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.