Katametreesi and Protometreesi

The process that begins with monas is katametreesi, but a similar process protometreesi runs alongside it. The monas i put on or against the shape or form to be measured in protometreesi , while the monas is placed on the form or shape to be measured in katametreesi.

The terminology is distinct, because protometreesi is about finding a common measure or metron. Thus several monads are involved and one has to be chosen as a common metron, that is "the monas" that unifies the shapes in "measurement process". We can then use one, single(monas!" method of measuring. The summetria and resonance in the terms is hard to express, but is clear i think.

The method of finding this common measure is the basis of many of the technical terms in Euclid book 7, relating ro Arithmos. Thus the general terms Meizonos, Elassos and Isos do not relate to any specific method of comparison, but the terms perissos, proto and artios do relate to the analytical method of deriving arithmoi.

Given any arbitrary shape, one may assign a monas /metron to that shape/solid. Whether that assignment is a useful one or not is the very question. If the metron as a monad fits perfectly onto the form it divides the form into an artios arithmoi, or by collecting the modads into this form and arrangement one constructs an artios arithmoi, or plethos of the form.

If i lay the metron on the form to find out if it fitted this is more exactly katametreesi, and if i build the exact form by putting the metrons/monads together this is protometrreesi. The end result must be sugkeima, that is covering the form by gathering together monads, forming a net or plethos..

Now in the case where the monas is too big for the form the arithmoi/plethos are perissos. Thus we need to find a smaller monas to "put on"(proto) or in or against the form . This proto monas will form proto arithmoi. If the proto monas fits the protoarithmos/plethos is artios if the proto monas does not fit the proto arithmos/plethos is perissos.

We clearly repeat the choice until we pick a proto monas that is artios.

One other test used for any protoarithmos that is artios is that it can be halved and the forms are still artios, and one can then easily select a proto monas that maintains this quality.This test is important because it introduces duality or isos into the method in a formal way.
http://en.wikipedia.org/wiki/Euclidean_algorithm
http://books.google.co.uk/books?id=RVQ8AAAAIAAJ&pg=PA535&lpg=PA535&dq=Euclids+method+common+measure&source=bl&ots=bGYVXRNtfu&sig=LvksZ5cr-BRPHPOl_bQgMykC3nU&hl=en&sa=X&ei=9-K9T7KqIYyC8gP54rFg&redir_esc=y#v=onepage&q=Euclids%20method%20common%20measure&f=false

IT IS PROBABLY FAIR TO SAY THAT PROTOMETREESI IS A TRIAL AND ERROR METHOD OF MEASUREMENT. This make proto metron, the trial metron and thus proto monas the "guess" monas and protoarithmos the guess net of monads(plethos) that cover the form or shape. Thus protoarithmos can be artios, in which it becomes artios arithmos, or perissos in which it becomes perissos arithmos. Thus perissos is always greater than the form by at least a monad and possibly more, but this relies on the algorithm to determine by how much when the common measure is determined.

And those , which by the method have a common measure of one are perissos arithmoi. and of course still protoarithmoi, but those which are artiosarithmoi have lost the distinction of being protoarithmoi through sheer joy of flowing onto the desired form as a net (plethos(!

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