168 7 ANCIENT NUMBER THEORYsimplest way to denote any integer would be repeating a symbol for 1 an appropri-
ate number of times. Thus, he said, the number 5 could be denoted aaaact. This
train of thought, if followed consistently, would lead back to a notation even more
primitive than the hieroglyphic notation for numbers, since it would use only the
symbol for units and discard the symbols for higher powers of 10. The Egyptians
had gone beyond this principle in their hieratic notation, and the standard Greek
notation was essentially a translation of the hieratic into the Greek alphabet. You
can easily see where this speculation leads. The outcome is shown in Fig. 1, which
illustrates triangular, square, pentagonal, and hexagonal numbers but using dots
instead of the letter a. Observe that the figures are not associated with regular
polygons except in the case of triangles and squares. The geometry alone makes it
clear that a square number is the sum of the corresponding triangular number and
its predecessor. Similarly, a pentagonal number is the sum of the corresponding
square number and the preceding triangular number, a hexagonal number is the
sum of the corresponding pentagonal number and the preceding triangular number,
and so forth. This is the point at which modern mathematics parts company with
Nicomachus, Proclus, and other philosophers who push analogies further than the
facts will allow. As Nicomachus states at the beginning of Chapter 7:
The point, then, is the beginning of dimension, but not itself a
dimension, and likewise the beginning of a line, but not itself a
line; the line is the beginning of surface, but not surface; and the
beginning of the two-dimensional, but not itself extended in two
dimensions... Exactly the same in numbers, unity is the beginning
of all number that advances unit by unit in one direction; linear
number is the beginning of plane number, which spreads out like
a plane in one more dimension. [D'ooge, 1926, p. 239]This mystical mathematics was transmitted to Medieval Europe by Boethius.
It is the same kind of analogical thinking found in Plato's Timaeus, where it is
imagined that atoms of fire are tetrahedra, atoms of earth are cubes, and so forth.
Since the Middle Ages, this topic has been of less interest to mathematicians. The
phrase of less interest—rather than of no interest—is used advisedly here: There
are a few theorems about figurate numbers in modern number theory, and they have
some connections with analysis as well. For example, a formula of Euler asserts that
oo oo
J](l-Xfc)= £ (-l)""^3 ""^1 )/^2.
fc=l n=-oo
Here the exponents on the right-hand side range over the pentagonal numbers for
ç positive. By making this formula the definition of the nth pentagonal number for
negative n, we thereby gain an interesting formula that can be stated in terms of
figurate numbers. Carl Gustav Jacobi (1804-1851) was pleased to offer a proof of
this theorem as evidence of the usefulness of elliptic function theory. Even today,
these numbers crop up in occasional articles in graph theory and elsewhere.
2.2. Euclid's number theory. Euclid devotes his three books on number theory
to divisibility theory, spending most of the time on proportions among integers and
on prime and composite numbers. Only at the end of Book 9 does he prove a theo-
rem of a different sort, giving the method of searching for perfect numbers described
above. It is interesting that Euclid does not mention figurate numbers. Although