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Greek geometry is written in a style adapted to oral teaching. Mathematicians memorised theorems the way bards memorised poems. Several oddities about how Euclid’s Elements is written can be explained this way.
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Greek geometry is oral geometry. Mathematicians memorised theorems the way bards memorised poems. Euclid’s Elements was almost like a song book or the script of a play: it was something the connoisseur was meant to memorise and internalise word for word. Actually we can see this most clearly in purely technical texts, believe it or not. It is the mathematical details of Euclid's proofs that testify to this cultural practice. That sounds almost paradoxical, but I’m sure I will convince you.
The surviving documentation about ancient Greek geometry consists almost entirely of formal treatises. Very stilted and dry texts. Definition, theorem, proof. Pedantically written. Highly standardised, formalised. Completely void of any kind of personality. Where is the flesh and blood, the hopes and dreams, the lived experience of the ancient geometer? It’s as if they were determined to erase any traces of all of those things, and leave only a logical skeleton.
But it’s not as hopeless as it seems. At first glance it looks as if these texts have been scrubbed of all humanity. But, in fact, if we read between the lines we can extract quite a bit of information. There are implicit clues in these texts that reveal more than the authors intended.
That’s our topic for today: How these seemingly purely logical texts actually say quite a lot about the social context in which they were produced.
One thing we learn this way is that we should think of the Greek geometrical tradition as spoken geometry, not written geometry. Today we think of written texts as the primary manifestation of mathematics. When mathematicians disseminate their ideas, the published article is the official, definitive, primary expression of those ideas. The mathematician crafts a written document with the expectation that reading the text on paper is going to be the primary way in which people will access this material.
Not so in antiquity. Oral transmission was considered the primary mode of explaining mathematics. Written documents were a last resort when personal contact was not possible. And the written document was not meant to be a primary exposition in its own right. Writing was merely the oral explanation put down on paper (or papyrus, rather).
At least it must have been like that in the early days. Many conventions of Greek mathematical writing only make sense from this point of view. They must have been formed in an oral mathematical culture. Probably in later antiquity the situation was not so clear cut. Writing probably gradually became more of a thing in its own right, rather than merely a record of oral exposition. But even then, the conventions of written mathematics remained largely fixed. Greek mathematics never liberated itself from these conventions that had been set in an oral culture. They lived on. Perhaps in part due to tradition and conservatism, but probably also because the oral element remained a significant part of mathematical culture, perhaps especially in teaching.
Here’s an example of this, which I have taken from Reviel Netz’s book The Shaping of Deduction in Greek Mathematics. Consider the equation A+B=C+D. Here’s how the Greeks expressed this in writing: THEAANDTHEBTAKENTOGETHERAREEQUALTOTHECANDTHED. This is written as one single string of all-caps letters. No punctuation, no spacing, no indication of where one word stops and the next one begins.
A Greek text is basically a tape recording. It records the sounds being spoken. There is a letter of the alphabet for each sound one makes when speaking. The scribe just stenographically puts them down one after the other. From this point of view there is no distinction between upper or lower case letters: a letter just stands for a sound and that’s it. And there is no punctuation or separation of words, because those are not spoken sounds. And of course no mathematical symbols such as plus or equal signs, because that also does not exist in spoken discourse.
The only way to understand a text like that is to read it out loud. You have to read it like a child who is just learning to read: you sound it out letter by letter, and then interpret the sounds, rather than interpret the writing directly.
So the Greeks had a very limited conception of writing. They thought of writing only as a way of recording speech. They completely missed the opportunities that writing provides when embraced as a primary medium in its own right. Writing is a better way of representing equations, for example, than speech. But the Greeks completely missed that opportunity because they were stuck with the limited notion of writing as merely recorded sounds.
I like to compare this with early movies. Think of those classic movies from, say, the 1950s or so. They are basically recorded stage plays. There are limitations inherent in the medium of theatre. The actors have to speak quite loudly, articulately, to be heard by the audience in the back of the theatre. And the scenery on stage cannot easily be changed or moved. In a play you better stick to one or two sets, such as the interior of a room. That you can set up carefully with furniture and all kind of stuff on the walls and so on. But because you can’t change it easily, you have to have to have large parts of the play take place in that single setting.
These technical limitations constrain the artistic freedom of the playwright. You have to come up with a story where all the various characters have some reason or other to come and go into a single room, and once there to have loud conversations that drive the plot. All emotional depth and so on must be conveyed in this particular form.
These things became second nature to writers. So when film came around they kept doing the same thing even though that was no longer necessary. Many treated film as simply a way of recording plays. So in early movies you still have a lot of these static scenes with a fixed camera at one end of a room, and characters coming and going, having loud conversations.
Film affords new artistic possibilities. You are no longer limited to a static camera showing a fixed set, the way the audience of a theatre would be looking through the “fourth wall” of a room. You have many more options to convey things visually, instead of being limited to strongly articulated stage dialogs as the only driver of the plot.
But many early movies didn’t take advantage of that. They just kept doing what they had always been doing at the theatre and just recorded that. They saw the new medium of film merely as a way of “bottling” existing practice. It’s just a storage medium. They didn’t consider that the new medium was in some ways better than the old one and enabled you to do completely new things.
It was the same with writing in antiquity. Writing was merely for storing speech. They failed to take advantage of the ways in which writing could not only preserve existing cognitive practice but in fact transform it and improve it. Such as working with equations symbolically.
Here is another consequence of this: the absence of cross-referencing. If a mathematical text is like a tape recording, you can’t easily access a particular place in the tape. The only way to make sense of the text is to “hit play,” so to speak, and translate it back into sounds. Only then can it be understood. You can “fast forward” and “rewind”—that is to say, start reading at any point in the manuscript. But you can’t turn to a particular place, such as Theorem 8.
Modern editions of Euclid’s Elements are full of cross-references. Each step of a proof is justified by a parenthetical reference to a previous theorem or definition or postulate. But that’s inserted by later editors.
There is no such thing in the original text. Because it’s a tape recording of a spoken explanation. Referring back to “Theorem 8” is only useful if the audience has a written document in front of them. If they are merely listening to a long lecture, or a tape recording of a lecture, then there is no use referring back to “Theorem 8”, because the audience has no way of going back specifically to that particular place in the exposition.
For this reason, oral mathematics involves committing a lot of material to memory. In the arts, people memorise poems and song lyrics. Actors memorise the dialogues of plays. Ancient mathematics was like that as well. You would learn to recite theorems the same way you learn to sing along to your favourite song.
This aspect of the oral culture thoroughly shaped the way ancient mathematical texts are written. Euclid’s Elements and many other texts follow a certain stylistic template that at first sight seems quite irrational, but which starts to make sense once we consider the oral context.
Consider for example Proposition 4 of Euclid’s Elements. This is the side-angle-side triangle congruence theorem. It’s completely typical, I’m just picking a theorem at random. Let’s look at the text of this proposition. First we have the statement of the theorem in purely verbal terms. It goes like this:
“If two triangles have two sides equal to two sides, respectively, and have the angle enclosed by the equal straight lines equal, then they will also have the base equal to the base, and the triangle will be equal to the triangle, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles.”
Ok, so: two triangles have side-angle-side equal, the it follows that they also have all the other things equal. Namely the remaining side, the remaining angles, and the area. “The triangle will be equal to the triangle,” says Euclid: this is his way of saying that they have equal area.
After Euclid has stated this, he goes on to re-state the same thing, but now in terms the diagram. “Let ABC and DEF be two triangles having the two sides AB and AC equal to the two sides DE and DF, respectively. AB to DE, and AC to DF. And the angle BAC equal to the angle EDF. I say that the base BC is also equal to the base EF, and triangle ABC will be equal to triangle DEF, and the remaining angles subtended by the equal sides will be equal to the corresponding remaining angles. ABC to DEF, and ACB to DFE.”
This is exactly the same thing that he just said in words. But now he’s saying it with reference to the diagram. He always does this. He always has these two version of every proposition: the purely verbal one, and the one full of letters referring to the diagram.
For simple propositions you can understand the value of both formulations. But quite soon, when the material gets more technical, it often happens that the verbal version becomes so abstract that it’s quite impossible to follow. This happens quite soon already in Euclid. Ken Saito has a recent paper on this, “traces of oral teaching in Euclid’s Elements.” He takes as an example Proposition 37 from Book 3 of the Elements. I’ll read it to you just to convince you how convoluted and unnatural it is to state theorems in this purely verbal form. Here it is, Euclid’s statement of this proposition:
“If a point be taken outside a circle and from the point two straight lines fall on the circle, and if one of them cut the circle, and the other fall on it, and if further the rectangle contained by the whole of the straight line which cuts the circle and the straight line intercepted on it outside between the point and the convex circumference be equal to the square on the straight line which falls on the circle, the straight line which falls on it will touch the circle.”
That’s very difficult to follow. Of course, as always, Euclid immediately goes on to state the same thing, but in terms of the diagram. That part is much easier to follow, and it turns out to be a pretty straightforward claim. The theorem is a kind of formula for the length of a tangent; how far it is to the point of tangency from a given point outside the circle. But you would hardly know that by reading the verbal statement only.
For some reason the Greeks insisted that the verbal formulation should be one single, rambling sentence. No matter how complicated your theorem is, you have to cram all the conditions and all the consequences, everything you want to say, into one single sentence.
This is taken to absurd lengths in Apollonius for example. Let me read to you an example from the Conics of Apollonius. This is Proposition 15: one of the earliest. It only gets worse from there, but this is bad enough, I’m sure you will agree when I read it to you. The proposition is a kind of change-of-variables theorem for ellipses: it tells you the equation for an ellipse in a new coordinate system conjugate to the first. So it has to specify what the equation of the ellipse was in the first coordinate system and what the assumptions for that was, then how the change of coordinates is defined, and then what the equation of the ellipse is in the new coordinate system. And it has to do all of that purely verbally, and in one single sentence, one big “if ... then ...” statement. So you get this crazy monstrosity of a sentence, it goes like this:
“If in an ellipse a straight line, drawn ordinatewise from the midpoint of the diameter, is produced both ways to the section, and if it is contrived that as the produced straight line is to the diameter so is the diameter to some straight line, then any straight line which is drawn parallel to the diameter from the section to the produced straight line will equal in square the area which is applied to this third proportional and which has as breadth the produced straight line from the section to where the straight line drawn parallel to the diameter cuts it off, but such that this area is deficient by a figure similar to the rectangle contained by the produced straight line to which the straight lines are drawn and by the parameter.”
What’s going on with this crazy stuff? Were the Greeks some kind of aliens with brains that could understand that type of thing? No. When encountering a theorem like this, they surely did not try to parse a sentence like that in the abstract. Instead they would turn to the diagram explication for help. Just as Euclid always does, so also Apollonius always goes on to restate the theorem in terms of labelled point in a diagram. And this explanation is not one big crazy sentence, but nicely broken into small steps. Much easier to follow.
At a certain point you may ask yourself: Why even include the purely verbal formulation at all? It’s so abstract, so difficult to follow. Surely any reader or listener will be lost before you have even gotten halfway through a sentence like that. And since you’re going to restate the theorem immediately anyway, why bother? You might as well only do the diagram version of the theorem. That’s the one you are going to use for the proof anyway.
That’s something of a puzzle in itself, but here’s the real kicker though. Not only does Euclid insist on including the abstruse verbal formulation of every theorem, he actually includes it twice! This is because, at the end of the proof, his last sentence is always “therefore ...” and then he literally repeats the entire verbal statement of the theorem. It is literally the exact same statement, word for word, repeated verbatim. You say the exact same thing when you state the proposition and then again when you conclude the proof. Copy-paste. The exact same text just a few paragraphs apart.
Astonishing. What a waste of papyrus and scribal effort. This was an enormous cost back then. There were no printing presses. You had to copy all of this by hand. Writing materials were expensive, copying was expensive, preservation was expensive. They had every incentive to cut and keep things minimal, yet they included this massive redundancy of repeating the rambling verbal statement of every proposition twice in short succession.
You may recall that an important treatise by Archimedes was scrubbed off its parchment because the parchment itself was so valuable even when recycled. And medieval scribes were big on minimising writing. Think of “etc.”, “e.g.”, “i.e.”: we still use those shortened versions of Latin expressions. They were invented back when people were writing and copying manuscripts by hand. Very understandable.
Yet despite all of that, for some strange reason, including the entire verbal statement of the proposition twice was somehow found valuable enough to warrant the enormous cost.
In the case of the side-angle-side theorem for example, the verbal statement of the theorem takes up about 15% of the total text of the proposition and proof. And then another 15% for the redundant recapitulation. So that’s 30% of the total text that could simply be cut. The remaining 70% of the text would still contain the full statement of the theorem in its diagram form, and the complete proof.
You’d think the temptation would be great to cut at least those last 15% of pure recapitulation. Even the standard English edition of the Elements by Heath simply writes “therefore etc.” at the end of the proofs, instead of repeating the full statement like the original did.
So what was the value of this very expensive business of repeating the statement of the proposition? The oral tradition explains it. The verbal statement of the proposition is like the chorus of a song. It’s the key part, the key message, the most important part to memorise. It is repeated for the same reason the chorus of a song is repeated. It’s the sing-along part.
In a written culture you can refer back to propositions and expect the reader to have the text in front of them. Not so in an oral culture. You need to evoke the memory of the proposition to an audience who do not have a text in front of them but who have learned the propositions by heart, word by word, exactly as it was stated, the way you memorise a poem or song.
This is why, anytime Euclid uses a particular theorem at a particular point in a proof, he doesn’t says “this follows by Theorem 8” or anything like that. He doesn’t refer to earlier theorems by number or name. Instead he evokes the earlier theorem by mimicking its exact wording. Just as you just have to hear a few words of your favourite chorus and you can immediately fill in the rest. So also the reader, or listener, of a Euclidean proof would immediately recognise certain phrasings as corresponding word for word to particular earlier propositions. They would have memorised the earlier propositions not only in terms of content but in terms of the exact verbal phrasing, almost melodically, rhythmically. Just hearing the first few words of such a formula repeated would trigger the full memory to flow out naturally and unstoppably, like singing along to the chorus of a song you love.
You can see an example of this already in Euclid’s Proposition 5. We already discussed his Proposition 4, the side-angle-side triangle congruence theorem. Euclid applies this result twice in the course of the proof of Proposition 5. However, he really only needs part of the theorem. Remember that Proposition 4 concluded several things: that the remaining sides were equal, that the remaining angles were equal, and that the areas of the two triangles are equal. Areas are completely irrelevant to Proposition 5, which is a statement purely about angles. Yet each time Euclid applies the side-angle-side theorem he spells out the full conclusion. Including the needless remark that the areas are equal.
In one case it is even irrelevant that the remain sides are equal as well, but Euclid still needlessly remarks on this pointless information in the course of the proof of Proposition 5 even though it has no logical bearing on the proof. Go look up Euclid’s proof if you want to see this nonsense for yourself. Ask yourself why Euclid points out that “the base BC is common” to both triangles the second time he applies the side-angle-side theorem in the proof of Proposition 5. It’s completely redundant and worthless. He could have just omitted that remark, and it wouldn’t have affected the logic of the proof at all.
But from the oral point of view it makes sense. Applying a theorem is a kind of package deal. You get the whole thing whether you need it or not. Once you’ve triggered the memory of the previous theorem with the appropriate key phrases, then the whole conclusion comes blurting out. Once you’ve committed to singing the chorus there’s no going back. You can’t sing only the part of the chorus you need. The whole thing goes together. You have memorised it in one flow. Once you hit play on that memory you automatically run through the whole thing.
This is why Euclid is needlessly talking about areas in the proof of Proposition 5, even though that serves no logical function whatsoever. He is mimicking word for word the phrasing of the previous proposition, filling in the specifics of the case at hand as he goes along. You sing the “chorus” of the side-angle-side theorem and you “fill in the blanks” as it were. The purely verbal statement of the side-angle-side theorem spoke of sides and angles and so on in the abstract. To apply the theorem is to repeat that exact same phrasing, but inserting AB, BCF, and so on, into that formula to specify what the sides and angles are in the particular case at hand.
It’s like singing “happy birthday”: it has a fill-in-the-blank part. Just as you would go: “Happy birthday dear Euclid”, so also you would go: “If the side AB equals CD, then the angle is ...” and so on, something like that.
Here’s maybe another consequence of this: Euclid’s odd formulation of the side-side-side triangle congruence theorem. This is Euclid’s Proposition 8. As we saw, in the side-angle-side case, Euclid drew all the possible conclusion: about sides, about angles, about area. So the theorem became a mouthful, and led to the introduction of superfluous remarks any time the theorem is applied, because you have to repeat all the conclusions whether you need them or not.
To avoid this problem it might be tempting to state theorems in less general form. And this is exactly what Euclid does with the side-side-side theorem. He introduces an asymmetry in the statement of the theorem. Instead of three sides, he speaks of two sides and a base. And his statement of the conclusion is that one particular angle (the angle between the two “sides”) is equal in both triangles. Of course it is completely arbitrary which side you designate as the “base.” And of course you could just as well have concluded that the other angles too correspond to each other in these congruent triangles. Yet Euclid choses to arbitrarily limit the generality of his theorem, and introduce arbitrary specificity and asymmetry. You’d think that would be anathema to a mathematician.
But if we think of the downsides of the way he formulated the side-angle-side case, we can understand why he went with this non-general formulation in the side-side-side case. Any time you are going to apply the side-side-side theorem, you probably want to conclude something about a specific angle, not all three angles of a triangle. So if you formulated the theorem generally, then every time you applied it you couldn’t stop yourself of course from reciting the entire chorus and hence you would end up with one conclusion that you actually needed, about one angle, and then needless spelling out two other conclusions about the other two angles that you don’t want at all. So this way you will only clutter your proofs with needless and irrelevant remarks. So the strangely specific, non-general formulation of the side-side-side theorem is actually well chosen given this constraint that you have to repeat the full theorem verbatim any time you apply it.
It’s pretty fascinating, I think, how textual aspects that appear to be purely technical and mathematical, such as a few barely noticeable superfluous bits of information in the proof of Proposition 5, can open a window like this into an entire cultural practice. The oral tradition must have been there, and the best proof of this is hiding in the ABCDs of Euclid’s formal text. It’s the beauty of history that historical texts can be read on so many levels. They carry so much hidden information about the culture that produced them. You would think Euclid’s ultra-formalised proofs would be the last place to find such clues, but here they are. We’re just a few proposition into the Elements and from the smallest technical quirks we have already recreated a rich picture of the ancient singing geometers and the strange culture in which they worked.
