The Sagacity of Circles: A History of the Isoperimetric Problem - Early History of the Problem

The isoperimetric problem as it is generally understood was first articulated by the Greek mathematician Zenodorus. Usually placed sometime after Archimedes in the 2nd century B.C.E., historians know of Zenodorus and his treatise On Isoperimetric Figures (now lost), through the 4th century C.E. commentaries of Theon of Alexandria and Pappus [Nahin, 47]. Theon of Alexandria begins with the following assertion, taken from Book I of Ptolemy’s Syntaxis:

In the same way, since the greatest of various figures having an equal perimeter is that which has most angles, the circle is the greatest among plane figures and the sphere among the solid [Thomas, 388].

He then develops this idea, with a summary of the proofs presented by Zenodorus in On Isoperimetric Figures. According to Theon, Zenodorus did not initiate his discussion of isoperimetry with the circle. Rather, he stated that “Of all rectilinear figures having an equal perimeter – I mean equilateral and equiangular figures – the greatest is that which has the most angles” [Thomas, 388-89]. In more modern language, the proposition is stated as follows: “Given two regular n-gons with the same perimeter, one with n = n₁, and the other with n = n₂ > n₁, then the regular n₂–gon has the larger area” [Nahin, 47]. Following this, Zenodorus was able to arrive at the proposition that “if a circle have an equal perimeter with an equilateral and equiangular rectilinear figure, the circle shall be the greater” [Thomas, 391].

As Heath notes in his History of Greek Mathematics [Heath, 209-212], Zenodorus chose to base his proof of this proposition on the theorem already established by Archimedes that “the area of a circle is equal to the right-angled triangle with perpendicular side equal to the radius and base equal to the perimeter of the circle.” From here, Zenodorus proceeded on the basis of two preliminary lemmas: first that “if there be two triangles on the same base and with the same perimeter, one being isosceles and the other scalene, the isosceles triangle has the greater area;” second that “given two isosceles triangles not similar to one another, if [one constructs] on the same bases two triangles similar to one another such that the sum of their perimeters is equal to the sum of the perimeters of the first two triangles, then the sum of the areas of the similar triangles is greater than the sum of the areas of the non-similar triangles.” It is at this point, however, that the difficulties of studying ancient mathematical texts comes to the forefront, for neither the text of Theon of Alexandria nor of Pappus contains a clear indication of the direction of the remainder of Zenodorus’ proof. Both commentators seem to hint that it will be covered in subsequent chapters, but as Heath bemoans “in the text as we have it the promise is not fulfilled.”