Page:The Algebra of Mohammed Ben Musa (1831).djvu/64

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taken eighty-one times.”^[1] Computation: You say, ten less thing, multiplied by itself, is a hundred plus a (35) square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.

If some one say: “I have purchased two measures of wheat or barley, each of them at a certain price. I afterwards added the expences, and the sum was equal to the difference of the two prices, added to the difference of the measures.”^[2]

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1. ${\begin{aligned}(10-x)^{2}&=81x\\100-20x+x^{2}&=81x\\x^{2}+100&=101x\\x={\tfrac {101}{2}}-{\sqrt {{\tfrac {101^{2}}{4}}-100}}&=50{\tfrac {1}{2}}-49{\tfrac {1}{2}}=1\end{aligned}}$
2. The purchaser does not make a clear enunciation of the terms of his bargain. He intends to say, “I bought $m$ bushels of wheat, and $n$ bushels of barley, and the wheat was $r$ times dearer than the barley. The sum I expended was equal to the difference in the quantities, added to the difference in the prices of the grain.”