Archimedes gave us the fundamental principles of Statics and Hydrostatics  but somehow managed to avoid founding the science of dynamics though, as a practising civil and military engineer, he must have had to deal with the mechanics of moving bodeis. The Greek world-view, that which has been passed down to us anyway, was essentially geometrical and the truths of geometry are, or were conceived to be, ‘timeless’ : they referred to an ideal world where spherical bodies really were perfectly spherical and straight lines perfectly straight.
Given the choice between exact positionaing and movement (which is change of position) you are bound to lose out on one or the other. But science and technology must somehow encompass  both exact position and ‘continuous’ movement. So how did Newton cope with the slippery idea of velocity ?  From a pragmatic point of view, supremely well since he put dynamics on a firm footing and went so far as to invent a new form of mathematics tailor-made to deal with the apparently erratic motions of heavenly bodies — his ‘Method of Fluxions’ which eventually became the Differential Calculus. Strangely, however, Newton completely avoided Calculus methods in his Principia and relied entirely on rational argument supplemented by cumbersome, essentially static,  geometrical demonstrations. Why did he do this? Probably, because he felt himself to be on uncertain ground mathematically and philosophically when dealing with velocity.
If you are confronted with steady straight line motion you don’t need Calculus — ordinary arithmetic such as even the ancient Babylonians and Egyptians employed is quite adequate. But, precisely, Newton was interested in the displacements of objects subject to a force, thus, by definition, not in constant straight line motion. And when the force was permanent, as was the case when dealing with gravitational attraction, the consequent motions of the boldies were never going to be constant (if change of direction was taken into account).
Mathematically speaking, speed is simply the first derivative of displacement with respect to time, and velocity, a vector quantity, is ‘directed speed’, speed with a direction attached to it. The modern mathematical concept of a ‘limit’ artfully avoids the question of whether a ‘moving’ particle actually attains a particular speed at a particular moment : it is sufficient that the difference between the ratio distance covered/time elapsed and  the proposed limit can be made “smaller than any finite quantity” as the time intervals are progressively reduced. This is a solution only to the extent that it removes the problem from the domain of reality where it originated. For the world of mathematics is an ideal, not real world though in some cases there is a certain overlap.
Newton was not basically a pure mathematician, he was a mathemartical realist and a hard-nosed materialist (at least in his physics). He was obviously bothered by the question that today you are not allowed to ask, namely “Did the particle attain this limit or did it only get very close to it?” 
It is often said that Newton did not have the modern mathematical concept of limit, but he came as close to it as was possible for a consistent realist. He speaks of “ultimate ratios” “evanescent quantities” and, unlike Leibnitz, tends to avoid infinitesimals if he can possibly manage to. He sees that there is indeed a serious logical problem about these diminishing ratios somehow carrying on ad infinitum and yet bringing the particle to a standstill.

“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. But by the same argument, it may be alleged that a body arriving at a certain place, is not its ultimate velocity: because the velocity, before the body comes to the place, is not its ultimate velocity; when it has arrived, is none.”
Newton, Principia    
Translation Andrew Motte

Note that Newton speaks of ‘at a certain place’ and ‘its place’, making it clear that he believes there really are specific positions that a moving particle occupies. He continues :

“But the answer is easy; for by ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place and with which the motion ceases. And in like manner,  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.”
  Newton, Principia     Translation Andrew Motte

      But this implies that there is a definite final velocity :  

        “There is a limit which the velocity at the end of the motion may attain, but not exceed. This is the ultimate velocity.”

         Well and good, but Newton now has to meet the objection that, if the ‘ultimate ratios’ are specific, so also, seemingly, are the ‘ultimate magnitudes’ (since a ratio is a comparison between two quantities). This would seem to imply that nothing can properly be compared with anything else or, as Newton puts it, that “all quantities consist of incommensurables, which is contrary to what Euclid has demonstrated”.  

     “But,” Newton continues, “this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of the 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 in infinitum.”   (Newton, Principia     Translation Andrew Motte)

       The last phrase (’till the quantities are diminished in infinitum’) seems to be tacked on. I was expecting as a grand climax, “to which they approach nearer than by any given difference, but never go beyond, nor in effect attain” full stop. This would make the ‘ultimate ratio’ something akin to an asymptote, a quantity or position at once unattained and unattainable. But this won’t do either because, after all, the particle does pass through such an ultimate value (‘limit’) since, were this not the case, it would not reach the place in question, ‘its place’. Bringing in infinity at the last moment (‘diminished in infinitum’ ) looks like a sign of desperation.
A little later, Newton is even more equivocal
“Therefore if in what follows, for the sake of being more easily understood, I should happen to mention quantities as least, or evanescent, or ultimate, you are not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end.” 

But what meaning can one give to “quantities….diminished without end” ?  None to my mind, except that we need such quantities to do Calculus, but this does not make such concepts any more reasonable or well founded. The issue, as I said before, has ceased to worry mathematicians because they have lost interest in physical reality, but it obviously did worry Newton and still worries generations of schoolboys and schoolgirls until they are cowed into acquiescence by their elders and betters. The fact of the matter is that to get to ‘its place’ (Newton’s phrase) a particle must have a velocity that is the ‘ultimate ratio’ between two quantities (distance and time) which are being ‘endlessly diminished’ and yet remain non-zero.
In Ultimate Event Theory there is no problem since there is always an ultimate ratio between the number of grid positions displaced in a certain direction relative to the number of ksana required to get there. When doing mathematics,  we are not going to specify this ratio, supposing even we knew it : it is for the physicist and engineer to give a value to this ratio, if need be, to the level of precision needed in a particular case. But we know (or I do) that δf(t)/δt has a limiting value (Note 1)which we may call df(t)/dt if we so wish. Note, however, that the actual ‘ultimate ratio’ will almost always be more (or less) than the derivative since there will be non-zero terms that need to be taken into account. Also, the actual limiting value will vary according to the processes being studied since manifestly some event-chains are ‘faster’ than others (require less ksanas to reach a specified point).  Nonetheless, the normal derivative will usually be ‘good enough’ for practical purposes, which is why Calculus is employed when dealing with processes that we know to be strictly finite such as population growth or radio-active decay.   S.H.

Note 1 :      Why must there always be a limiting value?  Because δt can never be smaller than a single ksana — one of the basic assumptions of Ultimate Event Theory.

(The opening image is from a painting by Jane Maitland)         S.H.     5/08/12     

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