Page:The Algebra of Mohammed Ben Musa (1831).djvu/122

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and add to it the four shares. Then you have the capital and one-third of one-fifth of the capital, which are equal to five shares and one-fifth. Reduce this to one capital, by subtracting from what you have the moiety of one-eighth of it, that is to say, one-sixteenth. Then you find the capital equal to four shares and seven-eighths of a share. Assume now thirty-nine as capital; one-third of it will be thirteen, and one share eight; what remains of one-third, after the deduction of that share, is five, and one-fifth of this is one. Subtract now the one, which was excepted from the legacy; the remaining legacy then is seven; subtracting this from the one-third of the capital, there remain six. Add this to the two-thirds of the capital, namely, to the twenty-six parts, the sum is thirty-two; which, when distributed among the four sons, yields eight for each of them.

If he leaves three sons and a daughter^[1], and bequeaths to some person as much as the share of a

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1. Since there are three sons and one daughter, the daughter receives ${\tfrac {1}{7}}$, and each son ${\tfrac {1}{7}}$ths of the residue.
   If the 1st legacy=$v$, the $2d=y$, and therefore a daughter’s share=$v$,
   ${\begin{aligned}1-v-y=7v;{\tfrac {1}{5}}+{\tfrac {1}{6}}={\tfrac {11}{30}}\\\therefore \;{\tfrac {5}{7}}+{\tfrac {2}{7}}-v-{\tfrac {11}{30}}[{\tfrac {2}{7}}-v]=7v\\{\text{i.e.}}\;{\tfrac {5}{7}}+{\tfrac {19}{30}}[{\tfrac {2}{7}}-v]=7v\\\therefore \;{\tfrac {5}{7}}+{\tfrac {19}{15\times 7}}=[7+{\tfrac {19}{30}}]v\\\therefore \;{\tfrac {94}{7}}={\tfrac {229}{2}}v\;\therefore \;={\tfrac {188}{1603}}\\\end{aligned}}$
   The 2d legacy =$\;\therefore \;y={\tfrac {99}{1603}}$