424 14. EQUATIONS AND ALGORITHMSÇ—
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i—ÇJFIGURE 3. Al-Khwarizmi's solution of "square plus 10 roots equals 39 dirhems."such cases in problems of inheritance, which occupy more than half of his Algebra.
Here is a sample:
A man dies, leaving two sons behind him, and bequeathing one-fifth
of his property and one dirhem to a friend. He leaves 10 dirhems in
property and one of the sons owes him 10 dirhems. How much does
each legatee receive?Although mathematics is cross-cultural, its applications are very specific to the
culture in which they are used. The difference between the modern solution of this
legal problem and al-Khwarizmi's solution is considerable. Under modern law the
man's estate would be considered to consist of 20 dirhems, the 10 dirhems cash on
hand, and the 10 dirhems owed by one of the sons. The friend would be entitled to
5 dirhems (one-fifth plus one dirhem), and the indebted son would owe the estate
10 dirhems. His share of the estate would be one-half of the 15 dirhems left after
the friend's share is taken out, or l\ dirhems. He would therefore have to pay l\
dirhems to the estate, providing it with cash on hand equal to 12^ dirhems. His
brother would receive l\ dirhems.
Now the notion of an estate as a legal entity that can owe and be owed money
is a modern European one, alien to the world of al-Khwarizmi. Apparently in al-
Khwarizmi's time, money could be owed only to a person. What principles are to
be used for settling accounts in this case? Judging from the solution given by al-
Khwarizmi, the estate is to consist of the 10 dirhems cash on hand, plus a certain
portion (not all) of the debt the second son owed to his deceased father. This
"certain portion" is the unknown in a linear equation and is the reason for invoking
algebra in the solution. It is to be chosen so that when the estate is divided up,
the indebted son neither receives any more money nor owes any to the other heirs.
This condition leads to an equation that can be solved by algebra. Al-Khwarizmi