- MEDIEVAL EUROPE 181
round, and the first child will start the following round as number 3. The problem
is to see which child will be the last one remaining. Obviously, solving this problem
in advance could be very profitable, as the original Josephus story indicates.^15 The
Japanese problem is made more interesting and more complicated by considering
that half of the children belong to the couple and half are the husband's children
by a former marriage. The wife naturally wishes one of her own children to inherit,
and she persuades the husband to count in different ways on different rounds. The
problem was reprinted by several later Japanese mathematicians.
The eighteenth-century mathematician Matsunaga Ryohitsu (1718-1749) dis-
cussed a variety of equations similar to the Pell equation and representations of
numbers in general as sums and differences of powers. For example, his recipe for
solving the equation x^3 — y^3 = z^4 was to take æ — ôç^3 —ç^3 , then let χ = mz, y = nz.
But he also tackled some much more sophisticated problems, such as the problem
of representing a given integer k as a sum of two squares and finding an integer
that is simultaneously of the forms y^2 + 69j/i +15 and y\ + 72y^2 + 7. Matsunaga
gave the solution as 11,707, obtained by taking y\ = 79, yi — 78.
7. Medieval Europe
In his Liber quadratorum (Book of Squares) Leonardo of Pisa (Fibonacci, 1170-
1250) speculated on the difference between square and nonsquare numbers. In the
prologue, addressed to the Emperor Frederick II, Leonardo says that he had been
inspired to write the book because a certain John of Palermo, whom he had met
at Frederick's court, had challenged him to find a square number such that if 5 is
added to it or subtracted from it, the result is again a square. This question inspired
him to reflect on the difference between square and nonsquare numbers. He then
notes his pleasure on learning that Frederick had actually read one of his previous
books and uses that fact as justification for writing on the challenge problem.
The Liber quadratorum is written in the spirit of Diophantus and shows a keen
appreciation of the conditions under which a rational number is a square. Indeed,
the ninth of its 24 propositions is a problem of Diophantus: Given a nonsquare
number that is the sum of two squares, find a second pair of squares having this
number as their sum. As mentioned above, this problem is Problem 9 of Book 2 of
Diophantus.
The securest basis of Leonardo's fame is a single problem from his Liber abaci,
written in 1202:
How many pairs of rabbits can be bred from one pair in one year
given that each pair begins to breed in the second month after its
birth, producing one new pair per month?(^15) Josephus tells us that, faced with capture by the Romans after the fall of Jotapata, he and
his Jewish comrades decided to commit mass suicide rather than surrender. Later commentators
claimed that they stood in a circle and counted by threes, agreeing that every third soldier would
be killed by the person on his left. The last one standing was duty bound to fall on his sword.
According to this folk legend, Josephus immediately computed where he should position himself
in order to be that last person, but decided to surrender instead of carrying out the bargain.
Josephus himself, however, writes in The Jewish Wars, Book 111, Chapter 8 that the order of
execution was determined by drawing lots and that he and his best friend survived either by
chance or by divine intervention in these lots. The mathematical problem we are discussing is also
said to have been invented by Abraham ben Meir ibn Ezra (1092-1167), better known as Rabbi
Ben Ezra, one of many Jewish scholars who flourished in the Caliphate of Cordoba.