Every circle has the property that the distance around it is just over three times the distance across. This has been “common knowledge” since the earliest times. But mathematicians do not trust common knowledge; they demand proof. Who was first to prove that all circles are similar, in the sense that the ratio of circumference C to diameter D has the same value for all?
Slicing a disk to estimate pi (Image Wikimedia).
You might expect to find a proof in Euclid’s Elements of Geometry, but a search there would be in vain. It is likely that Euclid would have liked to include the result as a theorem but, with the axioms or assumptions that he used, he could not prove it, and he made no mention of the ratio.
Euclid XII.2
Euclid’s Proposition XII.2 says that the areas of circles are to one another as the squares of their diameters. In symbolic form, if two circles have areas A1 and A2 and diameters D1 and D2 , then
A1 / D12 = A2 / D22
We would expect to find an analogous theorem: “The circumferences of circles are to one another as their diameters” or, in symbolic form,
C1 / D1 = C2 / D2
but we do not find this anywhere in Euclid.
There is a simple “heuristic” argument that all circles are similar: if we scale up a circle by a factor K, then all lengths should increase by this factor. Therefore, the ratio of two lengths should remain unchanged. However, this is not a rigorous argument. We need to define the length of the circumference; but how do we define the length of a curve? There is no recipe anywhere in the Elements nor any ruler-and-compass procedure to determine the length.
Another objection to the heuristic argument is that while all lengths scale by the same factor not all properties of the circle behave this way. For example, the curvature of the circle is given by the reciprocal of the radius and clearly this is diminished by a factor K under the scaling. So we must be careful.
Archimedes and Arc Metrics
It required the genius of Archimedes, who lived a few generations after Euclid, to prove that C / D is constant, and he needed to introduce axioms beyond those of Euclid to achieve this. There is an excellent account in an article by Richeson (2015), including details of the additional axioms.
The problem of proving that C / D is constant brings us to the question: how do we define the length of a curve or, in particular, the length of a circular arc. At the time of the ancient Greeks, there was no theory of arc-length. They had no means of constructing a line segment equal in length to a given curve using only straight-edge and compass. Aristotle had stated that it was impossible to compare the lengths of curves and straight lines. But Archimedes was undeterred: he approximated the circumference of a circle to arbitrary degree of precision by a sum of straight-line segments.
Archimedes determined π accurately by considering polygons inscribed within a circle and polygons surrounding it. A regular hexagon fitting snugly within a circle of unit diameter has length 3. Clearly, its perimeter is shorter than the circumference π of the circle, so π is greater than 3. A less obvious derivation shows that a hexagon drawn around the circle has length twice the square root of 3 or about 3.46, so π is less than this.
Polygons approximating a circle.
To prove this, Archimedes had to introduce additional axioms. He then doubled the number of sides four times, ending up with polygons having 96 sides. These were very close to the circle, allowing him to get a much more precise estimate, between 223/71 and 22/7, or 3.141 and 3.143, which was remarkable given the limitations of the Greek system of numerals. Archimedes’ estimate was not bettered for centuries.
In his work Measurement of a Circle, Archimedes proved that the area of a circle is equal to the area of a right-angled triangle with one of the legs of length equal to the radius R of the circle and the other leg of length equal to the circumference C. In symbols, Archimedes’ theorem is
A = ½ R C
Archimedes’ Theorem (Image Wikimedia).
Note that while Archimedes used a triangle with one side equal to C, he did not say in what way this triangle might be constructed. Indeed, he could not have provided any method of construction within the confines of classical geometry. Such a rectification of the circle was not possible.
The Main Result
Archimedes’ theorem, together with Euclid’s Proposition XII.2, leads directly to the result that C / D has the same value (which we now call π) for every circle. Archimedes did not state this explicitly, but he must have been aware of it. It also follows immediately that the “area constant” associated with Euclid XII.2 is also π:
A / R2 = C / 2 R = C / D = π .
Moving to higher dimensions, Euclid did prove that V / D 3 has the same value for every sphere. However, he did not give any indication of the value of this constant. Moreover, he had nothing to say about the constancy of S / D2 (V is the volume of the sphere of diameter, and S is its surface area).
There are several constants, all of which turn out to have the same numerical value; there is the circular circumference constant, the circular area constant, the spherical surface constant and the spherical volume constant, and they may be defined as follows:
ΠC = C / D , ΠA = A / R2, ΠS = S / 4 R2, ΠV = 3 V / 4 R3.
Euclid’s proposition XII.2 (that A / D 2 is the same for all circles) implies the area constant. But Euclid never compared the lengths of a line segment and the arc of a circle. Archimedes had the remarkable insight that additional assumptions or axioms were required to complete his demonstration, and he set down these extra axioms in his work On the Sphere and Cylinder.
Archimedes showed that all four geometric constants are equal:
ΠC = ΠA = ΠS = ΠV = π .
The problem of rectifying the circle, that is, constructing a straight line with length equal to its circumference, is equivalent to the problem of squaring the circle, that is, constructing a square with area equal to that of a given circle.
A much more complete account of Archimedes’ work is given in the article by Richeson (see below).
Source:
Richeson, David , 2015: Circular reasoning: who first proved that C/d is a constant? College Math. Journal, 46, 162-171. Preprint (2013) online at: https://arxiv.org/abs/1303.0904v2 .
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