NICOMACHUS 
103 
RATIOS GREATER THAN UNITY 
RATIOS LESS THAN UNITY 
' 
Where the more general form 
1+ m , instead of 1 + ** 
has to be expressed, Nicoma- 
chus uses terms following the 
third plan of formation above, 
e.g. 
1-|- = rptfreTrtTre/xTrros 
ly = T€TpaKLcre(f)el38ofjiO'5 
ly = TrevrttKifreTrevttros 
and so on, although he might 
have used the second and called 
these ratios e7n,rpt7rep,7rros, &c. 
4. (a) General 
TroXXaTrXacrteTrqxoptos 
(multiplex superparticularis) 
This contains a certain mul 
tiple plus a certain submultiple 
(instead of 1 plus a submultiple) 
and is therefore of the form 
m + - (instead of the 1 + - of 
n n 
,, , , , mn + 1 
the e7Ttp,opios) or 
(b) Particular 
2y = 8i7rXacrie^)77p.io'DS 
(duplex sesquialter) 
2g- = StTrAacneTTirptros 
(duplex sesquitertius) 
34 = rptTrXcun.eTrtTrep/Trros 
(triplex sesquiquintus) 
&c. 
4. (a) General 
v7ro7roXXa7rXacrie7rtp,optos 
(submultiplex superparticularis) 
of the form —• 
mn +1 
The corresponding particular 
names do not seem to occur in 
Nicomachus, but Boetius has 
them, e. g. subduplex sesquialter, 
subduplex sesquiquartus. 
5. (a) General 
TroXXa—XaaxeTrqxeppi 
(multiplex superpartiens). 
This is related to e~tp,epr/s 
[(3) above] in the same way as 
7roXXa7rXacm7n-p,op(,os to e7rtp,optosj 
that is to say, it is of the form 
m (n + l)m + w 
P+ , or — 
m + n m + n 
5. (a) General 
v ir 07T oXXfi-ir kaa iti—if leprjs 
(submultiplex superpartiens), 
a fraction of the form 
m+n 
{p+1) m +n