In science, numbers stand to the geometry
like the letters of the alphabet stand to words;
whoever confuses or identifies these with that
does not perceive the soul of things in the world.
Devoid of computers, the ancients had to stick to the use of manual tools, able however to give credit to the sublime geometry which, as one should argue, does not need numbers and calculations to perfectly express all sorts of values and measures. . But yet…Technically, with 'tracing' one immediately adheres to the established rule, to do it using only a straightedge and compass with respect for certain rules.
A standard dictated – in my opinion – not by the limited means of an era so far away from us, or by a mathematical view not yet refined, but by a greater tuning with the transcendent nature of all geometry, which manifests much more of the expressions of the Great Architect than any calculation or human numeral formulation, being able to offer us a circle without the need to declare π, which resides in gestures, as well as the golden 'Divine Proportion' of the Φ~φ administering multiple forms of living nature.
Whilst we will always be able to define the exact area of a square, the knowledge of current mathematics only allows us to approximate as much as possible to that of a circle, as in the amazing simulation of the π; but it might not be enough.
It would even have been questionable whether a square with an area absolutely identical to that of a circle could ever be created (even if the latter provides for a quadrangular symmetry), precisely because of the different structure, from static-linear to dynamic-isotropic, that not even the calculation infinitesimal could determine absolutely, but at the limit make it intuitive and plausible.
That the π needs to one but not the other, stands to premise it with sufficient evidence.
The most heterogeneous projects alternated for centuries presumptions and exhibitions with frustrations; until the day when the alleged demonstration of the transcendence of π came to their aid, from which derives the inapplicability of using only ruler and compass so coveted by their predecessors; regardless of what the compass had been able to do since before the invention of the wheel.
It is that his overwhelming emphasis referred to a π of human ideation, a kind of counterfeit that somehow alienated from the real goal; but no one noticed.
A constant manufactured by manipulating perimeters and areas of polygons, extreme to the point of being producible only numerically.
The year was 1882 when, after more than two millennia of unsuccessful attempts, the impossibility of squaring the circle drastically enunciated by Ferdinand von Lindemann (not the first, but certainly the most famous) sanctioned his [not] having solved the problem, once and for all and for all of us.