Within the first 2 statements of book seven Euclid lays out the Agenda for arithmetic.
Define units,(legetai, monad)
Aggregate (sugcheimenon) units to a form or structure, the side (plethos) (of the gnomon)
Appreciate the sequence of aggregation.
Apprehend the order of the aggregation to the form
Name the form(count) Arithmos.
In making this practical application of geometrical relations he moves from a general approach to a an increasingly specific and particular one. He goes from "algebra" to arithmetic.
Aggregating to a form allowed geometric relations to be employed, and besides it gave the sense of "perfect fit", or perfection.
The gnomons used were many and various, and thus the monads used were many and various. The one of interest is the curved gnomon or sector of a circle. by aggregation these momads to a quartew circle or a s3emi circle many relationships to do with the agonia ,the twisting or angling of things could be standardised.
Hipparchus is fered with having worked on the monads in the pythagorean relationships, which is probably why he raised the issues about the monads in tha arithmetic system there. Eudoxus however enabled the work to continue by effectively developing a teory that meant that monads were relative. In one sense his is the first theory of relativity. The definition of a monad or of the monas reflects this theory, and in fact establishes a fractal paradigm at the very heart of arithmetic. It also establishes the fundamental rule of algebra like is aggtregted with like, and only those monads that are the same may be summed to a subtotal form or a subform.
The notion of a modulus derives from this theory too and it is recursive
Modulus= scalar * monad.
By the theory the modulus may be perceived as a new monad and the fractal continues to grow or shrink if the monad is perceived as a modulus. In this way a fine structure is associated with every form in space.
Ptolemy is the next astronomer after Eudoxus who developed the arithmetical relationships of Hipparchu tri-gnomon-metry, by relating it to the arcs of the circle. he related the chords that segmmnted the arcs to ratios of the diameter. and was able to use the incredible relationship of chords subtended from the diameter to the circle, they always formed a right gnomon!
The Indians after the greeks occupied there country to some extent later worked on Ptolemy"s ideas and simplified and refined them to a great degree, but their biggest advance was in regularising calculation around a small modulus consisting of nine arithmoi. The notion of Shunya was crucial to their thinking Whenall 9 arithmoi were used the nex arithm was called "full" -Shunya! and another larger form was used. When that larger orm was used it took longer to fill, and it was only filled when ten of he previous modulii were used as monads. The growing form was geometric and enabled larger and larger arithmoi to be named and recognised.
This was in fact an idea that Archimedes had, but it fell to the Indians to make practical and commercial use of it. the facility with arithmoi was subtle and astounding, and the structure was applicable to every form. This was the beginning of the "real" arithmoi and the move from geometrical form to modulus structure. These ciphers were "free" from any specific form, and applicble to all forms, but they requred the forms to be specified, the units had to be specified or they were useless practically. They revealed another aspect of the generality of geometry, the fine structure of space that exists in all hings.
It would take a while for the trigonometry and algebra to be formulated as general relations, and in that time the geometric reference was lost between the 2, Geo,etry was slowly being sidelined as a department of some great building, when in fact it is the building , and we have divtned the fine structures of it.The diferential geometry is the culmination of this process.
The analogies Newton was able to use to explain relations of dynamism are the other hived off aspects of the search for a full apprehension of the experiential continuum. It is geometry or spaciometry however that makes sense of it all, the geometry of Pythagoras and the inquisitive mind of the philosophers, not the dead facsimile that some have served up on a cold plate to us of his generation.
Προτος αριθμος εστιν ο εχ μοναδι μονη μετρουμενος
11. A prime number is one (which is) measured by a unit alone.
Euclid declares protoi Arithmoi are single units.
Thus his notion of scales is based on the protoi Arithmoi. The natural order of scales is "prime"; therefore our appreciation of reality and what it looks like at different scales is governed by protoi Arithmoi.
Precisely, Euclid states: before you can get an arithhmoi you need to decide on a unit of measurement. Protoi Arithmoi are those unit measurement decisions, and they come "before" the arithmoi based on them. Thus "The Irrational "numbers" are protoi arithmoi in this sense.
Prime numbers make no sense, but protoi arithmoi make perfect sense because they are unique geometrical figures.
These are the unique forms required to measure forms related to them, that is to perfectly fit these forms. Measurement is about perfect fit, conforming to the form, perfect proportion.
Thus when we measure we assert this form is made up of these other forms precisely. These other forms are defined as monads, and also a protos Arithmos. Thus We sense that the concerns were not about counting but about fitting perfectly, about tessellation and therefore symmetry and all the affine transformations. When you look at an Escher print you are looking a Protoi Arithmoi! Some iterated funtion System fractals are pictures of protoi Arithmoi.
The "number " notion misconceives all of this and hides the simple joyful pursuit of forms that tessellate an enclosing form. By this means Euclid established the fractal link between the ideals in the first 6 books and the pragmatics of the "real" world. The basic drive is to fractalise: find a form that fits an unknown form and by this way apprehend the unknown form. The fractal mesh, the fine structures by which this is done is the basis of calculus.