Page:The Algebra of Mohammed Ben Musa (1831).djvu/134

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share will be two dirhems and two-nineteenths. Or, if you wish to exhibit the share distinctly, complete your square, and reduce it, when the dirhem will be eleven of the capital.

If he leaves five sons, and bequeaths to some person a dirhem, and as much as the share of one of the sons, and one-third of what remains from one-third, and again, one-fourth of what remains from the one-third after the deduction of this, and one dirhem more; then the computation is this:^[1] You take one-third, and subtract one share; there remains one-third less one share. Subtract herefrom that which is still in your hands, namely, one-third of one-third less one-third of the share. Then subtract also the dirhem; there remain two-thirds of one-third, less two-thirds of the share and less one dirhem. Then subtract one-fourth of what you have, that is, one-eighteenth, less one-sixth of a share and less one-fourth of a dirhem, and

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1. Let the legacy=$x$; and a son’s share $v$
   ${\begin{aligned}&1-x=5v\\&{\tfrac {2}{3}}+{\tfrac {1}{3}}-v-{\tfrac {1}{3}}[{\tfrac {1}{3}}-v]-d-{\tfrac {1}{4}}[{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d]-d=5v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d-{\tfrac {1}{4}}[{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d]-d=5v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{4}}[{\tfrac {2}{3}}[{\tfrac {1}{3}}-v]-d]-d=5v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{6}}-{\tfrac {1}{2}}v-{\tfrac {7}{4}}d=5v\\&\therefore \;{\tfrac {5}{6}}-{\tfrac {7}{4}}d={\tfrac {11}{2}}v\\\end{aligned}}$
   $\therefore \;{\tfrac {10}{66}}$ of the capital $-{\tfrac {21}{66}}$ of a dirhem = $v$
   $\therefore \;{\tfrac {16}{66}}$ of the capital $+{\tfrac {105}{66}}$ of a dirhem = $x$, the legacy.
   If the capital ${\tfrac {45}{2}}$ dirhems, or ${\tfrac {1}{3}}$ of the capital=$7{\tfrac {1}{2}}$ dirhems, $v={\tfrac {34}{11}}$=dirhems=$3{\tfrac {1}{33}}$ dirhems.