Page:The Algebra of Mohammed Ben Musa (1831).djvu/24

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whole root is equal to twenty, and the square which is formed of it is four hundred.

I found that these three kinds; namely, roots, squares, and numbers, may be combined together, and thus three compound species arise; ^[1] that is, “squares and roots equal to numbers;” “squares and numbers equal to roots;” “roots and numbers equal to squares.”

Roots and Squares are equal to Numbers;^[2] for instance, “one square, and ten roots of the same, amount to thirty nine dirhems;” that is to say, what must be the square which, when increased by ten of its own roots, amounts to thirty-nine? The solution is this: you halve the number^[3] of the roots, which in the present instance yields five. This you multiply by itself; the product is twenty-five. Add this to thirty-nine; the sum is sixty-four. Now take the root of this, which is eight, and subtract from it half the number of the roots, which is five; the remainder is three. This is the root of the square which you sought for; the square itself is nine.

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1. The three cases considered are,
   ${\begin{aligned}{\text{1st.}}\;cx^{2}+bx=a\\{\text{2d.}}\;cx^{2}+a=bx\\{\text{3d.}}\;cx^{2}=bx+a\end{aligned}}$
2. ${\begin{aligned}{\text{1st case}}:\;cx^{2}+bx=a\\{\text{Example}}\;x^{2}+10x=39\\x={\sqrt {({\tfrac {1}{2}})^{2}+39}}-{\tfrac {10}{2}}\\={\sqrt {64}}-5\\=8-5=3\end{aligned}}$
3. i.e the coefficient.