Page:The Algebra of Mohammed Ben Musa (1831).djvu/172

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three hundred dirhems; then the master dies.” Computation:^[1] Take the property left by the slave, which is three hundred dirhems, and subtract the ransom, which (111) is three hundred less thing; there remains thing, one half of which belongs to the daughter, while the other half returns to the master. Add the portion of the daughter, which is half one thing, to her inheritance, which is three hundred; the sum is three hundred dirhems plus half a thing. The husband receives the moiety of this; the other moiety returns to the master, namely one hundred and fifty dirhems plus one-fourth of thing. All that the master has received is therefore four hundred and fifty less one-fourth of thing; and this is twice as much as the legacy; or the moiety of it is as much as

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1. A. is emancipated by his master, and then dies, leaving a daughter, who dies, leaving a husband. Then the master dies.
   A.’s price$=a$; his property a. What he receives from the master $=x$.
   The daughter’s property $=d$
   A.’s ransom $=a-x$. The daughter inherits ${\tfrac {1}{2}}[\alpha -a+x]$, and ${\tfrac {1}{2}}[\alpha -a+x]$ goes to the master.
   ${\tfrac {1}{2}}[d+{\tfrac {1}{2}}[\alpha -a+x]]$ goes to the daughter’s husband and ${\tfrac {1}{2}}[d+{\tfrac {1}{2}}[\alpha -a+x]]$ to the master.
   Hence, according to the author, we are to make
   $a-x+{\tfrac {1}{2}}[\alpha -a+x]+{\tfrac {1}{2}}[d+{\tfrac {1}{2}}[\alpha -a+x]]=2x$
   

   $\therefore \;x={\tfrac {1}{9}}[3\alpha +a+2d]=200$
   Daughter’s share	$={\tfrac {1}{9}}[6\alpha -4a+d]=100$
   Husband’s	$={\tfrac {1}{9}}[3\alpha -2a+5d]=200$
   Master’s	$={\tfrac {1}{9}}[2\alpha +6a+4d]=400.$