Created equal: Euclid’s Postulates 1-4

The etymology of the term “postulate” suggests that Euclid’s axioms were once questioned. Indeed, the drawing of lines and circles can be regarded as depending on motion, which is supposedly proved impossible by Zeno’s paradoxes. Although whether these postulates correspond to ruler and compass or not is debatable, especially since Euclid seems to restrict himself to a “collapsible” compass in Proposition 2. Furthermore, why did Euclid feel the need to postulate that “all right angles are equal”? Perhaps in order to rule out non-flat surfaces such as cones.

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Opinionated History of Mathematics

Transcript

Euclid’s Postulate 4 is super weird. It says: “all right angles are equal.” What kind of a postulate is that? 90 degrees equals 90 degrees? A right angle is equal to itself?

Why would you need to state that as an axiom? And if you do need to state it as an axiom, why only right angles? Why wouldn’t you need other axioms starting that various things are equal to themselves? 10 degrees equals 10 degrees, 1+1 equals 2: Why don’t we need axioms like those? What’s so special about right angles? Why do they need to be singled out like that, in their very own postulate?

But Euclid knew exactly what he was doing. His postulate only appears crazy and weird. There’s a way to make sense of it. We must reconstruct the original context and intent of the postulate.

I say: Euclid included this postulate in order to rule out cone points. I will explain what I mean by this. But let’s note first the historical methodology we are using here.

There is nothing in Euclid, and in fact nothing anywhere in any ancient source, that actually says that this was the intent of the postulate. The interpretation that the postulate has to do with cone points is purely a hypothetical reconstruction by historians, formulated thousands of years after Euclid.

Yet the reconstruction is so convincing. It just has to be right. If it’s right, everything fits; everything makes sense. If it’s not right, then we can’t explain the postulate, and we just have to assume that Euclid hadn’t really thought it through all that carefully and just put the postulate down on kind of whim or whatever and it doesn’t mean all that much in the greater scheme of things.

This is the difference between a great text and an average one. Great texts in intellectual history, like Euclid’s, reward reflection. If something seems weird it’s because you haven’t understood it. There’s a reason behind every step of the text. The text is the tip of an iceberg. It’s built on a huge body of supporting thought. This is why the text rewards reflection. The text is not just whatever popped into Euclid’s head. It is the fruit of an intellectual culture where these ideas had been scrutinized and criticized forwards and backwards and inside and out.

This is why you should read great texts like Euclid’s. These are the kinds of texts that, every time you dig into even the weirdest parts you realize that, huh, that’s actually a good point. The more you probe the text, the more compelling it becomes. It’s the mark of a great text that when you scrutinize an apparent weakness, it turns out to be a strength.

Euclid’s right angle postulate is an example of this. It looks silly and weird at first sight, but when we think about it, it opens our eyes to new and unexpected perspectives and insights.

So, “all right angles are equal,” what’s the deal with that? First of all, what does “right angle” mean? Euclid defines it in Definition 10. Draw a line. Consider the space on one side of the line. Cut that space in half with another line. That’s a right angle. A right angle is half the space on one side of a line.

So what does “all right angles are equal” mean? It means: Suppose you have made a bunch of right angles. That is to say, you have drawn various lines and then cut the space on one side in half. So you have a piece of paper full of what looks like a bunch of copies of the letter capital T. There are a bunch of T’s scattered across the paper, at random angles and positions. “All right angles are equal” means: if you cut out one of those T’s and put it on top of one of the other ones, then it fits. All the different T’s align perfectly with each other, as far as angles are concerned.

So the right angle postulate is really a kind of homogeneity postulate. It effectively says that no part of the paper is different than any other. The space on the side of a line is the same anywhere.

A cone is an example of a surface where that is not the case. Like an ice cream cone. The cone is non-homogenous. It has an exceptional point, the cone point, the apex of the cone, which is different from the other points.

Euclid’s postulate is false on the cone. A right angle at the cone point is smaller than a right angle elsewhere.

You can see this if you think about how a cone is made. You can make a cone like this. Start with a circular piece of paper. Then cut out a wedge from it, like a pizza slice. Then grab the two sides of the cut and pull them together. Now you have a cone.

Think about the amount of space around each point. Most points are surrounded by the same amount of space as they were originally, on the paper we started with. 360 degrees’ worth of space, so to speak.

But the cone point is different. It’s surrounded by “less space” or “fewer degrees” than before. The pizza wedge you cut out took away some of the angle sum around this point. Not so for any other point. Even the points along the sides of the cut are still normal. They lost 180 degrees, but then you pasted another 180 degrees right back in. So they are back to normal. But not the cone point. It lost some of its angles and never got them back.

So the right angle postulate is false on the cone because right angles are smaller at the cone point. Since right angles are defined in terms of cutting the space on the side of a line in half, then if there’s less space around some points compared to others, then the right angles there will be smaller too.

I believe this is what Euclid had in mind when he wrote his postulate. We can’t prove that this is what Euclid meant, but it is the most satisfying explanation.

Here’s a little cultural sidelight. The Declaration of Independence of the United States starts in a kind of Euclidean manner. It says: “We hold these truths to be self-evident,” and then it lists a number of “truths” the first of which is “that all men are created equal.”

So the Declaration of Independence has self-evident axioms just like Euclid, and they sound the same too: “all right angles are equal”; “all men are created equal.”

That’s no coincidence. The founding fathers of the United States were obsessed with antiquity. They used the ancient world as a model all the time. As a model for their political system, of course. The senate, for instance, is straight up copied from Rome, and so on with many other things. Euclid was part of that package as well. A very conscious revival of ancient enlightenment.

So the founding fathers of the United States called their axioms “self-evident.” And of course many people have interpreted Euclid that way too. You don’t have to prove the postulates because they are immediately obvious. You can draw a line from any point to any point: yes, of course you can, it’s too simple to even prove, but it’s impossible to doubt. That’s one way to think about Euclid’s postulates. A common way.

But one could argue that it’s a bit more complex. This is suggested even by the word that Euclid uses: postulate. These simple and self-evident starting points are called postulates. But this term doesn’t suggest that these things are self-evident or impossible to doubt. To postulate is more of a demand or a request. So the term doesn’t seem to take assent for granted but rather the opposite: it seems to imply that some people might oppose these statements, no matter how obvious they might seem.

How could anybody deny that you can draw a line from any point to any point? In fact, some people in Ancient Greece did deny this, and they were not crazy; they had some very compelling reasons.

A useful book on this is The Beginnings of Greek Mathematics by Arpad Szabo. I will summarize it for you.

“Mathematics grew out of the more ancient subject of dialectic”—that is to say, philosophical debate. Just as we discussed before, the argumentative Greeks, they loved debating. Two philosophers passionately disagreeing and trying to poke holes in each others’ arguments in a lively disputation before an audience: that was their idea of a good time. Instead of “dinner and a movie” you would go to a philosophy debate.

So that’s “dialectic”—a debate with two warring sides. Terms such as axiom, postulate and many others seem to have originated in this setting. These terms were imported into mathematics from dialectic. Today only their mathematical meaning survives. Therefore to us these terms have rather different connotations than they did for the ancient Greeks. That’s how the idea that axioms or postulates are supposed to be self-evident has become associated with the terms even though that was not the original intent or meaning.

The terms axiom and postulate originally mean something like “concessions which the participants in a discussion have agreed to make.” “We know that the term aitema [=postulate] came from dialectic where it was used to denote a ‘demand’ about which the second partner in a dialogue had reservations.”

“Let us see whether there is any connection between this early meaning of the word and Euclid’s postulates. At first glance, Postulates 1-3 appear to be such simple, self-evident and easily fulfilled ‘demands’ that one is tempted to disregard the literal meaning of their name.” But no.

Euclid’s postulates arguably rely on motion. To draw a straight line from any point to any point: how do you do that? You put a ruler down and trace the line with a pen. The pen is moving: you put it at one point and move it to the second point. Same thing with circles: you draw them with a compass, which is also a moving instrument.

It’s quite possible to deny that such things can be done. In fact, you may have heard about the famous paradoxes of Zeno, which purport to prove that motion is impossible. One of them goes like this.

Suppose I have to walk from A to B. Before I can walk all the way to B, I first have to walk half the way to B. Then, when I’m at the halfway point, before I can get to B I have to walk half of what’s left. And so on. Whatever distance is left, I always first have to go half of it.

But this process never ends. There’s always “another half to go.” So to go from A to B you have to “do an infinite number of things,” so to speak.

You can think of it this way. When I have gone half the way from A to B, I say: one. Then when I have gone half again of what’s left, I say: two. I go half of what’s left: three. And so on. This implies that if in fact I can go all the way from A to B, I will have shouted out all the numbers that exist: one, two, three, four, five, … all of them.

So to say that you can go from A to B is to say that you can count through all the numbers in finite time. But of course you can’t. Nobody has ever counted through all the numbers. So therefore you can’t move either. Motion is impossible. It must be an illusion.

We only think we move. That’s feeble sensory “knowledge,” or so-called knowledge. We discussed before the extreme rationalistic tendency of Greek philosophy: reliance on pure deductive reason at the expense of all other forms of knowledge. Zeno’s paradox is an example of this. The senses say we can move, but deductive “reason” says we cannot.

We discussed before how the stage debate format incentivized philosophers to pick the side of reason in such cases, no matter how extreme and outrageous the conclusion may be. “All is water”, “all is fire”: the crazier the better. Proofs of radically unexpected conclusions is perfect for the stage debate setting.

Zeno’s argument is great way to dazzle an audience and to show how clever you are. Being reasonable and arguing that one can walk from A to B is boring. Who wants to hear that? You won’t become a blockbuster debate star by arguing for the obvious. You gotta have some signature absurdities that you claim to prove.

Zeno also had a second form of his argument that is equally amusing. Here’s how Simplicius describes it:

“The argument is called the Achilles because of the introduction into it of Achilles, who, the argument says, cannot possibly overtake the tortoise he is pursuing. For the overtaker must, before he overtakes the pursued, first come to the point from which the pursued started. But during the time taken by the pursuer to reach this point, the pursued always advances a certain distance; even if this distance is less than that covered by the pursuer, because the pursued is the slower of the two, yet none the less it does advance, for it is not at rest. And again during the time which the pursuer takes to clever this distance which the pursued has advanced, the pursued again covers a certain distance. And so, during every period of time in which the pursuer is covering the distance which the pursued has already advanced, the pursued advances a yet further distance; for even though this distance decreases at each step, yet, since the pursued is also definitely in motion, it does advance some positive distance. And so we arrive at the conclusion that not only will Hector never be overcome by Achilles, but not even the tortoise.”

So that’s another way to prove that motion is impossible. Those who believe in motion believe that Achilles can out-run a tortoise. But that contradicts reason, as we have just seen. Therefore those who believe in motion must be wrong.

Why did Zeno prove the same thing in two ways? Maybe he was just like: Hey guys, I thought of another funny one, it has a tortoise in it, I’m sure you’ll get a kick of it. Or is there more to it than that? Do Zeno’s two forms of the argument differ in substantial respects?

I think they are subtly different. You might say that the Achilles argument assumes the possibility of motion and derives a contradiction. It so to speak plays along with those who believe in motion for a bit, only to then trap them in a paradox.

The other argument—the dichotomy, or half half half argument—doesn’t really need to even presuppose motion at all. It derives the impossibility of motion more from the nature of length. It has more to do with the infinite divisibility of the continuum than with motion as such.

So in that respect the dichotomy argument is more “pure” as it were. Since it doesn’t need to use motion to refute motion.

But on the other hand it is less pure in another respect. It assumes metricity; that is to say, an absolute notion of distance. For the argument to work, it must be possible to talk about the half of something. But half involves quantification. You need to put a number on the full length before you can know what half of it is.

So Zeno’s opponents could say: Your argument doesn’t disprove my beliefs, because although I believe in motion I do not believe in metricity. I do not believe that numerical lengths can be assigned objectively to the paths between points. Therefore the whole business about halfs doesn’t work, and you haven’t really disproved motion after all.

If Zeno’s opponents tried to wiggle out from under the dichotomy argument along those lines, then Zeno could just hit them with the Achilles argument. Because the Achilles story doesn’t involve assigning numerical lengths to anything. It purely about relative positions: the tortoise is in front of Achilles. It doesn’t say by how much. The argument doesn’t need the notion of being in front to be quantifiable. It needs only relative positions. So in that sense the Achilles argument is the purer one.

Well, that’s fun to think about, but let’s get back to our original purpose. I brought up Zeno’s paradoxes because they are related to the issue of whether Euclid’s postulates are obvious or not.

“If we bear [Zeno’s paradoxes] in mind, it is easy to understand why Euclid’s first three postulates had to be laid down. They really are demands (aitemata) and not agreements (homologema); for they postulate motion [such as the motion of a pen that is drawing a circle], and anyone who adhered consistently to [Zeno’s] teaching would not have been able to accept statements of this kind as a basis for further discussion.”

So when Euclid is presenting his postulates, he doesn’t seem to be saying: surely you all agree with these statements; they are clear even without a proof. Instead Euclid seems to mean by postulate: these are assumptions that must be accepted for the sake of argument if we are to do geometry; if you don’t like them, then we just have to agree to disagree.

The same goes for Euclid’s Common Notions. “Our text of Euclid” has a separate heading called common notions, but this was not a well-entrenched term and these principles “obviously bore the name axioma in pre-Euclidean times,” and “the noun axioma, when used as a dialectical term, originally just meant a ‘demand’ or ‘request’.”

Indeed the common notions could be doubted. They “are assertions which are justified by practical experience and, in some cases, directly by sense-perception. [One of them] states that ‘things which coincide with one another are equal to one another’. It can literally be seen that plane figures which coincide are actually equal; hence this axiom is verified by sensory experience.” Therefore the common notions “could not have been accepted by [those] who required that all knowledge be obtained by purely intellectual means and without appealing to the senses.” And there were plenty of people like that. Just as Zeno’s argument implies: an extreme trust in purely intellectual reasoning, even when it goes flatly against even the most basic and immediate experience.

This is why “these principles were originally called demands (axiomata): because the other party in a dialectical debate had reservations about accepting them as a basis for further inquiry or, in other words, because their acceptance could only be demanded.” People would not have agreed that these things were self-evident; that’s why they had to be “demanded”, or postulated.

There’s yet another way to criticize Euclid’s principle that “things which coincide with one another are equal to one another.” Not only does it rely on sense evidence, it is also arguably conceptually incoherent. If “two” things coincide and are equal, doesn’t that mean that they are actually one thing? Does it even make sense “to speak of two things unless they can be distinguished from one another“?

So we see how Euclid’s axioms can be questioned in various ways. The Greeks loved to quarrel. Mathematics was born in this kind of climate. Everybody criticizing everything, trying to poke holes in it.

So that’s why Euclid’s text starts with “demands.” Many later readers were happy to accept them as self-evident, but Ancient Greek geometers could not have expected to get away with that.

So the terminology of “postulates” and “axioms” points to this ancient context. But the meaning of the terms morphed over time. In the very early days, mathematics lived within the dialectical tradition and was a subordinate part of it. But mathematics took on a life of its own and soon outlived dialectic.

Soon “the essentials of [the old] dialectic [context] were no longer very well understood; hence the ancient term axioma acquired a new meaning. Since it had always been used to refer to a group of principles which, from the viewpoint of common sense, were evidently valid, it came now to denote those statements whose truth was ‘accepted as a matter of course’.”

So that American phrase—“we hold these truths to be self-evident”—is perhaps not as Euclidean as Jefferson and those guys thought.

Here’s another interesting aspect of Euclid’s postulates. The first three postulates basically state that lines and circles can be drawn. That is to say, lines and circles can be taken to exist. That’s a primitive assumption of geometry.

Lines and circles are so to speak the Adam and Eve of geometry. In the beginning there are only these two, these male and female generative principles. You couldn’t get very far with just one of them, but together they combine to make rich offsprings. They eventually populate the entire Euclidean universe. Everything that ever happens in Euclid’s world comes from these two parents, the line and the circle.

The line and the circle are also embodied in physical tools: the ruler and the compass. To what extent is that important? Is this physical realizability important to the credibility of these postulates? Or is Euclid merely talking about lines and circles in the abstract, and it’s just a coincidence that they correspond to physical tools?

There is no simple answer. Euclid’s text is ambiguous in this respect. You can read it either way.

Insofar as we can say anything about what Euclid meant in this respect, we must infer it from the technical material later in the text. Euclid never tells us: “here’s my philosophy.” We can only read his proofs and ask ourselves what implicit assumptions appear to be made and what implicit philosophy might have guided the particular choices Euclid makes in technical arguments.

Already Proposition 2 is very interesting in this regard. Euclid shows in Proposition 2 how to transfer a length from one position to another, using only his postulates about line and circle, or ruler and compass.

In other words, somebody has drawn a line segment on a piece of paper, and now you want to draw an equally long line segment somewhere else on the paper.

Euclid accomplishes this by a very elaborate construction. It involves drawing numerous circles and an equilateral triangle. Very elegantly, this leads to exactly what you need: the given segment has been reproduced in the new position, with exact mathematical precision.

That’s all very neat, but it’s also weird, isn’t it? It seems totally out of touch with reality. If a craftsman or engineer or architect would need to transfer a length, surely they would not use Euclid’s absolutely baroque procedure.

First of all you might say: just use a ruler. Measure the given length. It’s so-and-so many centimeters. Then put the ruler wherever you want the length to go, and mark off the same number of centimeters there. Done. No need for drawing a bunch of circles and god knows what else.

Why doesn’t Euclid accept this and save himself some time? Actually it’s not so crazy. In a way you might it’s a mistake to think that length lives in the ruler. Actually, out of the two “parents” ruler and compass, length comes from the DNA of the compass, not the ruler.

We are so used to working with rulers, measuring things with rulers. It’s the prototypical manifestation of length. But think about it. Where do rulers come from? How do you make a ruler? How do you put the centimeter marks on it?

You do it with a compass. You set the compass to a fixed opening, and you mark off the size of that opening repeatedly along the ruler. Can you feel it? You start with a blank ruler, just a straight piece of wood. Now you take your compass and make it so to speak “walk” along the edge of the ruler. Left foot, right foot, left foot, right foot. The places where the compass “stepped” so to speak become the marks of the ruler. So when you use a ruler to measure things, you are really relying on the compass. Length is born from the compass.

This suggests that Euclid is on to something when he involves circles in his proof of Proposition 2. But it still doesn’t explain why it has to be quite so complicated.

A compass can solve the problem directly. Just open the compass to the length you want to move, then lift it and put it back down wherever you want the length to go. The length you wanted in the new position where you wanted it is directly manifested in the form of the distance between the two legs of the compass. Piece of cake. There’s nothing to it. You can move lengths directly with the compass without any hassle.

Euclid acts as if this is not possible. One might say that Euclid behaves as if his compass is “collapsible”: it stays at a fixed opening while drawing a particular circle but as soon as it is lifted from the paper it “collapses,” or closes up, so that the opening to which it was set is lost and cannot be used elsewhere.

Of course there are no collapsible compasses. It’s not a real thing. So you might say: aha! This proves that Euclid is in fact talking about lines and circles abstractly, maybe in the manner of Plato and his world of ideals. From that point of view Euclid’s proof is not problematic. It’s a dazzling intellectual construction. Great stuff. Hopelessly impractical, to be sure, but that’s just all the better of course as far as Plato is concerned.

Meanwhile, if you want to say that Euclid’s postulates correspond to actual rulers and compasses, then you have to bend over backwards and make up stories about “collapsible compasses,” which don’t exist.

So it seems we have a clear winner. Only the abstract, non-physical reading of Euclid makes sense.

But I’m not so sure. Maybe “abstract versus physical” is the wrong lens to use here. We can also make sense of Euclid’s peculiar proof from a different point of view that is independent of this issue of physical versus abstract.

This point of view is: assumption minimalism. Euclid masterfully reveals the minimum assumptions necessary for geometry. Remember: reduce, reduce, reduce. That seems to be Euclid’s mantra. That’s the philosophy of “reading backwards.” If you can avoid an assumption, then you should avoid that assumption.

This kind of minimalism or purism doesn’t depend on whether geometry is physical or abstract. Either way, if something can be proved rather than assumed, then that’s regarded as a win. This kind of reduction is about exploring and clarifying the ultimate foundations of geometry and the bedrock source of geometrical knowledge. It is applicable regardless of whether geometry is physical or abstract.

This perspective of minimalism demands that we do not allow lengths to be merely transferred directly by a compass. Even if we do think physical compasses are somehow important to geometry, we should still pursue this reduction. It is our duty to always reduce.

Just as a chemist reduces molecules to atoms. Of course molecules are great. The best level at which to explain many things is molecules, not atoms. But since they can be reduced, they must be reduced. It is our scientific duty to run the reduction as far as it goes. Of course we still retain the explanatory power of molecules. The reduction to atoms is just a supplement.

Maybe so also in Euclid. Maybe the physical compass should be seen as the operative tool throughout the Elements, just as molecules are the right level of analysis for many chemical phenomena. But even so it makes sense to show up front how it could, in principle, be reduced even further to more basic building blocks. We might say that Euclid does this in Proposition 2.

The fact that one can do away with the assumption that a compass can transfer length is an interesting foundational insight. Since Euclid can prove this, he does. This does not imply that he is opposed to the idea of a non-collapsible compass. One could simply delete Proposition 2 from the Elements and all the rest would still stand verbatim as a treatise about constructions with non-collapsible compasses.

So Proposition 2 can be viewed as an optional exercise in foundational minimalism within a paradigm otherwise fully based on physical compasses. Rather than as evidence of conceptions fundamentally at odds with such a physical point of view.

Analogous situations occur in modern mathematics all the time. For example, open any textbook on abstract algebra and turn to the definition of a group. The definition of a group says that any group has an identity element: anything multiplied by the identity stays the same. As far as this definition is concerned, there could potentially be several identity elements in any given group. However, all textbooks immediately proceed to show that the identity element is in fact unique. Other groups axioms imply that it must be unique.

These textbook authors could have made life easier for themselves by simply making the uniqueness of the identity element part of the definition. Then there would have been no need to prove it a separate theorem. But it is better to keep definitions and axioms as simple and minimalistic as possible, for instance in order to minimise the risk of inconsistency, or because proving properties instead of gratuitously including them in the definition illuminates fundamental relationships.

But note that one cannot infer from this that the uniqueness of the identity is somehow a secondary or less embraced aspect of the group concept. It is proven as a theorem rather than included in the definition solely because of the technical possibility of doing so, not because it was seen as less essential than the definitional group properties. This does not show that the fundamental conception of a group that mathematicians have in mind is ambivalent regarding the uniqueness of the identity. On the contrary, this is arguably a core aspect of the intuitive notion of a group that has, in itself, no less of a claim to being fundamental than the definitional properties. But if one tries to find the smallest set of key properties of a group to take as definitional, then one finds that uniqueness of the identity is a property that can most efficiently be made into a theorem.

In the same way, then, one might argue, Proposition 2 of the Elements does not show that Euclid’s fundamental notion of the circle-drawing constructions and postulates were divorced from a physical compass. It does not prove this any more than a modern textbook proves that the uniqueness of the identity is fundamentally divorced from the group concept.

Just as a modern algebra textbook would have nothing a priori against including uniqueness of the identity in the definition of a group, so Euclid may very well have had nothing a priori against assuming a non-collapsible ruler. Just as the modern algebra textbooks nevertheless arrives at the conclusion that it is better to make the uniqueness of the identity into a theorem because that enables the minimisation of definitional properties overall, so Euclid may very well have decided to assume only a non-collapsible ruler purely for reasons of axiomatic minimalism. If so, it would be a mistake to infer from this proposition that he didn’t care about physical tools like the compass.

Even if you’re not familiar with group theory I’m sure you have encountered a similar aesthetic elsewhere. For example, some people, when they cook pasta, they save a few spoonfuls of the cooking water and toss it into the dish. To make the pasta less dry.

I always thought it’s a little pretentious when TV chefs do this. Obviously you could achieve the same result various other ways. Instead of adding some of the cooking water, you could add other water, oil, make your sauce a bit runnier, etc. I’m sure nobody could tell the difference.

But it’s cool somehow to use the actual cooking water. It makes you feel creative and spontaneous. Almost spiritual: it’s like you’re in synch with the universe like some ancient Indian who lived in harmony with the land. Making use of everything, every part of the pig, even the cooking water. It takes skill and true understanding to use things for something other than their intended purpose. It’s a rock ‘n’ roll move. Anybody can cook the way it says on the tin, but I’m such a creative rebel that I use the very cooking water itself.

Euclid’s Proposition 2 is a bit like that. Of course you could accept the transfer of lengths as a separate assumption, or implied by the compass. But it’s cooler if you can do without it, and instead use what is already at hand in an unexpected new way. Euclid uses the cooking water, so to speak. He uses the assumptions from the postulates that were already necessary anyway. By cleverly combining these, he shows that you don’t need anything else. It’s satisfying in the same way the pasta trick is satisfying.

I think this is enough to explain why Euclid wanted to include Proposition 2. So we don’t need to attribute to Euclid any anti-compass agenda. It’s enough that he thought this was a cool trick.

So the question is still open then whether Euclid meant his postulates to correspond to ruler and compass or not. We will have to keep reading to find out more. Let’s do that.