continued product added to any one of them gives a
square_ (V. 21). Book VI. contains problems of finding rational
_right-angled triangles_ such that different functions of their parts
(the sides and the area) are squares. A word is necessary on
Diophantus' notation. He has only one symbol (written somewhat like a
final sigma) for an unknown quantity, which he calls [Greek: arithmos]
(defined as "an undefined number of units"); the symbol may be a
contraction of the initial letters [alpha][rho], as [Delta]^[Upsilon],
[Kappa]^[Upsilon], [Delta]^[Upsilon][Delta], &c., are for the powers
of the unknown ([Greek: dynamis], square; [Greek: kubos], cube;
[Greek: dynamodynamis], fourth power, &c.). The only other algebraical
symbol is [graphic: /|\] for minus; plus being expressed by merely
writing terms one after another. With one symbol for an unknown, it
will easily be understood what scope there is for adroit assumptions,
for the required numbers, of expressions in the one unknown which are
at once seen to satisfy some of the conditions, leaving only one or
two to be satisfied by the particular value of x to be determined.
Often assumptions are made which lead to equations in x which cannot
be solved "rationally," i.e. would give negative, surd or imaginary
values; Diophantus then traces how each element of the equation has
arisen, and formulates the auxiliary problem of determining how the
assumptions must be corrected so as to lead to an equation (in place
of the "impossible" one) which can be solved rationally. Sometimes his
x has to do duty twice, for different unknowns, in one problem. In
general his object is to reduce the final equation to a simple one by
making such an assumption for the side of the square or cube to which
the expression in x is to be equal as will make the necessary number
of coefficients vanish. The book is valuable also for the propositions
in the theory of numbers, other than the "porisms," stated or assumed
in it. Thus Diophantus knew that _no number of the form 8n + 7 can be
the sum of three squares_. He also says that, if 2n + 1 is to be the
sum of two squares, "n must not be odd" (i.e. _no number of the form
4n + 3, or 4n - 1, can be the sum of two squares_), and goes on to
add, practically, the condition stated by Fermat, "and the double of
it [n] increased by one, when divided by the greatest square which
measures it, mus