kitchen table math, the sequel: Euclid's Postulates and Definitions

Wednesday, October 21, 2009

Euclid's Postulates and Definitions

Euclid's Elements has 13 books. Book 1 starts with definitions.

Here are the first few:

1. A point is that which has no part.
2. A line is a breadthless length.
3. The extremities of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.

The main problem is that modern language and his language are a bit confusing. We don't use "extremities".

The definition of straight line above might not be clear unless you already know what a straight line is.
But in total, Euclid defines: point, line, straight line, surface, plane, plane angle, right angle, perpendicular angle, obtuse angle, acute angle, boundary, figure, circle, center, diameter, semicircle, equilateral triangle, isosceles triangle, scalene triangle, square, rhombus, trapezoid, parallel.

The 5 postulates are (in his words)
1. [One can] draw a straight line from any point to any point
2. [One can] produce a finite straight line continuously in a straight line.
3. [One can] describe a circle with any center and distance.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, athe two straight lines, if produce indefinitely, meet on that side on which are the angles less than the two right angles.

In modern terminology, we usually list the first postulate as "two points determine a line". The fifth one is often rewritten to say "parallel lines don't intersect"; it's the most famous for being the one that makes the geometry described here planar--remove that postulate, and look at the lines of longitude on the globe.

So why are these postulates not definitions? The fifth one is complicated enough to be phrased as an "if-then", so it's probably reasonably clear why that's not a definition.

But why do we need to postulate a line or a circle, rather than define it?

Postulating a circle means we postulate its existence. We could just define a circle or a line, but it doesn't mean that such a thing with those properties as defined exists.

And we can't prove that those things exist, either. We have to assume them. This may be the oddest thing to a young student. "WHAT do you MEAN a circle doesn't EXIST? Look, I can draw one." (The standard answer to this is that any drawn circle is actually imperfect--doesn't have constant radius, doesn't have a boundary of no thickness, etc. This is typically a deeply unsatisfying answer. You could then tell him about the REAL constructivists: mathematicians who tried to construct all of math using the actual constructed "non platonic" geometry they could realize...but that's for another day.)

But this is where constructions come from, and this is what they were for. The answer to the above complaint is that you can actually CONSTRUCT everything you're asked to prove using just three things: a straight edge, a compass, and a pencil.

That is what it means to say "postulate a circle": it means "give yourself a compass to create a circle out of thin air." Same with "postulate a line: "give yourself a straight edge", and "postulate a line segment" : "give yourself a pencil to mark points on your line.

Then, the postulates have real meanings, because you can create everything else from those devices. Equilateral triangles are defined, but their existence is proved--and can be proved by construction from just the above three tools. Same with regular polygons, various congruent angles and lines, etc.

The idea that you start with 24 postulates is really really depressing, since it undermines everything beautiful about geometry that could be seen by construction.

Here's a link to Euclid's books. It contains all of the propositions and hyperlinks to the proofs.

Sometimes it's difficult to see the great truth he proved geometrically because of his wording and naming conventions. (Would you recognize his proof of the factoring of (x^2 -1) ?) But reading this might show you what you need to know and what's superfluous. And it will show you how to prove 19 of those postulates...


OrangeMath said...

You may find the Heath translation most appropriate. Commentary is needed because of small errors in Euclid (unstated axioms or definitions). For example, I had great trouble with "betweeness." You may find it useful to read the many comments to a book based on the Elements: before reading the Elements.

MagisterGreen said...

"The main problem is that modern language and his language are a bit confusing. We don't use "extremities"."

Neither does Euclid. The Greek word he uses is "perata", which is the plural of the word which means "end, limit, boundary." So you could just as easily say "The ends of a line are points." He uses this particular word to ensure his reader understands that he means the furthestmost points of any line, instead of some other points on the line.

I don't mean to be picky, but as a Latin teacher I always tell my students not to allow their English translations to influence how they understand the original language. Translations are so messy at times :) .

SteveH said...

"straight line"

I always thought this was redundant.

" can actually CONSTRUCT everything you're asked to prove using just three things: a straight edge, a compass, and a pencil."

The most vivid memory of geometry I have was my determination to find a way to trisect an angle. I found it, but since I had to do it with real instruments, I would first ask for an error tolerance.

Lsquared said...

Hmmm. As a geometry purist, I think I need to quibble with you about the proving of the postulates. Hilbert was good. There aren't any of his postulates that aren't needed. (Google Hilbert's axioms for geometry).

People who don't see the need for the extra postulates, just have a bunch of postulates that they are assuming but never prove. This includes Euclid. The clearest example of this is Euclid's proof of SAS. To do this he says: we can apply one triangle to the other so that two sides and an angle match up. He's assuming that you can pick up a triangle from one part of your "plane" and move it to another part of your "plane", and it will lie flat. (I put plane in quotation marks because it is an undefined term. A plane can be anything that satisfies the axioms.) Now, what Euclid is assuming is that a plane is homogeneous--that it's the same everywhere. That's a great assumption, but it needs a postulate to make it part of what something has to be like to have a plane. If you want an example of a "plane" that's not homogeneous, look around your house for a book that got wet , and the pages wrinkled. There's a part of the page that's still flat, and a part that's wrinkled. Now, if you cut a triangle out of the flat part, and tried to put it on a part of the page where it's transitioning from flat to wrinkled, it won't fit, because it doesn't have the right kinds of wrinkles (in mathematician-ese the part where the wrinkles happen is has negative curvature, and the place where there are no wrinkles has 0 curvature). There needs to be some way of saying a plane is the same everywhere, or we can't prove much about geometry: so we need a postulate to say this. The usual way (ala Hilbert) is to make SAS a postulate (you can prove the other congruence theorems from SAS, but SAS can't be proved from Euclid's axioms alone).

Saul Stahl did a very thorough cleaned up version of Euclid (staying very close to the original) in his book: Geometry from Euclid to Knots, in which he inserts just a minimal number of extra postulates, one of which he calls "Application" after Euclid, which fills in the gap in SAS, and another of which is "Separation" (that lines and circles and such separate the plane into two parts). Things which Euclid used but didn't put into his postulate list, perhaps because he didn't have anyone looking over his shoulder coming up with counterexamples.

Coming soon: trisections

Lsquared said...

Bleah--I need to find a better way to edit myself before I hit "publish".

People who don't see the need for the extra postulates, just have a bunch of postulates that they are assuming but never _state_.

Anonymous said...

but as a Latin teacher I always tell my students not to allow their English translations to influence how they understand the original language

Great advice. I only deal with two languages (English and Japanese), but I've long been of the opinion that there's really no such thing as "translation" per se, just interpretation.

I was always taught that lines are endless.


Michael Weiss said...

Euclid's geometry -- as distinct from what is called, in the modern day, "Euclidean geometry" -- is all about what can be drawn, and what property those drawn things can provably have. So "infinite lines" don't exist in Euclid. In Euclid's terminology, "line" means what we would call today "a curved segment", and "straight line" means what we would call today a "line segment". That's why Def. 3 states that the endpoints of a line are points -- every line has to start and stop somewhere. But Euclid also says that lines can be extended to make longer lines, and that this extensibility is limitless (Post. 2).

I disagree, slightly, with Allison when she says that the postulates ensure that circles and lines exist. On the contrary, one of the "improvements" in Hilbert's axioms is that they do ensure the existence of points, lines, and circles, whereas Euclid's don't. But that is not, in my opinion, a defect -- it is a feature. Euclid's first three postulates are stipulations of what kinds of figures may be produced from existing figures. In modern parlance, they specify a production system. So, if you have two points, you can produce a line from one to the other. If you have a line, you can extend it to make a longer line. If you have a center and a radius, you can construct a circle. If you don't have any of those things to start with, Euclid's postulates don't help you.