Book 5 is a Game changer!
The following videos explain the background to why!
A quick overview by Norman of Book 5
But Norman misses the connection between books 5 and 6, the Eudoxian theory of Proportions or rather Analogia! Eudoxus lays out the linea; basis for Analogous thinking,
Then books 7 to 10 recast the ideas of books 1 to 4 in terms of the Arithmoi, the general notion of mosaic grids or nets made from any of the ideas discussed in books 1 to 6!
Euclids Stoikeia thus represent n introductory course to Platos theory of Ideas/Forms in Books 1 to 6 finishing the first year of a 2 year course in introductory Pythagorean principles. The second year begins with Book7 and introduces the Gematria of the Pythagorean schools, the Numerology, the combinatorial philosophy of the Pythagoreans. This continues through to book 10.
Finally books 11 to 13 form a more philosophical discussion of the real dynamic world, and how the preceding principles in the 2 year course apply to the older spherical and circular schemes of proportions and spaciometry. Basing this discussion or dicourse on the Stereos, that is shadow casting forms, the projective geometrical I deas of Eudoxus in book 6 , come to the fore in the projection of the circle and the sphere.
Within the circle regular polygons wee constructed i book 4 , these were projections from the regular solids in book 13 onto the plane, the Epiphaneia, that is the light catching surface! The epipedos or Mosaic surface was explored in book 7, and what connects them all are the point and the Good line!
The notion of a good line as a constructed dual pointed line is fundamental to the connected structure of the Stoikeia. The gramme is officially defined as that greaness of magnitude that has no plates! this is literall a line drawn by a sharp implement, a cut into a surface!
Plates derives from the root plassos, which, briefly is the concept of plasticity as in clay, plastacene or some substamce like putty. When squeezed this substance yields and spreads out . When worked its form transforms continually, so ones initial apprehension of its magnitude or greatness changes. We are lead into more and more descriptive notions of the distinctions appearing before our eyes! This behaviour is summed up in plates, which is consequently more than the simple translation of width, just as mekos is more than the simple translation of length.
Mekos, plates and bathos are the three descriptive notions for a solid form. They do not mean length breadth and height of length width and depth as some have it. That is to say they refer to the conception from which we draw these 3 dimensions as instances. Mekos alone is sufficient to define length and girth, and our immediate and natural response to any form in the instant of us perceiving it. the other 2 arise as a consequnce of empirically interacting with the form. Thus plates , as explained above expresses this transforming girth or spread of a form as it is worked. Bathos arises from the experience of being in the form . It is easy to see that bath carries this idea into our everyday use very well. The scal e of the bath is hinted at by our terminology. Tus a bird bath, a swim bath are all baths of different sizes and ratios to one another. such ratios when precisely copied or scaled are called analogues. That is he forms are said to be in analogue to each other ,
Why avoid the word proportion? because the notion of analogue is far more powerful and useful. For example few realise that Logarithms are Analogues > By using the principles of Eudoxus in book 5 and 6, and even the basic terminology of Eudoxus Napier clarified the Analogue relationship between the sines and the arcs of the sectors of those sine ratios written as fractions , in long hand to at least ten places of digits. The 2 lengths napier refers to are the length of the sine on the perpendicular and the length of the arc on the circle. This length was obtained by rolling the circle on a flat plane without slipping.
Eudoxus defines proportionality using the good line. this good line is obtainable only by use of dual points marked off from the centres of 2 circles as ppoint of intersection of the perimeters. This is why they are called dual points because 2 circles are required to determine them. They cannot be place freehand, at least not unless you are as skilled as Michaelangelo! As Newton points out , the underpinning mechanics of this fact reveals that geometry is derived from and improves upon in recursive relation, the practices of mechanics!
This held clearly in mind, the length of the circumference of a circle was no real mechanical problem. The word kuklos means disc, and the disc is simply but carfully rolled on the pane to gramme or draw out its mark which remains fixed, not transformed by spread or depth. . One revolution enables the mechanic to represent the perimeter by a straight line segment , marked of in the epiphaneai. Eudoxus, and all greek pilosophy therefore deals with curvature pragmatically. They straighten it out by some mechanical means.
The issue of proportions that is analogies depends therefore on comparing the same things! thus the magnitude of a line, even though it is a straight line cannot be assumed!. The straight lie drawn by rolling a circle is a different magnitude to a straight line drawn against the side of a cube!. A straight line drawn with a mark cut across it is not the same as a straight line not segmented, because the cut means that the line is in fact a rectilinear form!(book 2)
however, what Eudoxus taught was hat by reducing all forms to points and lines, certain methods that were common to all forms could be deduced and studied (compared, analogised. In doing this, it becomes vital that the kind of things compared is noted. thus if we analyse circumferences as straight line representations, then any results strictly apply to perimeters of circles and not necessarily generally to any lineal magnitude.
This does not stop us from comparing differnt kinds of magnitudes, and in fact this is what Napier did in formulating his Logos: Arithmos, or logarithms. The ratios in the sines are compared with the ratios in the arc lengths that correspond. Thus the logarithms are an analogue system. However, because they are of different kinds we cannot use the equal sign or the dual concept. What we use is the Analogue concept of Eudoxus.
The reiteration of these principles from book 7 onwards is to teach the student that what applies to lines can be generalised through using the 2 gnomons, the parallelogrammic gnomon and the curved Gnomon in a circle used in the proof of Thales theorem. The parallel lines are crucial in preserving or transforming a shape. The rotation is crucial in preserving any shape but changing its orientation, and the projection from a point is crucial in scaling and transforming a shape.
With these 3 projections in or onto a plane : the circular, the parallel, and the perspective plus the analogous thinking of Eudoxus the dynamic reality around the student could be apprehended, sudied and utilised to make wise judgement as to the appropriate Kairos. After all this is what an Astrologer is expected to be able to do, to judge times and seasons for the opportune time to engage in any action. Thus Kairos is the fullest extent of the astrologers art, ot for rhetoric, but for pragmatic and wise livng in a dynamic universe.