THE SEVERAL MEANS DISTINGUISHED 
87 
Nicomachus 1 and Pappus 2 ; their accounts only differ as 
regards one of the ten. If a>h>c, the formulae in the third 
column of the following table show the various means. 
No. in No. in 
Nicom. Pappus. 
1 1 
Formulae. Equivalent. 
a—b a b c 
t = - = -=- a + c = 2b (arithmetic) 
0 — C Ob o c 
2 
3 
2 
3 
a — b_aV_b 
b—c ~ 6L c . 
a — h a 
b — c ~ c 
ac = 0 2 (geometric) 
11 2 ., 
—i— — 7- (harmonic) 
CC C 0 
4 4 
5 5 
6 G 
a — h c 
b — c a 
a — h c 
b — c b 
a — b h 
b — c~ a 
a 2 + c 2 _ ^ (subcontrary to 
a + c harmonic) 
a = h + c-r 
} [ (subcontrary 
7 CL 2, I to geometric) 
c = a + h — j 
a — c a 
(omitted) = - 
c 2 = 2 ac — ah 
8 9 
9 10 
a — c_a 
a — b c 
a — c _ h 
b — c c 
a, 2 + c 2 = a {b + c) 
b 2 + c 2 = c (a + h) 
10 
a — c_h 
a — b c 
(omitted) 
a—c_a 
a — b b 
a — b + c 
a 2 = 2 ah—he 
The two lists together give five means in addition to the 
first six which are common to both; there would be six more 
(as Theon of Smyrna says :5 ) were it not that ^ is 
illusory, since it gives a—h. Tannery has remarked that 
1 Nicom. ii. 28. 
3 Theon of Smyrna, p. 106.15, p. 116. 3. 
Pappus, iii, p. 102.