ARITHMETIC AND LOGISTIC 
15 
mathematics, 
at it was well 
XoyurTLKrj in 
sems to be far 
ulosophy, nay 
to treat even 
succeeds even 
e classification 
Giat XoyicyTLKTj 
Variae Gollec- 
liast to Plato’s 
r s Geminus, 3 is 
heory of plane 
t investigates, 
hey are succes- 
fiane numbers, 
on to the third 
. by themselves 
mt with refer- 
he applies to 
d, calling some 
7Xov, an apple, 
? (from (pidXr], 
Her still; 5 
.inhered things, 
in its essence, 
iered object as 
a decad, and 
irticular) cases, 
me hand what 
the other hand 
Ring to bowls, 
said “ apples ”); 
»ers of sensible 
eiW). Its sub- 
Its branches 
thods in multi 
decompositions 
, &c. 
0. 5-5. 
,pter II, pp. 52-60. 
of fractions; which methods it uses to explore the secrets of 
the theory of triangular and polygonal numbers with reference 
to the subject-matter of particular problems.’ 
The content of logistic is for the most part made fairly 
clear by the scholia just quoted. First, it comprised the 
ordinary arithmetical operations, addition, subtraction, multi 
plication, division, and the handling of fractions; that is, it 
included the elementary parts of what we now call arithmetic. 
Next, it dealt with problems about such things as sheep 
(or apples), bowls, &c.; and here we have no difficulty in 
recognizing such problems as we find in the arithmetical 
epigrams included in the Greek anthology. Several of them 
are problems of dividing a number of apples or nuts among 
a certain number of persons ; others deal with the weights of 
bowls, or of statues and their pedestals, and the like; as a 
rule, they involve the solution of simple equations with one 
unknown, or easy simultaneous equations with two unknowns; 
two are indeterminate equations of the first degree to be solved 
in positive integers. From Plato’s allusions to such problem^ 
it is clear that their origin dates back, at least, to the fifth 
century b.c. The cattle-problem attributed to Archimedes 
is of course a much more difficult problem, involving the 
solution of a f Pellian ’ equation in numbers of altogether 
impracticable size. In this problem the sums of two pairs 
of unknowns have to be respectively a square and a tri 
angular number; the problem would therefore seem to 
correspond to the description of those involving £ the theory 
of triangular and polygonal numbers’. Tannery takes the 
allusion in the last words to be to problems in indeter 
minate analysis like those of Diophantus’s Arithmetica. The 
difficulty is that most of Diophantus’s problems refer to num 
bers such that their sums, differences, &c., are squares, whereas 
the scholiast mentions only triangular and polygonal numbers. 
Tannery takes squares to be included among polygons, or to 
have been accidentally omitted by a copyist. But there is 
only one use in Diophantus’s Arithmetica of a triangular 
number (in IV. 38), and none of a polygonal number; nor can 
the rpiycorovs of the scholiast refer, as Tannery supposes, to 
right-angled triangles with sides in rational numbers (the 
main subject of Diophantus’s Book VI), the use of the mascu-