Page:The Algebra of Mohammed Ben Musa (1831).djvu/163

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the legacy to the first slave, whose price is three hundred dirhems, thing. His ransom is three hundred dirhems less thing. The legacy to the second slave of a price of five hundred dirhems is one thing and two-thirds, and his ransom five hundred dirhems less one thing and two-thirds (viz. his price being one and two-thirds times the price of the first slave, whose ransom was thing, he must pay one thing and two-thirds for

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Let $x$ be that which the master gives to A. in emancipating him.
A.’s ransom is $a-x$; and his property, minus his ransom, is $\alpha -a+x$.
A.’s daughter receives ${\tfrac {1}{2}}[\alpha -a+x]$, and the master’s heirs receive ${\tfrac {1}{2}}[\alpha -a+x]$
Hence the master receives altogether from A.,
$a-x+{\tfrac {1}{2}}[\alpha -a+x]={\tfrac {1}{2}}[\alpha +a-x.]$
B.’s ransom is $b-{\tfrac {b}{a}}x$
The master’s heirs receive from A. and B. together ${\tfrac {1}{2}}[\alpha +a+2b]-{\tfrac {1}{2a}}[a+2b]x$; and this is to be made equal to twice the amount of the legacies to A. and B., that is, ${\tfrac {1}{2}}[\alpha +a+2b]-{\tfrac {1}{2a}}[a+2b]x=2{\tfrac {a+b}{a}}x$
$\therefore \;x=a{\tfrac {\alpha +a+2b}{5a+6b}}={\tfrac {1700}{15}}=113{\tfrac {1}{3}}$
The master’s heirs receive from A., ${\tfrac {2a[\alpha +a+b]+3\alpha b}{5a+6}}=293{\tfrac {1}{3}}$
A.’s daughter receives $[a+b]{\tfrac {3\alpha -2}{5a+6}}a=800\times {\tfrac {600}{4500}}=106{\tfrac {2}{3}}$
The legacy to B. is $b\;{\tfrac {\alpha +a+2b}{5a+6b}}=188{\tfrac {8}{9}}$; his ransom $b{\tfrac {4a+4b-\alpha }{5a+6b}}=311{\tfrac {1}{9}}$
The master’s heirs receive from A. and B. together $2[a+b]{\tfrac {\alpha +a+2b}{5a+6b}}=604{\tfrac {4}{9}}$.