Similarly, the other rational **integers** must be distinguished from the corresponding cardinals.

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**.

We find that fractions follow certain laws corresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were **integers**.

The solution in **integers** of the indeterminate equation ax+by=c may be effected by means of continued fractions.

If we take aq-bp= +1 we have a general solution in **integers** of ax+by=c, viz.