The History of Mathematics: A Brief Course

1. HELLENISTIC GEOMETRY 319

Â

FIGURE 1. Two theorems of Zenodorus. Top: When two regular

polygons have the same perimeter, the one with the larger number

of sides is larger. Bottom: A circle is larger than a regular polygon

whose perimeter equals the circumference of the circle.

Zenodorus' proof that a circle is larger than a regular polygon whose perimeter

equals the circumference of the circle is shown at the bottom of Fig. 1. Given

such a polygon and circle, circumscribe a similar polygon around the circle. Since

this polygon is "convex on the outside," as Archimedes said in his treatise on

the sphere and cylinder, it can be assumed longer than the circumference. (Both

Archimedes and Zenodorus recognized that this was an assumption that they could

not prove; Zenodorus cited Archimedes as having assumed this result.) That means

the circumscribed polygon is larger than the original polygon since it has a larger

perimeter. But then by similarity, Ç Ê is larger than ÈË. Since the area of the

circle equals half of the rectangle whose sides are its circumference and HK, while

the area of the polygon is half of the rectangle whose sides are its perimeter and

ÈË, it follows that the circle is larger.

1.2. The parallel postulate. We saw in the Chapter 10 that there was a debate

about the theory of parallel lines in Plato's Academy, as we infer from the writing

of Aristotle. This debate was not ended by Euclid's decision to include a parallel

postulate explicitly in the Elements. This foundational issue was discussed at length