ΣΤΟΙΧΕΙΩΝ for a long long time have been misrepresented. Maybe not misunderstood but certainly mistaken for something they are not.
When one begins to teach, especially children, one first gathers together material to form the content of that teaching, to form the information or DATA one wishes to transmit to the student, the disciple and the learner haphazardly or inconsistently or confusingly, at least not intentionally. The material is worked into some kind of order or sequence. An arrangement is chosen o decided upon that links the material together coherently and cogently and engages the students supposed faculty for learning and memory. At all costs an impression is sought to be made on the mind of the interested audience.
ΣΤΟΙΧΕΙΩΝ therefore is not "elements" as if in reference to elemental Gods, but simply ordered teaching material starting from the easiest notions and progressing to the more complex.
I see therefore a consistent notion of providing for the public good ,and benefiting the general populace within the conception of the ΣΤΟΙΧΕΙΩΝ, an education in the sense of leading out of the darkness of ignorance into the light of knowledge through ordered, sequenced and systematic instruction.
ΣΤΟΙΧΕΙΩΝ should therefore start with the ABC's and the 1 2 3 .., but in greek they are identical. They are identical because the first lesson in greek thought is sequence, order and linear arrangement, consecutivity, and sequentiality.
After this ordinal straight-jacket, then comes letters and numbers.
But Euclid does not start there. He assumes the reader has this knowledge already. Instead he starts at a more profound beginning place: the semeion.
By any standards therefore the sequential teaching material is not designed for kids, but for thinking adults. This is teaching material for adult education classes!.
Adults do not need to know about the simple order of human language culture, they need to know the simple order of the world around them, the patterns of sequences they observe in phusis, the leaves, the birds, the rolling waves, the clouds.
So why start at the semeion? This is the mark of mystery, of divine interaction with the ordinary every day world experience.
The gramme is the human response to the edge or boundary of any form. It is a physically or mentally drawn line, but that lie is surrounded by semeion! The perata is the surrounding extremities, and a gramme is declared formless, without any surrounding edges, and thus surrounded by semeion. It carries the mere notion of greatness, of extension, without the substance of it.
Every edge of every object in the known universe is describable by gramme.. but there is a "good gramme" for every form, and this is a subjective experience. However Euclid chooses the "right" gramme to be one which has a self similar relationship with itself and all the semeion that lie on this self similar curve.
This line was chosen because of invariance, and the first invariant concept is a straight line. Euclid has narrowed down the field to straight lined objects, but only temporarily. He has many other invariant forms to show us, all in good time and i the correct order.
Euclid did not exclude "non euclidean spaces" in fact he relished them, but his course of materials was not heading in the direction of considering every detail but rather in that of precisely those fundamental forms that ere most useful for artisans and engineers.
We will see that ΣΤΟΙΧΕΙΩΝ contain within the materials insights on spacial geometries when they are aggregated, These insights it turns out are only a part of a greater calculus called an algebra.
It is not just an algebra it is a vector algebra. And thus every geometry has an associated vector algera which in turn embeds within it an arithmetic of real , rational and integral multiple of a vector quantity.
The Elements consists of thirteen books. Book 1 outlines the fundamental propositions of plane geometry, includ-
ing the three cases in which triangles are congruent, various theorems involving parallel lines, the theorem regarding
the sum of the angles in a triangle, and the Pythagorean theorem. Book 2 is commonly said to deal with “geometric
algebra”, since most of the theorems contained within it have simple algebraic interpretations. Book 3 investigates
circles and their properties, and includes theorems on tangents and inscribed angles. Book 4 is concerned with reg-
ular polygons inscribed in, and circumscribed around, circles. Book 5 develops the arithmetic theory of proportion.
Book 6 applies the theory of proportion to plane geometry, and contains theorems on similar figures. Book 7 deals
with elementary number theory: e.g., prime numbers, greatest common denominators, etc. Book 8 is concerned with
geometric series. Book 9 contains various applications of results in the previous two books, and includes theorems
on the infinitude of prime numbers, as well as the sum of a geometric series. Book 10 attempts to classify incommen-
surable (i.e., irrational) magnitudes using the so-called “method of exhaustion”, an ancient precursor to integration.
Book 11 deals with the fundamental propositions of three-dimensional geometry. Book 12 calculates the relative
volumes of cones, pyramids, cylinders, and spheres using the method of exhaustion. Finally, Book 13 investigates the
five so-called Platonic solids.
In Euclids teaching material, it is clear that he introduced the aeithmoi some way into the course. This is clearly because the arithmoi as a conception clearly required the preceding 6 books to form a basis of comprehension, It is customary to say as easy as ABC, or 1,2,3… .However clearly Euclid did not think that arithmoi were in that category. One has to dissociate the concept of number from arithmoi. They are nt numbers, but special geometrical forms and relationships.
Formally Euclid defines Arithmoi in terms of Monas. Quite blatantly then this makes the content of the first 6 books a working definition of the concept of Monas.
however this is not a final definition of Monas, but rather an inductive definition. "Later Dudes" like Appolonius and Archimedes etc extneded the figures and relationships within Monas. Euclid himself , having explored the concept of monas in a Surface, any surface i might add then goes on to explore the concept in the solid or 3 dimensional world. I might also add, that by this stage in the books he had already introduced the reader to multiple dimensions measured with multiple parameters.
The fractal nature of space, and phusis is what has been hitherto sidelined. Thus clearly when Euclid explains similarity and shortens his general demonstrations because of it , it is because self similarity is so evident that it needs only to be pointed out. However, in terms of monas, each similar figure is an entirely new monas. Thus we patently have not one dimension but an infinite relationship between dimensions and a taxonomy of different dimensions, The categorisation of these relaionships and differences is the subject matter of the ΣΤΟΙΧΕΙΩΝ. Euclid provides a Taxonomy, an ontology, a categorisation, and a course of logical deductive and inductive thinking in "Kairos" or "Logos" that is a proportional or ratioed manner, that is a considered or rational approach.
Plato extended the notion of Monas, by exploring other forms and relationships, but it was Euclid's Corse structure that he most admired, and it tis that course structure that underpins the entire western Academia.
I have to draw attention to the Arabic influence . Greek thought, or Hellenization is a worlwide phenomenon.But its most powerful influence came from the Arabic refotmation. The greek influence had spread as far as India and the borders of china where apparently it was resisted by land and cultural defenses. However. trade between china and India opened the door to a commercial influence that was profound on Eastern Asia.
The Chinese do not like to admit this much, but their contact with the west on the borders of China is documented in skeletal remains, and the legendary progenitors of China have an interesting origin in the west. However, it is clear that India had the biggest influence on Chinese Astro;ogy and astronomy, and that is a pure greek stream scented by indian perfumes.
However the Arabs gathered together all the knowledge i the world and produced a Renaissance of wisdom and learning. In so doing they amalgamated all the different sreams into what they exprienced as the most useful, and thus the Indian computational development of the Arithmoi, and their philosophy of the monas and the semeion was interwoven into the Euclidean by Al Khwarzim in a book he called Al Jibr. It was ahomage to Indian Arithmoi development, the vedic sutras that extended the subtle relationships of these geometric forms, and intoduced a modulus or modulo structure to accounting that made the ciphers from 0 to 9 have a limited, boring but eminently practical use. Calculationwas divorced from philosophy and metaphysics; gematria did not need to be considered when accounting: lucky or unlucky combinations did not need to be avoided or sought,
This is not to say that lucky numbers went away! The reason why "negative" numbers were so resisted is because Brahmagupta called them misfortunate numbers! WE have never quite recovered from that term, just as we have never quite shaken off the prejudice against "imaginary" numbers.
What Al Khwarzim did was to insert another field boundary within Euclid's ΣΤΟΙΧΕΙΩΝ, it came to be known as Algebra, and later as Tensor algebra,. It is only just being recognised as Geometric Algebra, and therefore as something that was wholly within the conception of Euclid as he devised his Course.
The man who has had the biggest influence on the conception of the Monas is BEnoit Mandelbrot. When he dre attention to and named Fractal geometry he released the power that is pent up within the Euclidean teaching materials. The man who is Euclid born for our age is Hermann Gunther Grassman, but without Hamilton running before him and clearing the path like john the Baptist, we would never have appreciated that.