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Science                                  167
the course of the work, it was necessary to multiply 15 by 13,
the Egyptian proceeded as follows:
/IXIS =   15    The multipliers which added up to 13
2X15 =   30    were ticked off and the corresponding
/4.X 15 =   60   products added together.
/8xiS = 120
13X15= 195
The method of proportion was frequently employed, but
never explicitly formulated. A typical method was to assume
a trial result and then find what alteration was necessary to
fulfil the requirements of the problem. Thus, to quote an
actual case: 'What number added to its one-seventh part gives
19?' To 7 is added its seventh part. Result—8. The figure 19
is then divided by 8 and the result is multiplied by 7 giving the
correct result i6J- J, or i6f as we should now write it.
This is the method of 'false position' (regula falsi or fositio
falsa) much used by Diophantus of Alexandria (c. A.D. 250). It
continued in use in our early arithmetical text-books, until dis-
placed by algebraic methods.
With the exception of f and f no mixed fractions were used,
but only those with unity in the numerator—unit fractions.
A fraction which we should now write as ^ was expressed by
the unit fractions J J written side by side, implying addition,
just as we now write if. Tables were drawn up for fractions
with 2 in the numerator, with their equivalents in two or more
unit fractions. A typical entry in such a table is:
Division of 2 by 89      :     ^   5^   rfl   gfe
If, in the course of the work, it was necessary to double ^ the
writer would refer to the table and set down the series of unit-
fractions given above.
In this way, the Egyptian avoided the trouble of evolving a
more complex fractional notation, which would greatly have