $x^n$." ("Let the Quantity $x$ flow uniformly, and let it be proposed to find the fluxion of $x^n$.") Clearly $x$ and $x^n$ are numerical quantities.
• "Parts" and "points" are not the same word ("partium" vs. "punctorum"); in fact, Newton is contrasting the two. Parts are pieces you divide something into. Points are points, although since you're thinking of them as moving, you might say point particles. I'm sure there's more to say about what a point is according to Newton.
• In is interesting that Newton deleted in the first phrase of the book the expresions "indivisibilibus vel" and "vel infinite", the later referring to infinitely small, the former being a bit more agnostic about size. – arivero Aug 21 '15 at 14:30
• Berkeley's Analyst appeared in 1732, and Newton's work on calculus dates back to mid 1660s, Principia came out in 1687. So he could not have been motivated by Berkeley's critique. Neither infinitesimal, nor kinematic versions of calculus were worked out at the time, so Newton was not in a position to choose between the two, and there was algebraic calculus of Descartes that did not use infinitesimals. Apposition of points does not need infinitesimals either. So avoiding them does not explain why Newton "chose" kinematic version, and that choice does not explain avoiding apposition of points. – Conifold Aug 22 '15 at 21:59
• @Conifold: Issac Barrow might have influenced Newton; because he was the one in his "Geometrical lectures", states on different methods of production of magnitudes; and even in the preface, he states that he has sent a copy of his book to Newton. – Immortal Player Aug 28 '15 at 13:23

Although Newton was certainly trying to avoid infinitesimals there is some distance between "no infinitesimals" and "no points", after all we now have calculus with points but without infinitesimals. Moreover, Newton's goal was not to invent calculus, but to analyze motion, so his reasons were somewhat deeper. Zeno's paradox Arrow argues that if time consisted of instants then flying arrow would never move, because within an instant there is no difference between arrow in flight and arrow at rest. Zeno, and Plato after him, concluded that motion is but an illusion and should be banished from geometry, position loyally followed in Euclid's Elements, but obviously unacceptable to Newton.

Aristotle offered an alternative resolution that "saved" motion: time does not consist of instants, a magnitude can only be broken into smaller magnitudes, but never into points. This was the price of making the arrow fly, see Aristotle's objection to Zeno's paradox. After Euclid some prominent geometers, notably Archimedes and Apollonius, considered curves generated by combining linear and circular motions. Newton was very familiar with such curves, as well as with Aristotle and Euclid. Fermat and others tried to get around the issue with infinitesimals, but motion itself provided Newton with a natural alternative, what we now call limits, and what he called "first and last ratios". As he wrote in Principia:"Quantities and ratios of quantities which in finite time converge continually to equality and before the end of that time approach nearer to each other than by any given difference, become ultimately equal".

Newton's definition sounds similar to the modern explanation of limits, but to make it a definition today we use three nested quantifiers $\forall\varepsilon>0\,\exists\delta\,\forall x: |x-x_0|<\delta\Rightarrow|f(x)-a|<\varepsilon$. Aristotle's syllogistic, the only logic known to Newton, had no quantifiers, and while $\forall x$ can be simulated with variables, $\exists x$ with constants, and even $\forall x\exists y\,P(x,y)$ with implicitly defined function $y=f(x)$, the combination $\forall\exists\forall$ needed to express convergence to limits is beyond it. But it is not beyond kinematic interpretation, which simulates the missing quantifier dependence, but requires that lines and other magnitudes be generated by motions in the first place. So the quote in the OP is essentially Newton's definition of "quantity". Newton's "first and last ratios" manage to express the modern limit concept without quantifiers, but at Aristotelian expense of sacrificing decomposition of magnitudes into points.

This is explained by Friedman (pp. 477-481) Kästner in his influential mathematics textbooks, remarkably for an 18th century German author, "argues that Newton's fluxions are in some respects clearer and more perspicuous than Leibniz's differentials. Further, he explicitly applauds Collin Maclaurin's attempt, in his monumental Treatise of Fluxions (1742), to develop the calculus on the basis of a kinematic conception of the limit operation". And "when the kinematic interpretation was explicitly criticized by mathematicians like D'Alembert and l'Huilier in the late eighteenth century, it was not on grounds of coherence and consistency but because it was thought to import a "foreign" or "physical" element into pure mathematics".

• I find the statement "the combination $\forall\exists\forall$ needed to express covergence to limits is beyond it." very interesting; do you happen to know a source where this impossibility-claim is given a precise form and proved? (What is the "it" in the statement? Skolemization + something, I suppose?) – Peter Heinig Jan 22 at 10:47
• @PeterHeinig This is discussed in detail in the linked Friedman's paper (see pp. 464-6 and 476-7 especially). I wouldn't say "proved", he is a philosopher :) As far as I know, the originator of this interpretation and Friedman's source is Hintikka. He might have it spelled out more technically, but, unfortunately, I can't say where. – Conifold Jan 22 at 21:51

Seeing as Newton developed a lot of what we now define as Calculus I, it makes sense that he would consider lines as continuous/differentiable, as opposed to his contemporaries who might have only considered them a finite collection of smaller line segments or points.

Calculus is, after all, the analysis of change of "mathematical quantities" at infinitesimally small distances. My interpretation: Newton is claiming that he works with lines under the assumption that there is a "continuous motion" or differentiability at every point.

• Thank you for the answer. "Newton is claiming that he works with lines under the assumption that there is a "continuous motion" or differentiability at every point." But, why he wants to use the assumption of math quantities described by motion rather than by apposition of points? Please see the edited question to avoid confusions. – Immortal Player Aug 20 '15 at 11:23
• Dear friend, I am sorry, personally I didn't find answer in your answer. I really appreciate your further participation in understanding the truth by dialectics. +1 for your efforts. – Immortal Player Aug 20 '15 at 11:43
• I am reading the book now, in some places, I find Newton trying to give reason for choosing "motion" view, but everywhere it is blurry, I didn't understand his deep roots in reasoning untill now. For example, in 2nd paragraph, he says "I sought a method of determining quantities....I fell by degrees upon the method of Fluxions...". And in 12th paragraph, "..and I was willing to show that, in the method of fluxions, there is no necessity of introducing figures of infinitely small in geometry...." – Immortal Player Aug 21 '15 at 5:08
• I tend to agree. Zeno paradox is still there. And geometry had its version, Democritus cone/cylinder paradox. – arivero Aug 22 '15 at 12:14

Comment (in the answer window for the sole reason of being able to include the orignial quote from Descartes' book):

This thread still seems to lack the basic remark that probably Newton's reasons were much less a negative attitude towards building a line from parts than rather a positive attachment to the long tradition of favoring contructibility in geometry (not only in the Greek authors, but also in Descartes, whose work influenced Newton strongly). In 'Livre premier' of La Géométrie, Descartes (1) declares that in his opinion, it is required to find a method of drawing a curve which one already knows as a set of points, (2) reproaches Pappus of not having sufficiently 'described' curves which he had indirect determined. I submit that normative statements like Descartes' "il est aussy requis de [...] tracer la ligne" have had an effect on Newton.

(Source: Descartes: La Géométrie, first edition, facsimile in a 1925 edition by 'The Open Court Publishing Company')

As translations are available, I don't translate the above page. (Except for the archaic orthography, this is easy French, still understandable if one knows contemporary French.)