A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

'
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
_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
or
Sa + 9d = 7(2a + d),
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 2
+ y 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
442 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
arithmetical epigrams contained in the Greek Anthology. Most
of these appear under the name of Metrodorus, a grammarian,
probably of the time of the Emperors Anastasius I (a.d. 491-
518) and Justin I (a.d. 518-27). They were obviously only
collected by Metrodorus, from ancient as well as more recent
sources. Many of the epigrams (46 in number) lead to simple
equations, and several of them are problems of dividing a number
of apples or nuts among a certain number of persons, that
is to say, the very type of problem mentioned by Plato. For
example, a number of apples has to be determined such that,
if four persons out of six receive one-third, one-eighth, onefourth
and one-fifth respectively of the whole number, while
the fifth person receives 1
for the sixth person, i.e.
apples, there is one apple left over
2;X + ±X + %x + ~x + 10 + 1 — x.
Just as Plato alludes to bowls ((f>id\ai) of different metals,
there are problems in which the weights of bowls have to
be found. We are thus enabled to understand the allusions of
Proclus and the scholiast on Charmides 165 E to fi-qXiTai
and (jytaXiraL dpi6/xoi, 'numbers of apples or of bowls'.
It is evident from Plato's allusions that the origin of such
simple algebraical problems dates back, at least, to the fifth
century B.C.
The following is a classification of the problems in the
Anthology. (1) Twenty-three are simple equations in one
unknown and of the type shown above; one of these is an
epigram on the age of Diophantus and certain incidents of
his life (xiv. 126). (2) Twelve are easy simultaneous equations
with two unknowns, like Dioph. I. 6 ; they can of course be
reduced to a simple equation with one unknown by means of
an easy elimination. One other (xiv. 51) gives simultaneous
equations in three unknowns
and one (xiv.
# = 2/ + §z, y = * + £«% z=10+§2/>
49) gives four equations in four unknowns,
x + y = 40, x + z=45, x + u = 36, x + y + z + u = 60.
With these may be compared Dioph. I. 16-21, as well as the
general solution
of any number of simultaneous linear equa-