As far as I know, the terms "conjugate" and adjugate were termed by Bombelli, in the context of the Italian competition to find the general cube root. The word root was used but it is translated from radix, the original Al Khwarzimi rhetoric meant something akin to rod, that is a measuring stick. The idea was to find a measure that made any shape equal to a square, and in this instance a cube .
The solutions to these rhetorical propositions were methods of thought based on a combination of Indian and Greek notions of duality of form and cipher. This sometimes intense rhetorical style and process was aided by terms, terminology and marks and notation, to which the audiences attention were constantly and consistently drawn. This style came to be known as Al Jibra, the brain or mind twist!
Bombelli was the first to produce an algebraic text book introducing these methods to students in the west, as well as his own innovative thoughts and notation. Thus, in relating his take on the work of Tartaglia and Cardano he introduced the first rules for the strange formulae or rather binomials or tri nomials called the impossible resolutions, or roots. Later, Descartes was to call them both impossible and imaginary. In introducing these rules, which came about as a consequence of his own "secret" method of solution, using geometrical constructions with a method called neusis. In moving hs set square into positions on the drawings he noted there were always 2 solutions. This attribute he called "Conjugare". Thus conjugate refers to that binary relationship between two objects in and of space where they exist as a consequence of each other. You can't have one without the other.
However, Bombelli also noted that the formulary, or terminology was constructed from distinct parts. Thus the conjugate formulae consisted of parts and these parts he called "Adjugare", that is these parts, within the context had a separate existence, but combined to form the conjugate.
Thus the synthetic relationship between conjugate and adjugate was formulated in the context of polynomial descriptions of a form. These ppolynomial descriptions were literally polynomial models of the form, but some conventional notation was neede to give the algebraic form we have today. The heroically model was full and explanatory, and derived from Eulidean propositions and definitions. It may therefore be fairly said that the onions of addition and multiplication were derived in part from Bombellis formulation of the rules of such for directed quantities.
Of course Brahmagupta preceded him by about roughly a millennia, but the rhetorical style was undergoing a radical ymbolic hange in Bombellis era, and this symbolic change continued in fits and starts up to the time of the Prussian Empire in the early 19th century when it seemed all intellectual developments were being concentrated in the germanic people's in competition with France, sain, Italy and Great Britain.
We go, therefore, as Hamilton clearly shows, from the notion of form/idea and it's analysis into pollapleisios or multipleform straight to the notion of gnomic " multiples" and thus multiplication in modern translations of Euclid, whereas Euclid goes from form to unit, to standard unit descriptions of form, to recognition of a multipleform standard description of a larger form by a smaller or lesser form, to factorisation of a form by its multiple form, and then various special syntheses of these factors, each given a distinguishing name, and certain special multipleforms identified as a spanning group called proto arithmoi.
From these considerations, and prior ones in book 6 Euclid/Eudoxus describes the algorithm for commensurability, by which common factors may be derived for any two compared forms(ratios), and upon which the principle of exhaustion is based, leading directly to the fundamentals of Calculus.
The notion of conjugacy precedes the factor algorithm, just as the notion of commensurability is coequal with the notion of factor. When I focus on a region, some internal and unconscious process factors Shunya to a commensurable scale. This means that the focus region is internally held as a conjugate factor of the entire universe. The same holds true for any form selected as a focus region, and for focus regions within forms. Thus inherently I factorise space, I consider Shunya as commensurable in some undefined way.
Conjugation is a powerful psychological limiting concept on all of my experience of reality. It automatically makes me assume that there are boundaries to Shunya, which I can somehow apprehend. In fact I can't apprehend it except by an undefined concept called the whole, or entirety. To actually get a handle on this concept I have to live a little. Thus it is an inductive experiential and empirically based iterative process. The notion can only be approached by iterative experiential steps of increasing magnitude. The consequence of this is the onion layer effect. At each iteration I push back the " boundary of extent" and shrink the lower boundary of intensiveness .
Therefore, I do not have a foundational basis for any quantity of magnitude beyond my own will and assent. The logical consequence of that is that all things may be analysed indefinitely to a "point " at which I choose to cease. From this point however, all quantities may nascently arise! It is therefore by cultural consensus that my experience of possibility is shaped, for indeed all things are possible!
The psychological impact of this situation coupled with the social dynamic makes for interesting possibilities!
The Euclidean notion of factor is that of a part of a whole. Of course metas can just be translated as part, but the notion of me ras is involved in the context of Elasoon and Meizonoon, least and greatest parts. While it is natural to compare it with fraction, it is in fact an older idea than fraction . The metas is a conjugate part of a whole. Thus a whole form is broken, shattered or otherwised divided into meroi . The very first metros conjugates the whole. This conjugat with the remaining part are adjugates of the whole. One of the things we naturally do is proportion. We compare the meroi . Thus the very first meroi are treated as factors of the whole.
Within this notion of metros/ factor/ conjugate/ adjugate, is the notion of comparative scale or commensurability. Thus, if the part can be shown to to measure the remaining part, that is exactly cover the remaining part in a process of Katameetresei, that is measuring by sugkeimai, gathering copies of the part to precisely cover by contiguous boundaries, then the part is a cofactors of the whole with the remaining part. This relationship is recorded by two separate notations. One involving the 2 parts adjugated into the whole, this is aggregation, the other involving the measuring part as a metron with the external count. Two notations therefore represent the whole.
We loose sight of this conjugation by confusing the notation using numerals. The adjugation of the parts is the foundation of the notion of synthesis by aggregation , the conjugation of the parts is the notion of the whole being formed by a multiple of a part. Since Euclid in book 7 is introducing the factorisation algorithm, and Richmond are securely based on this algorithm as standard pollapleisiioi it is clear that adjugate synthesis is of secondary or less theoretical interest. Again, this supports a natural tendency to compare in order to find co factors of a form, and to comply with an inherent need(psychological) for commensurability.
Conjugacy is therefore solidly founded on inherent tendencies and formally on the commensurability or factor algorithm. This means that designating a form by its cofactors is a mixture of notations representing the factor and count.
The representation of a form by cofactors is a symmetrical piece of rhetoric. If we have a factor that is counted into the whole , we have a whole that is counted by that factor. If we have a factor that is counted into the whole we can factor that factor and use the new lesser factor to count into the whole. This now gives not 2 factors but 3 factors and 3 counts. Pretty soon the counts become confuse able with factors of the whole, when in fact they remain distinct as multiple forms of the whole.
When reading Euclid this distinction is clear, but when reading a modern translation this distinction is blurred. A conjugate of a form therefore consists of 2 parts.these parts are adjugates. When an adjugate is a factor of the other conjugate/ conjugate then it can represent the whole as a multiple form in which it is notated with its count rhetorically.. This notation is called factorisation, but it is also derived from a conjugation of the form into these parts as factors.
There is a confusion of terms here, and it is often deliberate. But the way to crystallise the 2 concepts of conjugate and adjugate, with the related notions of metros or part/ factor was to provide a flexible tool. This tool was a piece of string!
The history of rope making is interesting, but suffice it to say here, that it represents one of the key elements in the myths about the creation of the universe. The inchoate mass that is primordial matter is like a ball of wool, and from it a thread is pulled and twisted into Hekate, the flowing god Rhea, the writhing chaos monster.
From this thread other threads are segmented off into the Henads. At each iteration more segments are branched off until a fine web connects every element of the universe to each other flowing Back to Hekate like roots to the trunk, and from Hekate into the inchoate mass, the Monad.
Similarly, from Shunya a division proceeds until as in the Taijitu of Lhao Tsu we end up through the 5 elements with the Myriad of things. And from the myriad of things all returns to the whole, the Pole, Shunya.
So we see the mythology like a rope coils all things back into a single cord. This single cord therefore is representative of an entire universe.
Euclid st arts however with the point of all possibilities. From it stretches a chord, and by revolving the cord sweeps out a sphere of influence. Every part of that sphere can have arbitrarily many points with their own cords and their own spheres. It is when those spheres and cords tangle that we get the beginnings of the circular planes and the cones that define them, and within those planes the good lines thatj lie within them these good lines in the planes of entangled spheres are where Euclid starts.. These cords are like the lines used to make the ropes that anchored ships, that twined to make hunting nets, that spun to make fine threads for weaving garments.
Wherever you look string as a woven or twisted together synthesis has an analogy in our experience.
Euclid makes one essential definition in book 2. The thread used to mark out the perimeter of a rectilinear form, with parallel sides would be represented by a segmented thread.mthis was after he introduced the symbols or schematic of proportion called the gnomon. What he demonstrates in book 2 is how the segmented string can be used to analyse the properties of these proportions in rectilinear form, particularly in conjunction with the gnomon. It was absolutely crucial that everything was a type of gnomon, because the parallel properties preserved proportions.
In spheres it was found that radiating cords cut out sectors that scaled forms. The proportions changed, but the comparisons using the factors remained constant. Spheres and their properties were fundamental to making proportionate relations in a fractal reality. Parallel surfaces crucial for preserving certain quantities, even if the parallels were straight or "good" and carried the quantities away out of view(aphoria).
The notational device, or terminology had to be simple to capture all these complex relations and the good line/ cord suggested itself as a suitable candidate for many reasons, not the least being the mysterious wonders of the cast shadow!
There is a satisfying analogy between a cord stretched taut and a good drawn line, both for artist and sculptor. There is the satisfaction that a cord can then be used to bind, tie an object; to weave a net to capture it or a cloth or sheet to cover it.b
Such a utility and analogy makes a string an adept symbol of creative interaction withn Shunya.
I notice, as I meditate on this layers of confusion in my own mind at how the ideas interface with the notation or terminology we use today, but I awake to find a clear set of statements based on the notion of a finite cord
Let a finite cord be labelled U
Conj(U) is labelled eM for elassoon and Meizonoon , lesser part and greater part(metros). These are factors of U
e and M are just placed next to each other as conjugates, a binary factorisation of U
They are labels that mimic the partitioning of U into factors, and focussing on one conjugates U by the other.
But now focusing on U makes the labels and their respective parts / factors adjugates. Now I can mentally place an action between them, that is sugkeimai, gathering them together to cover U. The action is aggregation of the factors .
Now I can perform another action that is Katameetresei, a single act of placing one factor on the other. This single act is the root of comparison and also the first step in a sequence of actions called measuring.
This simple act enables me to determine which factor is to be labelled e for the lesser, and which M for the greater. If it should happen that they are the same then I would label them I for isos meaning dual and
Conj(U) would be I I
We now have the comparison which reveals dual factors or different factors. From here we divide our explanatory notation into types that go after the dual conjugation and do so through the factor algorithm, and types that simply accept the factors as adjugates and aggregates them. The action of katametresei introduces ratio comparison as apriori to any notion of a multiplication process.
Ratio comparison is a form of division, and this division is comparison of factors. The factor algorithm is the formal definition of the proceeses we gaily call subtraction, division and multiplication today.
There is one other process. Iteration.
Book 7 is a work that reiterates and refines its terminology at each reading. Thus it's definitions are tautological. We experience tht sensation of shifting referents not because the language is general, but because it is Adjectival. It is adjectival to the factor algorithm. The whole of book 7 is a meditation on the factor algorithm, describing terms that initiate the algorithm, and then define themselves as adjectives of the algorithm, some are clearly adverbs of the algorithms process. We understand the terms by iterating through the algorithm.
We now have essentially 2 approaches involving the factors of U, and the simplest binary factorisation of U, the simplest multipleform may now be fractally divided into smaller factors,
This is an intensive analysis of U. We may now use this structure to extend U to a larger piece of string!
How long is a piece of string?