The circle theorem in book 3 of the Stoikeioon, which deals with intersecting chords and the rectangles made from their segments is the basis of factorisation. It is also the classic description of logos Analogos.
By means of proposition VI.16, the statement, "the rectangle AE by EC equals the rectangle DE by EB," may be converted into one about ratios, namely, "AE : EB = DE : EC." This proposition is not used in the rest of the Elements.
The basis of book 3 is suggestive of extensive work done bynApollonius of Perga.mit is known that Apollonius wrote a critique of Euclid's Stoikeia, and the manuscripts we hold may be the redaction envisaged by Apollonius. On ny case , book 3 is a general introduction to the ircl, Thales theorem and Hamonic logoi and logos analogos framework rhetoric fomalised in book 5.
Several types of incommensurables are introduced.
The first introduction of similar sectors is introduced based on dual rotations.
A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them. Here a sector BAC is illustrated. The angle BAC encloses the sector. Note that the remainder of the circle would not be considered a sector by Euclid since the angle at the center would be greater than two right rotations.
The things to note here is that the words figure , angle and circumference are technical terms for a pattern of lines that intersect or meet, a rotationnandna piece of the perimeter of a circle that is between two points, ie an arc.
Having defined an arc and a sector and a rotation as congruent similar segments renthenndefined
While presented as a definition of the notion of similar it relies upon the fundamental similarity between all circles and the similarity of all sectors with the same rotation. This requires the experience of concentric circles to demonstrate this. the demonstration is apparently so immediate that it is given as a definition for similarly arranged segments. The amazing demonstration that any point on this segments arc storms an identical rotation of straight lines joined to the base of the segment or the larder arc on which it sits is left as a demonstration(3:21)
While a straight line, and a pair of parallel straight line segments, is defined by construction of dual points in a circular disc plane, constructed by mutual ly intersecting spheres, the direct relation of parallel straight lines to parallel circular lines as concentric circles is a striking demonstration of these notions.
Now, without Katametresee, that is without measurement or Metron or counting certain dualities are demonstrated as contingent on spaciometric attributes alone. The notion of multiple form is a metric notion, and this is sinimilarity has to be defined in terms of duality.
Katameetresee is in fact repeated application of duality, that is recursive or iterative process of dialling to determine where dealing ceases. Thus dealing bynitting things on topmofbeach other as yo be fully explored and understood prior to extension to the concept of measurement.
We can see that segmenting a straight line in the context of a circle is in fact factorisation of the line into fundamental magnitudes with an inherent logos analogos framework. This framework underpins all models of Phusis or physicsvasba spaciometric reality. We believe we can measure space in and by these logos analogos frameworks derived in the context of the sphere.
By this proposition we can define the extreme and mean magnitudes as we go around the perimeter of the circle . The extremes are the first and last nd the means are the middle 2, in fact a maximum and minimum is demonstrated in this book..
While 2 dual lines are demonstrated either side of the least segment, it is the arbitrary logos analogos structure or framework that is the goal of this book. In this context duality of rectangles and squares are demonstrated. Thus in the context of a circle we can dual a rectangle with its corresponding square, or dual 2 rectangles in logos analogos relationship..
It is worth emphasising that this demonstration is not defined in terms of multiple forms.,thus it is demonstrated as inherently dual. On that basis Eudoxus constructs hid theory of logos analogos, knowing that it will be inherently dual no matter what Metron is chosen or required.. Also commensurability is not an issue where this inherent duality exists. In every sense, these magnitudes would be considered protos Arithmos, where protos and deuterons are the sequential positions in a logos.. This would make a logos a segmented line magnitude, or a pair of similar tpe magnitudes placed in line to be compared.
How do we get from this construction to multiplication?
We do not.n
In every case we get to multiple forms by factorisation, that is a division process. It is only after the multiple form has been established by this division process called factorisation, that dealing and counting allows us to develop a notion called a multiplication bond, usually expressed in table form.
This table form is in point of fact a tabulation of factors of related magnitudes by this division method. By memorising these factors we create the constructed notion of multiplication.
For most things the multiplication construct is a speedy way to arrive at comparisons, but it is next to useless for theoretical constructions of Algebras or spaciometric models. For that we need the notion of Logos Analogos and the attributes of magnitudes within a spherical context.
Considering the demonstration we can see that commutativity is not a consequence of duality alone.mit requires a fixity of terminology to form, that is a fixity of correspondence. Then given A line a cut mutually by another line B into segments a1,a2 and b1, b2, then a1x2 is dual with b1 b2 where this form oes not mean multiplication, but the logos arrangement on the lines, by which rectangles may be constructed contained by these 2 segments of the line. In the context of the circle the duality is between the rectangles! And this duality exists irrespective of the order in which the logos for each line is written.
Thus we derive the notion of commutativity, and by extending to the sphere and mixing the logos t 2 different kinds we can extend the principle to associativity..
Thus, what we have come to recognise as multiplication is in fact a rhetorical notation or the logos on a line cut into 2 segments within the circular context. Likewise associativity is an extension to the spherical context.