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I am interested in this question:

Who was the first to show that for every circle the fraction $$\frac{\text{circumference}}{\text{diameter}}$$ is always constant?

I am not interested in $\pi$ approximations.

By "proof" I mean any argument that could be "fixed" in some way to work.

(because I suppose that an ancient proof would not fit the modern standards)

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It must be old at least as much as the first written Egyptian and Babylonian approximations : if you do not think that it is a "specific" number, why try to compute it ?


Written sources :

Euclid's Elements , Book III, Def.1 :

Equal circles are those whose diameters are equal, or whose radii are equal.

Book III, Prop.26 :

In equal circles equal angles stand on equal circumferences whether they stand at the centers or at the circumferences.

Book XII, Prop.2 :

Circles are to one another as the squares on their diameters.

Archimedes' Measurement of a Circle : Prop.1 :

The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.

Thus, if we assume that two circles of equal radii have equal area (by Euclid, XII.2), then the two circumferences must be equal.

Prop.3 is the well-known Archimedes' approximations of $\pi$ :

The ratio of the circumference of any circle to its diameter is greater than $3\tfrac{10}{71}$ but less than $3\tfrac{1}{7}$.

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  • $\begingroup$ Many people "think" and risk that something is true but since when was it a theorem that we are searching for a constant? $\endgroup$ – Konstantinos Gaitanas Oct 13 '15 at 20:30
  • $\begingroup$ We know so little about Egyptian mathematics that any sort of attribution of intentions seems a very unreliable exercise. $\endgroup$ – Chappers Oct 20 '15 at 16:19
  • $\begingroup$ ${10 \over 71}$? How did he come up with such a denominator?? $\endgroup$ – SF. Oct 29 '18 at 23:01
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Before ancient Greeks no notion of "proof" existed, mathematical works such as Babylonian tablets and Egyptian papyri, gave practical recipes. "Circles are to one another as the squares on their diameters" is proposition 2 of Book XII of Euclid's Elements, but is broadly believed to be due to Eudoxus of Cnidus (c.400 - 350 BC), the inventor of the method of exhaustion, which is used to prove it.

Euclid (and likely Eudoxus) did not consider the ratio of circumference to diameter. The invariance of that ratio, along with bounds for it, is proved in Archimedes's Measurement of a Circle, also by method of exhaustion. He had to use additional postulates however, in particular the famous "line segment between two points is shorter than any other path between them".

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  • $\begingroup$ Does Archimedes actually prove the invariance? It sounds to me more like he assumed invariance and put numerical bounds on the ratio. $\endgroup$ – Ben Crowell Nov 21 '15 at 15:52
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    $\begingroup$ @Ben Crowell For regular polygons the invariance of perimeter to diameter ratio follows from similarity, since circumference is bound by perimeters of inscribed and circumscribed polygons to any precision the invariance of its ratio to diameter follows as in Euclid for areas. The bounding argument requires the additional postulates which appear in On Sphere and Cylinder. $\endgroup$ – Conifold Nov 24 '15 at 18:58
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It seems that the answer is here.The author has the same question with me.

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    $\begingroup$ Link-only answers are discouraged on stackexchange. Answers are supposed to be as self-contained as possible. Also, the paper is paywalled, so it's difficult to tell what it actually says. $\endgroup$ – Ben Crowell Nov 21 '15 at 15:56
  • $\begingroup$ The article is "Circular reasoning: Who first proved that C divided by d is a constant?" by David Richeson in The College Mathematics Journal 46 (2015) 162--171. It is available through JSTOR (which has a free "borrowing" system), Taylor & Francis Online, and was reprinted in Princeton University Press's Best Writing in Mathematics 2016. $\endgroup$ – Brian Hopkins Oct 24 '18 at 21:48
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Old Kingdom Egyptian architecture featured engaged columns--i.e., columns embedded into walls--that were fluted around their exposed circumferences. Analysis of the columns showed that if they were freestanding columns, they would have 22 flutings of equal width around their circumferences. The diameter of the column from tip of fluting to tip of fluting would be very nearly equal to 7 units of fluting width. It would appear that ancient Egyptian architects were aware of a practical circumference-to-diameter constant ratio of 22/7.

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  • $\begingroup$ Just because they made circular things with circumference/diameter near 22/7 does not mean they were aware of anything, does it? (How could they not have?) $\endgroup$ – Francois Ziegler Oct 24 '18 at 1:08
  • $\begingroup$ They were aware of the fact that after they used their dividers to mark out a circular plan of the "drum" of the column, they would have to inscribe a 22-gon inside the circle to mark the placement of the flutings. The sides of the 22-gon were taken as 1/7 of the diameter of the circle. $\endgroup$ – R. Parker Oct 24 '18 at 1:45
  • $\begingroup$ I mean, couldn’t you say that of anyone who ever drew a circle with circumference/diameter near 3.14? $\endgroup$ – Francois Ziegler Oct 24 '18 at 1:48
  • $\begingroup$ Only if they inscribed a 22-gon inside it ;-) $\endgroup$ – R. Parker Oct 24 '18 at 2:18

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