Page:The Algebra of Mohammed Ben Musa (1831).djvu/136

( 120 )

of the share. Subtract now one-fourth of what you have, namely, one part less one-sixth of a share; and remove also one part on account of the one dirhem; the remainder, then, is two parts less half a share. Add this to the two-thirds of the capital, which is fifteen (dirhems). Then you have seventeen parts less half a share, equal to five shares. Reduce this, by removing half a share, and adding it to the five shares. Then it is seventeen parts, equal to five shares and a half. Divide now seventeen by five and a half; the quotient is the value of one share, namely, three dirhems and one-eleventh; and one-third (of the capital) is seven and a half (dirhems). (89)

If he leaves four sons, and bequeaths to some person as much as the share of one of his sons, less one-fourth of what remains from one-third after the deduction of the share, and one dirhem; and to another one-third of what remains from the one-third, and one dirhem; then this legacy is determined by one-third.^[1]

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1. Let the 1st legacy be $x$, the 2d $y$; and a son’s share = $v$
   ${\begin{aligned}&1-x-y=4v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {1}{3}}-v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]-d-{\tfrac {1}{3}}[{\tfrac {1}{3}}-v+{\tfrac {1}{4}}({\tfrac {1}{3}}-v)-d]-d=4v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {1}{3}}-v+{\tfrac {1}{4}}({\tfrac {1}{3}}-v)-d]-d=4v\\&{\text{i.e.}}\;{\tfrac {2}{3}}+{\tfrac {2}{3}}[{\tfrac {5}{4}}({\tfrac {1}{3}}-v)-d]-d=4v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {5}{18}}-{\tfrac {5}{6}}v-{\tfrac {5}{3}}d=4v\\&\therefore \;{\tfrac {17}{18}}-{\tfrac {5}{3}}d={\tfrac {29}{6}}v\\&\therefore \;{\tfrac {17}{87}}-{\tfrac {20}{58}}d=v\\&{\text{also}}\;{\tfrac {14}{87}}+{\tfrac {33}{58}}d=x\\&{\tfrac {5}{87}}-{\tfrac {47}{58}}d=y\end{aligned}}$