Page:The Algebra of Mohammed Ben Musa (1831).djvu/66

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the two measures, the arithmetical equivalent for which is six-and-twenty thirteenths, added to the difference of the two prices, which is two-thirteenths: both differences together being likewise equal to twenty-eight parts.

If he say: “There are two numbers,^[1] the difference of which is two dirhems. I have divided the smaller by the larger, and the quotient was just half a dirhem.”^[2] Suppose one of the two numbers^[1] to be thing, and the other to be thing plus two dirhems. By the division of thing by thing plus two dirhems, half a dirhem appears as quotient. You have already observed, that by multiplying the quotient by the divisor, the capital which you divided is restored. This capital, in the present case, is thing. Multiply, therefore, thing and two dirhems by half a dirhem, which is the quotient; the product is half one thing plus one dirhem; this is equal to thing. Remove, now, half a thing on account

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1. In the original, “squares.” The word square is used in the text to signify either, 1st, a square, properly so called, fractional or integral; 2d, a rational integer, not being a square number; 3d, a rational fraction, not being a square; 4th, a quadratic surd, fractional or integral.
2. ${\begin{aligned}&{\tfrac {x}{x+2}}={\tfrac {1}{2}}\\&x={\tfrac {x+2}{2}}={\tfrac {x}{2}}+1\\&{\tfrac {x}{2}}=1\;{\text{and}}\;x+2=4\end{aligned}}$