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.