Page:The Algebra of Mohammed Ben Musa (1831).djvu/159

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namely, one hundred and twenty. This is the legacy (to the slave,) and the ransom is one hundred and eighty.

“Some person on his sick-bed has emancipated a slave, whose price is three hundred dirhems; the slave then dies, leaving four hundred dirhems and ten dirhems of debt, and two daughters, and bequeathing to a person one-third of his capital; the master has twenty dirhems debts.” The computation of this case is the following:^[1] Call the legacy to the slave thing; his ransom is the remainder of his price, namely, three hundred less thing. But the slave, when dying, left four hundred dirhems; and out of this sum, his ransom, namely, three hundred less thing, is paid to the

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1. Let the slave’s original cost=$a$; the property he dies possessed of=$\alpha$; the debt he owes = $\epsilon$
   He leaves two daughters, and bequeaths to a stranger one-third of his capital.
   The master owes debts to the amount $\mu$; where $a=300;\alpha =400;\epsilon =10;\mu =20$.
   Let what the master gives to the slave, in emancipating him $=x$.
   Slave’s ransom=$a-x$; slave’s property—slave’s ransom=$\alpha +x-a$
   

   Slave’s property—ransom—debt=$\alpha +x-a-\epsilon$
   Legacy to stranger	$={\tfrac {1}{3}}[\alpha +x-a-\epsilon ]$
   Residue ...............	$={\tfrac {2}{3}}[\alpha +x-a-\epsilon ]$

   The master, and each daughter, are, by law, severally entitled to ${\tfrac {1}{3}}\times {\tfrac {2}{3}}[\alpha +x-a-\epsilon ]$
   The master’s heirs receive altogether $a-x+{\tfrac {2}{9}}[\alpha +x-a-\epsilon ]$ or ${\tfrac {7}{9}}[a-x]+{\tfrac {2}{9}}[[\alpha -\epsilon ]$, which, on the principle that $2$