The ratio is a constant that allies to many observable circumstances. Thus the ratio of the hemisphere to its circular base is the same as the ratio of demi hemisphere to its semi circular base, which is the same as a quadrant sphere to its circle quadrant base which is by this process of limits the same as the quarter arc to the radius. This constant ratio declares itself by the rotational relationship between each factor in the sequence. Thus by rotating the quarter arc on its base about the upright or right angled axle we obtain by degrees, or in stages each of the ratios . Seeing as rotation has made no material change except in the quantity of solid rotation, proportion demands that each rotational stage is proportionate to the "radian" of turn, and thus the ratio of the forms is equivalent to the ratio of the turns between them. But since we compare adjugates of the forms, that same ratio applies to each adjugate and thereby is discounted! Only , it is not forgot but set to the notion of identical, leading to the notion that the things or adjugates compared maintain a constant relation!!
The ease with which this notion is attained highlights the fundamental harmony , the Harmoniam Mensuram of Cotes, that underpins this STYLE of reasoning.
Now this style is in the Greek known as rhetoric, and reasoning is derived from Latin" ratio" which in the Greek is called proportioning or Kairos, a distinctive approach to communication which Greek philosophers began to label by the term Logos. Thus this style of reasoning is heavily underpinned by the reasoners Sagacity, pragmatism and Sophia or wisdom. It should come as no surprise then that at the very foundation of rhetorical skills is a deep mystical apprehension of personal congruence with ones surroundings.
I have not shied away from considering this important foundation to all knowledge including the so called mathematical knowledges. As a consequence I am not surprised to find that true innovation and vibrancy in the so called mathematical wisdoms goes hand in hand with a deep esoteric mystical experience in apprehending any reality, so called.
Our languages and our cultures are not divorced from our experiences. To attempt to sanitise or hygienist, tidy up or excoriate these " irrational" notions is to cut away the flesh from the arterial life support. One might think one is harvesting fruit from the plant, except that fruit contains the germ of the originator! If this germ is continually suppressed, then anything made from its fruit will grow into some tangled , knotted plant, impenetrable and unsightly.mthis is the state of so called mathematics today. Those that tend this plant are as gnarled and difficult to comprehend as the garden they so lovingly nurture!
On the other hand, those who like birds flit through the matted mess, sucking up the thwarted seeds, and fly away to deposit them in freezer pastures may survive the original beautiful intent. Occasionally a woodsman, like Grassmann may come along intent on burning the bush to the ground! The trolls that guard this monstrosity however successfully fended him off! Maybe their senses have grown dull with inbred thought, for Mandelbrot has managed to infect the bush with the most deadly virus. When it has done its work Grassmann's original strain may more easily be grafted onto the remaining stock.
Grassmann's strain is not pure, but it is hardy, and it may tolerate the purging by the Mandelbrot virus and grow to produce wholesome ripe fruit.
What can be said of pi, can also by analogy be said of i, but I is a pure label for a ratio, whereas pi is an impure label for the result of a metric process. Algebra has airways been about assigning pure labels to observables whether they are forms or ideas or processes. These pure labels have no metric content, and that is there freedom. But they must label there subject or else they become meaningless.
When one labels the subject in focus, one creates a set of adjugate labels that relate to each other. These adjugates are relational, but not metricated, without a metric, the pattern of labels may be found to apply to many circumstances, that is a wide range of subject foci. This is recognised as analogy, metaphor, simile or similarity. Thus describing a situation with these metric free labels may so organise ones apprehension of the structure of reality, that one is swayed away from continual vigilance and empirical observation. In particular, the introduction of a metron may ruin this sense of orderliness! Thus some who do not wish to be changing as Shunya continually changes may resist this continual life saving and life giving obligation to be empirical, to test the labels in the metricated subject focus, and so to clip the wings of vaulting flights of fancy and fantasy!
To some this may be onerous and an unwelcome experience, but to those who engage it it is a purging experience, pruning the plants in ones garden, the setFs, so that by this evolutionary process plants fit for purpose grow in the mind, bearing with them the DNA of all!
That one may survive whatever the prevailing environment is the true prize of this process. And to appreciate that one must look beyond the individual, constructed subjective self to the colonic environment from which it springs, and beyond that to the dynamics of the ceaselessly changing Shunya !