Page:The Algebra of Mohammed Ben Musa (1831).djvu/62

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by itself; it is five-and-twenty. Subtract from this the twenty-four, which are connected with the square; the remainder is one. Extract its root; it is one. Subtract this from the moiety of the roots, which is five. There remains four, which is one of the two parts.

Observe that, in every case, where any two quantities whatsoever are divided, the first by the second and the second by the first, if you multiply the quotient of the one division by that of the other, the product is always one.^[1]

If some one say: “You divide ten into two parts; multiply one of the two parts by five, and divide it by the other then take the moiety of the quotient, and add this to the product of the one part, multiplied by five; the sum is fifty dirhems;”^[2] then the computation is this: Take thing, and multiply it by five. This is now to be divided by the remainder of the ten, that is, by ten less thing; and of the quotient the moiety is to be taken.

(34) You know that if you divide five things by ten less thing, and take the moiety of the quotient, the result is

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1. ${\tfrac {a}{b}}\times {\tfrac {b}{a}}=1$
2. ${\begin{aligned}{\tfrac {5x}{2(10-x)}}+5x&=50\\{\tfrac {x}{2(10-x)}}+x&=10\\x^{2}+100&=20{\tfrac {1}{2}}x\\x={\tfrac {41}{4}}-{\tfrac {9}{4}}&=8\end{aligned}}$