‘ HAU ’-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: 
i-x + x = 19, 
§x + £x + j-x + x = 33, 
(x + §x) — f (x + §x) = 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 sub 
stituting 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 + id, a + 3d, a + 2d, a + d, a be the shares, then 
3cc 9 d = 7 (2a + cZ), 
or d = 5 \ a. 
Ahmes says, without any explanation, ‘ make the difference, 
as it is, 5f’, and then, assuming a = 1, writes the series 
23, 17f, 12, 6f, 1. The addition of these gives 60, and 100 is 
1| times 60. Ahmes says simply ‘multiply If times’ and 
thus gets the correct values 38f, 29f, 20, 10|f, If. 
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 1 + y 2 is thus found to be • 
In order to make 100, ff 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 solved in the Papyrus Rhind are just 
the kind of equations of which we find many examples in the