A history of Greek mathematics - Wilbourhall.org

484 DIOPHANTUS OF ALEXANDRIA
or fractional. It is possible, therefore, that Diophantus was
empirically aware of the truth of the theorem of Fermat, but
we cannot be sure of this.
Conspectus of the Arithmetical with typical solutions.
There seems to be no means of conveying an idea of the
extent of the problems solved by Diophantus except by giving
a conspectus of the whole of the six Books.
Fortunately this
can be done by the help of modern notation without occupying
too many pages.
It will be best to classify the propositions according to their
character rather than to give them in Diophantus's order.
should be premised that x, y, z . . . indicating the first, second
and third . . . numbers required do not mean that Diophantus
indicates any of them by his unknown (9)
; he gives his unknown
in each case the signification which is most convenient,
his object being to express all his required numbers at once in
terms of the one unknown (where possible), thereby avoiding the
necessity for eliminations. Where I have occasion to specify
Diophantus's unknown, I shall as a rule call it £, except when
a problem includes a subsidiary problem and it is convenient
to use different letters for the unknown in the original and
subsidiary problems respectively, in order to mark clearly the
distinction between them. When in the equations expressions
are said to be = u 2 , v 2 , w 2 , t 2 ... this means simply that they
are to be made squares. Given numbers will be indicated by
a, b, c ... on, n ... and will take the place of the numbers used
by Diophantus, which are always specific numbers.
Where the solutions, or particular devices employed, are
specially ingenious or interesting, the methods of solution will
The character of the book will be best
be shortly indicated.
appreciated by means of such illustrations.
[The problems marked with an asterisk are probably
spurious.]
It
(i)
Equations of the first degree with one unknown.
I. 7. x — a — m(x — b).
I. 8. x + a = m (x + b).