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Although the extant commentary of Theon of Alexandria gives only limited proof of Zenodorus’ aforementioned proposition, it does contain Zenodorus’ work on two other key aspects of the isoperimetric problem, namely the question of equilateral and equiangular polygons and the problem of the sphere. The commentary begins with the statement that “of all rectilinear figures having an equal number of sides and equal perimeter, the greatest is that which is equilateral and equiangular” [Thomas, 395]. The first portion of this proposition, namely that of all polygons of the same number of sides an equilateral polygon has the greatest area can be proven using the abovementioned lemma concerning the areas of isosceles and scalene triangles. First, assume that two sides of the “maximum polygon” *ABCDE*, shown here as *AB* and *BC*, are unequal (See Fig. 3). Draw the line *AC* and construct an isosceles triangle *AFC* such that *AF + FC = AB + BC*. By Zenodorus’s first lemma, the area of the triangle *AFC* is greater than the area of he triangle *ABC*, and thus the area of the polygon was “increased by construction,” creating a contradiction with the original hypothesis holding that the area was a maximum [Heath, 212].

Figure 3. “It is equilateral,” from [Heath, 212]

Similarly, the proof (see Fig. 4) that of all polygons of the same number of sides, an equiangular polygon has the greatest area makes use of Zenodorus’s second lemma, concerning the areas of similar triangles. As Heath summarizes, “let the maximum polygon *ABCDE* [shown to be equilateral above] have the angle at *B* greater than the angle at *D*. Then *BAC*, *DEC* are non-similar isosceles triangles.” Construct two isosceles triangles *FAC* and *GEC* similar to one another on bases *AC* and *CE* respectively such that the sum of their perimeters is equal to the sum of the perimeters of triangles *BAC* and *DEC*. Then, concludes Zenodorus, “the sum of the areas of the two similar isosceles triangles is greater than the sum of the areas of the triangles *BAC* and *DEC* [by the second lemma]” and the area of the polygon has been increased by construction, once again contradicting the original maximum hypothesis [Heath, 212].

Figure 4. “It is equiangular,” from [Heath, 212]

Beyond proving that “of all polygons of the same number of sides and equal perimeter the equilateral and equiangular polygon is the greatest in area” [Heath, 207] as was demonstrated above, Zenodorus also dealt briefly with the notion of the sphere. As Theon quotes, “Now I say that, of all solid figures having an equal surface, the sphere is the greatest.” Accordingly, Zenodorus made use of “theorems proved by Archimedes in his work *On the Sphere and Cylinder*” [Thomas, 395], thus extending the isoperimetric problem to three dimensions.

Jennifer Wiegert, "The Sagacity of Circles: A History of the Isoperimetric Problem - The Work of Zenodorus," *Loci* (July 2010)