One fundamental always addressed without deep thought is notation or terminology.
Justus Grassmann was autisticly strict on the syntax of notation or terminology, but so was Euclid. Aristotle was another strict notations list who categorised everything. Pedagogues very however from the strict to the anything goes! Typically all philoshers split hairs, because they are there to be split!
The common speech is not so. In fact Newton lamented this imprecision in common speech, and distinguished those who use it from those who use a sound philosophical method of rhetoric. Thus rhetoric is essential for distinguishing, recording and conveying or communicating to the inner mind or that of another certain "Truths".
The very nature of setting out such distinctions is inherently tautological, self reflexive and circular. We have come to learn to accept this, and to put to one side Aristotelian systems thinking . Such thinking is not applicable to a definition. Some, swooning before some ideology of logic pretend that axioms are intuitively obvious, that also a good definition is intuitive and clear, but this is a fiction.
Clarity on intuition was brought by Hulme but not accepted by all. Intuition is learned. We have to have experience of many things before we can speak or write a good definition. Take for example the task of defining a bed. Socrates would make such fun of all our definitions!. In the end no definition is useful without the assent and consent of others.
Seemia is the final state of all Analysis. Declaring this is meant to spark synthesis. But we have endless arguments over what precisely is a seemeioon?
To construct we need a common notion of Dual. This is not the same as equal. Euclid starts with dual then he introduces dual terminology, that is to say terminology that looks dual, but in fact describing quite different forms which nevertheless are analogous
Finally the German reformers also wanted to set down Sätze and thus the = sign took on the German notion of Werden, or becoming, and thus a 2 sided identity becomes a procedural law insisting on certain results. Under such laws lines and planes systematised into a variable terminology which being a rhetoric has confusedly bern called algebra.
So, within the ideology of Pythagoras, passed down by the Platonic Academy in Alexndria under Euclid was the distinct notion of " isos" meaning simply dual, but much much more. And the missing concept the epiphaneia and epipedos being the embedded mosaic on the floor. Onto this mosaic shadows are cast , and the lines draw taught about them, quickly before they moved. Or, using the fishermans skill full net drawn flat, they could make such a grid anywhere. Or, even with a single Knotted line or cord , each knot dually spaced from one on either side, they could mark out in the sand the semblance of their Arithmos.
This is the beginning of their Gematria, the practice of their calculus in the earth, preparing them to ratio the Stars in their Astrology.
To create such a mosaic it was important that a student could perform 4 things. draw a line segment. Extend a line segment straight and good, Draw a circle to any given radius. Construct dual Ortho rotations! This last one is a theorem by Thales. The fifth demand I think Apollonius added at this point, not that Euclid did not know it. Rather the drawing of very long straight lines that is good lines is harder to do than it is to think to do! Students were not required to draw long parallel lines because that is easy to do. What is harder is to draw a long good line! Even harder to draw two long good lines that must cross!
Hipassus would also have required it for accurate trigonometric ratios, so both would have argued it was a fundamental demand on those who aspired to be Astrologers. Both the Conics and the Trigonometry are well served by this demand.
On this Mosaic then, all kinds of dynamics were performed. Neusis was the general manoeuvring on that areal plane. No measuring stick was needed, the embedded forms provided all counting and measuring required. Calculus was done on the go. Duality evident by a glance, confirmed by a simple count. The trisecting of rotations done in an instant, the square of a rectangle quickly confirmed. But over time this marvellous simplicity was lost. Those who taught this knowledge on the mosaic floors, would no longer stoop or humble themselves to the ground, but recite long practiced and long remembered rhetoric. This was like Algebra to the uninitiated! The Mosaics gone, they could not apprehend the assured correctness of what they were taught.
Circles in the ratio 1:1.5 can be used to 2/3rd an arc. This enable the third of an arc to be constructed., but it requires a simple neusis of a fixed length. If this is denied, then it is impossible to third an arc! But such a conclusion is obvious, for a proportion requires that a scale be used. Without a scale the very concept of proportion cannot be had, let alone apprehended!
The concept of proportion is inherent in a mosaic, and from the outset, it is explored transformatively in Euclids Stoikeia. On the mosaic the forms are drawn, an the gnomonic proportions clearly illustrated in book 2. The proportions of the circle in book 3. Until at last we arrive in Book 6 at the definition of proportion in terms of similitude!