In the third century B.C., Archimedes of Syracuse created a special spiral-shaped curve by pulling the legs of a compass apart while turning it. By performing both actions at a steady rate, he found that the resulting spiral moved outward by the same amount with each turn of the compass. The groove in an old-style LP record is an example of such an Archimedean spiral.

The most significant mathematical use to which Archimedes tried to put his spiral was to create a better method of determining the area of a circle. Using a spiral to figure out the area of a circle seems a waste of energy today since anyone with a calculator can do so by pressing a few buttons. However, in ancient Greece either a physical measurement of the circumference of the circle had to be made or a critical factor in the...