- ANCIENT GREEK NUMBER THEORY 169
FIGURE 1. Figurate numbers. Top row: triangular numbers Tn —
n(n + l)/2. Second row: square numbers Sn — n^2. Third row:
pentagonal numbers Pn = n(3n - l)/2. Bottom row: hexagonal
numbers Hn = n(2n — 1).the Pythagorean and Platonic sources of Euclid's treatise are obvious, Euclid ap-
pears to the modern eye to be much more a mathematician than Pythagoras or
Plato, much less addicted to flights of fanciful speculation on the nature of the
universe. In fact, he never mentions the universe at all and suggests no practical
applications of the theorems in his Elements.
Book 7 develops proportion for positive integers as part of a general discussion
of how to reduce a ratio to lowest terms. The notion of relatively prime numbers is
introduced, and the elementary theory of divisibility is developed as far as finding
least common multiples and greatest common factors. Book 8 resumes the sub-
ject of proportion and extends it to squares and cubes of integers, including the
interesting theorem that the mean proportional of two square integers is an integer
(Proposition 11), and between any two cubes there are two such mean proportionals
(Proposition 12): for example, 25 : 40 :: 40 : 64, and 27 : 45 :: 45 : 75 :: 75 : 125.
Book 9 continues this topic; it also contains the famous theorem that there are
infinitely many primes (Proposition 20) and ends by giving the only known method