A history of Greek mathematics - Wilbourhall.org

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
# = 2/ + §z, y = * + £«% z=10+§2/>
and one (xiv.
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-