To experience magnitude is to conjugate Shunya . The comparison of any 2 conjugates of Shunya is defined as the Logos response by Eudoxus. Further, the comparison of any 2 logoi is laid out in a Sxesis of 4 magnitudes, and the 2 logoi are distinguished as logos and analogos. This is the basis of all Greek reasoning as taught by the Pythagorean school. From this springs the Aristotelian logic, as well as the Eudoxian rationality, an equally useful logic

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Book 5

Oroiαʹ. Μέρος ἐστὶ μέγεθος μεγέθους τὸ ἔλασσον τοῦ μείζονος, ὅταν καταμετρῇ τὸ μεῖζον. .

A part is

A magnitude belonging to a magnitude,

The lessert magnitude of the greater magnitude

Which measures by laying down upon the greater magnitude.

Eudoxus defines a process which in turn contextualises the meaning of the terms, and in this way defines them.

Thus a magnitude is not divided into a lesser and greater part, for what does part refer to? Instead a magnitude is demonstrated to the senses, that is experienced as a kind of greatness, and a lesser magnitude is conjugated from that greatness. Thus the greatness is termed the greater and the lesser is in reference to it, or relative to it.

This conjugation is the process and experience of a part or of focussing on a part.

The Katametressi requires that lesser part to be physically identified as a separate entity. This is done in 2 ways: making an exact copy of that part; cutting that part off from the greater magnitude. The second way destroys the initial magnitude, or transforms it into 2 distinct other magnitudes which combine to give the initial magnitude. This second way is explored as combinatorial Spaciometry. The first way explores metrical Spaciometry.

The first way finds an exact copy or, using dual initial magnitudes transforms one of them as in the second method and uses the parts as exact copies of the conjugated parts. Selecting the desired one, it is then laid upon the second of the dual magnitudes as a Metron, in a process called Katametresee, or measuring by laying a Metron on top of what is to be measured.

In establishing this definition Eudoxus establishes magnitude, lesser, greater, part , Katametresee, ans Metron. In addition Metron is to be placed on top of hat is to be measured?

This spatial sequencing and positioning I call first and second position respectively.

These terms I'll feature in what Eudoxus goes on to define.

βʹ. Πολλαπλάσιον δὲ τὸ μεῖζον τοῦ ἐλάττονος, ὅταν κα- ταμετρῆται ὑπὸ τοῦ ἐλάττονος.

A multiple form is thus, the greater magnitude relative to the lesser magnitude, when it has been measured by the laying down upon process, UNDER the lesser magnitude.

We can recast our thoughts about the greater magnitude after the process of measurement by the lesser magnitude. Not until we have done the process can we define a multiple form.

When we have done the process we can define a multiple form relative o the lesser.

The lesser here is used as a Metron and the greater is now a relative greater magnitude called a multiple form . This multiple form is precisely a multiple form relative only to the lesser magnitude: the greater magnitude is now relative o he lesser.

Before the process of Katametresee we could not say hat the greater magnitude was a multiple form. We could only say greater than the lesser. Now we can define it as a multiple of the lesser. The Greek uses a relative case, not a genitive case.

The process Katametresee generates multiple forms by placing the greater under the lesser. The greater goes in second position, the lesser in first position.

In his definition we have not yet considered the case when the greater is not an exact multiple form of the lesser.

γʹ. Λόγος ἐστὶ δύο μεγεθῶν ὁμογενῶν ἡ κατὰ πη- λικότητά ποια σχέσις.

δʹ. Λόγον ἔχειν πρὸς ἄλληλα μεγέθη λέγεται, ἃ δύναται πολλαπλασιαζόμενα ἀλλήλων ὑπερέχειν.

Let a Logos be 2 same genus magnitudes , same kind , same type, same source experiences of greatness which make (skesis) a relation pattern by placing down (peelikoteeta) magnitudes, sizes, experiences of greatness.

http://www.perseus.tufts.edu/hopper/morph?l=phlikothta&la=greek#lexicon

Logos being magnitudes " pros" (positioned before) one another ,let it be so defined, which let be capable of( dynamic in) being higher multiple form-constructs each of the other .

http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dsxe%2Fsis

Eudoxus introduces the notion of Logos as an experience of a process of experiences. These experiences are various and so he points to the same kind of experiences. The observer has to be able to distinguish the sameness of experiences, I particular that sameness that is genus, and thus to do with the source of that experience.

Genus relates directly to isos as a veinal , fibrous sameness, where the source is visibly dividing into parts usually copies of each other and copies on a smaller scale of the larger or greater . The greater precedes the lesser in this type of genus sequence.,

Given these kinds or sorts of experiences, these types of same but different apprehensions, Eudoxus rhetorically describes a process called Sxesis. Simply this is forming a pattern by placing these 2 identified experiences down on a surface and arranging them according to some scheme or some model or template. Thus a Sxesis like a sketch or a schematic is some constructed pattern that s representative of some observed pattern of arrangement or behaviour or both.

Eudoxus reduces our experiences of patterned behaviours or sequences of behaviours or arrangements to a notion of sketching some kind of representation subjectively and objectively by marks on some surface. It is these marks that represent the magnitudes we experience, placing these marks down onto a surface is placing likenesses of the observed magnitudes on the surface, and the arrangement of these likenesses is a representation of the observed arrangement.

Thus logos is an accurate description of an experience of interacting with our environment, which is initially nonverbal and subjective, and which then illicit a response to draw representations of the magnitudes of our experience, where draw is a process of placing magnitudes sequentially together in exact representation of the observed.

Alongside this initial logos response the audible and verbal response is collaboratively generated.. Thus Logos is a combined sensory motor response to a perceived pattern in experience, which necessarily consists in experiences.In apprehending this perception we express it dynamically , vocally and artistically. We scratch ot designs, we paint over regions, we arrange objects into patterns all in an attempt to apprehend a perception.

Eudoxus then switches to the terminology he will use to define these logoi. Very starkly he defines logos sequentially and thus visually and or auditoria lily. : magnitudes in a position of before are mutually before one another. This perception when it exists is a logos by definition.. This relation of magnitudes is perceived by sequence and juxtaposition, thus it is visually or auditor illy convyable.

The cooing of a dove immediately presents an auditory logos . Any 2 marks next to each other, any 2 objects next to each other, any 2 ideas juxtaposed etc all are defined by Logos, providing they are the same kind of magnitude.

If the magnitudes are not of the same genus we experience synaesthesia, which then allows us to define logos through the syn aesthetic response, which confuses and mixes up the genii and allows us to define hybrid geni.

Given they are the same kind of magnitudes, by induction we must be able to make multiple forms of them, and hence multiple forms of each by the other. Only with physical visual objects can we dynamically achieve this, by choosing a pair of objects as logos we can, as previously discussed measure each by the other, but in a mutual way. The lesser " covering" the greater, the lesser in first position.

As observed, Eudoxus has not yet discussed precise and approximate measures as multiple form, but in logos he indicates a part of the mutual process he will later clarify: each magnitude must be capable of making multiple forms " extensively".Uper here means higher in the sense of more extensive.

Intensive measurements are taken when a larger unit is used to measure a smaller unit magnitude. Extensive measurements are those where the smaller unit magnitude is used to measure the greater unit magnitude. A multiple form therefore could be made in 2 ways: using smaller pieces, or by using the larger piece repeatedly.

εʹ. ̓Εν τῷ αὐτῷ λόγῳ μεγέθη λέγεται εἶναι πρῶτον πρὸς δεύτερον καὶ τρίτον πρὸς τέταρτον, ὅταν τὰ τοῦ πρώτου καί τρίτου ἰσάκις πολλαπλάσια τῶν τοῦ δευτέρου καὶ τετάρτου ἰσάκις πολλαπλασίων καθ ̓ ὁποιονοῦν πολλα- πλασιασμὸν ἑκάτερον ἑκατέρου ἢ ἅμα ὑπερέχῃ ἢ ἅμα ἴσα ᾖ ἢ ἅμα ἐλλείπῇ ληφθέντα κατάλληλα.

Magnitudes which are in the logos itself should be thus defined; first pros second, and also third pros fourth, precisely when that which goes into the first and the third position are dualed multiple forms; and with reference to that which goes into the second and Fourth positions are dualed as to multiple forms, laying them down by any method of construction of multiple form constructs one to its corresponding one, which could be(reap) an overflow , which could be a dual, or which could be a shortfall by a leephthenta ( a piece taken away, a divisor, a remainder), as set down each by the other.

http://www.perseus.tufts.edu/hopper/morph?l=Auto&la=greek#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=elleiph&la=greek#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=elasson&la=greek#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=Lhfqenta&la=greek#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=ekateron&la=greek#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=poi%3Dos&la=greek&can=poi%3Dos0&prior=o(poi=os#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=poih&la=greek#lexicon

http://www.perseus.tufts.edu/hopper/morph?l=AMA&la=greek#lexicon

I have completed the above translation, and draw attention now to Eudoxus introduction of isos and isakis. These concepts are around duality .The noun adjective and the verb adverb The whole method is based on making duals by recasting magnitudes into multipleforms of each other.

http://www.perseus.tufts.edu/hopper/morph?l=Isakis&la=greek#lexicon

The reader would be advised to note, that although Eudoxus defines third and fourth position, the text so far is speaking relative to first and second position. It defines the process of dulling a logos. When third and fourth position are involved, the third magnitude has to already be a Fula of the first and so also for the fourth.. Eudoxus says no more about this case here, leaving that for the next text and beyond.

Normally a greater would be made into a multiple form of a lesser. However, this cannot always be done precisely. Consequently higher multiple form constructs are used to find a duality. Thus a certain skesis of one magnitude is constructed by making a pattern with that magnitude and then the other magnitude is placed repeatedly on top until a fit is found. If no fit is found, a new larger pattern or Sxesis is formed and the process repeated. The coal is a duality between the 2 multiple forms. This is the method of uperexee.

Eudoxus mentions 2 other methods: issue(dual) an elleipee( lessened, made less). The uperexee is an extensive method, in which bigger and bigger multiple forms are used, made by laying down duals of the initial magnitudes. In elleipee the idea is to use smaller and smaller parts of the original magnitudes. This is an intensive method, and is pragmatically more time consuming, because to determine the parts you have to perform Katametresee with them, which is essentially the method of uperexee but with reducing magnitudes.

However, theoretically this is not allowed to be a problem, and in fact elleipee is the basis of Euclids algorithm, so called, in book 7.

In practice, elleipee would be done by using tables of standard multiple forms.

Where do such tables come from? The method of Isa or dual. At first this does not seem a likely candidate as a method! But in fact, as Eudoxus points out, the goal of both the other methods is duality of multiple form. So Eudoxus asks the question: what does duality of multiple form look like?

It turns out that it looks like any dual magnitude shattered into any multiple form whatsoever!

Take 2 A4 sheets of paper, fold one 2 times , and the other one 11 times. The duality is not altered, even though there are pieces of the sheets which will never fit each other precisely! Yet I have an incontestable duality. If I can find a pattern of these fragments or fractions of the one part of the dual which exactly match the Sxesis or pattern of fragments or fractions of the second part of the dual, then I can experience duality, and know how these fragmentary magnitudes relate to each other.

Now I have introduced the notion of counting. All the while I have been careful not to place numbers where they do not exist. It is better to not use numbers at all, at least not until one realises what they are and how they arise. The notion of named quantities does not arise until book 7 . Even in book 7 we are not dealing with any Renaissance notion of number. Posis, that is formed magnitude or quantity cut out of a magnitude, and magnitude itself are the only fundamental conceptions.

So the method of isa leads us to the isaxis of multiple forms by uperexee or elleipee.

How does this relate to Sqrt 2?

There is no problem because we do not use numbers or numerals. The magnitude is constructive, and therefore the methods of Eudoxus can be applied. In addition, the triangular form that constructs this magnitude of length is itself a magnitude of area. Both length and area are again Renaissance concepts not found in the Stoikeia. The experiences in the Stoikeia are introductory to a philosophy of Forms and ideas as experiences.

Eudoxus definition of a first and second position in a logos, and subsequently a third and fourth position in a special case where the first and the third are in fact duals, and the second and the fourth are duals is the framework he needs to work out ( dunatai) all sorts of relationships between magnitudes. He later defines the Logos Analogos framework by which he compares, contrasts and relates all manner of magnitudes, and from which the Platonic schools derived their logic and reasoning.

We know, from Parmenides, this form of reasoning was not new, but the framework as laid out by Eudoxus is the best and fullest exposition of it that we urgently know.

ϛʹ. Τὰ δὲ τὸν αὐτὸν ἔχοντα λόγον μεγέθη ἀνάλογον καλείσθω.

Which because of magnitudes having the logos definition itself, should be called Analogos!

But that is another post!