The Sagacity of Circles: A History of the Isoperimetric Problem - Analytic Solution to the Problem

Other mathematicians attempted to tackle the isoperimetric problem from the analytic or calculus-based perspective. Using calculus, the isoperimetric problem can be stated as follows:

Find an arc with parametric equations x=x(t), y=y(t) for t in [t₁,t₂] such that x(t₁) = x(t₂), y(t₁) = y(t₂) (where no further intersections occur) constrained by

\[L = \int_{t_1}^{t_2} \sqrt{x^{{\prime}^2}+y^{{\prime}^2}}\,\,dt\]

such that

\[A ={\frac{1}{2}} \int_{t_1}^{t_2} \left({xy^{\prime}-x^{\prime}y}\right)\,dt\]

is a maximum [Weisstein, 1].

As Dunham observes in The Mathematical Universe, even Steiner’s elegant proof and the application of calculus did not end work on the isoperimetric problem. Although the circle is indeed the solution to the question as Zenodorus considered it, “it is conceivable,” writes Dunham, “that [one] might exceed a circle’s area by assembling not regular polygons…but parabolas, ellipses, or some other irregular curves” [Dunham, 112]. Indeed, it was over an aspect of isoperimetry that the mathematician brothers Jakob and Johann Bernoulli came to blows, a struggle which eventually contributed to the development of the calculus of variations [Dunham, 14-15, 112-13].

Although the isoperimetric problem has its origins in the ancient world, it is nonetheless a problem for the ages. Despite the fact that he was a member of a mathematical culture as yet unacquainted with calculus or the modern notions of extrema, Zenodorus developed a geometrical proof of a concept which proved so significant as to find its way into literary, philosophical, and, most importantly, modern mathematical texts. Ancient problems, such as that of isoperimetry, are not relics of an obsolete mathematical past, but rather important steps towards the development of contemporary mathematics. Problems posed by the ancients not only speeded the progression towards more rigorous, complete systems of mathematics, but also prompted later innovators to develop new systems to deal with these early questions. The isoperimetric problem thus demonstrates an important continuity in mathematical thought. From Zenodorus to Pappus and from Steiner to the mathematicians of the twenty-first century, isoperimetry has transcended its origins in ancient geometry to become a building block of more modern analytic systems of mathematics. In his preface "On the Sagacity of Bees," Pappus observed that God had given to man "the best and most perfect understanding of wisdom and mathematics" [Thomas, 589]. Indeed, it is this wisdom which led geometers not only to the solution of the isoperimetric problem but more importantly to a greater appreciation of the beauty and logic of the mathematical world.