Math and Magic: Euclid Defines Space
It is not clear who Euclid’s intended audience is, but at very least, he seems to be writing a book about the way things are. He has not necessarily invented these ways of thinking, he is writing them down, and there is something about writing ideas down that necessarily transforms the nature of the idea itself in some way. The reality of the idea becomes different than when it was an oral presentation, or lines drawn in the sand. Euclid has named his text “Elements” rather than “lines”, “shapes,” or “geometry 101”. Elements would seem to suggest a primary position before anything else. Euclid relies on the idea that these foundations of geometry — these “Elements” — are a part of the natural world that is being uncovered or revealed, rather than invented. To what degree might we in the present day say this is so, and not that geometry is merely another artifact of human consciousness? Euclid seems to find inherent truth as somehow self-contained, and a part of the natural world. In this case, it would seem there was something in the ancient mind connecting the elements to a sense of divine order — ie the perfection of a square as a representation of the fourfold earth — four directions, four elements. Euclid takes time, epistemically building one proposition upon the one before it, not arriving at the first quadrilateral figure until proposition 33, creating a four-sided figure by combining two triangles. Euclid is attempting, at least in part, to arrange the propositions so that they build off each other. The progression of circle to triangle to quadrilateral in itself hearkens to a sort of divine order with ascending levels of perfection, unfolding higher levels of complexity, in a sense mirroring the project of God or the gods in the pursuit of divine order.
In the modern sense of the word, “mathematics”, the manipulation of numbers is seen as primary, whereas Euclid’s Elements establishes geometry as the primary inquiry. From a phenomenological perspective, numbers would certainly be contained within some sort of subjectivity, but it’s not clear that geometry as the definition of real space, could be said to be the same. Nothing has changed about Euclid’s geometry to the present day. All of the propositions can be reproduced by a contemporary student, with accuracy equivocal to what Euclid demonstrated. Everything he demonstrates is correct by today’s reckoning as well. This could not be said of Lucretius, Aristotle, Galileo, or even Einstein.
There is something unique about geometry: it would seem to be demonstrable and self-evident, independent of the progress of human academic knowledge, or even language itself. When Euclid sets out to write Elements, he accepts the idealization of things such as lines, shapes and forms — already, he is beginning from a Platonic point of reference regarding the objectivity of this world, and the lack of an awareness of intersubjectivity. When Euclid asserts these propositions, they are presented as universal truths, and anyone who came along later and contradicted Euclid lacks the weight of Common Sense. There is something timeless about the Euclidean proofs that cannot be said about Lobachevsky — there is nothing intuitive about Lobachevsky, and the fact that diagrams cannot exactly be drawn to represent his ideas of parallelism shows that he has moved well beyond the comfortable sense of familiarity that is present in Euclid — the sort of intuitive “aha!” one feels when working through a Euclidian proposition.
Geometry, for me, was always just a subject in school, but it is clear that the ancients thought of geometry in a very different way. The idea of God as a geometer or architect writ large is sometimes attributed to Plato, but I imagine the idea is older than him. In the chaos of the natural world, humans seem always apt to assign meaning to it. Perfect squares and isosceles triangles don’t exactly exist in nature, but four sticks can present the imagination with a rough quadrilateral, and an ideal square can be inferred from that. Only the circle would seem to appear ready-made in nature — perceived in the sun and in the moon. Perhaps that is why Euclid begins with the circle and derives all other shapes from it.
The circle would seem to hold a space of primacy in most, if not all sacred ritual traditions. Hindu yantras, sacred mandalas, the Puebloan Kiva, the ritual circles of western symbolic ceremonial magic. Even Christian churches were once circular with congregations standing in circles well before the Gothic cross-shaped structures emerged. In Euclid’s first proposition, a line is inscribed and a circle constructed. He does not tell us how to construct the circle — images of Euclid usually depict him with some sort of ancient compass. A string would suffice. Whatever the method, Euclid seems to assume the means of constructing the circle to be common knowledge. The line, in his first proposition, is pre-existent. The circle is dynamic as it is constructed around the line.
Euclid gives us no introduction, no explanation of his project. He simply begins by stating, “a point is that which has no part.” Although we represent it visually with a dot, the point is an imaginary concept. In a certain way, the circle is a sort of projection of the nondimensional point into dimensional space, regardless of how large or small. Definition 16 tells us, “the point is called the centre of the circle,” which seems to suggest this relationship. Circles are constructed; other forms are constructed based on these circles. Triangle, square, presumably all shapes of any number of sides can be thus constructed from that which paradoxically has no sides.
Describing circles, the definition of space itself, would seem to relate to the Greek idea of tememos — a place set apart. Carl Jung speaks of tememos as a place both sacred and protective where high-level subconscious work can take place. To what degree might Euclid’s idea of defining space be informed by this sort of spiritual concept? There is no question that the text, in its simplicity, does seem to read like a work of Taoist philosophy. The Egyptians considered the perfect circle of Horus’ pupil to be the source of all generative power in the universe, and represented that with the hypocephalus which was always contained within a perfect circle and placed upon the bodies of the dead. They referred to it as a “hidden place” and as a geometric form that exists without straight sides and without angles, the circle is simultaneously the perfection of form and perfect formlessness. It is just as imaginary as the point and just as primary.
Paul Ricoeur credits Euclid with the “glory of having conceived of an infinite task of knowing”( Ricoeur 161). I wonder what this infinitude might mean in a practical way. Defining space is to connect points with lines. A line, like a point, has no actuality, and is merely an ideal form that cannot be represented, no matter how thinly we draw the line. Husserl tells us, “The Pythagorean theorem , indeed all of geometry exists only once, no matter how often or even in what language it may be expressed…” (Derrida 72) It would seem to be a relatively agreed-upon notion that geometry stands apart from all other human endeavors in that it exists beyond culture, beyond language, beyond human consciousness, even that geometric truth would somehow exist if humans did not.
Consciousness is not required for geometry to exist. Systems of sacred geometry question whether Consciousness itself might even proceed from geometry in some mysterious way. Whether geometry proceeds from human consciousness, or the other way around, Euclid’s first proposition claims to be in search of a perfect triangle, but essentially he begins by creating a vesica piscis: the circle is both eternity and supreme unity, and when a random point on the circle serves as center point for a second circle, the vesica piscis is formed. The circle is one long, eternally curving line; the vesica piscis is a two-sided form. Both are mystical forms, but are necessary to procure the first concrete form: the triangle. One plus two equals three. Just as he does not teach us how the circle is constructed, he slips the two-sided vesica in right at the beginning without mention, and names the three-sided shape as the first geometric form. When lines connect the apex of each side of the vesica to the point where the circles cut each other, the perfect isosceles triangle is formed. The mystical meanings and lore surrounding the vesica are encyclopedic. Essentially it is one of the most mysterious and powerful symbols in sacred geometry.
There is no way to know what esoteric teachings may have accompanied this text, but as I read more closely, I wonder if it wasn’t a magic book mistaken for a math book. The defining of space as a ritual and spiritual practice has such an important place in the ancient worldview, I can’t help but see Euclid under this lens and interpret his endeavors as not just showing something about the way things actually are, but also to describe an idealized version of reality, within the sacred/magical understanding of space defined.
