Is it possible that our ancestors have missed upon the capital principle of Euclidean geometry, “adplicatio”, the application or overlapping of the areas (in Greek “paraboli” – to divide; to split), such an important concept, which they attributed to the so called conical lines, meaning the hyperbole, the parable or the ellipse? Could it be, that in that moment the “rupture” was produced, and the interest for the ancient knowledge brought by Pythagoras from Egypt to have disappeared, the attention shifting primarily to analogue operations of sectioning the so called Appolonius from Perge cones? It was already known that this method to approach geometry was practiced by the Egyptian priests. After long explorations and tests, I came to the conclusion that the application of areas is the capital principle of Euclidean geometry. The geometry respects and follows the fundamentalist dualist theory about the world from ancient Egypt – all force or phenomenon should only be represented and considered with its’ inseparable pair. If we think like that the Egyptians had square fields as the starting base for their works, the shape that perfectly covers the area, we can imagine that this field also contained its’ unseen and inseparable pair. The Pythagoras confraternity, under the guidance of the master from Samos, continued to practice and study geometry, following this fundamental principle, and with a good reason. This principle, however simple in nature, represents the biggest secret of Pythagoreans and mystery of Euclidean geometry. Both the principle of areas application and the dots system resulting from it, along with the vision of the holistic geometry of the One, of the ‘whole’, must have constituted three grand golden rules, which must have been followed to respect the norms of Euclidean geometry.
On the other side, the essence of Euclid’s 13 books, known as “Elements”, is actually found in his fifth book, written by Eudoxus, one of Plato’s students. The book talks about the Pythagorean Arithmology and the theory of proportions, the latter being shared, as it is known, to geometry; arithmology; music, meaning to the Math science. The remaining book could be serving as a simple “aide memoire”, an instrument to help him in remaking the graphic shapes of proportions generation. The name itself ‘Elements’, of Euclid’s books was supposed to logically send us towards the idea of a unitary, holistic vision of geometry, vision which unfortunately remains unknown. I assume that our ancestors gained knowledge about the existence of an integralist, unitary, holistic geometry, and by this I am referring to the members of the Pythagorean confraternity, whom under his guidance, were thinking and practicing it. The inheritance of Euclid’s Elements in the form that is known to us, has apparently not been enough for the occidental culture, to remake or find a holistic shape of this exact science, not to mention the fact that it did not keep its’ graphic shapes from antiquity for exemplification. Rightfully, we are sympathetic to Descartes discontent with our ancestors, who did not send to us any method to reveal, to invent geometry, but only one of exposure, him, who wished to fulfill his youth dream through finding that universal science, expressed using the phrase of “Mathesis Universalis”.
The principle of geometric base construction, as well as that of application of areas, gave a whole new understanding to the search for a solution to the never-ending issue regarding the graphic overlapping of the theory of proportions. I am of the opinion that approaching geometry from a holistic perspective will fully resolve this controversial problem lasting for centuries, while it also identifies a true ‘Key of Proportions’.
The graphic method used to construct the geometric system for generating proportions, as well as its’ means of manipulation is relatively simple. From a mathematical perspective, both the system and the geometrical constructions, perfectly respond to any control method, while the generation of most of the geometrical elements, and that of the most important geometrical issues known in the Greek classical geometry will simply be owned to the intrinsic graphic construction. These elements will co-work within the system, in a truly ideal manner.
“Knowledge can only include relations and structures, the number and not the substance, represent the only, eternal reality” quotes Nicomach of Gerasa, a very well known Pythagorean follower. Therefore, any geometrical diagram is generated by a ratio; a number; a relation or structure, on a mandatory basis. Each element will imprint a unique and specific character to the diagram. The structure of each geometrical diagram, either representing an amplitude, a multitude, a combination, a flux or a suite of monads, will highlight, as it grown or decreases, not only the analogy or symmetry of shapes (in the ancient meaning of the word), but also the geometrical spots (places) or invariables, meaning those curves, which we call circle; ellipses; parables; cisoid; logarithmic spirals etc., which according to d’Arcy Thomson peg the road towards knowledge. I find it interesting that these invariables or geometrical spots will actually stand out, will truly be generated by growing structures, composed of successive homothetic parts and rightfully placed, either through unifying the corresponding points in the progression, or through connecting the points generated by the interlacing diagonales of the rectangles, which build and complete the whole system. These invariables, on a graphic bassis, will act in accordance with the laws of the number, and will continuously imprint this law throughout the whole system, like true milestones of knowledge. This is possible, be it through the crumbling or sectioning the amplitude or amplifying the generation of multitude indefinitely, or through the generation of new geometrical mediums ad libitum, and the maintenance of strict order within the system. Each ratio or raport will highlight these invariable, which, outside of the circle, though always a circle, will respond to the principle “the same or another”. As we shall see, these invariable will give solution to the grand Delian problem, known as ‘Doubling the Cube’, an issue which over the centuries, has remained one of actuality and which should have been solved, as is stated by Hippocrates of Chios, through the intercalation between two values, of two geometrical moieties. This fact is possible precisely because of the specificity of the holistic system, the information being accumulated in a multitude of continuous relations, which to us seem to correspond to an orderly kinetic movement. Perhaps this is the reason why many of our ancestors found that the solution to the doubling of the cube, was to find the two geometrical moieties between the two values, throughout a sort of instantaneous movement of multiple curves. It for this reason that multiple types of compasses were built, to be able to draw these invariables, constructions which, of course, are contrary to the basic principle of Euclidean geometry, meaning drawing only by using the ruler and the compass, principle so dearly held by Plato also. From our human understanding point of view to understanding a phenomenon that has to do with the world of continuum, we are tempted to see in this holistic geometry natural movements, analogous to the world of the sensitive. We shall try not to enter the trap of this vision and understand geometry as something that exists and not as becoming, so as to remain to the idea that it is truly an ‘unmoved continuum’, just as it was transmitted by Boethius.
“Volens nolens”, we have reached using simple geometrical constructions, which followed the principle of application or overlapping of areas, for the four divisions in the mathematical sciences, other known as the quadrivium, arithmetic and music on one side, geometry and astronomy, on the other side. To the first class, one may attribute the idea of ‘how many’, while to the second that of ‘how much’. From the first class I elected the arithmetic or pythagorean arithmology, that which easily grew up from the “Egyptian Field” of squares, and only within reduced measures I approached the musical proportions. From the second class I chose only the geometry of continuous and unmoved quantities, the astronomy not being part of the subject treated in this study.
This piece of writing will treat upon the main themes within the following order:
- The detailed presentation of the constructions based upon geometrical principles, which originated in the millenary Egyptian tradition, and which were followed by the Pythagoreans, specifically the principle of the application of areas (parable), an absolutely necessary operation to logically obtain proportions, as well as corresponding geometrical shapes. In the first part, I shall discuss upon the generally discret geometry obtained using a field of squares, starting from the idea that this is the eruption place of the entire Pythagorean arithmology. These constructions, as simple as they may be, will serve us greatly in understanding how to manipulate the generating proportions system. This system is by excellence one shaped out of dots, and we can say like Pacioli did, that these dots instead of being determined by numbers, shall be determined by lines or through surfaces. To declare that Thales, using his two theorems, created the basis of linear geometry, is very dangerous. In the Pythagoreans’ vision, the monad, the unity or dot, remains the basis element of Euclidean geometry. The line, surface or volume, shall keep on being the most faithful help for the monad. These elements support the monad in placing a system under value, or a perfect relations structure.
- The most passionate problem will be the discovery or disclosure of a richness of endless shapes. I have considered not to follow the road of such a mathematical demonstration, unless it is strictly necessary, partly because this is because geometrical figures do not present any difficulty for a medium level scholar, in this field of knowledge, and to maintain an overall vision of the holistic phenomenon.
- It will be followed by a short analysis of the contributions that each of the grand Greek thinkers brought or would have brought to the system, reserching upon the work of George Johnston Allman, “Greek Geometry” from Thales to Euclid.
The text, pretty limited in size, will be accompanied by numerous drawings, as they will disclose all the necessary information. By excellence I tackled upon the Golden Section, and in a smaller measure, the radicals of 2 and 3.
The aim of this paper in to bring more clarity upon the understanding the ancient geometry spirit, as well as the necessity to review the teaching and explaining system of it, in schools.
(Preface of the treaty entitled “Holistic Geometry”)