A history of Greek mathematics - Wilbourhall.org

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HAU
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'-CALCULATIONS 441
in terms of it are equivalent to the solutions of simple equations
with one unknown quantity. Examples from the Papyrus
Rhind correspond to the following equations
_2 />» l jL /yt I 1 rp _L ryi Q O
3 iAy T^ o *&/ i^ rj tAj ^ iAs — O O ,
(a? + §a?)--|(a; + §a?) = 10.
The Egyptians anticipated, though only in
an elementary
form, a favourite method of Diophantus, that of the ' false
supposition ' or ' regula falsi \ An arbitrary assumption is
made as to the value of the unknown, and the true value
is afterwards found by a comparison of the result of substituting
the wrong value in the original expression with the
actual data. Two examples may be given. The first, from
the Papyrus Rhind, is the problem of dividing 100 loaves
among five persons in such a way that the shares are in
arithmetical progression, and one-seventh of the sum of the
first three shares is equal to the sum of the other two. If
a + 4;d, a+3d, a + 2d, a + d, a be the shares, then
Sa + 9d = 7(2a + d),
or
d = 5ja.
Ahmes says, without any explanation, ' make
as it is,
the difference,
5-J',
and then, assuming a = 1, writes the series
23, 17},' 12, 6£, 1. The addition of these gives 60, and 100 is
If times 60. Ahmes says simply 'multiply If times' and
thus gets the correct values 38|, 29f, 20, 10§|, 1|.
The second example (taken from the Berlin Papyrus 6619)
is the solution of the equations
x 2 +y 2 = 100,
•
x :y = 1 :*|, or y = \x.
x is first assumed to be 1 , and
x 2 + y
2 is thus found to be f |
In order to make 100, f§ has to be multiplied by 64 or 8 2 .
The true value of x is therefore 8 times 1 , or 8.
Arithmetical epigrams in the Greek Anthology.
The simple equations eolved in the Papyrus Rhind are just
the kind of equations of which we find many examples in the