περατα is a notion of extremeties. But the notion is really concerned with boundaries.
If i say "this is a horse", when does it stop being a horse, or horselike, or of a horse. and require a new distinction?
Perata is an artisans word, not a abstract geometerists. An artisan draws a line to represent the limit or boundary of reference for an object. This creates logical issues. The line does not exist in the object but on am Epiphany, a lighted plane on which one draws. So the representation on the epiphany is false, and yet it contains information that is needed to distinguish thre boundary of an object.
When i examine the boundary of a real object i find a fractal pattern of smaller and smaller boundaries, not a straight line at all, and not even a line.It is just a fractal pattern that relies on my being able to distinguish differences in density and intensity and hue and contrast, and also refraction and reflection effects of incident light as well as any inherent light. I also have to cope with any lensing aberrations within my eye.
So i deal with all this by a perceptual processing trick: at a certain level i phase out the detail and compute a simpler envelope. Where the guidance for what envelope to compute comes from i hazard to guess as an evolutionary empirical solution.
Now the notion of semeion/shunya underlies all of this, and just as i compute a line as an output at a certain level i also compute small disconnected regions or contiguous regions as a limitbeyond which i cannot resolve an"image". This is solely an image thing, a visual distinction,but there is of course an anlogue in the auditory,kinaesthetic and gustatory systems. We call these regions points in the visual field, but the greek semeion is capable of including the auditoy kinaesthetic and gustatory experience, as a computed multiplexed experience or sysnaesthesia of this limit. Thus a semeion is an subjective experience of a limit, and indeed it draws attention to a limiting process. Perata refers to this limiting process.
Limiting processes are always linked to boundaries, necessarily and thus perata refers to the boundary arrived at by a limiting process.
Euclid therefore philosophises that a gramme in it limit is bounded only by semeion. This is another way of saying that a gramme is a fractal structure which limits to semeion. We explore hese limit structures in fractal theory, and particularly to determine connectedness or contiguity. Thus we can describe now whether a gramme is a contiguous semeion structure or a disconnected structure like a Cantor set.
Euclid could hardly of been thinking this when he defined his notion of a gramme could he? The answer is yes he was. W have no primacy in his type of thinking. These secreted worlds have along provenance. The pragmatics of them is still moot, but the logical necessity is there as ever.
Using this notion of limits, or if you like computed surface outputs, Euclid defined the boundaries between semeion and gramme and between gramme and epiphany, and shows he fractal nature of his constructed definitions by using exactly the same sentence structure.
I can deduce from this that the gramme was known to consist in aggregated semeion, and the subjective notion of lying evenly is what is needed to verify straightness that is a good or trueness of line direction, and plainness. Here Euclid introduces and acknowledges the skill of the master craftsmen and women, the Artisans.