( 57 )
itself restores the square;”[1] then the computation is this: If you subtract one-third and three dirhems from the square, there remain two-thirds of it less three dirhems. This is the root. Multiply therefore two-thirds of thing less three dirhems by itself. You say two-thirds by two-thirds is four ninths of a square; and less two-thirds by three dirhems is two roots: and again, two-thirds by three dirhems is two roots; and less three dirhems by less three dirhems is nine dirhems. You (41) have, therefore, four-ninths of a square and nine dirhems less four roots, which are equal to one root. Add the four roots to the one root, then you have five roots, which are equal to four-ninths of a square and nine dirhems. Complete now your square; that is, multiply the four-ninths of a square by two and a fourth, which gives one square; multiply likewise the nine dirhems by two and a quarter; this gives twenty and a quarter; finally, multiply the five roots by two and a quarter; this gives eleven roots and a quarter. You have, therefore, a square and twenty dirhems and a quarter, equal to eleven roots and a quarter. Reduce this according to what I taught you about halving the roots.
![{\displaystyle {\begin{aligned}\left[x-({\tfrac {x}{3}}+3)\right]^{2}&=x\\{\text{or}}\;[{\tfrac {2x}{3}}-3]^{2}&=x\\{\tfrac {4x^{2}}{9}}+9&=5\\x^{2}+20{\tfrac {1}{4}}&=11{\tfrac {1}{4}}x\\x=9,\;{\text{or}}\;2{\tfrac {1}{4}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4aa3667016dee93d22f4483e353d4f5e329020a)
i
