http://m.youtube.com/watch?v=YfY7RyAzHPEImportant to assimilate these distinctions.

http://m.youtube.com/watch?v=_c5v14ZIUO0How these distinctions actually should be applied in constructing our descriptions .

Strange as it seems length area and volume are constructed products, and sometimes we cannot construct them!

So the thing we need to apprehend is the Monas, the Metron, and the Logos Anologos methodology

( Proportion). To replace these by numbers is to misunderstand Arithmos fundamentally!

Arithmoi are precise mosaics using a specific Monas whether that mosaic tessellated the spce or not was Not the main aim of the mosaic. Epipedoi, or speripedoi were also mosaics, but the elements were not necessarily a Monas , any forms could be aggregated together.

Arithmoi were canonical in that the constituents are homologous or homogenous. The first rule for the logos analogos methods is that the magnitudes are homogenous!!

The idea of a Metron is that a form becomes a Monas or a standard unit. This standard unit thus imposes homogeneity on any Katmetresee ( measuring by laying the Metron down ) of a larger form.

So an Arithmos is a mosaic of standard units, we can use any regular form as a Monas so that it can act as a Metron of larger forms

Given this factorisation of space by a given Monas used as a Metron , one can then derive proportions ( logos analogos) descriptions of relationships.

When it became fashionable to place the count of the mosaic in place of the actual mosaic, the idea of a quantity as free standing became formed and gradually led to that modern confusion called Number!

Later a Persian cleverly showed that ratios could be written as fractions, and that a limited set of ruled governed these fractions . It took a while for the furore to die down, but eventually the powerful use of fractional measurements on rulers led to the measuring line concept. In particular the Indian 8, 9 or 10 cyclical systems, Vedic, Bahai, and Brahmin systems, under Wallis were powerfully integrated into the measuring line conception. This organised structure was later used By Dedekind et al. To derive the modern Number line system. In the mean time numbers had takn on a cardinal, or lordly position, and that was distinguished from their ordinal use, as well as their dimensional multiple use.

So to the circle and to the diagonal of the unit square.

A fairy tale is widely told about the Pythagoreans being thrown into consternation by irrational quantities.

Firstly these were known as incommensurable , and that meant that the divisor Algorithm could never be finished! The simple solution was to define these incommensurables as new units, that is "protoi "! Thus protoi Arithmoi are all about these incommensurable magnitudes! Prime numbers are how the Pythagoreans studied incommensurable or irrational proportions( that is through the application of Euclids algorithm).

The fundamental unit Monas was thus the unit diameter sphere!

All of the results for a square can be obtained using a unit diameter circle.

It is clear that the square tessellate the plane, but the circle leaves gaps. This was not a problem, but a clear indication of the incommensurability of space even in the plane. However one could choose a Metron to suit the required outcome .

Thus in attempting to describe the proportion the iterative / inductive method of Euclids algorithm is the classicl fractal process upon which all fractal processes fundamentally rely

When this algorithm is applied to the diameter versus the perimeter of a circle of unit diameter it was known to be incommensurable. That did not prevent greater proportions being sought! Today we still attempt to develop even greater proportions!

Was it ever hoped that the proportion would be found? No. The reason is homogeneity. A circle is not homogenous with a straight line!

Careful astrologers up to Newton never expected a curved line to be straightened . The inhomogeneity of th comparison is why no proportion will ever be finalised.

However, for pragmatic Mechanics and gear design proportions were approximted by truncation. For those who deified the watchmaker as a type of divine Mechanic, the implied perfection covered over the inherent error introduced by approximation. Within hose errors lie th chaos theory which now accounts for many unexpected behaviours in the divine clockwork!

So the trisecting of an arc was inherently an incommensurable outcome, because 3 is the third prime or proto Arithmos!

Nevertheless a multiple of any prime is never an issue and this is the way to approximate to the trisection of n arc by using proportion and the ultimate proportion form: the circle / sphere.