Page:The Algebra of Mohammed Ben Musa (1831).djvu/185

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remain four hundred dirhems less three things, equal to twice the legacy, namely, two things and two-thirds. (119) Reduce this, by means of the three things, and you find four hundred, equal to eight things and one-third. Make the equation with this: one thing will be forty eight dirhems.

“Suppose that a man on his sick-bed makes to another a present of a slave-girl, worth three hundred dirhems, her dowry being one hundred dirhems; the donee cohabits with her, and afterwards, being also on his sick-bed, makes a present of her to the donor, and the latter cohabits with her. How much does he acquire by her, and how much is deducted?”^[1] Com-

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1. We have here the only instance in the treatise of a simple equation, involving two unknown quantities. For what the donee receives is one unknown quantity; and what the donor receives back again from the donee, called by the author “part of thing,” is the other unknown quantity.
   Let what the donee receives $=x$, and what the donor receives $=y$.
   Then, retaining the same notation as before, according to the author, the donee receives, on the whole
   $x-y-[\alpha -{\tfrac {\alpha }{a}}x]+{\tfrac {\alpha }{a}}[x-y]=2y$
   and the donor receives, on the whole
   $a-x+y+[\alpha -{\tfrac {\alpha }{a}}x]-{\tfrac {\alpha }{a}}[x-y]=2[x+{\tfrac {\alpha }{a}}[x-y]]$
   Whence $x={\tfrac {1}{2}}{\tfrac {a}{4a^{2}+5a\alpha -\alpha ^{2}}}[3a^{2}+3a\alpha -2\alpha ^{2}]=102$
   $y={\tfrac {1}{2}}{\tfrac {a}{4a^{2}+5a\alpha -\alpha ^{2}}}[a^{2}-2\alpha ^{2}]=21$

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