**Comparison of the methods of Alexander of Aphrodisias,****according to Simplicius, Commentary on Aristotle's**,*Physics*pp. 55 line 25 - 57 line 29**and of Eudemus of Rhodes,**-
**according to Simplicius, Commentary on Aristotle's**,*Physics*pp. 60 line 22 - 67 line 30 **in their respective discussions of the squaring of lunules by Hippocrates of Chios**©

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

Return to Vignettes of Ancient Mathematics

Introduction to Hippocrates

Introduction to Lunules

Alexander's Method

Eudemus' Method

main diagram 2:1 lunule or main diagram 5:3 lunule

(5:3 diagram 1 or
still diagram or
2:1 lunule showing Eudemus segments) In his account of lunules, Alexander
constructs the lunule out of two equal sectors of circles, the outer, *A*, and the inner, *B*, and then TEARS OUT a common area. The two figures that result, a lunule and a rectilinear figure, must then be equal. In the case where they are not equal, the difference will at least be given. The particular cases can be easily generalized, as was shown in the discussion of Alexander's method. Crucial to the method is finding the two equal sectors in a configuration that will produce a lunule. This results from having the ratio of the squares on the radii of the two circles from which the outer sector *A* and the inner sector *B* are formed: R_{A}^{sqr} :
R_{B}^{sqr} = *A** : *A*, where A* is similar to B and A* < A.
Hence, the basic theorems are that sectors are to circles as their angles to 4 right angles, that circles are as the squares on their diameters (or radii),
so that that sectors of circles are the compound ratio of the squares on their diameters (or radii) and their angles. The two treatment of lunules in Alexander makes the outer arc a semicircle. It is unknowable whether this is fundamental to Alexander's understanding.

main diagram 2:1 lunule or main diagram 5:3 lunule

(diagram 2) Eudemus, on the other hand, starts with segments on the inner and outer arcs of the lunule. It is crucial to his construction that the outer and inner segments be equal (and not overlapping). To get these segments equal, it is necessary that the arcs be divisible into arcs, whose segments have areas inversely proportional to the number of arcs. He then ADDS an area in between the two sets of segments to the outer segments to form the lunule and to the inner segments to form the rectilinear figure. If the two sets of segments are not equal, the difference, at least, will be given. (diagram 3) As was clear in the discussion of Eudemus' method, the outer segments are equal to each other and the inner segments are equal, while all the segments are similar.
If n_{out} is the number of outer segments and n_{in} the number of inner segments, and
b_{out}^{sqr} equals the square the base of an outer segment, and b_{in}^{sqr} equals the square the base of an inner segment,
then the inner and outer segments will be equal if n_{out} : n_{in} = b_{in}^{sqr} : b_{out}^{sqr}. As given, the method requires that the ratios be of whole numbers. One could imagine allowing a more general procedure that allows the outer segments to be unequal, and the inner segments to be unequal. After all, the only requirement is that the two sets of segments not overlap and that the sum of each group of segments be equal to each other. If so, more elaborate possibilities would ensue. Crucial to this method is the theorem that similar segments in different circles are as the squares on their bases.

It is a minor issue that whereas Alexander starts with the inner circle and builds the outer segment and lunule on a segment of it, Eudemus starts with the outer circle and then builds the inner circle. Of much greater importance, the basic conception of adding to segments also is the foundation of the construction of the lunule and circle equal to a triangle and hexagon.

main diagram 2:1 lunule or main diagram 5:3 lunule

It is a mark of the two methods that because Alexander's tears out the overlapping area of two sectors, while Eudemus' adds a common area to equal sets of segments, to form the lunule and rectilinear figure, that the two rectilinear areas that result will be different.

(Alexander diagram or Eudemus diagram) The relation between the two squarings of circle and lunule is also important. Although the inner arc lunule in each case is 1/6 circle, the outer arc is different, 1/3 circle in the case of Eudemus (2 arcs 1/6 circle), a semicircle in the case of Alexander. The basic equality is therefore different, coming from different observations. Intuitively, it is obvious that one would want to look at lunules built on hexagons (were the radius of the circle equals the side of the hexagon). The question is what lunule one will construct. In the case of Eudemus, it is built on an equilateral triangle and the simplest ratio, the same ratio used in the first construction. We can even imagine that the author, whether Hippocrates, Eudemus or someone else, starts with the idea of investigating lunules, successfully squares a lunule with diameter as base, and the ratio of segments as 2 : 1, and then proceeds to the next figure as providing a base but with the same ratio.

We can imagine Alexander's as coming out of a different conception. It works with the outer arc as a semicircle and asks what happens when we tear out common areas of different sectors. The first figure examined is the lunule on the semicircle on the side of an inscribed square, the next is the semicircle on the side of the hexagon. Would anything interesting have happened in the case of a triangle or a pentagon. Not really. The construction on a triangle will result in 8 lunules on the equilateral triangle equal to a rectilinear figure and 1/3 circle. The construction on a pentagon will not yield anything simple since the ratio of the semicircle on the pentagon to the semicircle of the circle will be irrational. So the two reported by Alexander are elegant in ways these are not.

To see that the relation between the Alexander and Eudemus acccounts are very different, consider that the if we take the circle in which the hexagon is inscribed in the Alexander construction as the same as the middle circle of the Eudemus construction, in which the triangle of the equality is inscribed and lable it "circle_{2}", we find the following relations:

hexagon-in-the-circle

_{2}= 12 lunules_{Alex}- 3 lunules_{Eudemus}circle

_{2}= 24 lunules_{Alex}- 12 lunules_{Eudemus}

Keep in mind that the circle in the Alexander equality will be 1/4 circle_{2}, the circle of the inner arc of the Eudemus lunule is 3 circle_{2}, while the smallest circle is 1/6 circle_{2}, so that this relation is readily converted to other possible combinations of the two constructions. Perhaps underneath these relations, there is something more profound that would reveal something about a causal relation between the discovery of the two constructions. However, it is not immediately evident.