104 
PYTHAGOREAN ARITHMETIC 
RATIOS GREATER THAN UNITY 
(b) Particular 
These names are only given 
for cases where n = 1 ; they 
follow the first form of the 
names for particular eVt/Aepeis, 
e. g. 
2f = 8t7rXacrte7ri,8t/xepp9 
(duplex superbipartiens) 
&c. 
RATIOS LESS THAN UNITY 
Corresponding names not 
found in Nicomachus; but 
Boetius has sub duplex super 
bipartiens, &c. 
In c. 23 Nicomachus shows how these various ratios can be 
got from one another by means of a certain rule. Suppose 
that 
a, b, c 
are three numbers such that a:b = b :c = one of the ratios 
described ; we form the three numbers 
a, . a + b, a + 2b + c 
and also the three numbers 
c, c + b, c + 2 b + a 
Two illustrations may be given. If a = b = c = 1, repeated 
application of the first formula gives (1, 2, 4), then (1, 3, 9), 
then (1, 4, 16), and so on, showing the successive multiples. 
Applying the second formula to (1, 2, 4), we get (4, 6, 9) where 
the ratio is •§; similarly from (1, 3, 9) we get (9, 12, 16) where 
the ratio is f, and so on ; that is, from the TroXXairXdcnoL we 
get the eTTL/iopLoi. Again from (9, 6, 4), where the ratio is 
of the latter kind, we get by the first formula (9, 15, 25), 
giving the ratio If, an and by the second formula 
(4, 10, 25), giving the ratio 2f, a 7roXXanXacne7rLp.6pLos. And 
so on. 
Book II begins with two chapters showing how, by a con 
verse process, three terms in continued proportion with any 
one of the above forms as common ratio can be reduced to 
three equal terms. If 
a, b, c