Page:The Algebra of Mohammed Ben Musa (1831).djvu/132

( 116 )

them to the other shares. Then you have one and one-eighth of one-sixth of capital, equal to eight shares and one-fourth. Reduce this to one capital, by subtracting from the parts as much as one forty-ninth of them. Then you have a capital equal to eight shares and four forty-ninths. Assume now every share to be forty-nine; then the entire capital will be three hundred and ninety-six; the share forty-nine; the legacy (86) determined by one-fourth, ten; and the exception from the second share will be six.

 

---

 

On the Legacy with a Dirhem.

“A man dies, and leaves four sons, and bequeaths to some one a dirhem, and as much as the share of a son, and one-fourth of what remains from one-third after the deduction of that share.” Computation: ^[1] Take

---

1. Let the capital=$1$; a dirhem=$d$; the legacy=$x$; and a son’s share = $v$
   ${\begin{aligned}&1-x=4v\\&x=v+{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]+d\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{3}}-v-{\tfrac {1}{4}}[{\tfrac {1}{3}}-v]-d=4v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {3}{4}}[{\tfrac {1}{3}}-v]-d=4v\\&\therefore \;{\tfrac {2}{3}}+{\tfrac {1}{4}}-d=[4+{\tfrac {3}{4}}]v\\&\therefore \;{\tfrac {11}{12}}-d={\tfrac {19}{4}}v\end{aligned}}$
   $\therefore \;{\tfrac {11}{57}}$ of the capital $-{\tfrac {12}{57}}$ of a dirhem = $v$ and ${\tfrac {13}{57}}$ of the capital $+{\tfrac {48}{57}}$ of a dirhem=$x$, the legacy. If we assume the capital to be so many dirhems, or a dirhem to be such a part of the capital, we shall obtain the