Page:The Algebra of Mohammed Ben Musa (1831).djvu/166

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hems less one thing and a half, and this is twice as much as the sum of the two legacies of both, namely (107) two things, consequently as much as four things. Remove from this the one thing and a half: you find seven hundred dirhems, equal to five things and a half. Make the equation. One thing will be one hundred and twenty-seven dirhems and three-elevenths.

“Suppose that a man in his illness emancipate a slave, whose price is three hundred dirhems, but who has already paid off to his master two hundred dirhems, which the latter has spent; then the slave dies before the death of the master, leaving a daughter and three hundred dirhems.”^[1] Computation: Take the property left by the slave, namely, the three hundred, and add thereto the two hundred, which the master has spent; this together makes five hundred dirhems. Subtract from this the ransom, which is three hundred less thing

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1. The slave A. dies before his master, and leaves a daughter. His cost is $a$, of which he has redeemed $a^{\prime }$, which the master has spent; and he leaves property $\alpha$.
   

   Then the daughter receives	${\tfrac {1}{2}}[\alpha +a^{\prime }-a+x]$
   The master receives altogether	${\tfrac {1}{2}}[\alpha +a^{\prime }+a-x]$
   The master’s heirs receive	${\tfrac {1}{2}}[\alpha -a^{\prime }+a-x]$
   And ${\tfrac {1}{2}}[\alpha -a^{\prime }+a-x]=2x\;\therefore \;x={\tfrac {1}{5}}[\alpha -a^{\prime }+a]$
   Hence the daughter receives	${\tfrac {1}{5}}[3\alpha +2a^{\prime }-2a]=140$
   The master’s heirs	${\tfrac {1}{5}}[2\alpha -2a^{\prime }+2a]=160$
   The master receives, in toto,	${\tfrac {1}{5}}[2\alpha +3a^{\prime }+2a]=360$

   If the slave had not advanced, or the master had not spent $a^{\prime }$,
   the daughter would have received ${\tfrac {1}{5}}[3\alpha +3a^{\prime }-2a]=180$
   and the master would have received ${\tfrac {1}{5}}[2\alpha +2a^{\prime }+2a]=320$