This is an example of Greek ingenuity, based on greek philosophy and greek Knowledge. The Greeks viewed the magnitudes in the Ouranos as Gods, and the dynamism of their motions as ordinances and signs. Despite the flowery translations the greek were pragmatists and related each god or magnitude to a model or image of it. Thus where the Judeo Christiian ethic forbade the modeling of the gods in the heavens or on the earth making it tantamount to idolatry, the greek Pantheon was freely modelled and innovated upon. Thus from simple blobs to highly sophisticated automata, greek inspiration knew no bounds. Coupled with the unique Greek anti gods stance, in which they dared to compete with and against the god's, the cosequence was a daring innovative Teknos that peered more penetratively into the magnitudes and intensities of their everyday life.
I have already shown how the Greeks used their mythology to encode, organise and shape their knowledge, and in particular how the god Isis is of crucial importance to the development of systematic knowledge, but it is worth emphasising that the use of gods, sunthemata, and sumbola is crucial to the understanding of the encoding of pragmatic knowledge. Thus the mechanical and magnitudinal nature of the reality we experience was not lost on the greeks or the greek philosophers, it was just expressed in terms of what was generally understood. This was the principle of Kairos applied by Pythagoras par example, but followed by all greek philosophers and particularly hose after him.
One can only apprehend the greek notion of arithmos in the context of dynamic msgnitudes, for arithmos are magnitudes that flow together, gather together and fit together like gear teeth in a gear chain. The rhythm , rhyme and the rhetoric of their motion is sheer poetry and gives the kinaesthetic and auditory sense to the notion of meter, which will always be a fractal song in ur hearts, iterating out the details of our experience like a harmonious symphony or a discordant tone poem.
Arithmoi arise always from comparing dynamic magnitudes, and are symbols of change of magnitude.
Pythagoras's notions of arithmos and the harmony of the spheres denote in simply understood language deep principles of harmony and resonance relationships. However, this type of relationship is commonplace in fractal systems where principles are seen to apply at many scales.
it does no good,beyond hypebolae to attract attention, to compare the antikythera ddevice to a computer. The machine is in fact a comparison of dynamic magnitudes. This comparison is set in a system which can only be called proportional, many ratios. The system itself, after Eudoxus is called the Kosmos, and may rightly be called a Kosmos machine. The orderliness of the Kosmos is in fact the ratios within the kosmos that remain "just" for the data recorded over many millenia by the Babylonians and the Egyptians.
Thus these ratios of "dynamic" magnitudes could be modeled by Euclidean shapes and gear teeth in the shapes transmit the arithmetical (arithmoi) relationships. Thus the arithmoi, which are forms that "fit and flow together", both stylistically and mechanically naturally convey dynamic magnitude, fixidity, scale and ideal relationships.
Proto arithmoi are of course arithmoi distinguished from artios and perissos arithmoi. but the triumviate only make real sense in the context of a dynamic comparison process like the commensurability process. £ results from this process would be artios, ,perissos, and proto. Proto would be where the process continues ad infinitum. Thus these arithmoi are a class of arithmoi that can be used to measure, but cannot themselves be measured. this is the sense in which we use a prototype as a new design which is not commensurate with any other and so will form the model of a new type of design.
The antikythera machine illustrates one important cosmological point: the magnitudes are all related by a fractal pattern which emphasises that all spatial magnitudes are related by dynamiism, motion; and in particular rotational motion. The gearing of such rotational motions is what gives rise to our notions of arithmoi from which comes that unit of magnitude (arithmos) which we specify as monad/metron. This monad can equally be a gear tooth as a bowl or a standard rod!
I also observe that "tyme" is the tracking of the motion of the planets relative to the earth and the stars!
I may also hazard that the terms perissoss, artios and proto are technical terms of comparison. Thus the Eudoxian comparators Meizonos, Elassos and Isos, are further sharpened by processes that compare the details of magnitude difference. The comparison is made by placing one schema on another, keimee, covering, to see if it fits(prosthe). if the shema that is being used to cover is too big it is called perissos, if it is too small it is called protos, the one placed to fit is a "new" magnitude. The proto magnitude is also perissos but in the contra sense.
Perissos means that the schema placed on the given schema is not artios-perfect fit, flowing with the pattern or style, but rather surrounding the given form(perissos) wih a border.Protos means tha the schama place on does not fit and so does not make a border(perissos). However the given form makes a perissos with the proto schema. The proto schema is thus a candidate for a new arithmos and is called a prot arithmos.
The Technical definition of perissos gives further information about a process of comparison. By removing the iso schema the border is revealed, but we have no precise measure of it. however if we suppose it to be a monad, that is one arithmoi then we can compare it with the given form. This is where the arithmoi introduce the counting of measurement into th comparison, the rhythm into the dance of the process of comparison.
If this monad fits precisely, artios then we have defined perissos as artios with one extra. If it soes not then we have to reiterate the process with the new diference. The comparison of the new perissos with the old perisoss reveals a continued fractional process or a fractal pattern which only ends when the newest perrisos fits exactly dual from the old perissos. this method produces a common measure and defines commensurability, and i also defines incommensurability or proto arithmoi by a similar but contra method.
Clearly the method requires some process of comparing differing shapes(odd shapes), and this is done by using a standard shape (artios or fitting shape) into which the odd shape can be transformed by using the plane geometrical methods associated with parallel lines and circles. Arithmoi are therfore most likely to be equilateral or right triangles or squares and rectangles as standard forms. While circles can be utilised, the importance of he quadrature of the circle lies in this standardistion of comparators or arithmoi.
Proto arithmoi therfore represent new forms that can be used as monads for measuring larger forms, or multiple forms, but is too small to measure given form, and is commensurate with that form by a common unit form which does not allow the proto form to be halved and so dualled in parts.
Clearly a proto form is significant because it precludes artios, especially by halving, which is dualing on the next scale below, and he isis structure is dependent on duality.