# Did Euclid prove that $\pi$ is constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to show that the ratios for any two circles is the same. This is fairly trivial if we approximate the circles by regular n-gons and take the limit as n goes to infinity, what the ancients called Eudoxus' method of exhaustion. Archimedes used this method with great success, in finding the circumference and area of a circle, the volume and surface area of a sphere, area bounded by a parabola, etc.

My question is, did Euclid ever prove that Pi is constant in his Elements? In Book XII Proposition II, he proves that the ratio of the area of a circle to the square of its diameter is the same for all circles, but does he ever prove that the ratio of a circumference of a circle to its diameter is the same for all circles? In Book VI Proposition 33 he proves that for two circles of equal diameter, the length of an arc is proportional to the angle that subtends it, but does he ever relate the lengths of arcs on unequal circles to each other?

Even if Euclid didn't prove this result, is it at least an easy corollary of something he did prove?

Any help would be greatly appreciated.

Thank You in Advance.

EDIT: In this thread on MathOverflow, it's claimed that the result follows immediately from Book III Proposition 34 and Book VI Proposition 33, but I don't see how it follows at all. As I said above, Book VI Proposition 33 is about arc lengths for circles of equal diameter, so how do you get from that to a result about arc lengths for circles of unequal diameter? Book III Proposition 34, which is just about transferring angles from one circle to another, doesn't seem like it would suffice.

EDIT 2: I think there's a proposition that the result is even more likely to follow from than Book VI Proposition 33: Book III Proposition 27, which says that equal arcs on equal circles correspond to equal angles. Is there any way to use that proposition to prove that arcs on two UNequal circles corresponding to equal angles are proportional to the diameters of the circles? That is to say, if S1 and S2 are arcs subtended by equal angles on circles of diameter D1 and D2 respectively, then S1/S2 = D1/D2.

EDIT 3: I should make clear that Euclid may not have viewed "the ratio of a circumference of a circle to its diameter" as meaningful, but I think he would have found meaningful the statement I gave in my previous edit: if S1 and S2 are arcs subtended by equal angles on circles of diameter D1 and D2 respectively, then the ratio of S1 to S2 is equal to the ratio of D1 to D2. I should also mention that Euclid's definition of the equality of two ratios is Eudoxus' theory of proportion, a precursor to the Dedekind cut construction of the real numbers.

EDIT 4: It occurs to me that just as Euclid believed that a straight line and a circular arc were not magnitudes of the same kind, he may have believed the same thing about circular arcs in circles of unequal diameter, i.e. he may have thought that it's meaningless to ask whether a circular arc from one circle is longer or shorter than a circular arc from another circle if the circles don't have the same diameter. Can anyone confirm or deny this?

• See this: mathoverflow.net/questions/72792/… – Casteels Oct 28 '13 at 14:44
• If you accept that ratios of lengths are invariant under dilation, then the constancy of $\pi$ follows. Maybe something like this is a postulate of geometry. – alex.jordan Oct 28 '13 at 14:47
• The polygons themselves are considered similar to each other based on triangle decomposition: But the similarity of two (or more) triangles is itself an axiom. And it's all ultimately based eye-sight; on the observation that when things are farther away from us, they are bigger, and smaller when closer, yet, despite the variance of their perceived absolute size on our retina, the proportions (or ratios) of their constituent elements are the same. – Lucian Oct 28 '13 at 14:49
• @Casteels The only claim in that thread of Euclid proving it is someone who says that it follows immediately from Book III Proposition 34 and Book VI Proposition 33, but I don't see how it follows at all. As I said in my question, Book VI Proposition 33 is about arc lengths for circles of equal diameter, so how do you get from that to a result about arc lengths for circles of unequal diameter? Book III Proposition 34, which is just about transferring angles from one circle to another, doesn't seem like it would suffice. – Keshav Srinivasan Oct 28 '13 at 15:28
• I didn't mean to imply that the link answered your question, just that it was the same question with many interesting answers. – Casteels Oct 28 '13 at 15:47

## 5 Answers

"This is fairly trivial if we approximate the circles by regular $n$-gons and take the limit as n goes to infinity." You describe one of the most sophisticated techniques in ancient Greek mathematics. It was most assuredly not fairly trivial.

Study the Elements closer and you will learn Euclid's style. Ancient Greek geometry had a completely different view of mathematics, from the foundations up. Any of the ancient Greeks would find the modern notion of a limit absurd. They would reject it on the grounds of a completed infinity, and it would never appear in any of their polished works. I'm not saying I side with them; but that's what they'd say.

The answer, though anticlimactic, is simple: The question you ask is totally alien to the language and spirit of the Elements. In Euclid's work, $\pi$ does not yet appear. Sure you can tease it out using modern interpretations. But overall, Euclid is far more interested in the special ratios involved in constructing structures, rather than describing them.

Studying the history of mathematics, the unprepared student is particularly prone to hindsight bias. Sadly so are many of the popular expositors of the subject!

• I meant trivial from our point of view, since we take calculus for granted. Certainly the method of exhaustion was a groundbreaking discovery by Eudoxus in ancient times. – Keshav Srinivasan Oct 28 '13 at 15:17
• Of course Pi didn't actually appear in Euclid's work, and ratios weren't even viewed as numbers. (Although Eudoxus came up with his theory of proportions that inspired Dedekind in his Dedekind cut construction.) But Euclid was able to find out that the ratio of the area a circle to the square of its diameter was the same for all circle, so it wouldn't be inconceivable that he would similarly find that the ratio of a circumference of a circle to its diameter is the same for all circles. – Keshav Srinivasan Oct 28 '13 at 15:23
• @KeshavSrinivasan I appreciate your opinion that it's not inconceivable. However, that result is not in Elements, so presumably if he did conceive of it then either he was unable to formulate it in his formal language (arguably likely), unable to prove it (unlikely), or didn't think it was important (most likely). – TBrendle Oct 28 '13 at 15:25
• First of all, I don't know whether it's in the Elements or not, which is why I asked the question. Second of all, Euclid could easily have formulated it in his language, using the language of either Book VI Proposition 33 or the language of Book XII Proposition 2. – Keshav Srinivasan Oct 28 '13 at 15:37

In XI.2 Euclid proved that circles are proportional to squares on their diameters. It follows from alternation (V.14, something Euclid used in XII.2) that the ratio of a circle to the square on its diameter is constant, and so is the ratio of a circle to the square on its radius a constant.

That constant is the same constant that you're calling $\pi$.

Euclid did not show that the ratio of a circle to the square on its radius is equal to the ratio of the circumference of a circle to its diameter. Archimedes did that.

That similar figures, and circles in particular, are proportional to squares on their corresponding parts was a general principle that predates Euclid by over a thousand years. Euclid proved that for triangles and polygons in Book VI, and for circles in Book XII.

• As I discuss in the edits to my question, Euclid showed that arcs subtended by equal angles on circles of equal diameter are equal are equal, and arcs on circles of equal diameter are proportional to their angles that subtended them. – Keshav Srinivasan Dec 26 '13 at 2:09
• True. Unfortunately that doesn't help in comparing circumferences of circles of different radii. Concepts not mentioned in the Elements are required for that. – David Joyce Dec 26 '13 at 2:18
• So how do you know that he did not show that arcs subtended by equal angles on circles of unequal diameter are proportional to the diameters? Is it just because you've spent so much time translating the Elements that you've become intimately familiar with it? (By the way, your site is the source of my Euclid knowledge.) Because if did show that then the proportionality of circumference to diameter would be an easy corollary. – Keshav Srinivasan Dec 26 '13 at 2:19
• What concepts are required for comparing arcs on circles of different diameters? – Keshav Srinivasan Dec 26 '13 at 2:21
• – David Joyce Dec 26 '13 at 15:05

Euclid's Elements is a collection of pretty much most of the known Mathematics of his era. That is circa 300 BC. However, it does not mean that it was all his Mathematics, and in particular the exhaustion method (μέθοδος τῆς ἐξαντλήσεως) is believed to have been discovered many years earlier by the great mathematician of Plato's Academy Eudoxus (408-355 BC). It is noteworthy that in Apostol's Calculus (and not only there) the method for defining Riemann integrability and integral, where the idea is also based on the exhaustion method, is attributed to Riemann-Eudoxus.

• Thanks for the interesting information, but I can't agree that "it is noteworthy" that Apostol mentions Eudoxus. Calculus book writers are generally poor historians. For example, many of them claim (contrary to fact) that Cauchy gave a formal epsilon, delta definition of continuity, and other ahistorical claims of this sort. – Mikhail Katz Dec 26 '13 at 13:52
• That's very true, unfortunately, but what Apostol wanted to point out is that the method of Riemann, who approximated the integral by upper and lower sums, in gradually thinner partitions, is clearly reminiscent of Eudoxus's exhaustion method. – Yiorgos S. Smyrlis Dec 26 '13 at 13:56
• If so, it should be possible to cite a reputable historian making such a claim on Eudoxus. What is "noteworthy" is not that Apostol said something but that Eudoxus said something. – Mikhail Katz Dec 26 '13 at 15:39

Certainly, Euclid was aware of this. Probably, it is not in the Elements because he had no rigorous method of describing the length of a curved line. The same was true of the area of a curved surface. The method of exhaustion works well for areas of plane figures, and also volumes, but curved lines and surfaces are much more subtle. Archimedes demonstrated how to extend the method of exhaustion to convex lines and surfaces.

• Euclid was certainly able to describe the length of a curved line. Look at Book VI Proposition 33, where he proved that if you have two circles of equal diameter, than the length of an arc is proportional to the angle that subtends it: aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI33.html The method he uses is Eudoxus' theory of proportions, which basically says (in modern language) that two ratios are equal if and only if the same rational numbers are less than, greater than, and equal to them, which was the inspiration for Dedekind's construction of the reals via Dedekind cuts. – Keshav Srinivasan Oct 28 '13 at 16:40
• So if Euclid was able to describe the ratio of arc lengths on circles of equal diameter, then why wouldn't he be able to do the same for circles of unequal diameter? – Keshav Srinivasan Oct 28 '13 at 17:22
• Here's a good exposition of Eudoxus' theory of proportions, by the way: public.iastate.edu/~lhogben/RealHandout.pdf It also discusses Dedekind cuts as a point of comparison. – Keshav Srinivasan Oct 29 '13 at 0:32
• @KeshavSrinivasan Are you trying to get a question answered, or start a debate? The latter is admirable but this site is not the place for it. – TBrendle Nov 1 '13 at 20:30
• @TBrendle No, I'm not trying to start a debate. I was just correcting the claim that Euclid wasn't able to describe the lengths of curved lines. – Keshav Srinivasan Nov 2 '13 at 2:04

There is a proof that pi is well-defined as ratio of circumference to diameter in the text by E. Moise "Elementary Geometry from an Advanced Standpoint"; he applies results on similarity from Book VI to the polygons approximating the circle.