Queen Dido and the Isoperimetric Problem

Consider a loop of thread lying on a table. How can we distort the loop, without stretching the thread, so that it encloses the maximum area?

The problem appears in a story told by the Roman poet, Virgil, in his epic poem, The Aeneid (Virgil, 19 BC):

These verses refer to the legend of Queen Dido who fled her home because her brother, Pygmalion, had killed her husband and was plotting to steal all her money. She ended up on the north coast of Africa, where she was given permission to rule over whatever area of land she was able to enclose using the hide of only one bull. She cut the hide into thin strips, tying them together to form the longest loop she could make, in order to enclose the largest possible kingdom. Queen Dido seems to have discovered how to use this loop to maximize the area of her kingdom: using straight coastline as her side border, she enclosed the largest area of land possible by placing the loop in the shape of a semi-circle.

Queen Dido's story is now the emblem of the so-called isoperimetric problem: for a fixed perimeter, determine the shape of the closed, planar curve that encloses the maximum area. The answer is the circle. Aristotle, in De caelo, while discussing the motion of the heavens, displays some knowledge or intuition of this result (Aristotle, 350 BC/1922, Book II):

Again, if the motion of the heavens is the measure of all movement … and the minimum movement is the swiftest, then, clearly, the movement of the heavens must be the swiftest of all movements. Now of the lines which return upon themselves the line which bounds the circle is the shortest; and that movement is the swiftest which follows the shortest line.