Before Euclid, from “The Unreasonable Influence of Geometry”

From the article The Unreasonable Influence of Geometry by Doug Muder

The geometry that we all know from high school is Euclidean geometry, which goes back to the book Elements of Geometry written by Euclid in about 300 BC, just after the time of Alexander the Great. But Euclid did not invent his geometry, he collected and systematized the geometry that was already known in his time. Plato, who lived a couple of generations before Euclid, clearly knew a great deal of geometry. He probably learned it from the Pythagoreans, a secret society of mystic mathematicians who were active in Greece at that time. Think about that for a minute: a secret society of mystic mathematicians. Already that should tell you that the world worked very differently in those days.

The Pythagoreans trace their origin back to Pythagoras, a semi-legendary figure who lived sometime before 500 BC – later than Homer, but earlier than Socrates. The Pythagorean theorem is named after him. Pythagoras is supposed to have learned his geometry during his travels in Egypt. This may or may not be true. Compare, for example, the story of Atlantis. We know the story from Plato. He is supposed to have heard it from a friend, who heard it from his grandfather. The grandfather is supposed to have heard it from Solon (another semi-legendary figure), who learned it during his travels in – guess where? – Egypt. In short, the Greeks thought of Egypt as the source of all things wise and mysterious – somewhat like our image of Tibet.

Geometry in Egypt

Summary: To understand how the Egyptians thought about geometry and geometers, you need to understand something very important about the mindset of the ancients: science, religion, and magick had not yet become separate things. Science/religion/magick taught that there is an invisible order behind the visible disorder of the world. The success of geometers in redrawing boundaries after the floods of the Nile was evidence for the power of that invisible order. Geometry, then, was the special (and probably secret) knowledge of a priesthood.

A lot of what the Greeks said about the history of geometry is probably mythical, but let’s take their story seriously enough to begin our inquiry in Egypt. What would geometry have meant to the Egyptians? One clue comes from the word itself. Geo-metry literally means “earth measurement” in Greek. In other words: surveying. It makes sense that the Egyptians would have been master surveyors. We know they were master builders. Also, their agriculture depended on the annual flooding of the Nile, and floods have a nasty way of knocking boundary markers aside. A person who could redraw the boundaries after a flood would be a very valuable and important person.

How would the Egyptians have looked at him and his art? To understand this you need to appreciate one fundamental way in which the ancient world was different from ours: science, religion, and magic had not yet separated. Science/religion/magic in the ancient world was a way of thought teaching that underneath the superficial aspects of the world there is an invisible order. The visible world, the world that our senses perceive, is constantly in flux. But the underlying invisible order is eternal. Someone who has been initiated into the secrets of the invisible order can work wonders that common sense would say are impossible.

Science, magick, and religion still teach the existence of an invisible order. But now they each teach it in their own way, and we usually don’t even notice the commonality. Science teaches us that all these apparently solid objects are actually clouds of tiny particles whose motions are controlled by invisible forces. Magick teaches the existence of invisible correspondences between objects and their symbols, and of spirits who can be invoked to manipulate events. And what about religion? William James wrote: “Were one asked to characterize the life of religion in the broadest and most general terms possible, one might say that it consists of the belief that there is an unseen order, and that our supreme good lies in harmoniously adjusting ourselves thereto.”

To us these all sound very different, and it matters greatly to us whether the invisible entity we are dealing with is an abstract physical law like gravity, a capricious spirit like the loa of voodoo, or a universal moral force like justice or the Tao. The ancient mind did not often make such distinctions. Evidence for the existence and usefulness of one part of the invisible order was evidence for the existence and usefulness of all of it.

The Greek mathematician Proclus wrote: “This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes the oblivion and ignorance which are ours by birth.”

And so, the surveyor who could redraw the boundary lines when all visible markings had been washed away was a miracle worker, just like the healer who cast out demons, or the priest who prepared your body for the afterlife. Knowledge of geometry would most likely have been the guarded secret of a priesthood.

Pythagoras and the Pythagoreans

Summary: The Pythagoreans were the first known secret society. They seem to have been some kind of graded order, in which each successive initiation gave the seeker access to more of the secret doctrine. Their worldview, as best we can reconstruct it, was based on an elaborate system of geometric and numerical correspondences – a surprising number of which survive to this day in figures of speech like “a square deal”. Through these correspondences, a geometric figure or an arithmetic calculation could have an allegorical significance and express philosophical mysteries.

The discovery of irrational numbers was a huge philosophical shock to the Pythagoreans. In some ways this shock prefigures the shock that non-Euclidean geometry had on the world of the Enlightenment.

In the context of the ancient world, it makes perfect sense that Pythagoras would construct his school as a secret society. (Some Masonic lodges claim Pythagoras as an ancestor. Historians regard this as unlikely.) Because the Pythagoreans were so secretive, our knowledge of their teachings is limited. But we do know that they studied great deal more than just geometry. They were musicians who understood the relationships between the lengths of strings and the tones they made when plucked. Strings whose lengths had certain ratios were harmonious, while other ratios produced dissonance. To the Pythagoreans this meant that the ratios themselves were beneficial or harmful. This seems to have been the basis of an elaborate system of numerology. Most of the system is lost to us, but we do know (because Plato tells us) that they associated the concept of justice with the number four and the figure of the square. This association lives on in many of our modern expressions, such as a square deal or squaring a debt. It is no coincidence that the angles of a square are right angles. This association between abstract concepts and geometric figures or ratios meant that a geometric construction could also be a philosophical argument, in the same way that an allegorical story or song can relate some higher truth.

The Pythagoreans were the first people to call the universe a cosmos. The word cosmos, which comes from the same root as cosmetic, refers to the kind of beauty which arises from harmonious order. The word cosmos, then, does not just refer to the universe, it makes a statement about it – that the universe is harmonious and orderly and therefore beautiful.

The Pythagoreans are also believed to be the first people who knew about irrational numbers. This was a great mystery and consternation to them. Previously, the Greeks had believed that numbers, fractions, and lengths were all the same things. When it was discovered that the ratio between the diagonal of a square and its side was the square root of two, and that this could not be expressed as any fraction of integers, a philosophical crisis ensued. It was a great shock for people who based their worldview on numbers to realize that something as simple as the diagonal of a square could not be represented by anything they recognized as a number.

“The discovery of incommensurable ratios is attributed to Hippasus of Metapontum. The Pythagoreans were supposed to have been at sea at the time and to have thrown Hippasus overboard for having produced an element in the universe which denied the Pythagorean doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” – Morris Kline, Mathematical Thought from Ancient to Modern Times

The mathematician Jonathan Holden tried to capture this jolt in his poem The Fall of Pythagoras.

Once all his calculations came out
like peas. You could pop them
whole from the pod, perfect spheres.
And they were sweet. The number Two
tasted the way deep fields taste
after rain. You could take the trees
apart, find godlike shapes
if you didn’t look too hard,
which is why I feel sorry for
Pythagoras. He tried to take a pure
square apart, he was tinkering
around too much. One hard look
at the diagonal, and it was too
late. The quality of light
had already changed. It lent the wind
forbidden possibilities, the clouds
this odd, this brooding weight;
for the diagonal was the square root
of Two. And what Two was made of –
this hard, sweet pea – what all
these leaves, these animals and clouds,
the sun, even these ugly dreams
he’d been having recently at night
were made of, would not add up.
The harder you tried to add them
up, the finer they became, the faster
sifted through your fingers
the incommensurable parts of everything.


We are told that Plato put a sign above the door to his academy saying “Let no one enter who is unfamiliar with geometry.” Our modern universities might do well to remember this admonition, because the geometry student faces many of the same conceptual difficulties that re-appear more abstractly in Plato’s theory of forms. Trying to understand Plato without understanding geometry is like fighting with one hand tied behind your back.

For example: nearly every geometry student at some point wants to prove a theorem by measuring something in the diagram. The teacher must go to some difficulty to explain that the diagram is just a picture of the problem, not the problem itself. Aristotle wrote: “The geometer bases no conclusion on the particular line which he has drawn being that which he has described.” The lines that we can draw in diagrams are not true lines, and the points in diagrams are not true points. They are merely imperfect representations of the true points and lines, which exist merely as ideas, and have never been seen by anyone. And yet, to the geometer, these ideal invisible points and lines are more real and more perfect than anything that can be drawn on paper.

This relationship between the invisible objects of geometry and the visible objects of the real world, including diagrams, is key to understanding the relationship between the Platonic forms and the real world objects that embody them. Your high school geometry class – whether you knew it then or not – was actually an introductory course in Platonic mysticism.

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