Here we have indeed a knotty problem.
Our text-books tell us
spheres are similar, and that similar solids are as the cubes of
Therefore, as the circumferences of the two
were one foot and two feet respectively and the cubes of one and two
added together make nine, what we have to find is two other numbers
cubes added together make nine.
These numbers clearly must be
Now, this little question has really engaged the
attention of learned
for two hundred and fifty years; but although Peter de Fermat showed in
the seventeenth century how an answer may be found in two fractions
a denominator of no fewer than twenty-one figures, not only are all the
published answers, by his method, that I have seen inaccurate, but
has ever published the much smaller result that I now print.
(415280564497 / 348671682660) and (676702467503 / 348671682660) added
together make exactly nine, and therefore these fractions of a foot are
the measurements of the circumferences of the two phials that the
required to contain the same quantity of liquid as those
eminent actuary and another correspondent have taken the trouble to
out these numbers, and they both find my result quite correct.
If the phials were one foot and three feet in
then an answer would be that the cubes of (63284705 / 21446828) and
(28340511 / 21446828) added together make exactly 28.
Given a known case for the expression of a number as the sum
difference of two cubes, we can, by formula, derive from it an infinite
number of other cases alternately positive and negative.
starting from the known case 13 + 23
= 9 (which we will call a
fundamental case), first obtained a negative solution in bigger figures,
and from this his positive solution in bigger figures still.
an infinite number of fundamentals, and I found by trial a negative
fundamental solution in smaller figures than his derived negative
solution, from which I obtained the result shown above.
That is the
We can say of any number up to 100 whether it is
express it as the sum of two cubes, except 66.
Some years ago I published a solution for the case of
of which Legendre gave at some length a "proof" of
impossibility; but I
have since found that Lucas anticipated me in a communication to