Plato considered to geometry and the numbers like most concise and essential, and therefore the ideal of the philosophical languages
VICTOR MANUEL GUZMAN VILLENA 
Geometry is one of the most represented sciences in the ceremonies and symbols of the initiatory schools and the one that is considered the more essential. Geometry means “Earth measurement”. In old Egypt, from Greece inherited this study, the Nile overflowed its margins every year, flooding the Earth and erasing the methodical drawing of the parcels and the zones of cultivation. That annual flood symbolized for Egyptian the cyclical return of the original watery chaos, and when the waters retired the task began of redefining and of reestablishing the edges. This work was called geometry and was considered like the reestablishment of the principle of the order and the law on the Earth.
Every year, each measured zone was a little different. The human order was changeable, and that is reflected in the Earth ordering. The astronomer of the temple could say that certain celestial configurations had changed, and therefore the direction or the location of a temple had o adjust to it. Therefore, the drawn of the parcels on the Earth had, for the Egyptians, a Metaphysical dimension as physical and social. That activity “to measure the Earth” became the base of a science of the natural laws such and as they are incarnated in the archetypes forms of the circle, squared and the triangle.
The fourth element of that important program of four parts, the Quadrivium, was the study of the harmony and music. The universal laws that defined the relation and the interchange between the temporary movements and celestial events and the space order and the development on the Earth by other.
Soon they added the grammar, that teaches to express the ideas with the own rules of the language; the rhetoric, the adornments and beauty of the spoken style; the logic to form exact judgments of the things; three more elements formed what it is called the Liberal Arts, that are seven and that are represented in the Seven masters necessary to form a correct lodge, regular and perfect.
The implicit objective of that education was to allow the mind to become a channel through what the “earth” (the level of the indicated form) could receive the abstract thing, the cosmic life of skies. The practice of geometry was an approach to the way in which the universe is become ordained and it is sustained.
The geometric diagrams can be contemplated like moments of immobility that reveal a continuous and timeless universal action generally hidden to our censorial perception. Of that way a mathematical activity apparently so common can become a discipline for the development of the intellectual intuition and spiritual.
Plato considered to geometry and the numbers like most concise and essential, and therefore the ideal of the philosophical languages. But it is not but by virtue of his operation at certain “level” of reality that geometry and the numbers can become a vehicle for the philosophical contemplation.
The Greek philosophy defined that notion of “levels”, so useful in our thought, distinguishing the “type” of the “archetype”. According to indications that we see in the reliefs of Egyptian murals, arranged in three registries - the superior, the meddle and the inferior -we can define a third level, “ectopic” located between the archetype and the type.
In order to see as each of them works, we take an example from something tangible, as a bridle or brake that is used to control a horse. That bridle can have certain number of forms, materials, sizes, colors, utilities, and all of them are bridle.
The bridle thus considered is a type: exists, is diverse and variable. But in another level it is the idea or the form of bridle, the model of all the bridles. This is an idea non indicated, pure and formal, and that is ectopic. And over that is still the ectopic level, that is of the principle or power-activity, that is to say a process that the ectopic form and the example of type of jewel only represents. The archetype has to do with the universal processes or dynamic models that can be considered independent of any structure or forms material.
The modern thought has difficult access to the archetype concept because the European languages require that the verbs and the action are associated with nouns. Therefore we do not have linguistic forms whereupon to imagine a process or an activity that does not have a material vehicle. The old cultures symbolize those pure and ether processes like Gods, that is to say, powers or attachment lines through which the spirit laid down in energy and matter. The bridle is related then to the archetype activity by means of the handle function: the principle of which the energies are controlled, specified and modified by means of the effects of the angles.
Therefore, we frequently see that the angle -that is essentially a relation between two numbers- would have been used in the old symbolism to designate a group of fixed relations that control complexes or interactive models. Of that way, the archetypes or Gods represent dynamic functions that link each other the superiors worlds of the interaction and the constant process, and the real world of the concrete objects.
We see, for example, that an angle of 90º or 45º, the same form, the geometric optics reveals that each substance reflects the light in characteristic form in its own particular angle, and is that angle the one that gives our more precise definition us of the substance. In addition, the angles of the patterns of union between molecules determine to a large extent the qualities of the substance.
In the case of the bridle, that relation or angular group is show in the relation between the mouthful and the strap of the bridle or between the mouthful and the inclination of the neck or the jaw and the biceps of the rider. Staring of the level of the archetype or active idea, the principle of the bridle can be applied metaphorically to many fields of the human experience.
Working then at archetype level, geometry and the numbers describe fundamental and causal energies in their weave and eternal dance. It is that way to see the one that underlies under the expression of cosmological systems and geometrical configurations. For example, the most honoured of all the tantrics diagrams, the Sri Yantra represents the active necessary functions in the universe by means of nine interlaced triangles. To sink in a geometric diagram of that nature is to enter a species of philosophical contemplation.
For Plato, the reality consisted of pure essences or archetypes ideas of which the phenomena that we perceived are only reflected (the Greek word “Idea” also is translated as “Form”). These ideas cannot be perceived by the senses, but only for the pure reason. Geometry was the language that recommended Plato like clearest model to describe that Metaphysical kingdom.
“Perhaps you do not know that the geometry scholars use the visible forms and speak of them, although it is not about them but those things of which they are a reflection and they study squared in itself and the diagonal in itself, and not the its image of the draw? And so on in all the cases. What they really look for is to be able to glimpse those realities that only can be contemplated by the mind”. Plato, the Republic, VII.
The Platonism considers our knowledge of geometry like innate in us, acquired before being born, when our souls were in contact with the kingdom of the ideal being.
“All the mathematical forms have a first permanence in the soul; in such a way before the sensible thing, this one contains numbers with its own dynamics; vital figures before the apparent ones; harmonic reasons before the harmonic things, and invisible circles before the parts that move in the circle”. Thomas Taylor.
To enter a temple constructed in its totality according to invariable geometric proportions is to enter the kingdom of the eternal truth. Thomas Taylor says: “The geometry allows to his devotee one, like a bridge, to cross the dark of the material nature, as if it was a dark sea towards the luminous regions of the perfect reality. However, one is not absolute about an automatic event that just by happens to take a geometry book. As it says Plato, the fire of the soul must gradually be revived by the effort: “What grace you do to me, those that seem worried because I impose impractical studies to you".
Is not own solely of the mediocre spirits, but all the men have difficulties to be persuaded that is through those studies, used as instruments the eye of the soul is purified, and is caused that a new fire burns in that organ that were growing dark and like extinguished by the shades of other sciences, an organ more important to conserve than ten thousand eyes, since is the unique one with which we can contemplate the truth”. The Republic, VII (Mentioned by Tenon of Izmir (century II) in its useful Mathematics to understand Plato.
Geometry deals with the pure form, and philosophical geometry reconstructs the development of each form from another previous one. It is a way to make visible the essential creative mystery. The passage of the creation to the procreation, of the pure idea, formal and not manifested (here down), the world that arises from that divine original act, can draw up by means of geometry, and be experienced by means of the practice of geometry.
Inseparable of this process it is the concept of the number and for the Pythagorean, the number and the form at idea level were only one. But in this context the number must be understood of special way. When Pythagoras said “Everything is ordered around the number”, he did not think about the numbers in the ordinary enumerator sense.
Besides the simple amount at ideal level the numbers are impregnated by a quality, in such a way that “duality”, the “Trinidad” or the “Tetra” for example, are not simples composed of 2, 3 or 4 units, but are a whole or a unit in themselves, each of them with their corresponding properties. The “two”, for example are considered like the original essence from which it comes and on which the power of duality founds its reality.
r.a. Schwaller of Lubiez proposes an analogy by which can be understood this universal sense and archetype of the number. A revolving sphere appears to us with the notion of an axis. We think about that axis like in an ideal or imaginary line that crosses the sphere. It does not own objective existence, and nevertheless we cannot but be convinced of its reality; and to determine any thing related to the sphere, as its inclination or its turnover rate, we must talk about that imaginary axis.
The number in its enumerated sense corresponds to the measures and movements of the outer surface of the sphere, while the universal aspect of the number is analogous to the immovable principle, not manifested and functional of its axis.
We take our analogy now to the bi -dimensional plane. If we took a circle and square and we give its value 1 to the diameter of the circle and also the side to the square, then the diagonal of the square one always will be (and this one is an invariable law) "unmeasurable" “irrational” number. That number can be prolonged in an infinite number of decimal without never get a resolution. In the case of the diagonal of the square, that decimal is 1,4142…. And is called squared root of 2.
With the circle, if we give to the diameter value 1, the circumference always will be of unmeasurable type: 3,14159…, that we know like the Greek symbol, pi.
The principle continues being the same in the inverse case, if we give the fixed value and that unique transaction in which the heard vibration becomes in seen form; and its geometry explores the relations of the musical harmony. Although interrelated in its function, our two main intellectual senses, the sight and the hearing, they use our intelligence in two completely different forms.
For example, with our optical intelligence, to form a thought we compose an image of our mind. On the other hand, the ear uses the mind in an immediate answer and without image whose action is expansive and evokes an answer to associate with subjective experiences, emotional, aesthetic or spiritual.
We tend to forget that also it takes part when the reason perceives invariable relations. Therefore, when we centered our censorial experience in our auditory capacity, we can realize that is possible to listen to a color or a movement. This intellectual capacity is very different from the "visual" analytical or sequential that normally we used. It is this capacity, associated with the right hemisphere of the brain, the one that recognizes patterns in the space or sets of any type. It can perceive the opposite ones simultaneously and get functions that before the analytical faculty seem irrational.
It is in fact the perfect complement of the visual and analytical capacity of the left hemisphere, since it absorbs space orders and simultaneous. That innate intellectual quality resembles much which the one called by the Greeks the pure reason, or what in India called the heart-mind.
The old Egyptians had for it a beautiful name, the intelligence of the heart, and to reach that quality of understanding was the implicit goal of life. The practice of geometry, although also makes use of the analytical faculty, uses and cultivates that auditory aspect and intuitive of the mind. For example one undergoes the fact of the geometric growth through the image of the square whose diagonal forms the side of a second square. Is a certainty not reasoned get by the mind from the real experience to execute the drawing. The logical is contained in the lines of paper, that can not be drawn in another form.
Like geometry scholars equipped solely with compasses and rules, we entered the bi-dimensional world of the representation of the form. A bond is establish between the kingdoms of the most concrete thought (forms and measured) and most abstract. In the search of the invariable relations that govern and interrelate the forms, we put ourselves in resonance with the universal order. When reproducing the genesis of those forms we tried to know the principles the evolution. In that way, when elevating our own pattern of thought at those archetypes levels, we caused that the force of those levels penetrates our mind and our thought. Our intuition cheers up, and perhaps as Plato says, the eye of the soul can be purified and be ignited again, “because only through it we can contemplate the truth”.
One of the fundamental assumptions of the traditional philosophies resides apparently in which the purpose of the intellectual faculties of the human is the one to accelerate our own evolution surpassing the limitations of the biological determinism which they restrict all the other alive organisms. The methods like yoga, the meditation, the concentration, the arts, crafts, are technical psycho physicals to approach that fundamental goal. The practice of geometry is one of the essential techniques of self-realization.
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