# Why is calculus missing from Newton's Principia?

I'm not suggesting that Newton did not discover calculus - the question is written this way to express my surprise that the Principia does not use the methods of calculus (or 'fluxions'). He instead uses plane and conic geometry; and of course the methods of differential calculus are implicit in it in the way that Euclid's demonstration of the area of a circle contains a limiting argument that expresses the notion of integration.

Now, even if he chose not to use calculus so as to get the widest possible audience it seems strange not to introduce it in an addendum or appendix to show that the same results can be deduced with greater conceptual clarity, and shorter proofs.

Are there lesser known published or unpublished works in which the novel techniques were advertised?

I mean for example the notation that is introduced in classical mechanics of $$f'$$ for the derivative of function $$f$$, or with a dot above it $$\dot{f}$$; which I recall reading was ascribed to Newton; whereas the notation $$\frac{df}{dt}$$ is usually ascribed to Leibniz.

There are too separate issues here. The method of fluxions and fluents, Newton's version of calculus, is amply represented in Newton's extant papers, starting with 1669 On Analysis by Equations with an Infinite Number of Terms sent as a letter to John Collins, and disseminated by him to multiple correspondents, including Leibniz. The dotted shorthand was only invented in 1691, after the publication of Principia (1687), and Newton did publish his account of calculus in 1693. Before that his methods were mostly known in Europe from letters sent through Collins and Oldenburg.

As to why he did not use calculus in Principia the answer is controversial. To begin with, it is not clear that calculus in its original cumbersome form would have introduced greater clarity or shorter proofs. It could instead compound the difficulty of understanding new mechanics by the difficulty of understanding new mathematics. There is also evidence that Newton worked out Principia in the very form that they were written, Euclidean, rather than back translated it from a calculus version as he later claimed. Whiteside gives an illuminating discussion.

• One of Newton's earliest submissions on the calculus to a journal was rejected, I believe ( plato.stanford.edu/entries/newton ). That must have influenced him in his decision to use standard geometric methods of proof in the Principia. Jun 3 '15 at 21:44
• Also see Needham's comments on the topic in his intro to his book Visual Complex Analysis. Aug 9 '15 at 0:51

Although this question and the answers now have some age to them, I suggest that it's important not to overlook the mythical character of the assumption that underlies this question. The question explicitly supposes that 'calculus is missing from the Principia'. But that is not true: it is not missing, not only have skilled commentators from the 17th to the 21st centuries clearly recognised the content of calculus in the work, but also one can point directly to many arguments and demonstrations in the work itself that definitely employ principles belonging to the field of calculus.

What is practically absent from the Principia is a largely different matter: it is mainly about notation. As to this, it is worthwhile to bear in mind the late Clifford Truesdell's assessment that "... a modern mathematician, according little respect to those who confuse notations with notions, finds the Principia a book dense with the theory and application of the infinitesimal calculus." (Essays in the History of Mechanics, 1968, 99 (at n.4)).

Truesdell was not alone in his view. Published for example already in 1696 was the book "Analyse des infiniment petits" (Infinitesimal analysis) by the Marquis de l'Hospital (or l'Hôpital): this was an exposition of Leibniz's form of the differential calculus. In its preface, after due praise of Leibniz, one reads (translating from the French): "... credit is also due to the learned M. Neuwton, as M. Leibnis himself (in the Journal des Sçavans of 30 August 1694) has acknowledged: that he too" [i.e. Newton] "had found something similar to the differential Calculus, as appears by the excellent book entitled 'Philosophia naturalis principia Mathematica', which he gave us in 1687, of which almost all is of this calculus" ['lequel est presque tout de ce calcul']. "However, the notation of Mr. Leibnis makes his much easier and more expeditious ...". (L'Hospital's last-quoted phrase, his assessment of the convenience and utility of Leibniz's notation, is of course an assessment echoed many times since 1696, applying also to modern descendant forms of Leibniz's notation, and understandably attracting wide agreement, as Newton's expository choices and forms have been regarded by many as a hindrance.) As part of the background to the acknowledgement of Newton by Leibnizians it is worth recalling, that at the relevant period no quarrel had yet been fomented between Newton and Leibniz, and each of them had made courteous acknowledgement of the other. As A.Rupert Hall put it, in 'Philosophers at War' (1980), for example "the [earlier] triangular exchanges of letters between Wallis, Leibniz, and Newton consistently suggest esteem and amity".

Coming now to more detailed mathematical considerations, a study by Bruce Pourciau (2001), in Historia Mathematica 28, 18-30, investigates "Newton’s understanding of the limit concept through a study of certain proofs appearing in the Principia." Pourciau finds "that Newton, not Cauchy, was the first to present an epsilon argument, and that, in general, Newton’s understanding of limits was clearer than commonly thought. We observe Newton’s distinction between two properties easily confused, namely f/g --> 1 and f - g --> 0, [and] we resolve a problem created by a spurious translation appearing in Cajori’s revision of Motte’s original translation, ...". Pourciau specially points out three important Lemmas in the Principia, "Lemma I on the limit of a difference [sic, this must be a slip for 'ratio' which appears in Lemma 1], Lemma II on the existence of the integral, and Lemma XI on the second derivative" and discusses how "their statements and proofs most clearly reveal Newton’s grasp of the limit process." But "to read these lemmas", says Pourciau, "requires a double translation, not only a first translation from the original Latin into English (for which we rely on [I B Cohen's 1999 translation of the Principia]), but then a second translation as well, for the lemmas come to us packed in the Principia’s unique blend of Euclidean geometry and limits, a sort of geometric calculus, and we cannot sort out what the lemmas really say without doing some unpacking. But any translation disturbs meaning, and we must take great care to minimize that disturbance, to preserve as far as possible Newton’s original intent." Thus Pourciau begins to show that while the notation of calculus as we now know it was largely absent from the Principia, the content is indeed to be found, expressed in a geometric form of infinitesimal calculus, often based on limits of ratios of vanishing small quantities.

One should also go directly to the source on such a question. Immediately after Newton's Definitions and Laws of Motion, the opening section 1 in Principia's Book 1 has a number of lemmas all about "the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow." The 'first' and 'last' ratios are explained as the limits of ratios of quantities either growing from zero ('nascent') or decaying to zero ('evanescent'). These lemmas include those discussed by Pourciau (2001) already cited. Then in the body of the work we find inter alia Propositions 1, 6, 10, 11: in Proposition 1 the argument proceeds by assembling a finite series of triangles expressing by their equal areas the increments of motion occurring after a finite series of discrete impulses, then Newton writes "now let the number of those triangles be augmented and their breadth diminished in infinitum", thus he expresses an argument of limits clearly belonging to the field of calculus, and draws his conclusion about a curved trajectory and its relation to a continuous force both expressed by reference to the results in the limit of a process involving an indefinitely increasing number of (individually) indefinitely diminishing elements. There are arguments of limits in the mentioned later propositions too, sometimes expressed more briefly, and it may be, loosely, as when Newton writes first of a geometrically-defined 'solid' expressed by a ratio in which one of the factors, in both numerator and denominator, depends on a distance PQ, and then of 'that magnitude which [the solid] ultimately acquires when the points P and Q coincide'. The context and the 'ultimately' both indicate that with the phrase 'when the points ... coincide', Newton is referring to the limit of the ratios developed as the points approach each other, in the way discussed in the opening section on the 'method of first and last ratios'.

It is worth mentioning that the notation of fluxions amounted to only one notational form, essentially the latest, among Newton's various expressions of his ideas in this field. He seems to have held a view that any particular notation was relatively unimportant compared to the idea 'which may be without it'. One may certainly quarrel with Newton's assessment and choice of notation and exposition, but his ideas and work applying them are attested in the various sources cited and discussed in the references already given.

In short, calculus is clearly not 'missing' from the Principia: and while this mistaken idea has become one of many myths about Newton, it tells more about the history of commentators and commentaries than about Newton's actual work.

In your question you implicitly assume that Newton wrote only Principia. Which is strange. Look at the Mathematical Works of Newton: http://www.newtonproject.sussex.ac.uk/prism.php?id=147 which contain abundant evidence of his mathematical discoveries.

• Hi. I wasn't assuming that he only wrote the Principia; but I did assume that it was his most famous work - and I think I'm reasonably justified in that opinion. May 22 '15 at 21:12
• Perhaps it is most famous. But his other work (optics, invention of calculus, and other contributions to mathematics) are also famous. In any case there is no doubt about the invention of calculus, and this answers your question. May 22 '15 at 21:35
• I don't think it does; I wasn't suggesting that he didn't invent the calculus - I mention it in the first line of the question; and nor am I suggesting that his other works aren't justly famous - for example there is a brief description of his experiments with a prism in a note that he submitted to the Academy (and which I read in the site you referenced). May 23 '15 at 0:11

Newton wanted to lay out his theory of gravity so that people would accept it. But his new methods of calculus were not yet widely accepted or known, and would therefore cast doubt on his physics. For that reason, Newton used classical geometry, the mathematical language of the ancient masters, to avoid having people attack his ideas based on his use of calculus.

Newton was well aware of the logical difficulties of his calculus, and struggled without success over the course of his career to explain the "ultimate ratio" of the difference quotient. He did not want to expose his theory of gravity to attacks based on those issues.

There's some more background here, supporting the idea that Newton understood the logical issues with his calculus and wanted to base his physics on math that everyone would believe.

The other reason was to deliberately make his work more difficult. As Newton said: To avoid being baited by little smatterers in mathematics, I designedly made the Principia abstract; but yet as to be understood by able mathematicians ...

We note in passing that were Newton to come back today and be introduced to the Internet, he'd be right at home. He suffered no fools, gladly or otherwise, and could flame with the best of them.