14 
INTRODUCTORY 
calculation) was a fundamental one in Greek mathematics. 0 f f rac f 
It is found elsewhere in Plato, 1 and it is clear that it was well the thee 
established in Plato’s time. Archytas too has AoyLarLK-q in to the s 
the same sense ; the art of calculation, he says, seems to be far ^l ie 
ahead of other arts in relation to wisdom or philosophy, nay clear b 
it seems to make the things of which it chooses to treat even ordinar 
clearer than geometry does; moreover, it often succeeds even plicatio 
where geometry fails. 2 But it is later writers on the classification include« 
of mathematics who alone go into any detail of what XoyLaTLKrj Next i 
included. Geminus in Proclus, Anatolius in the Variae Gollec- ^ or a pj. 
tiones included in Hultsch’s Heron, and the scholiast to Plato’s recogni 
Charmides are our authorities. Arithmetic, says Geminus, 3 is epigran 
divided into the theory of linear numbers, the theory of plane are p ro 
numbers, and the theory of solid numbers. It investigates, a certai 
in and by themselves, the species of number as they are succes- bowls 
sively evolved from the unit, the formation of plane numbers, ru ] e th 
similar and dissimilar, and the further progression to the third unknov 
dimension. As for the AoyianKos, it is not in and by themselves two are 
that he considers the properties of numbers but with refer- j n p 0s if 
ence to sensible objects; and for this reason he applies to it j s c ], 
them names adapted from the objects measured, calling some century 
(numbers) ingXirp^ (from /j.fjhov, a sheep, or ¡xrjXov, an apple, is 0 f c 
more probably the latter) and others (piaXirrjs (from (pidXrj, solutioi 
a bowl). 4 The scholiast to the Charmides is fuller still; 5 imprac 
‘ Logistic is the science which deals with numbered things, of unk 
not numbers; it does not take number in its essence, angula] 
but it presupposes 1 as unit, and the numbered object as corresp 
number, e.g. it regards 3 as a triad, 10 as a decad, and 0 f trial 
applies the theorems of arithmetic to such (particular) cases. allusioi 
Thus it is logistic which investigates on the one hand what ruinate 
Archimedes called the cattle-problem, and on the other hand , 
melites and phialites numbers, the latter relating to bowls, enmeui 
the former to flocks (he should probably have said “ apples ”); bers su 
in other kinds too it investigates the numbers of sensible the 4ch 
bodies, treating them as absolute (coy rrepl reXeicov). Its sub- Tanner 
ject-matter is everything that is numbered. Its branches have h 
include the so-called Greek and Egyptian methods in multi- on j v 0 
plications and divisions, 0 the additions and decompositions nurnbe 
1 Cf. Gorgias, 451 b, c ; TheaeteUis, 145 A with 198 A, &c. , the rpt 
2 Diels, VorsoJcratiker, i 3 , p. 337. 7-11. . , , 
3 Proclus on Eucl. I, p. 39. 14-20. 4 lb., p. 40. $-5. ngiit-a 
5 On Charmides, 165 e, 6 See Chapter II, pp. 52-60. main s