A little thought will make it clear that the answer must be fractional,
and that in one case the numerator will be greater and in the other case
less than the denominator. As a matter of fact, the height of the larger
cube must be 8/7 ft., and of the smaller 3/7 ft., if we are to have the
answer in the smallest possible figures. Here the lineal measurement is
11/7 ft.--that is, 1+4/7 ft. What are the cubic contents of the two
cubes? First 8/7 × 3/7 × 8/7 = 512/343, and secondly 3/7 × 3/7 × 3/7 =
27/343. Add these together and the result is 539/343, which reduces to
11/7 or 1+4/7 ft. We thus see that the answers in cubic feet and lineal
feet are precisely the same.
The germ of the idea is to be found in the works of Diophantus of
Alexandria, who wrote about the beginning of the fourth century. These
fractional numbers appear in triads, and are obtained from three
generators, a, b, c, where a is the largest and c the smallest.
Then ab + c² = denominator, and a² - c², b² - c², and a² - b² will be
the three numerators. Thus, using the generators 3, 2, 1, we get 8/7,
3/7, 5/7 and we can pair the first and second, as in the above
solution, or the first and third for a second solution. The
denominator must always be a prime number of the form 6n + 1, or
composed of such primes. Thus you can have 13, 19, etc., as
denominators, but not 25, 55, 187, etc.
When the principle is understood there is no difficulty in writing down
the dimensions of as many sets of cubes as the most exacting collector
may require. If the reader would like one, for example, with plenty of
nines, perhaps the following would satisfy him: 99999999/99990001 and
19999/99990001.