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I have an elementary question about the history of $\pi$. I thought the answer would be easy to find. But, to the contrary, after quite a bit of searching and after consulting math historians, I have been unable to find a satisfactory answer.

Who first proved that $C/d$ is independent of the choice of circle ($C$ and $d$ are the circumference and diameter, respectively)?

Or equivalently:

Who first proved that given two circles with circumferences $C_1$ and $C_2$ and diameters $d_1$ and $d_2$, that $C_1/C_2=d_1/d_2$? (Or, as I imagine Euclid would have written it: the circumferences of circles are to one another as their diameters.)

Most accounts of the history of $\pi$ spend a lot of time talking about how this fact has been "known" for a long time (giving Egyptian, Babylonian, biblical, etc. approximations to the value). But they never say who first proved it. I expected it to be in Euclid's Elements, but was surprised to find that it isn't. Can I take that to mean that it hadn't been proved by then? I would be very surprised if the proof was known to Euclid and he had not included it in Elements.

Note: Euclid does contain Eudoxus's proposition that $A_1/A_2=d_1^2/d_2^2$, where the $A_i$ are the areas of the two circles (Elements XII.2: Circles are to one another as the squares on their diameters.). This implies that the value of $A/d^2$ is independent of the choice of circle.

If we jump ahead a few years from Euclid we find the fact that $C/d$ is constant given implicitly in Archimedes's Measurement of the Circle. First of all, he finds bounds for $C/d$ (it being between $223/71$ and $22/7$). So presumably he knew that it was a constant. But also, it follows logically from his result that $A=rC/2$, where $r$ is the radius of the circle (Archimedes says that the area of a circle is equal to the area of a triangle with height $r$ and base $C$): if we take Eudoxus's proposition as saying $A=kd^2$ (for some constant $k$) and Archimedes's result as $A=dC/4$, then setting them equal we get $kd^2=dC/4$, or equivalently $C/d=4k$ (i.e., $k=\pi/4$).

So, my question is: who first prove this fact? Was it Archimedes? I've read that the version of the Measurement of the Circle that we have may be only a part of what Archimedes actually wrote. Do people conjecture that it was proved and stated explicitly in the missing part of this document?

This all seems very mysterious to me. I would be a little surprised to discover that the answer to this question is lost to history since it is such a major mathematical result (but maybe that is so). I would be surprised if it took until Archimedes to get a proof of this; if it was "known" empirically for the entire Greek period (which I assume it was), one would imagine that a rigorous proof would be highly sought after. One imagines a proof would have been within Eudoxus's reach. Finally, whether the answer the answer to the question is known or not known, I have been very surprised that no one has written about this fact (or at least not that I've found).

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Interesting question. I guess you have already read www-groups.dcs.st-and.ac.uk/~history/HistTopics/…, in particular "The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable." –  Martin Brandenburg Aug 12 '11 at 20:51
    
Yes, that's typical of most accounts of the history of $\pi$. But again, I think they're referring to the empirical fact that it is a constant slightly larger than 3. I doubt the Greeks would have been satisfied with that. I'd think they'd demand a rigorous proof. –  Dave Richeson Aug 12 '11 at 20:59
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Naive question: isn't it a corrollary of "all circles are similar"? –  Niels Diepeveen Aug 13 '11 at 2:54
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Are we only asking about independence of C/d from the size of the circle? Or also who first proved C/d is independent of the location of the circle's center? And its plane's angle relative to the horizontal? The circle's color? Etc. Perhaps the ancients found these independences just as obvious as we do ("It's the same calculation regardless"), and thus didn't seek formalizations in which to "prove" them. [Just a proposal; it's probably false, but one oughtn't overlook the possibility] –  Sridhar Ramesh Aug 13 '11 at 5:00

9 Answers 9

The question wouldn't necessarily have made sense to the ancients, mainly because the ancient notion of number wasn't the same as ours. The question never uses the word "real number," but it uses expressions like $C_1/C_2$ that are clearly understood by a modern reader as referring to operations in the real number system. The ancients didn't have a real number system.

Before Euclid, the notions of measurement were never formalized in the sense of an axiomatic system, so in the context of, say, ancient Egypt, there is no way to say what it would mean to provide a formal proof of such a statement. "Knowing" $C_1:C_2::d_1:d_2$ is completely different from proving it. It doesn't make sense to say that the proof of this fact is lost in the mists of time, because the whole idea of proof in an axiomatic system only dates back to Euclid, and Euclid is not lost in the mists of time. (By Euclid I mean a certain school of thought, not the individual mathematician.)

Euclid's notion of measurement is that number is proportion, and measurement is similarity. If I say, "Joe is 6 feet tall," what it means is this. I have line segment J which is Joe's height. I have line segment F which is my standard foot. Somewhere else I have my number line, on which I have a segment 1 that I have arbitrarily singled out, and a segment 6 that I've constructed by duplicating 1 additional times to make a total of six copies of it laid end to end. The statement that "Joe is 6 feet tall" is translated as saying that when I superpose F on J, I get a figure that is similar to the one in which 1 is superposed on 6.

This whole system of assigning numbers to measurements only works if similarity exists and if figures can be dilated arbitrarily. From a modern point of view, it falls apart if the parallel postulate fails, because then it becomes impossible to arbitrarily dilate figures. From the Euclidean point of view, it's not possible for $C_1:C_2::d_1:d_2$ to fail, because that's just a description of how we do measurements: measurement is similarity. But it is possible for the whole metrical system to fail, which it does if the parallel postulate fails.

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Of course, the old Egypts had no formal proof of the linearity, but they knew that $\frac{\pi}{4} = (\frac{8}{9})^2$ is a very good approximation, independent of the radius of the circle. The harpedonaptae could prove that fact by measuring smaller and larger circles using their ropes. And certainly they have done it. The quadrature of the circumference of the circle by Dinostratus (400 BC) using the Quadratrix invented by Hippias of Elis (420 BC) also presupposes that $C/d$ is a constant. –  Rhett Butler Mar 9 '13 at 17:43
    
Finally Thales' theorem is independent of the radius of the circle. Thales certainly knew already that the sides of the right angle, as a rough approximation of the half circumference, depend linearly on the radius. –  Rhett Butler Mar 9 '13 at 17:44
    
@Rhett Butler: Very interesting! Rope-stretching would of course not be considered a valid method of proof by Euclid or anyone later. I'm not an expert on Greek mathematics, but it sounds to me like Thales, Hippias, and Dinostratus were part of a Greek mathematical tradition that was in the process of building up to the synthesis represented by Euclid, and that earlier tradition and its methods are now mostly lost because the Elements were what was seen as worthy of study and preservation. This is why I say, "By Euclid I mean a certain school of thought, not the individual mathematician." –  Ben Crowell Mar 9 '13 at 18:09
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To a mathematician of Euclids time one could say : "take a circle and from two points A B extend the tangent lines meeting at C. Then I claim about lengths that AC=BC and AB<AC+BC" and she would reply "sure, those are theorems!" But if you added the claim "also the arc AB has length intermediate between AB and AC+BC" then she might say "I own a string, I know what you mean, but that is not a mathematically meaningful statement" –  Aaron Meyerowitz Mar 9 '13 at 18:35
    
@Aaron Meyerowitz: The statement $C_1:C_2::d_1:d_2$ doesn't involve comparing curved lengths with straight ones. $C_1:C:2$ compares a curve with a similar curve. In any case I also think Euclid would have considered curved lengths comparable to straight ones in a well-defined sense. See Book 1, defs 2 and 4, and how they distinguish "line" from "straight line." The method of exhaustion predates Euclid and was an established technique for comparing non-similar figures that could not be compared for exact equality by dissection. E.g., I don't think Euclid would have protested $C>d$ as a theorem. –  Ben Crowell Mar 9 '13 at 20:15

My first impression when reading this question was that it rather should read "who was the first to prove that $C/d$ need not be a constant?" Before Gauss, Bolyai, Lobachevsky linearity and transformation-invariance had always been assumed as basic to geometry like $n + 1 = 1 + n$ in arithmetic. Small wonder that nobody in ancient Egypt, Babylonia, India, or China would bother to prove that $C/d$ is a constant for every circle or that $n + 1 = 1 + n$ for every natural number.

However, as early as about 430 BC Hippocrates of Chios for the first time explicitly mentioned that similar segments of circles are in the ratio of the squares on their bases and proved this by proving that the squares on the diameters have the same ratio as the (whole) circles. Compare the lunes of Hippocrates which until now belong to the curriculum of schools.

We know this from a comment to Aristotle's Physics, written by Simplicius who quotes Eudemus of Rhodes (the pupil of Aristotle who compiled the first catalogue of mathematicians, not Eudemus of Cyprus after whom Aristotle named his famous text) as reporting this in his lost History of Geometry.

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I should have written Eudoxus, rather than Euclid. My reply, thus, assumes that it was Archimedes who capped the proof. You're right that it seems to be too late a development. I I'm allowed some guesswork, I'd say that some versions of the proof must've been around but it wasn't completely sure which fact relied on which so a suspicion of circularity was always present: it wasn't clear whether the area constancy was to be proved from the length identity or the other way around. Many may have thought that one of them had to be accepted as a postulate (this could be the case of Eudoxus). Then, Euclid comes along, his main goal seems to have been to provide a unbroken chain of arguments, thus dispelling the possibility of circularity. Accordingly, he does not use the C/d constancy as a postulate. He proves what he can at the time without circularity: A/r^2 constancy. He doesn't seem able to prove a length result independently (I'm assuming there's no "lost proof"). It's Archimedes who comes up with a proven length formula. Archimedes himself (not so worried about proving from first principles) may have assumed the C/d constancy but his proof does not use it. For later commentators, the issue was solved with this Euclid/Archimedes combination, so no further development was added.

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Nice question! I was wondering something similar. The way I see it is that Euclid showed that A/r^2 was independent of the choice of the circle. Archimedes showed that A = Cr/2. Your proposition follows from the combination of those two facts. This is the first recorded proof of this fact, but it was split between two different texts! I guess nobody thought it worthwhile to write a manuscript just to make this association explicit, which explains why there's no theorem stating it and thus, no name attached to it.

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It seems that the first published proof of the result in the title is due to the Banu Musa, two brothers from Baghdad living in the 9th century. In their book on The measurement of plane and solid figures composed around 850, Section V has the title

"The ratio of the diameter of any circle to its circumference is one (that is, the same for all circles".

The corresponding extract is available in Pi: A source book by Berggren, Borwein & Borwein.

See also p. 450 in Encyclopedia of the history of Arabic science vol 2, by R. Rashid, and R. Morelon (available on google books).

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I suggest the article A Circular Argument (Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162.) It may be relevant to your questions. It suggests that (a variant of) the limit $\lim_{x\to 0}\frac{\sin{x}}{x}=1$ is important to the area result of Archimedes which you mention and that the reasoning may be ... circular. Here is: a freely available version.

revised version I think that it is a bit subtle. The right question might be: Who first treated the question as one which could make sense. The answer to that is probably Archimedes. Once you have that (in an acceptably defined way) the result may not be that hard.

Consider first questions simply of inequalities. If a circle is inscribed in a square the Euclid would agree that the area of the circle is less than that of the square because the whole is greater than the part. But Euclid never says that the perimeter is greater than the circumference because they are different kinds of things. Mark Saphir notes that in Book VI Proposition 33, Euclid proves that in circles of equal radii the lengths of two arcs are in equal proportion to the (central) angles cutting them off. Just sticking to one circle for now with center $O$ we understand what it would mean to say that $\angle AOB < \angle COD$ or that $\stackrel{\frown}{AB} < \stackrel{\frown}{CD}$ and also what it would mean to say that one is twice the other. And hence we have that proposition: $\frac{\angle AOB}{\angle COD}=\frac{\stackrel{\frown}{AB}}{\stackrel{\frown}{CD}}$ (But $\frac{\angle AOB}{\stackrel{\frown}{AB}}=\frac{\angle COD}{\stackrel{\frown}{CD}}$ would not make sense.) Again, Euclid could describe the situation that the radius of one circle is twice that of another. And would even agree that the area of the second is four times that of the first. However he would not say that the circumference of the second was larger than that of the first (let alone twice as much.)

Archimedes introduces the concept of concavity and the postulate:

If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D.

This is intuitive (as befits a postulate) but is not obvious. With this in hand he can say that for a circle of diameter d, the circumference C is something such that p<C<P where p and P are the perimeters of polygons (of some number of sides, he used 96) inscribed and circumscribed about a fixed circle. If this is granted then p/d < C/d < P/d and, because we know the bounds are independent of d (thanks to similarity of polygons), we have that his bounds are independent. Implicitly, letting the number of sides increase, we have that C/d must be similarly independent.

Here we see the idea of arc length (for convex curves) as the limit of the length of inscribed polygonal paths (or perhaps the common limit, if it can be demonstrated, of inscribed and tangential paths.)

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This interpretation chimes vaguely with what little I remember from some History of Mathematics lectures as an undergraduate –  Yemon Choi Aug 13 '11 at 2:38
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Thanks for your comments and the link to this article. Very interesting and helpful. It had occurred to me that understanding the question may involve knowing how they defined arc length. According to the commentary here aleph0.clarku.edu/~djoyce/java/elements/bookI/… Euclid viewed arcs of circles as magnitudes (like line segments), but they could only be compared to arcs of circles with the same radius. This had changed by the time of Archimedes because, as you wrote, he compares the length of the circumference to the perimeters of the inscribed and circumscribed polygons. –  Dave Richeson Aug 13 '11 at 17:18

In fact, if you look at Euclid's proposition 34 in book 3, and proposition 33 in book 6, this is immediately implied. @Mark Sapir: I am quite sure that people regarded the statement obvious from the beginning of time, but we all know (and the Greeks did too) that obvious is not the same as trivial.

A good version of Euclid's Elements can be found here:

http://farside.ph.utexas.edu/euclid/Elements.pdf

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@Igor: It is basically one of the Euclidean postulates (the one about circles). In fact it is easier than most other postulates (the statement that there exists at most one line containing two distinct points is much less trivial). The argument involving dilations was known to Shumers, Indians, Chinese and whomever first invented pizza. The fact that it is in Elements (which it is not by the way) does not mean that Euclid was the first to discover it. –  Mark Sapir Aug 14 '11 at 22:49
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I disagree (see my revised answer). I don't see that Euclid ever compares arcs from circles of unequal radii. Euclid compared areas of assorted shapes but it took Archimedes to introduce a postulate to allow comparison of lengths of (convex) curves. –  Aaron Meyerowitz Aug 14 '11 at 23:37
    
@Mark: I did not look at the postulates (perhaps foolishly), but since the OP claimed it was NOT in Euclid, my answer was to point out that this was not correct. I certainly don't think Euclid was the first to discover it, but actually looking at the Elements (as I did because of the question for essentially the first time), I noticed that Euclid went to great lengths to prove statements which must have seemed obvious to him. Arguably, postulates are the hardest (they are obvious, but no one figured out an argument...) –  Igor Rivin Aug 15 '11 at 13:23
    
@Aaron: I disagree. The references I cite seem to be directly on the subject. –  Igor Rivin Aug 15 '11 at 13:24
    
@IgorRivin Can you clarify how those two propositions imply that the ratio of the circumference of a circle to its diameter is the same for two unequal circles? –  Keshav Srinivasan Oct 24 '13 at 2:28

The question is badly formulated because it excludes the most reasonable answer that nobody proved it first. I think that the statement was considered obvious, similar to, say Euclidean postulates (who proved first that every point belongs to a line and every two distinct points belong to a unique line?). Indeed, since every two circles are obtained from each other by dilation, and every dilation obviously changes all distances by the same factor, the statement follows.

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My understanding is that the Greek mathematicians (and philosophers) were aware of the dangers of "proof by appeal to obviousness". My knowledge of Euclid (and those whose work he was collating) is rather poor: do they take "every two circles are obtained from each other by dilation" as a postulate? –  Yemon Choi Aug 14 '11 at 0:47
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@Yemon: Why do you think that this fact was discovered by Greek mathematicians? Most probably it was known to Egyptians, Chinese, and everybody else starting with the ancient mathematician called Australopitek (or if you like Greek sounding names more Australopitekes). His first name is not known to historians, but most probably he was called Steve. –  Mark Sapir Aug 14 '11 at 2:37
    
Terrible answer, Mark. –  Andres Caicedo Dec 30 '13 at 18:42

I think this has to do a lot with what you mean by proof. In particular, how are you defining the arclength of a circle without calculus?

For me, it seems hard to say that anyone proved $C/d$ is independent of the choice of circle before Newton and Leibniz. Once the framework of integration is in place, however, this fact is a triviality, and doesn't seem like a great advance at the time calculus was invented. The great advance was rigorously defining the length of a parameterized arc.

Of course, Archimedes famous theorem about the ratio between the volumes of a sphere and a circumscribing cylinder was essentially proved using integration, although Archimedes didn't have the framework of a limit to phrase the argument. If you're satisfied with that type of reasoning, then perhaps the answer is different.

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I would not be so quick to write off the Greeks, nor to indulge in Whig history. Archimedes's method of inscribed and circumscribed polygons looks to me like an attempt to prove, within a framework where one assumes the circle has a property called length that satisfies certain natural requirements, that $C/d$ is independent. –  Yemon Choi Aug 13 '11 at 5:40
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@Yemon: Somehow I believe that Archimedes would not approve your comment. –  Mark Sapir Aug 14 '11 at 2:43
    
Ratios come from inequalities. Archimedes defines convexity for curves and introduces a postulate that says some are longer than others. Then he is off and running. –  Aaron Meyerowitz Aug 14 '11 at 23:57

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