The only case [im] possible in which the sum of n consecutive cases would cor­re­spond to the length of the arc they form, would be that one in which the units in question were indeed sides long only two 'points', a purely dialectical but unlikely supposition, which I suppose unattainable even by the infinitesimal calculus.

In practice, I argue that even when a set has reached its exhaustive ap­prox­i­ma­tion, in one sense or another, the same factor, transcendent or not, will al­ways in­sist on justifying the curvature, for the realistic principle that a pe­rim­e­ter or area they only make sense if they are made up of tangible, if not also us­a­ble, units.
From what has been ascertained, it can likely be deduced that even at the limit of geometric counterfeiting, the microscopic arc of circumference in place will al­ways be greater than the underlying chord, just as the whole circle will be great­er than the polygon it contains, at least for three thousandths.

Compared to the current focus of pi, if we really want to maintain the usual mode, we get a – let's say – a 'Coefficient of Curvilinear Constancy'
[3.1446÷ 3.1416] 1,000955 from whose presence it is given to deduce the threshold beyond which the calculation is lost in the abstract of infinity, eluding the faculty of a “real” measurement.
The natural prowess of this value denounces the limit of the theoretical frame­work, comparing the 'official' π with the effective π, just detected and which else cannot to be; and the most exciting aspect is that it is precisely ruler and compass that take it out of the oven, ready for use.

I would not exclude that if our ancestors had known this triangle and its a­fore­men­tioned properties, Golden Section in primis, they would not have hesitated to use the π that it contains and manifests, thence move on to squaring the cir­cle with all the ease that will follow shortly, saving many torments; but this does not dem­on­strate what only an instrumental survey, precise beyond the .003 would confirm, assuming this need is felt. Probably only a cylindrical ves­sel and a cube of proper proportions, filled with the same quantity of stable liquid, could reserve that satisfaction. …

Self-criticism is science

However, this analysis suggests that the π measurement may differ from the calculated one, since this cannot be deduced from a linear dimensioning without requiring a precise and dedicated rectification, precisely because that direct su­per­po­si­tion of perimeters and areas cannot take place in an elastic way; although this is all that has been compulsively pursued since the very be­gin­ning.
After all we are only a few thousand years old, and we like to think that they are who knows how many; (nay, we are going through the worst phase of our his­to­ry) .

Do you really think that π could be an entity disconnected from the most se­duc­tive harmonies of space and time? only the fulcrum of an autonomous car­ou­sel, entrusted entirely to a range of speculative arrangements, even if so far justified?
the maximum vehicle of structural equilibrium together with the hinge of the Gold­en Section, to be reduced rounded by a heap of fragments, as the unique chance for the methodology?