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