Infinity in Greek Mathematics

Perhaps the most interesting—and most modern—feature of Greek mathematics is its treatment of infinity. The Greeks feared infinity and tried to avoid it, but in doing so they laid the foundations for a rigorous treatment of infinite processes in 19th century calculus. The most original contributions to the theory of infinity in ancient times were the

theory of proportions

and the

method of exhaustion

. Both were devised by Eudoxus and expounded in Book V of Euclid’s

Elements

. The theory of proportions develops the idea that a “quantity” π (what we would now call a real number) can be known by its position among the rational numbers. That is, π is known if we know the rational numbers less than π and the rational numbers greater than π. The method of exhaustion generalizes this idea from “quantities” to regions of the plane or space. A region becomes “known” (in area or volume) when its position among known areas or volumes is known. For example, we know the area of a circle when we know the areas of the polygons inside it and the areas of polygons outside it; we know the volume of a pyramid when we know the volumes of stacks of prisms inside it and outside it. Using this method, Euclid found that the volume of a tetrahedron equals 1/3 of its base area times its height, and Archimedes found the area of a parabolic segment. Both of them relied on an infinite process that is fundamental to many calculations of area and volume: the summation of an infinite geometric series.