Roots of Unity

Roots of Unity

Mathematics: learning it, doing it, celebrating it.

What’s the Deal with Euclid’s Fourth Postulate?



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:



3) To describe a circle with any centre and distance.

4) That all right angles are equal to one another.


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.


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:





5. The whole is greater than the part."


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.


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.

Heath writes,


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:


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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