Diophantus devotes only the first book of his Arithmeticato the
solution of determinate equations. The remaining books extant
treat mainly of indeterminate quadratic equationsof the form Ax²+
Bx+C=y²,or of two simultaneous equations of the same form. He
considers several but not all the possible cases which may arise in
these equations. The opinion of Nesselmann on the method of Dio-
phantus, as stated by Gow, is as follows: "(1) Indeterminate equations
of the second degree are treated completely only when the quadratic
or the absolute term is wanting: his solution of the equations Ax²+
C=y²and Ax²+Bx+C=y²is in many respects cramped. (2) For
the 'double equation' of the second degree he has a definite rule only
when the quadratic term is wanting in both expressions: even then
his solution is not general. More complicated expressions occur only
under specially favourable circumstances." Thus, he solves Bx+C²
=y², B ₁ x+C ₁ ²=y ₁ ².

The extraordinary ability of Diophantus lies rather in another di-
rection, namely, in his wonderful ingenuity to reduce all sorts of
equations to particular forms which he knows how to solve. Very
great is the variety of problems considered. The 130 problems found
in the great work of Diophantus contain over 50 different classes of
problems, which are strung together without any attempt at classi-
fication. But still more multifarious than the problems are the solu-
tions. General methods are almost unknown to Dipohantus. Each
problem has its own distinct method, which is often useless for the
most closely related problems. "It is, therefore, difficult for a modern,
after studying 100 Diophantine solutions, to solve the 101st." This
statement, due to Hankel, is somewhat overdrawn, as is shown by
Heath. ¹

That which robs his work of much of its scientific value is the
fact that he always feels satisfied with one solution, though his equa-
tion may admit of an indefinite number of values. Another great
defect is the absence of general methods. Modern mathematicians,
such as L. Euler, J. Lagrange, K. F. Gauss, had to begin the study of
indeterminate analysis anew and received no direct aid from Dio-
phantus in the formulation of methods. In spite of these defects
we cannot fail to admire the work for the wonderful ingenuity ex-
hibited therein in the solution of particular equations.

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