Despite having already invented the Calculus (which he called the *Theory of Fluxions*), Newton did not use it in his *magnum opus*, the *Principia Mathematica, *probably because he felt uneasy about its logical basis. Instead he employed cumbersome strictly geometrical arguments without even employing co-ordinates ─ which makes the *Principia * almost unreadable for the modern student. Feynman, one of the greatest mathematical physicists of all time, confessed that he could not follow Newton’s proof that planets must follow elliptical orbits and instead offered his own geometrical proof (see *Feynman’s Lost Lecture *by Goodstein and Goodstein).

However, it is not true, as is often said, that Newton had no concept of limits. The very first section of Book I is entirely given over to eleven ‘Lemmas’ about Limits which he needs in order to show, amongst other things, that planets and other heavenly bodies verify an inverse square distance law. The key limit is the first:

“LEMMA I

*Quantities, and the ratios of quantities, which in any 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.*

* *If you deny it, suppose them to be ultimately equal, and let *D* be their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference *D*; which is contrary to the supposition.”

Newton, *Principia *(Motte/Cajori translation * *p. 29)

In particular, this Lemma leads on to the all-important Lemma VII which states that “*the ultimate ratio of the arc, chord and tangent, any one to any other, is the ratio of equality”*.

So, what are we to make of Lemma I? On the face of it, it sounds foolproof. Either diminishing ratios that converge to unity, attain their goal or they do not ─ exclusive sense of ‘or’. In practice, of course, this will not do; essentially Calculus wishes to have it both ways, to make such ratios attain equality when this is convenient and have them *not* attain equality when this is embarrassing. At least Newton grasps the nettle: by this all-round Lemma he affirms that the limit is* attained*.

Or, does he? In the *Scholium *(Commentary) which concludes the section, Newton admits that there is a conceptual problem, at any rate when we consider *speed*. Why so? Because *speed *is not an independent entity but rather a ratio of *distance *to *time*, and, in dynamics, we desire to know a body’s speed at a particular moment of time. In such a case, is there, or is there not, such a thing as an ‘*ultimate ratio’* of distance/time? Newton writes:

“*Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none”*.

Newton’s reply to this objection is interesting:

“*By the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish…..There is a limit which the velocity at the end of the motion may attain, but not exceed.”*

This is all very well but contradicts lemma I since Newton says in the above passage that this ‘ultimate ratio’ *‘may be attained’ *─ in which case it would constitute a difference *D *that is not supposed to exist according to lemma I.

And, a little further on, Newton even contradicts what he has just said since he now denies that this ‘*ultimate ratio’* is in fact attained:

* *“*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, but never go beyond, nor in effect attain to, till the quantities are diminished ad infinitum*” (p. 39 Motte/Cajori).

The contradiction remained like a worm in the apple of Calculus until the radical reworking the latter underwent at the end of the 19^{th} century. The definition of a ‘limit’ that every mathematics student encounters today neatly sidesteps the problem ─ without resolving it. Mathematically speaking, it is immaterial whether a sequence or series actually attains the proposed limit; the only issue is whether the absolute value of the difference between all terms after a given point and the proposed limit can be made “*smaller than any positive quantity*” (**Note 1**).

The mathematics student of today is discouraged, sometimes even specifically prohibited, from asking the question that every enquiring person wants to pose: *Does the function or sequence actually attain the limit? *In most cases of any interest in Calculus and Analysis the answer is that it does not. (The sequence 1, 1/2, 1/4, 1/8….1/2^{n} for example does not ever attain the obvious limiting value of zero.) The adroit way in which the limit is defined, originally due to the 19^{th} century mathematician Heine, means that, mathematically speaking, we get what we want, namely a clearcut test of whether or not a function ‘*tends to a limit’* while avoiding altogether situations where, for example, we might find ourselves tempted to ‘divide by zero’.

However, Newton, despite being the greatest pure mathematician this country has produced, was a physicist first and a mathematician second, which is why the modern ‘solution’ to the problem of limits, even had he thought of it, would probably not have appealed to him. I am afraid that I, as a philosophic empiricist, at any rate with regard to applied mathematics, am not at all satisfied by the sleight of hand; in cases of obvious physical importance I want to know whether a function or mode of behaviour generally actually does attain the proposed limit or not. However, Newton’s lemma VII which makes the “*ultimate ratio of the arc, chord and tangent….the ratio of equality” *does not convince me any more than it convinces any contemporary mathematics student.

So, what to do? The solution is quite simple and, I contend, perfectly valid mathematically ─ even though it will arouse howls of protest and derision from the *aficionados *of modern Analysis. We simply excise Newton’s lemma I and replace it by a positive statement:

“LEMMA I

*Quantities, and the ratios of quantities, which in any finite time converge continually to equality, do not in general become ultimately equal but differ from strict equality by a small but finite amount.** *

* *Now, it is true that in general we do not know what this ‘*small amount’* is ─ although in most applications it either *is *or *could conceivably be *ascertained. We now know that all energy interactions are quantised and that the inevitable inefficiency (because of friction and similar considerations) of an actual machine can be (and often is) estimated. Not only that, calculus is *already *used in situations where we know the value of the independent variable cannot be arbitrarily diminished. For example, *dn *in molecular thermo-dynamics cannot be made smaller than the size of a single molecule and *dx* in population studies cannot be smaller than a single living person. This does not matter too much since we are dealing with millions of entities, although it must be said that in more accurate work, the tendency these days is to not bother with calculus but to slog it out numerically with computers to the degree of precision required.

This drastic pruning of calculus does not make Analysis, and all that depends on it, altogether redundant since there is often no great difference in practice between assuming that *dx* has an ‘ultimate’ final value and letting it go as near to zero as we wish ─ the *dx *terms and *a fortiori *second and third order terms will most likely in end up by being discarded anyway. Nonetheless, one can and should question whether the assumptions of Analysis, especially infinite divisibility, are *realistic*. I believe they are not. There is a growing movement amongst physicists (e.g. Causal Set Theory, Loop Quantum Gravity &c.) that even spacetime, the last refuge of the devotees of the continuous, might be ‘grainy’. *SH 25/02/20*

**Note 1 **The technical definition for a function is:

“ *f(x) *tends to a limit *l *as *x *tends to *a*, if, given any positive number *e* (however small), there is a positive number *d* (which depends on *e*) such that, for all *x*, except possibly *a *itself, lying between *a − **d* and *a *+ lies between* l − **e* and * l + **e*”. The definition of the limit of a sequence is similar.

Such a definition will probably not mean much to the non-mathematical reader but the idea behind it is a sort of guessing game. I claim that some sequence or function tends to a limit *l* and my opponent challenges me to show that I can produce terms of my sequence or function that get me closer to this limit than some arbitrarily small quantity such as *10 ^{−6 }= 0.000001*. If I succeed, my opponent chooses an even smaller difference and so the contest goes on. The point is that this difference, though it

*can*be reduced to zero in some cases,

*need not*necessarily go to zero. For example, I might claim that the diminishing sequence

*1; 0.1; 0.01; 0.001; 0.0001;*and so on, has zero as a limit. My opponent asks me to get within 1/1000 of my limit, i.e. to make the difference

*d*smaller than, say, 1/1000. I do this easily enough by presenting him with

*0.00001*which is a term in the sequence but is smaller than 1/1000 (since

*0.00001 =*1/10000). Moreover, since this is a strictly diminishing sequence, all terms further down the line will also have a smaller difference than the one I have to better. If my opponent ups his challenge, I can easily meet it since if he comes up with

*1/10*for some positive integer

^{N}(*N*) I can get closer simply by adding more zeroes to the denominator. Yet, in such a case, if actually asked to produce a term in my sequence that makes the difference zero

*exactly*I cannot do so ─ since any term 1/

*10*, however large

^{N }*N*is, is still a positive quantity albeit a small one. But this does not matter, the limit still holds since I can get as close to it as I am required to.