**Constant Convergence Arguments**©

by Henry Mendell (Cal. State U., L.A.)

Return to Survey of Infinitary Arguments

Proof of the theorem in Euclid, Elements XII 2:

Constant convergence arguments must have been typical in early
Greek mathematics, for which the evidence comes mostly from non-mathematical
sources (such as Zeno the Eleatic). The indirect evidence
is that it is hard to see how else one could come to the theorems
of Euclid, __Elements__ xii. In fact, this difficulty
leads to a common complaint about Greek mathematics in the 17th
century, that they hid their methods of discovery. This
example is based on __Elements__ xii 2.

Lemma 1 (Euclid, __Elements__ v 12): If however many magnitudes
are proportional, as one of the leading terms is to one of the
following terms, so are all the leading terms to all the following
terms.

If A_{1} : B_{1} = A_{2} : B_{2}
= ..., then A_{1} : B_{1} = (A_{1}+A_{2}+
...) : (B_{1}+B_{2}+ ...)

Note: It is common to think of this theorem as presupposing
a finite number of terms (Euclid's proof is certainly finitely
conceived, but a similar point may be made about Archimedes' application
of the ratio of theorem at On
Spheroids and Conoids 1 to Method
14)

Lemma 2 (__Elements__ xii 1): If similar polygons
are inscribed in circles, their ratios are as the squares of the
diameters of the circles.

Theorem: Circles are to each other as the squares of their respective diameters.

Version 1: (using a sequence of figures approximating the circles)

Version 2: (gradually filling up the area between a collection of figures and the circles--version to may be the better reconstruction)

Difference between the two versions

(diagram 2) Clearly, the squares are similar and are as the squares of their diameters.

(diagram 3) If we bisect the squares, the resulting triangles are all similar and are as the squares of the diameters. Hence, the sum of the previous figure (the square) and the 8 triangles, i.e., the octagons, are similar and are as the squares of the diameters.

(diagram 4)
As we repeat this process, we get nearer and nearer to the circle.
Hence, as the number of sides of the polygons goes to infinity,
it is easy to see that the circles will be in the same ratio,
as the squares of the diameters.

It is actually more reasonable to suppose that the last step of such an argument, at some period of Greek history, would be expressed:

(diagram 2) Clearly, the squares are similar and are as the squares of their diameters.

(diagram 3) If we bisect the squares, the resulting triangles are all similar and are as the squares of the diameters. Hence, the sum of the previous figure (the square) and the 8 triangles are as the squares of the diameters.

(diagram 4)
As we repeat this process, we get nearer and nearer to the circle.
We keep repeating this (ad infinitum).

1. Hence, as square and all the triangles
in one circle are to the square and all the triangles in the other,
so is one circle to the other.

2. As square and all the triangles
in one circle are to the square and all the triangles in the other,
is as square of one diameter to the other.

3. Hence, the circles are as the
squares of their diameters.

The major claim (1) is based on the observation that the square
and triangles of a circle *fill up* the circle.

Version 1 conceives of the circle as the limit of the approximating polygons, as if the circle were an infinite-gon. Nonetheless, the conception of each polygon as built out of the previous and maintaining the same ratio as the previous polygon is visually very intuitive, for which see Antiphon.

Version 2 merely conceives of the circle as filled up by the square and the infinity of triangles. This version is tidier than the first, and fits more closely than Version 1 to the argument in Euclid, Elements xii 2.