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.
