Page:The Algebra of Mohammed Ben Musa (1831).djvu/158

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computation is this:^[1] Call the legacy to the slave thing: He has to return the remainder of his price, after the deduction of the legacy, or three hundred less thing. This ransom, of three hundred less thing, belongs to the master. Now the slave dies, and leaves thing and a (103) daughter. She must receive the moiety of this, namely, one half of thing; and the master receives as much. Therefore the heirs of the master receive three hundred less half a thing, and this is twice as much as the legacy, which is thing, namely, two things. Reduce this by removing half a thing from the three hundred, and adding it to the two things. Then you have three hundred, equal to two things and a half. One thing is, therefore, as much as two-fifths of three hundred,

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1. Let the slave’s original cost be $a$; the property which he dies possessed of, $\alpha$; what the master bequeaths to the slave, in emancipating him, $x$. Then the net property which the slave dies possessed of is $\alpha +x-a.{\tfrac {1}{2}}[\alpha +x-a]$ belongs, by law, to the master; and ${\tfrac {1}{2}}[\alpha +x-a]$ to the slave’s daughter. The master’s heirs, therefore, receive the ransom, $a-x$, and the inheritance, ${\tfrac {1}{2}}[\alpha +x-a]$; that is, ${\tfrac {1}{2}}[\alpha +a-x]$; and on the same principle as the slave, when emancipated, is allowed to ransom himself at two-thirds of his cost, the law of the case is that $2$ are to be taken, where $1$ is given.

   $\therefore \;{\tfrac {1}{2}}[\alpha +a-x]=2x\;\therefore \;x={\tfrac {1}{5}}[\alpha +a]$
   The daughter’s share of the inheritance	${\tfrac {1}{5}}[3\alpha -2a]$
   The master’s heirs receive .............	${\tfrac {2}{5}}[\alpha +a]$

   If, as in the example, $\alpha =a$, $x={\tfrac {2}{5}}a$; the daughter’s share = ${\tfrac {1}{5}}$a; the heirs of the master receive ${\tfrac {4}{5}}$a.