The notation introduced in Book 2 has had a generally powerful shaping effexpct on algebraic combinatorics, especially through the work of the master Geometers Descartes,De Fermat and Vieta. Wallis made no mean contribution to notational synonymy, matching labelling to the actual physical structures of the object. So Newton had a great convention or consensus of how to label Euclid's forms algebraically to draw on.
However, Euclid's distinctions were not always observed, and factorisation is a case in point. Napier's logarithms observe the distinction in the Euclidean notation , nd is the reason why I have chosen logarithms to represent the process of iteratively conjugating an object, using a factor that is commensurate as a metron.
The notion or idea of combining a whole universe of things into a flowing, lengthening twist of string is a powerful one. It invests twisting threads with magical, symbolic power. It invests a Seemeioon as a indicator with infinite possibility, and a gramme as a type of string with incomprehensible potential, and possibility. It ravels up space and unravels the potential of Shunya. It binds, it bonds, it bounds all relationships in space. It covers and it captures and it emulates all forms.
When I choose a focus region in Shunya I conjugate Shunya into 2 labelled adjugates. Suppose one of those conjugates/adjugates is a piece of string. Why is a focus region that is a piece of string already complex? Because I am already complex! I want to see the whole as combined adjugates, a synthesis, bu t I want to analyse the whole into conjugates, a binary analysis. Then I want to compare and contrast, this factorises the pieces or parts or factors, and begins an algorithmic , iterative process of factorisation, that determines commensurability. In carrying out this process I fractalise the whole iteratively and methodically, and I create or resynthesise the whole as a multiple form of commensurate factors, if possible. In this sense possible means pragmatic, which means able to be completed. What is pragmatic changes with the tools we invent, also through iterative processes. Computers now make many processes that were pragmatically infinite now do-able.
So, my innate iterative conjugation processes fractalise my experience of reality, factorise reality and make reality commensurate with some chosen factor of it as a metron/Monas. And then all these fractal regions become adjugates that I can aggregate into the whole by a creative process of synthesis. The simplest aggregation structure, synthesised by twisting threads together is a piece of string.
String, by it's nature symbolises an unending process of twisting threads into strings which in turn become threads for cords, which in turn become threads fore ropes, and so on, until galaxies are constructed from huge threads of gas, and the whole cosmos from threads of dark matter.
Taking from a piece of string a drawn line, conjugating it to give 2 segments, Euclid defines, in conjunction with the spherical relationships of parallelism the notion of a representation of a rectilinear form. Further, each segment takes on a symbolic role for the factors of this rectilinear form, but also another role as a symbol of a count. Thus the segmented line, used as a symbol of a rectilinear form loses its status as 2 conjugates/adjugates and becomes one conjugate/adjugate with its count. It's count arises from the comparison algorithm, and ultimately the factor algorithm
In making this notational synonym, Euclid provides mnemonic help in a complex process. It is these tricks and observations that aid the fluid mind in safely navigating the complexities of detail. But they also obscure what is actually happening to the uninitiated. When Napier devised his " logos arithmos" it was precisely by doing the same notational trick. He too took a segmented line and let the segment that was the factor stand as the logos, and the other segment stood for the count or arithmos. The complexity of his system was thus notation ally simplified, but the iterative algorithm was as sequential and iterative as the Euclidean factor algorithm. The difference was that the metron did not remain constant, the factor was itself conjugated at each stage, producing smaller and smaller conjugated factors. These factors were pair wise ratios, but any scheme of ratios is known as a proportion, again after Euclid . This scheme of proportions were linked to a greater or increasing count, and this in essence represents the scaling process that we innately perform as we concentrate on smaller regions of focus.
Our scaling is logarithmic, and all scalars are logarithmic. We hide this fact behind the p-adic number system the Indians created out of polynomial forms. What we traditionally call a number, is in fact one unending polynomial to a fixed logarithmic base. As I have just intimated, this logarithmic base is structured by Euclid's segmented tring notation from book 2
Finally one other use of this segmented string notation was derived by Hermann Grassmann in his Ausdehnungslehre. He directly uses segmented string/ lines to represent the radial , dynamic cords of circles, and thus also spheres. These segments became symbols fo vectors in Hamilton's mind, but in Grassmann's they remained true to Euclid, and he always called them Strecken.
Using Euclid's notational device the segmented string, long ith it's parallel process lines that together synthesised an arbitrary parallelogram, Frassmann revealed a facility vailable to the Greeks, but hidden away from the modern mind by the artifices of Descartes. Descartes was not one to particularly give credit to any predecessor, making out that his meditational praxis gave him superior insights. Certainly he gave small credit tp Al Kwharzimi, whose rhetorical algebra underpins all algebra, and no credit to Bombelli whose book on algebra inspired so many engineers and geometers for so long. Rather he set his sights on Vieta, hoping to demonstrate French superiority over the Italian bombast. In so doing he neglected to mention hs teacher Harriot, something Wallis with meticulous research brought to the attention of his students in Cambridge.
Wallis too was a Euclidean scholar, and an enemy of plagiarism, something that survives today in all academic institutions. Thus when Descartes introduced the literal notation for lines and points, it was not of his own invention, neither was the use of ordinate and abscissae, or coordinates. These were well founded Greek ideas but written rhetorically. Descartes was not really the inventor of his Cartesian coordinates, or even a great promoter of them. These refinements belong to De Fermat, Herriot and Wallis, and the general pragmatics of the printing block!
Nevertheless he was credited with it due to hs fame in French intellectual circles, and the importance of his philosophical ideas about space and god.
Wallis determined, that if the coordinate method was going to be useful, then the co ordinate needed to be measured against a fixed axle. That is to say, the general geometer knew well enough that for any line and any point off that line any number of arcs intersecting the line from the point could be drawn, but there was only one perpendicular line that went through that given point. Thus if the position of a moving object relative to that line is RePresented by points, that is its locus is presented as this collection of points, then perpendiculars from each point could be drawn to the line.
These perpendiculars and their relationship to one fixed point called the focus was the way that a particular locus called a parabola were defined. Geometrically this produced a lovely fan like diagram and a couple of defining ratios, related to each of these perpendiculars. Descartes did show thst if the point was represented by 2 labels, x and y then a considerable simplification occured. Wallis showed that if you used a pair of fixed axles, called axes, then this simplification leads to a Characteristic equation.
The difference is profound. What wallis showed is that merely by looking at the form of this characteristic equation you could sketch or imagine what the curve would look like against these axles(axes in later translations). Wallis went on to defoine the conic equations algebraically using his system. Of course he could not claim originality for the axle framework, and neither could Descartes. Nevertheless it was given to Descartes by default, despite Wallis's evidence to the contrary,
In fact we owe the system to Euclid, who in book one lays out all it's principles and in book 2 demonstrated its application. Do not be fooled by scholars who like to claim that Euclid did not have a symbolic presentation of the subject. The symbols were more apt than any yet devised. They were the segments of a string.
The sophistication of Euclids notation escaped students in the west for centuries. Many commentators on Euclid puzzled over its meaning. It was not until grassmann that it became clear that it's meaning was hidden in plain sight! The so called Cartesian system was one derived application of it, mathematical notation was another directly derived notion from it and finally grassmann realised thst the line segments (Strecken) were themselves the symbol of what thy represented. They tautologically represented themselves.
The notion of a vector immediately strikes one as a tautology. By changing the name, even the language one can fool ones processing into downplaying the tautology. This is a modern predilection. In Greek , the Greek mind seemed to revel in tautology!
The nearest English equivalent I know are the definitions of Newton in the Principia. The English translations are rich in tautology.
Now I naturally balk at this, because my English language avoids self reflexive language as much as possible. Thus my very language culture makes me uneasy with tautology. However, several European languages are very much inhere in reflexive or tautological language structures. Thus grasping this aspect of Euclid ought to have been self evident to some, but Euclid was always presented in translation in these times, and thus all were denied access to the originals. In fact some Greek versions are translations back from the Latin into the Greek, doubly confounding he notions!
It was not until original Greek texts were found that any intimation of the mental warping hat had occurred was picked up. Nevertheless, the genius of Newton is he saw through the translators treachery to the authors intent, and the same understanding cme to grassmann by his linguistic abilities.
When one reads an original text rather than a translation, the mind grip is quite unique! What such an experience communicates is often extra to the words sounded, or the marks viewed on the page. Literally it can be like tuning the brain to a particular frequency which then pulls these resonant ideas out of the surrounding space!
Grassmann realised at an early stage that not all lines are the same! Some lines are powerful symbols around wich ideas form or in which information is encoded. So it was thst he got the Euclidean sense of string! Strecken were like stretched string, which could bend around points to form segmented lines which were parts of parallelograms. The conjugated string therefore was a powerful notation for a parallelogram. Using any parallelogram. Any point in space could be uniquely referenced.
The beauty of Grassmann's realisation is that it starts with the triangle,mand expands stepwise to the rectilinear figures and beyond, and then, by fixing the rectilinear as unique forms a nother line out of the plane takes him to die Raume, where all things become possible. At each stage grassmann adds another line, another dimension, but creating another facet rather thn some science fictional hyperdimensional space. Grassmann only ever creates faceted objects in der Raum
Using Euclid's segmented string to create nested triangles and parallelograms is how Euclid established methods of drawing parallel lines in any orientation and collections of vertices which lie on a line in any direction. This is fundamental to our understanding of space.
The direct use of nested parallelograms and nested triangles was as a precise reference frame. It is important to note that the segmented string is used only to act as a reference for orientstion( confused often with direction, or rather stacked underneath the notion of direction). Thus the segmented string and its whole paraphenalia of parallel lines takes on a new interpretation: it is a description of the extension of our experience in and interacting with space.
This use of the segmented string to draw out regions in and of space, and to fractalise the segments further by an iterative conjugation and compare and contrast procedure, provided the ancient Greeks with the methods of trigonometry!
It is a moot point if the Greek Hypatchus " invented" it or whether he revealed the temple knowledge of the priests of ancient Harappian and Mesopotamian priests, along with ancient Afro Egyptian civilisations. Nevertheless Thales is credited with the fundamental theorem of trigonometry, which reveals the ancient knowledge of spherical trigonometry.
The reference frame of Euclid's day was more sophisticated than the so called Cartesian coordinates. There is a case for the simplification the axle system brought, but there is also a case against the teachers and pedagogues who buried the sophisticates trigonometric knowledge , and the rhetorical algebraic representation under mounds of turgid, turdlike symbols.
It is apparent that the schematics of solids and the models of segmented line relationships easily generalise to the sphere by iterative processes. And from the sphere one can quickly reduce to many solid forms. This kind of apprehension of forms in space requires a particular use of the segmented string: the string must become multiple from the centre of nested spheres. The string is not only " split" from this centre, combining at the centre to produce this single cord which is the fundamental orientation; but they must also be labelled to distinguish them. The use of these labels is the basis of so called complex valued algebras.