April 21, 2014 | 4
In February, I wrote about Euclid’s parallel postulate, the black sheep of the big, happy family of definitions, postulates, and axioms that make up the foundations of Euclidean geometry. I included the text of the five postulates, from Thomas Heath’s translation of Euclid’s Elements:
“Let the following be postulated:
1) To draw a straight line from any point to any point.
2) To produce a finite straight line continuously in a straight line.
3) To describe a circle with any centre and distance.
4) That all right angles are equal to one another.
5) That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”
The first three postulates have a similar feel to them: we’re defining a few things we can do when constructing figures to use in proofs. Those postulates say that if we want to, we can connect two points by a line, draw lines that continue indefinitely, and draw circles wherever we want and of whatever size we want. Fair enough.
But why the heck do we need a postulate that says that all right angles are equal to one another? You probably remember learning in a middle or high school geometry class that right angles are 90 degree angles, and two angles are congruent if they have the same degree measure. We don’t need a whole postulate that says this. It’s just part of the way we define angles. Why not a postulate that says that all 45 degree angles are equal to one another? Or all 12 degree angles? The fourth postulate seems a bit bizarre. But Euclid knew what he was doing, so there must be a reason for this postulate.
To understand what it would have meant to Euclid, we need to go back and look at Euclid’s treatment of angles. In the beginning of the book, he includes a few definitions relating to angles. Definition 8 states, “A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.” Definition 10 says, “When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.” Definitions 11 and 12 are for obtuse and acute angles, which are defined as being greater than or less than a right angle, respectively. Intuitively, we can all imagine what greater and less mean for angles: angle A is greater than angle B if it’s “more open” than angle B. It’s less if it’s “more closed.” We know it when we see it.
But Euclid never tells us exactly how to compare two angles. He never discusses degrees, radians, or how to measure an angle using a protractor. Contemporary Greek astronomers and mathematicians used degrees, and Euclid was probably aware of them, but he doesn’t use them in the Elements. Without a way to measure angles, what might Euclid have meant by angles being equal?
The axioms might shed some light. Again, from Heath’s translation:
“1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.”
On its face, Axiom 4 seems to say that a thing is equal to itself, but it looks like Euclid also used it justify the use of a technique called superposition to prove that things are congruent. Basically, superposition says that if two objects (angles, line segments, polygons, etc.) can be lined up so that all their corresponding parts are exactly on top of each other, then the objects are congruent.
For example, in Book 1, Proposition 4, Euclid uses superposition to prove that sides and angles are congruent. Proposition 4 is the theorem that side-angle-side is a way to prove that two triangles are congruent. In Oliver Byrne’s translation, which I think is a bit more poetic on this point than Heath’s, the proof starts, “Let the two triangles be conceived, to be so placed, that the vertex of the one of the equal angles shall fall upon that of the other…” In other words, Euclid seems to describe physically placing one triangle on top of the other one. When he does this, he shows that all their parts line up and concludes that they are congruent.
Now it makes a little more sense that Euclid would want a postulate that states that right angles are congruent. We need to know that creating a pair of right angles on one piece of paper is the same as creating them on another piece of paper. We need to be able to put the pieces of paper on top of each other and have the angles line up exactly. In effect, the fourth postulate establishes the right angle as a unit of measurement for all angles. Although Euclid never uses degrees or radians, he sometimes describes angles as being the size of some number of right angles. In this light, Euclid’s fourth postulate doesn’t seem quite so bizarre.
But if you are a bit put off by the fourth postulate, you are not alone. Proclus, a 5th century CE Greek mathematician who wrote an influential commentary on the Elements, thought that the fourth postulate should be a theorem and provided a “proof” of it in his commentary. But his proof relies on assuming that angles “look” the same wherever we are in space, a property that Heath referred to in his 1908 commentary as the homogeneity of space. Basically, Heath states that Proclus’s proof replaces the fourth postulate with a different, unstated, postulate.
“While this Postulate asserts the essential truth that a right angle is a determinate magnitude so that it really serves as an invariable standard by which other (acute and obtuse) angles may be measured, much more than this is implied, as will easily be seen from the following consideration. If the statement is to be proved, it can only be proved by the method of applying one pair of right angles to another and so arguing their equality. But this method would not be valid unless on the assumption of the invariability of figures, which would therefore have to be asserted as an antecedent postulate. Euclid preferred to assert as a postulate, directly, the fact that all right angles are equal; and hence his postulate must be taken as equivalent to the principle of invariability of figures or its equivalent, the homogeneity of space.”
Even if we do want accept the postulate without proof, Proclus would prefer that we call it an axiom, rather than a postulate. He thought the postulates should be about construction—something we do—while the axioms should be self-evident notions that we observe. (The axioms are sometimes called “common notions.”) But Heath sees a good reason that the fourth postulate should be placed where it is.
“As to the raison d’être and the place of Post. 4 one thing is quite certain. It was essential from Euclid’s point of view that it should come before Post. 5, since the condition in the latter than a certain pair of angles are together less than two right angles would be useless unless it were first made clear that right angles are angles of determinate and invariable magnitude.”
As a side note, I found Heath’s interpretation of the difference between axioms, which he calls common notions, and postulates interesting:
“As regards the postulates we may imagine [Euclid] saying: ‘Besides the common notions there are a few other things which I must assume without proof, but which differ from the common notions in that they are not self-evident. The learner may or may not be disposed to agree to them; but he must accept them at the outset on the superior authority of his teacher, and must be left to convince himself of their truth in the course of the investigation which follows.’”
In 1899, the German mathematician David Hilbert published a book that sought to put Euclidean geometry on more solid axiomatic footing, as the standards and style of mathematical proof had changed quite a bit in the two millennia since Euclid’s life. Hilbert uses a different set of definitions and axioms, and in his formulation, the equality of right angles is a theorem, not an assumption. But with Euclid’s original set of postulates and axioms, the fourth postulate is necessary. In effect it establishes the right angle as the universal ruler for angles. It’s not what we’re used to now, but it works just as well as degrees or radians.
To explore Euclid’s Elements further, check out David E. Joyce’s page. You can read the commentaries of Proclus and Heath on Google Books, and if you just can’t get enough axiomatic geometry, Hilbert’s Foundations of Geometry (pdf) is on Project Gutenberg. I’d like to thank Colin McKinney of Wabash College for his help with some of the details of this post. All errors are mine.
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