To Fix: cindyJS is currently not showing linelabels present in the Cinderella originals. Considerwhether or not the mathematical device called the Limit – as used, for example, in differential and integral calculus – may validly be applied to projectively defined curves, such as the conic, or the path curve. Take a nondegenerate conic, such as shown below. 
We have in mind to progressively ‘reduce’ (shrink) intervals, such as H_{0}H_{1}, here indicated in yellow, on black line b so as to have the corresponding (red) chords c on the conic move to become, “in the limit”, tangents to the conic, each in place, everywhere on the conic. That is, we are to see a set of straightline, chordal segments, “in the limit”, merge and, in some sense, sum, or integrate, In other words, we seek the projective geometric equivalent of integration.—There seem to be three possibilities— Possibility 1: If the chordal segments are to “sum”, they must link end to end, and we then have that –All the intervals on line b, such as the one between points H_{0} and H_{1} on line b, are generated It follows at once that if, for the purposes of establishing a limitcondition in respect of one interval, we shrink that interval, then all the other intervals must correspondingly and simultaneously shrink, and given that line b and the conic remain ‘fixed as shown ’ (invariant) throughout such an operation, the only way to bring about the shrinkage of any interval is to have directrix D_{u} approach directrix D_{d} on the conic. If that is done, and if we elect to keep point H_{0} fixed in place (invariant) on line b, then while point H_{1} approaches H_{0}, H_{2} will approach H_{1}, and H_{3} will approach H_{2}, .... and H_{n+1} will approach H_{n}, .... and so on, for as many intervals as there may be. Try it! Click and drag either directrix, D_{u} or D_{d}, above, toward the other. In other words, all intervals on line b will approach the first interval, H_{0}H_{1}, and all will, “in the limit”, be zero – that is, the intervals will all be zero when D_{u} actually coincides with, and lies in, D_{d} – and the intervals will all lie at point H_{0}. All the chords similarly must approach each other and all eventually pass through the point on the conic chosen to remain invariant, and these chords will indeed then all become tangents to the conic – but all at the same place on the conic. So, having approached and reached the “limit”, we surely have all our tangents, We conclude that a projectivelyspecified curve, such as a conic, 
Possibility 2: Intervals, with their corresponding chords and chordal segments, are defined seperately and oneatatime,

Once again, given invariance of the disposition of conic relative to line b, all intervals depend simultaneously and only on directrices D_{u} and D_{d}, and all shrink as D_{u} and D_{d} are made to approach each other – but all shrink in place. Try it! Click and drag either directrix, D_{u} or D_{d}, above, toward the other. Accordingly, in the limit, we do get distributed tangents, as many as we have chosen to specify, but at the expense of linkage (because the segments need not be endtoend), so the segments, in the limit, do not automatically sum to the conic. And we see that, if the segments are to be forced to go endtoend in order to sum one to the next, we must have recourse to possibility (1), above, which, as we saw, does not go to a limit with the desired or anticipated effect. So neither possibility in fact works, and we may say that conics, and by extension, path curves, are not the result of a limit process. 
Possibility 3: The shrinkage is to be brought about by “iterative division”.That is, each and every interval in a set is split into at least two, equal subintervals, N times over, where N is an integer “made” to increase, eventually to ∞. This procedure
Now the only way to ensure the projective, geometric identity of a set of intervals is to fashion them all in the same geometric way, which here requires that all the intervals in the set be referred to the same pair of directrices. This in turn means that a given interval can, at each division, be divided by just two (2). There is no explicit, strictlygeometric way to divide an interval by three (3), or more. So, 'by two' turns out to be both the minimum and the maximum by which an interval may be divided using a strictly geometrical method – which means, of course, a method strictly in accord with projectivegeometric axioms. But actually finding and ‘markingin’ the successive divisionsbytwo is not an especially straightforward business, as may be guaged from the diagram below, where it has been done once, for just one interval. 
It involves finding
This new directrix could then be used with D_{u} and D_{d} to “distribute” all the intervalbisections  which would need to happen at least ∞ times (i.e., once for each of the ∞ intervals needing bisection). And all this has to happen again at least ∞ times for the next general bisection, then again for each of at least ∞ further general bisections. However, just by working in this way,
So even in this third, apparently most promising, case, we find that the Limiting Process is not properly available. We may now say that shrinkage must be continuous, and based only on the approach of directrices. The corollary is that uncollapsed transformation must be quantised.This is to say that, because the chords cannot go everywhere tangential, they must stay as chords, and the transformation
cannot proceed "continuously", It must leave irreducible, “unvisitable” gaps. 
In the final analysis, 