In book ! Euclid introduces, amongst his common judgement or notions Isos as a concept of duality, and as a chain link for duality. In book 5 however Eudoxus inroduces carefully a more general notion of Duality based on Logos: Analpgos!
Whereas Isos is derived naturally from dividing a magnitude into veins or fibres, the veins being Isos to one another, relatively, logos is a man made division of magnitude, a division made consciously in a context of sequenced forms, specifically within a laid mosaic.. The defined logos is in fact a partition of a foursome. The full concept of logos is only partially intuited if defined in a twosome. The full concept is logos -Analogos. It is a fractal concept and it is a definition of a different type of duality.
In the case of Isos , the prior magnitude is not specifically defined. Two magnitudes are indicated as prior notions, and then these two magnitudes are compared dynamically, often spatially by simply placing on on the other untiil it fits, The neusis involved in fitting is not defined, but generally one rotates wiggles reorients until a match is found. However, this cannot be done for a fixed form, such as a laid down mosaic. Thus Eudoxus addresses the issue formally.
Any motion or rotation must be done prior to laying down in the mosaic. once laid in the mosaic one mys proceed algorithmically to determine duality.
Of course this is not a necessary process for just 2 or 3 magnitudes. But for four it becomes an incresingly efficient way of comparison and distinguishing duality relationships between the four, if they exist.
This algorith makes use of combinatorial concepts and in fact defines combinatorial order and sequence. There is much much more this algorithm captures in its formalism, giving a "rigorous" basis to common concepts of analogy. However, this is extending it beyond its design purpose into general everyday language and rhetoric. It is useful to see how when we try to pin down a referrent or concept rhetorically, in words, it still manages to wriggle out of control!
Isos and Analogos are therefore 2 different ways of determining duality which in fact represent a fractal or rather recursive relationship. Because of this relationship. mathematics so called is actually based on the Analogical form of duakity more than on the Isos form of duality which has a natural home in Spaciometry. The natural home of Analogical duality, because of its rhetorical definition is Algebra.
In book 7 the topic is revisited in terms of the Arithmoi, formal mosaics. These are then used in the 2 dualiies/ When Isos is used as the duality the arithmoi formally derive Arithmetic , through counting the monads, When Analogos is used as duality the Arithmoi formally derive Algera by comparing the like magnitudes separately.
Thus , as Pythagoras claimed, the Arithmoi are prior to both geometry and music or harmonics , a kind of Analogia or algebra, The word harmonic is still used today in relation to ratios and proportions.
Pythagoras, reputedly went on to discover that logs could be distinguished by tone and analogos by a relate tone called an Octave. This alternative way , alternative to the visual mosaic, of apprehending ratio and proportion, logs and analogos has often intuitively informed many mathematical relations and concepts.
Of course a Mosaic is a visual construct, and the natural evaluation of a logos analogos went down the visual route. The Oimi schematoon eugrammaton represent a construction of good lines which are similarly arranged. The similarly configured straight lines reveal the visual evaluation of Logos Analogos. By looking at such constructs in the plane we develop the notion of form.
The first notion of form is with parallel lines or lines that meet.
The second notion of form is rotation of these relative simple forms
The third notion of form is the closure or enclosure of space in the surface and eventually in 3d when the forms are made up from simpler forms in a plane or in space which involve relative rotations in every aspect of the closed or open spaces, and in special cases the parallel lines and parallel surfaces create a spatial regularity or Lattice which is the mosaic itself rotated relative to itself.
Logos Analogos also relates to non straight line magnitudes in precisely the same set of notions, corresponding to that kind of magnitude. So in particular circular lines in concentric or meeting mode ; the relative rotation of these is rich and symmetrical; and the form has its own natural closure that generates curved regions within and without, and particular regularities that scale relative to a centre.
These are just some of the result of similarly configured lines and surfaces.