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)
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 $3\tfrac{1}{7}$.
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".
It seems that the answer is here.The author has the same question with me.
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.
