172 7. ANCIENT NUMBER THEORYTo give another illustration of the same method, we consider the problem fol-
lowing the one just discussed, that is, Problem 9 of Book II: to separate a given
number that is the sum of two squares into two other squares. (That is, given one
representation of a number as a sum of two squares, find a new representation of
the same type.) Diophantus shows how to do this using the example 13 = 2^2 + 3^2.
He lets one of the two squares be (ς + 2)^2 and the other (2ò - 3)^2 , resulting in the
equation 5ò^2 - 8ò = 0. Thus, ò = §, and indeed (^)^2 + (±)^2 = 13. It is easy
to see here that Diophantus is deliberately choosing a form for the solution that
will cause the constant term to drop out. This amounts to a general method, used
throughout the first two books, and based on the proportion
(a+Y): X = X :(a-Y)
for solving the equation X^2 + Y^2 = a^2.
The method Diophantus used to solve such problems in his first two books was
conjectured by Maximus Planudes (1255 1305) and has recently been explained in
simple language by Christianidis (1998).
Some of Diophantus' indeterminate problems reach a high degree of complexity.
For example, Problem 19 of Book 3 asks for four numbers such that if any of the
numbers is added to or subtracted from the square of the sum of the numbers, the
result is a square number. Diophantus gives the solutions as
17,136,600 12,675,000 15,615,600 8,517,600
163,021,824' 163,021,824" 163,021,824' 163,021,824"3. China
Although figurate numbers were not a topic of interest to early Chinese mathe-
maticians, there was always in China a great interest in the use of numbers for
divination. According to Li and Du (1987, pp. 95-97), the magic square
4 9 2
3 5 7
8 1 6appears in the treatise Shushu Jiyi (Memoir on Some Traditions of the Mathe-
matical Art) by the sixth-century mathematician Zhen Luan. In this figure each
row, column, and diagonal totals 15. In the early tenth century, during the Song
Dynasty, a connection was made between this magic square and a figure called the
Luo-chu-shu (book that came out of the River Lo) found in the famous classic work
J Ching, which was mentioned in connection with divination in Chapter 1. The
/ Ching states that a tortoise crawled out of the River Lo and delivered to the
Emperor Yu the diagram in Fig. 2. Because of this connection, the diagram came
to be called the Luo-shu (Luo book). Notice that the purely numerical aspects of
the magic square are enhanced by representing the even (female, ying) numbers
as solid disks and the odd (male, yang) numbers as open circles. Like so much of
number theory, the theory of magic squares has continued to attract attention from
specialists, all the while remaining essentially devoid of any applications. In this
particular case, the interest has come from specialists in combinatorics, for whom
magic squares and Latin squares form a topic of continuing research.
Another example of the use of numbers for divination comes from the last
problem (Problem 36 of Chapter 3) of the Sun Zi Suan Jing. The data for the
problem are very simple. A woman, aged 29, is pregnant. The period of human