Logos and the Primes

The first mention go measurement is in Stoikeioon 5 when logos and Analogos are defined.

The concepts of greater , lesser are also introduced, and the concept of magnitude. Magnitude it is clear is an impression of greatness, and there are different kinds of magnitude. The concept of Isos is noticeably not part of the greater or lesser division of magnitude. It is introduced in defining Analogos.

A logos is defined as a structured sequence of 2 magnitudes. This structure we call order, and the order is protos deuteros . This is the first definition of protos as applied to protos Arithmos. From this we learn that Arithmos were defined by the logic of logos Analogos duality.

The magnitudes were arranged in this order. There was no rule that thi had to also be the greater before the lesser or vice versa. The important principle was that here was a protos position and a deuterons position in which order magnitudes were placet arbitrarily.

However, experience revealed that certain magnitudes were best placed in the protos position. What thes magnitudes were depended on the notion of division of the greater by the lesser. When this was done ,factorisation into multiples is what has been done, multiples of the lesser. This is usually confused with multiplication, which is an entirely different process based on rote learning.

The concept of division is called Katametressi, that is the process of using the lesser as a Metron, to count how many times it must be laid down upon the greater to cover the greater. There are rules about this use of a Metron, principally that the Metrons should be contiguous.

Using a Metron does not imply using it continuously, but unless yo did you could not determine the notion of isos for these 2 magnitudes. Recall in book 1 the notion of isos was defined as one thing( undefined, so arbitrary) "fitting" " upon or on top of " a thing" of the same kind( again not specified except by adjectival ending ) . The concept of a perfect or artios fit and that of a perisos fit derive from these circumstances. Thus Isos is associated with artios.

Perisos is a bit more complex because could be greater than the greater one or lesser than greater one. Later in book 7 Euclid defines these notions, but in the opening part of book 5 Eudoxus is concerned about a different relationship called logos. He consequently highlights the leapfrogging of magnitudes associated with getting the two magnitudes to " fit" .
Thus the magnitudes were not to be restricted except in position, they had to be capable of exceeding each other until an exact, artios , multiple of the lesser was achieved.

Thus the magnitude in the protos position was doubled, trebled etc until the magnitude in the second position divides the first or protos position precisely. In achieving this we arrive at the standard form. However, students are usually taught to cancel or cast out multiples or common factors to achieve this. This is a much later definition of a ratio based not on magnitudes nd measurement but in numeral relationships learned bt rote in tables, and confused with the Arithmoi.

Thus we se that the protos position helps the process by keeping track of the multiples of the greater or lesser depending on which occupies that position . It soon became clear that certain magnitudes placed in protos position had no " common" multiples and thus could easily be rationalised by using themselves as the multiple for the second position .

Now because there was no rule about the occupant of the first position in book 5 later philosophers imposed one in defining the Arithmoi. The proto Arithmoi therefore by convention take the first position in the Logos, when the logos Analogos logic is used. These proto Arithmoi have to be defined and learned by rote. They became extremely important in the factorisation process, when it was found how they simplified multiple forms descriptions and manipultions.

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