Page:The Algebra of Mohammed Ben Musa (1831).djvu/176

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five hundred dirhems; and remember that the price of the slave is one hundred dirhems. Call the legacy of the donee thing. The emancipation of the slave, whose price is one hundred dirhems, has already taken place. He has bequeathed one thing to the donee. Add the dowry, which is one hundred dirhems less one-fifth thing. Then in the hands of the heirs are six hundred dirhems less one thing and one-fifth of thing. This is twice as much as one hundred dirhems and thing; the moiety of it is equal to the legacy of the two, namely, three hundred less three-fifths of thing. Reduce this by removing the three-fifths of thing from three hundred, and add the same to one thing. This gives three hundred dirhems, equal to one thing and three-fifths and one hundred dirhems. Subtract now from three hun-

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Hence, according to the author, we are to make
$a-x=2[\alpha +x]$; whence $x={\tfrac {a-2\alpha }{3}}$
And her ransom is ${\tfrac {2}{3}}[a+\alpha ]$
But if a male slave be at the same time emancipated by the master, the donee must pay the ransom of that slave.
If his price was $b\;,b-{\tfrac {b}{a}}x$ is his ransom.
Hence, according to the author, we are to make the sum of the two ransoms, viz. $a-x+b-{\tfrac {b}{a}}x=2[\alpha +x]$
$\therefore \;a+b-2\alpha =[3+{\tfrac {b}{a}}]x\;\therefore \;x=a{\tfrac {a+b-2\alpha }{3a+b}}=125$
The donee pays ransom, in respect of the slave-girl $(a-x)=375$ and he pays ransom for the male slave $b-{\tfrac {b}{a}}x=75$.