The Sagacity of Circles: A History of the Isoperimetric Problem - The Work of Pappus

The isoperimetric problem reached its greatest expression in the ancient world in the commentary of Pappus, a Hellenistic mathematician usually placed in the first half of the fourth century C.E. The Mathematical Collection, Pappus’s largest and most extensive work, was a summary and consolidation of the geometric theorems and formulae of the time. As Burton notes, it was designed to “give a synopsis of the contents of the great mathematical works of the past and then to clarify any obscure passages through various alternative proofs and supplementary lemmas” [Burton, 221]. Pappus’ most famous contribution to the isoperimetric problem is also one of the most famous passages in mathematics, the brief preface to Book V entitled “On the Sagacity of Bees.” As discussed earlier, Pappus’s bees display a clear understanding of the isoperimetric problem. As Nahin notes, the insects address “the ancient question of how to tile the plane (how to divide an infinite two-dimensional surface into congruent n-gons)” [Nahin, 51]. According to Pappus,

They would necessarily think “that the figures must all be adjacent to one another and have their sides common, in that nothing else might fall into the interstices and so defile their work. Now there are only three rectilinear figures which would satisfy the condition, I mean regular figures which are equilateral and equiangular, inasmuch as irregular figures would be displeasing to the bees. For equilateral triangles, squares, and hexagons can lie adjacent to one another and have their sides in common without irregular interstices [Thomas, 591].

Knowing, presumably through the proofs of Zenodorus, that “of all rectilinear figures having an equal perimeter … the greatest is that which has most angles,” the bees choose the hexagon in order to achieve the maximum amount of honey storage.

Beyond merely collecting the proofs of Zenodorus and philosophizing on their significance in his preface, Pappus also contributed the notion of the circumference of semi-circles to the body of work on isoperimetry. He first proposed that “of all circular segments having the same circumference the semi-circle is the greatest” [Heath 390-91]. Pappus based his proof on two preliminary lemmas, then proceeded to show that the area of a full semicircle ABC (see Fig. 5) is greater than the area of another circular segment DEF of equal circumference.

Figure 5. “Semi-circle and circular segment” from [Heath, 392]

One first constructs the semicircle ABC centered at G and a second circular segment DEF such that the circumference of ABC is equal to that of DEF. Construct H as the center of the circle DEF and draw EHK and BG perpendicular to DF and AC, respectively. Finally, draw DH and the line LHM parallel to DF. As summarized in [Heath, 393], Pappus arrives at the following conclusions:

LH: AG = (arc LE): (arc AB)

= (arc LE) : (arc DE)

= (sector LHE) : (sector DHE)

and, because circles are as the squares on their radii,

LH²: AG² = (sector LHE) : (sector AGB).

Therefore, it follows that

(sector LHE) : (sector DHE) = (sector DHE) : (sector AGB).

In a preliminary lemma, Pappus had shown that

(sector EDH) : (half segment EDK) > (right angle) : (∠ DHE).

Note that this is equivalent to the modern statement that

\[{\frac{2\alpha}{2\alpha -\sin 2\alpha}} > {\frac{\pi}{2\alpha}}\quad{\rm or}\,\, {\rm to}\quad{\frac{\theta}{\theta -\sin \theta}} > {\frac{\pi}{\theta}},\]

where \(\alpha\) = ∠ DHE, \(\theta = 2\alpha,\) and \(0 < \theta < \pi\). (You can convince yourself of the truth of the inequality by plotting both sides on a graphing calculator.)

Then

(sector EDH) : (half segment EDK) > (∠ LHE) : (∠ DHE)

> (sector LHE) : (sector DHE)

> (sector EDH) : (sector AGB).

“Therefore,” concludes Heath, “the half segment EDK is less than the half semicircle AGB, whence the semicircle BC is greater than the segment DEF” and the semicircle is established as the figure of maximum area. Pappus’s treatment of the isoperimetric problem was, therefore, an important step not only in developing ancient mathematical philosophy, but also in preserving and adding to the body of work already existing on Dido’s now famous problem.