16
INTRODUCTORY
line showing that only rpiyœvovs dpidpovç, triangular num
bers, can be meant. Nevertheless there can, I think, be no
doubt that Diophantus’s Arithmetica belongs to Logistic.
Why then did Diophantus call his thirteen books Arithmetica 1
The explanation is probably this. Problems of the Diophan-
tine type, like those of the arithmetical epigrams, had pre
viously been enunciated of concrete numbers (numbers of
apples, bowls, &c.), and one of Diophantus’s problems (V. 30)
is actually in epigram form, and is about measures of wine
with prices in drachmas. Diophantus then probably saw that
there was no reason why such problems should refer to
numbers of any one particular thing rather than another, but
that they might more conveniently take the form of finding
numbers in the abstract with certain properties, alone or in
combination, and therefore that they might claim to be part
of arithmetic, the abstract science or theory of numbers.
It should be added that to the distinction between arith
metic and logistic there corresponded (up to the time of
Nicomachus) different methods of treatment. With rare
exceptions, such as Eratosthenes’s kovkivov, or sieve, a device
for separating out the successive prime numbers, the theory
of numbers was only treated in connexion with geometry, and
for that reason only the geometrical form of proof was used,
whether the figures took the form of dots marking out squares,
triangles, gnomons, &c. (as with the early Pythagoreans), or of
straight lines (as in Euclid YII-IX) ; even Nicomachus did
not entirely banish geometrical considerations from his work,
and in Diophantus’s treatise on Polygonal Numbers, of which
a fragment survives, the geometrical form of proof is used.
• (/3) Geometry and geodaesia.
By the time ofc‘ Aristotle there was separated out from
geometry a distinct subject, yecoSauria, geodesy, or, as we
should say, mensuration, not confined to land-measuring, but
covering generally the practical measurement of surfaces and
volumes, as we learn from Aristotle himself, 1 as well as from
a passage of Geminus quoted by Proclus. 2
1 Arist. Metaph. B. 2, 997 b 26, 81.
2 Proclus on Eucl. I, p. 89. 20-40. 2.
