That which has no part: Euclid’s definitions

Euclid’s definitions of point, line, and straightness allow a range of mathematical and philosophical interpretation. Historically, however, these definitions may not have been in the original text of the Elements at all. Regardless, the subtlety of defining fundamental concepts such as straightness is best seen by considering the geometry not only of a flat plane but also of curved surfaces.

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“A point is that which has no part.” What a bonkers way to start a book. But that’s Euclid for you. Let’s start the whole thing off with a negative, Euclid apparently told himself. He’s like: Let me tell you what a point is. Think of things that have parts. It’s not that. It’s the other stuff. Stuff that doesn’t have a part. Pretty weird that the first thing you introduce is actually defined by exclusion, in terms of what it is not. But anyway, never mind that. There are important interpretative issues at stake here.

The first two lines of Euclid’s Elements are the most misunderstood. They define the concepts of point and line. “A point is that which has no part” and “a line is a length without breadth.” We might interpret this as saying that a line is 1-dimensional, and a point is 0-dimensional.

Here’s how people misunderstand this. They say: Aha, told you! Geometry is not about physical things; it’s about objects in some ideal realm, just like Plato said. Because if you draw a line with a pen for example, it will always have some breadth, no matter how thin it may be. No physical object can ever be a “breathless length.” This proves that Euclid is not talking about physical space.

But that is a terrible argument, which makes no sense. It is demonstrably false. Yet you hear it repeated again and again. Some ancient philosophers made this argument. Aristotle mentions it in the Metaphysics (998a). Still today, many modern scholars walk into this fallacy all the time. But don’t worry, I’m here to save you from this mistake.

There is no inconsistency between Euclid’s definitions and a physicalist view of geometry. On the contrary, these kinds of idealisations are an essential part of any physical theory. Ptolemy, the astronomer, treats the moon as a point for the purposes of many of his demonstrations, for instance. Obviously no one would infer that he is therefore believes the moon is a mathematical point with no extension. The convention of treating the moon as a point is simply a common-sense idealisation that is the only sensible thing to do for many mathematical purposes, regardless of what one’s estimation of the actual body of the moon may be.

It is the same for instance in Archimedes’s work on levers, where the lever arm is a weightless mathematical line and the weights are applied at mathematical points. Since such idealisations are unequivocally used all the time without further ado in applied mathematics, it makes no sense to take them to be inconsistent with a physicalist view of geometry. On the contrary, such idealisations are exactly the standard assumption one would expect in physicalist geometry, just as one invariably finds it any other mathematical theory pertaining to the real world.

So if this argument is right, that Euclid’s definitions prove that his geometry is divorced from reality would, then it is equally true that the Greeks did not intend their astronomy or their statics to apply to the real world either, which is obviously absurd. So it’s madness to infer from Euclid’s definitions that he thinks geometry is non-physical.

It is more plausible to read these definitions as specifications of idealisations made in geometry, rather than as claims about the ultimate nature of geometrical objects. Indeed you can find support for this in ancient sources. Heron, for example, clearly takes such a view. He writes:

“Already in ordinary language use we have the notion of a line as something which has only length, but not at the same time width and thickness. For we say: a road of 50 stades, as we concern ourselves with the length only, but not at the same time its width.”

Here the identification of geometry with everyday physical objects is evident. The allegedly Platonic or ontological aspects of the definitions is merely a common-sense matter of simplifying assumptions and directing attention only to the relevant aspects of the situation.

Proclus makes the same point as Heron. He also uses the example of a road. And he attributes this view to “the followers of Apollonius.” In other words, Proclus puts this view right at the mainstream of Greek geometry at its peak. Apollonius is at the heart of the mathematical establishment. Heron was also a mathematical author. So mathematicians were the ones who thought a road was a good example of a line. Meanwhile, those who tried to use Euclid’s definition to drive a wedge between mathematics and physical reality were philosophers.

It’s typical, of course, that philosophers focus on the first two lines of Euclid and try to dismiss the relevance or status of geometry on that basis. Perhaps they never made it past the first page of the Elements. How convenient that they immediately found an excuse to dismiss geometry based on the first two definitions. How convenient that their objective analysis just happend to justify ignoring all technical mathematics.

Such a motivation is quite transparent in at least one of these philosophical authors, Sextus Empiricus. He gives probably the most extensive articulation of this idea that the first definitions of Euclid undermines the credibility of mathematics. The very title of his work is Against the Mathematicians. “The mathematicians talk idly,” he accuses, “for the straight line shown to us on the board has length and breadth, whereas the straight line conceived by them is ‘length without breadth’.” Gotcha, huh? You can decide for yourself if you think Sextus Empiricus is a razor-sharp philosophical mind who has outsmarted all the mathematicians, or whether he’s a guy who doesn’t like mathematics and wants to rationalize his own ignorance.

Those of us who read Euclid beyond the first page quickly realize that there is a further compelling argument for why one must not make too much of the alleged ontological import of Euclid’s definitions of point and line. Namely, that these definitions are the most extraneous part of the Elements.

The Elements is obviously a very carefully constructed logical theory, where almost every statement is carefully formulated to correspond precisely to the justification of specific inferences in deductive proofs. Obviously postulates and propositions are of this type, and so are many definitions, such as the definition of a circle which is used already in the very first proposition to infer that since two line segments are radii of the same circle, they must be equal.

However, the definitions of point and line are not of this type. These definitions serve no direct role in the deductive structure of the theory. They are effectively ornamental. They are arguably the most inconsequential parts of the entire Elements, since they are never actually used in any proof. Yet these are the very lines always cited as virtually the only textual evidence in mathematical sources of alleged anti-physicalist tendencies in Greek geometry. Madness.

In fact, these definitions may not even have been part of Euclid’s original text of the Elements at all. The version of the Elements we have has been edited, unfortunately. When Euclid wrote it, it was a sophisticated analysis of the foundations of geometry. It’s readers were high-level mathematicians. Later it became a textbook for schools. Editors interfered to make it more accessible. Possibly adding the first couple of definitions for example.

This is especially clear with respect to Definition 4 of the Elements, the definition of a straight line. Here’s what it says: “A straight line is a line which lies evenly with the points on itself.” This definition is meaningless drivel. What does it even mean to “lie evenly with itself”? How can such a masterful work, which is clearly written by a top-quality mathematician, open with such junk?

There’s a compelling answer to this conundrum, proposed by Lucio Russo. It goes as follows. Euclid didn’t define “straight line” at all. His focus was on the overall deductive structure of geometry, and for this purpose the definition of “straight line” is essentially irrelevant. Indeed, the utterly useless Definition 4 is never actually used anywhere in the Elements.

Archimedes agreed with Euclid as far as the Elements were concerned, but in the course of his further researches he found himself needing the assumption that among all lines or curves with the same endpoints the straight line has the minimum length. He therefore stated this property of a straight line as a postulate in the context where it was needed.

The Hellenistic era, which included Euclid and Archimedes, was one of superb intellectual quality. Unfortunately it did not last forever.

Heron of Alexandria lived about 300 years later. This was a much dumber time. Fewer people were capable of appreciating the great accomplishments of the Hellenistic era. But Heron was one of the best of his generation. He could glimpse some of the greatness of the past and tried his best to revive it.

To this end Heron tried to make Euclid’s Elements more accessible to a less sophisticated audience who didn’t have the background knowledge and understanding that Euclid’s original readers would have had. He therefore wrote commentaries on Euclid, trying to explain the meaning of the text.

To these new, more ignorant readers of geometry, it was necessary to explain for example what a straight line is. Heron realised that Archimedes’s postulate about the line as the shortest distance captured well the essence of a straight line. However, it is not itself suitable as a definition, because a line should have the property of being the shortest distance between any two of its points, not be phrased in terms of only two fixed points, as Archimedes. Remember, Archimedes was not trying to define a straight line, only to make explicit an assumption about straight lines that was particularly relevant in a particular work of his.

To adapt Archimedes’s idea into a definition, Heron therefore explained that, and now I quote him: “a straight line is [a line] which, uniformly in respect to [all] its points, lies upright and stretched to the utmost towards the ends, such that, given two points, it is the shortest of the lines having them as ends.”

In this passage, the phrase “uniformly …” obviously refers to the universality of the shortest-distance property. The point of this phrase is to highlight that this property applies to any two points on the line.

This is what later becomes Euclid’s phrase “evenly with the points on itself.” The original purpose of this phrase was to say that the distance-minimization property of the straight line holds for any pair of points on the line: that is to say, the property holds “uniformity” or “evenly” across the entire line. Not only for the endpoints.

The definition in the Elements is a mutilated version of what Heron said. Heron’s point is that no matter which two points on the curve you pick, the straight line is always the shortest path between them. The mutilated version ignores the part about shortest distances, and distorts the part about it applying across all points into the vague phrase about evenness of all points.

How did that happen? To understand this we need to fast-forward another 300 years. Intellectual quality has now plunged deeper still. Geometry is in the hands of rank fools. Euclid’s Elements, which was once written for connoisseurs of mathematical subtlety, is now used by schoolboys who rarely get past Book I and “learn” that only by mindless rote and memorisation. A time-tested way (still in widespread use today) to “teach” advanced material to students who do not have the capacity to actually understand anything is to have them blindly memorise a bunch of definitions of terms.

In this context, therefore, there is a need for an edition of the Elements which includes many definitions of basic terms, which must be short and memorisable, and which don’t need to make mathematical sense.

In this era of third-rate minds, some compiler set out to put together an edition of the Elements that would satisfy these conditions. Heron’s commentary on the Elements is appealing in this context since it affords opportunities to focus on trivial verbiage instead of hard mathematics. But Heron’s description of a straight line is still too complicated. It’s too long to memorise as a “soundbite” and the mathematical point it makes is moderately sophisticated.

The compiler therefore makes the decision to simply cut off Heron’s description after the bit about “uniformly in respect to [all] its points.” This solves all his problems: the definition becomes shorter and easier. The only drawback is that the “definition” becomes utter and complete nonsense. But since the whole purpose of it is nothing but blind memorisation anyway this doesn’t matter anymore.

This is how the ridiculous Definition 4 ended up in “Euclid’s” Elements. It’s a mutilated version of what was once a very good definition. According to Russo’s hypothesis, which is compelling.

As Russo also observes, in the works of other great Greek mathematicians such as Archimedes and Apollonius (who “belong to the same scientific tradition” as Euclid) “there is nothing analogous to the pseudo-definitions of fundamental geometrical entities contained in the Elements. The introduction of terms implicitly defined through postulates is instead frequent.” So this supports the hypothesis that the Elements was corrupted due to its association with introductory teaching. While these more advanced works remained less tampered with.

If we want a definition of a straight line consistent with Greek geometry, I would propose defining it as follows: a straight line is the path of a stretched string. In other words, a straight line is a curve that doesn’t change shape when you pull its endpoints.

This is closely related to the notion of the shortest distance between two points. Related, but not equivalent. To get to the bottom of the notion of straightness it is useful to consider not only the usual plane but also other surfaces. Euclid’s geometry is the geometry of a flat plane, a flat piece of paper so to speak. Other surfaces have other geometries. A cylinder, for instance, like a Pringles can. It has its own geometry. Pringles lines, Pringles triangles.

To appreciate the geometry of a surface we should forget for a moment that it is located in three-dimensional space. We should look at it through the eyes of a little bug who crawls around on it and thinks about its geometry but who cannot leave the surface and is unaware of any other space beyond this surface. Think of for example those little water striders that you see running across the surfaces of ponds. They know the surface of the pond ever so well. They can feel any little movement on it. But they are quite oblivious to the existence of a third dimension outside of their surface world. This makes the water strider an easy prey for a bird or a fish that strikes it without first upsetting the surface of the water.

It is instructive to think about the intrinsic geometry of surfaces in this way. It forces us to realise that many things we take for granted as “obvious” objective truths in geometry are really a lot more specific to our mental constitution and unconscious assumptions than we realise. In some ways we are as ignorant of our own limitations as the water strider.

Let’s transport ourselves into the cylinder world to practice seeing geometry from a different point of view. On a cylinder there are stretched-string curves that are not the shortest path between its two endpoints. Wrap a shoelace around a Pringles can. You can make various spirals that are stretched strings. Or a helix as it’s called, a corkscrew curve. So these are straight lines, according to my definition. But they are not the shortest distances between their endpoints. Even if you have to stay on the surface of the cylinder, you can still get from one endpoint to the other more directly than by a spiral that winds around and around an excessive number of times.

So “stretched string lines” and “shortest distance lines” are not the same thing, as this example shows. It is arguably the stretched string that gets it right. It makes straightness a “local” property.

We can alter the distance characterisation of straightness to be local too. Then we would say: a curve is a locally shortest path if, for any given point on the curve, there is a neighborhood around that point such that the distance along the curve between any two points on the curve in that neighborhood is the shortest possible distance between those points. This picks out the same straight lines as the stretched string definition. Being a stretched string is the same thing as being a locally shortest path: it’s the shortest path between points on the line when you zoom in, but not necessarily between points on the line that are far apart.

Straight lines can also be defined as curves possessing half-turn symmetry about every point: a curve has half-turn symmetry if, for any given point P on the curve, there is a neighbourhood around that point such that when this neighbourhood is rotated about P by half the angle-measure around P then the curve ends up on top of itself. More loosely, a curve is straight if it always “cuts angles in half”; it “leaves the same amount of space on either side.” To test for this kind of straightness on surfaces one can use the “ribbon test”: if a ribbon or band can be laid flatly along the curve without creasing on either side, then the curve is straight.

Try it on your Pringles can. You can use a measuring tape for instance, for instance those free paper ones you can get at hardware stores or furniture stores. That’s your “ribbon.” Try wrapping it around the Pringles can. Some ways of wrapping it makes it lay flat against the surface; those are straight lines. Other ways of wrapping it makes it crease up on one side or the other; those are not straight lines because they don’t leave the same amount of space on either side.

Here’s a fun thing to investigate and think about. We have now defined straight lines on a Pringles can in two different ways: one in terms of a stretched string, like a shoelace, and one in terms of a flat ribbon, like a measuring tape. Are they the same? Are there some lines that are “shoelace-straight” but not “ribbon-straight” or the other way around? I’ll leave that to you to explore.

So we have two notions of straightness, and both of them get at something very fundamental:

The stretched string highlights the idea of straightness as minimization, or as a tight fit. This idea is reflected in many real-world occurrences of straightness. For instance, the path of a cross-Atlantic flight. You know that when you look at the path on a map, in the flight tracker, it looks curved. It looks like you’re flying from Paris up toward the North Pole, and then back down again to get to New York. Why not go “straight across” instead? Of course the path is in fact straight. It looks curved only because the map is an imperfect representation. If you have a globe you can stretch a string between Paris and New York and feel for yourself that the shortest path indeed goes “up” toward the North Pole. But that path is straight, according to the stretched string definition.

But we also have the second idea of straightness: that of straightness meaning “the same amount of stuff on both sides.” This is also reflected in various familiar situations. For instance, when you fold a piece of paper, the edge is straight. Why is that? This doesn’t have to do with stretched strings or least distances. Instead it has to do with the sameness of both sides. To fold something you match up points on one side with points on the other. Folding is only possible if the two halves are precisely equal.

There is also a kind of three-dimensional version of this. Namely the axis of rotation when a solid body is rotated. For example a döner spit at a Middle Eastern restaurant, or a basketball spinning on your finger tip. The axis of rotation is a straight line. Why? This is again because of sameness on all sides. The moving parts have to fit into each others’ space. So they have to be equal on either side.

Here’s an example from engineering. Mirrors are made flat by rubbing two of them against each other face-to-face, with a fine sand or other polishing agent applied between them. This too embodies the idea of flatness or straightness as equivalent to sameness on both sides.

Another example is rowing a boat. You go straight in a rowboat if you apply equal force to each oar. This is again symmetry-straightness, not stretched-string straightness. It’s not built into the very rowing process that this necessarily corresponds to the shortest distance between the endpoints of the journey. But it is built into the very act of rowing this way that you leave equal amounts of space on either side.

Light rays are straight. But this is more like the stretched string again. Light “cares” about minimizing the time of travel, so to speak. Just like the airline. The airline stretched a string across the globe to find out how to fly from Paris to New York. They also tightened their purse strings, so to speak, with the same move, because the shortest path is also the cheapest path. Light is a bit of a penny-pincher too, it would seem; or it is impatient, perhaps. Because it chooses the quickest path. For instance, if it has to go from point A to point B via a flat mirror, then it chooses to bounce off the point on the mirror that makes the total distance as short as possible.

You can reproduce this path with a stretched string. Suppose A and B are two points on a wooden table. Let’s hammer two nails into those points. One of the edges of the table we regard as the mirror. Take a vertical metal bar and put it against the edge of the table. Now wrap a string from A, around the metal bar at the end of the table, and then to B. Now pull the string as tight as you can. The metal bar forces the string to go to the edge of the table and back. But the bar can move along the edge of the table. When we pull the string we force the bar into a particular position, namely the position that minimizes the total distance. The path of the string is the same as the path of light between these points via a mirror at the edge of the table. You can try it out with a laser pointer if you don’t believe me.

So light is like stretched strings. Indeed artists use this sometimes. The pull strings to simulate light rays in order to get vantage points and perspectives just right.

I’m trying to emphasize with these examples how thinking about what straightness means is connected to many aspects of culture and experience. Isn’t it fascinating how the mathematical notion of straightness is a sort of root of all these diverse phenomena? Once you’ve read the Elements you see geometry everywhere. Flight paths, döner spits, spinning basketballs, light and mirrors, rowboats, Pringles cans––henceforth, anytime you encounter these things you will go: ah, of course, that reminds me of Euclid’s Definition 4!

The idea of straightness as corresponding to stretched string also generalizes well to other surfaces that are not homogenous. So far we have mentioned the plane, the cylinder, and the sphere. These surfaces are all homogenous in that every point is alike. If you cut out a piece of the surface, it fits on top of any other part of the surface.

Some surfaces are not like that. For example, the surface of the human face. It has regions of different curvatures, as we say. A flat piece of paper has zero curvature: it’s not curved at all. A ball has positive curvature: it curves the same way in all directions. A saddle has negative curvature: it curves in different ways in different directions. A saddle for riding a horse. It curves “upwards” along the spine of the horse, and “downwards” where your legs go. Opposite directions of curving. This is what makes the curvature negative.

The human face has both negative and positive curvature. Some parts are like a saddle. For instance the side of the nose, or the area just below your mouth. If you put your finger there and run it top-to-bottom, then it curves one way. But if you ruin it side-to-side, it curves the others way. So those are regions of negative curvature. They are like a saddle.

Other parts of the face have positive curvature, like a ball. For instance the chin and the cheeks. There the surface curves the same way no matter which direction you run your finger.

Felix Klein, a 19th-century mathematician, thought this might be the key to a mathematical analysis of the elusive concept of human beauty. Since the face has regions of positive curvature and regions of negative curvature, there’s a diving line running between them. Between the cheek and the nose, between the lips and the chin, and up again on the other side.

So Klein drew this line of zero curvature on a classical sculpture. You can google it, Felix Klein Apollo Belvedere, and you can see photos of this. Klein was hoping that a simple pattern would emerge that would “explain” the beauty of this face. But it didn’t work. No such pattern was discernible.

Still it makes for a good story. It’s also a good piece of “first date mathematics.” You can explain this idea to your date over some glasses of wine. And of course slowly reach out and sensually trace these curves on their face and so on. Great stuff.

But where were we? I wanted to discuss how the notion of straightness extends to these other surfaces. Surfaces with variable curvature. We can still say that straight lines are stretched strings. We often call them geodesics rather than straight lines in such cases. But the stretched-string idea is still the same.

Here are some examples. Think of bandaging an injured limb. The bandage needs to be tightly wrapped. This means that it must follow a geodesic path, a stretched-string path. The bandage is a “straight line” in the sense that it is a stretched string. In other words, it always takes the locally shortest distance. Of course not the shortest distance overall, since it wraps around and around. But the shortest distance between any two nearby points on its path, because otherwise it would create slack which you would never do of course.

Another example: The heart beats through the contraction of muscular threads across its surface. These muscular threads must be geodesics. They must be stretched-string paths. Because the heart beats by contracting these threads. If these muscular threads were not positioned along geodesic paths, then when they contracted they would just slide around on the surface of the heart instead of contracting it. The human heart is carefully designed with this geometry in mind. And if it wasn’t we would all die very quickly. So the stretched-string notion of straightness is truly a matter of life and death.