4 - Indivisibles

Summary

Every mathematical subject advances thanks to imaginative conjectures. One of the earliest examples of such risk-taking in calculus is due to Democritus (Greece, born c. 460 bce),who lived about 200 years before Archimedes. He is credited with a claim such as the following:

If two solids are cut by a plane parallel to their bases and at equal distances to their bases, and the sections cut by the plane are equal, and if this is true for all such planes, then the two solids have equal volumes.

Although this claim does not directly state that solids are composed of infinitely many two-dimensional slices, it certainly toys with the idea. One might ask, for example, what becomes of the topmost slice of a pyramid, at its tip. Do we jump from two dimensions to only one? Is the jump sudden, or gradual? In fact, Democritus himself skeptically inquired if two infinitely thin slices of a solid could be neighbors. Yet despite puzzles like this, mathematicians used the statement above to compare the volumes of cylinders, cones, prisms, pyramids.

Inspired leaps leading to truth — it is no wonder that some have claimed that revealing truths in mathematics takes as much creativity as in the arts and letters. In this chapter, we see how European mathematicians engaged in this pursuit.