the Arabs who translated or
commented on Diophantus ever had access to more of the work than we now
have. The difference in form and content suggests that the _Polygonal
Numbers_ was not part of the larger work. On the other hand the
_Porisms_, to which Diophantus makes three references ("we have it in
the Porisms that ..."), were probably not a separate book but were
embodied in the _Arithmetica_ itself, whether placed all together or, as
Tannery thinks, spread over the work in appropriate places. The
"Porisms" quoted are interesting propositions in the theory of numbers,
one of which was clearly that _the difference between two cubes can be
resolved into the sum of two cubes_. Tannery thinks that the solution of
a complete quadratic promised by Diophantus himself (I. def. 11), and
really assumed later, was one of the Porisms.

Among the great variety of problems solved are problems leading to
determinate equations of the first degree in one, two, three or four
variables, to determinate quadratic equations, and to indeterminate
equations of the first degree in one or more variables, which are,
however, transformed into determinate equations by arbitrarily
assuming a value for one of the required numbers, Diophantus being
always satisfied with a rational, even if fractional, result and not
requiring a solution in integers. But the bulk of the work consists of
problems leading to indeterminate equations of the second degree, and
these universally take the form that one or two (and never more)
linear or quadratic functions of one variable x are to be made
rational square numbers by finding a suitable value for x. A few
problems lead to indeterminate equations of the third and fourth
degrees, an easy indeterminate equation of the sixth degree being
also found. The general type of problem is to find two, three or four
numbers such that different expressions involving them in the first
and second, and sometimes the third, degree are squares, cubes, partly
squares and partly cubes, &c. E.g. _To find three numbers such that
the product of any two added to the sum of those two gives a square_
(III. 15, ed. Tannery); _To find four numbers such that, if we take
the square of their sum [+-] any one of them singly, all the resulting
numbers are squares_ (III. 22); _To find two numbers such that their
product [+-] their sum gives a cube_ (IV. 29); _To find three squares
such that their