THE SEVERAL MEANS DISTINGUISHED 
89 
No. in 
Nicom. 
8 
9 
No. in 
Pappus. 
Formulae. 
Solution 
in terms of 
a, ¡3, y. 
Smallest 
solution. 
8 
a — c a 
a — 2a + 3 ¡3 + y 
a = 6 
a — b b 
b = a + 2 /3 + y 
b = 4 
c — 2(3 + y 
c = 3 
9 * 
a — c a 
a — a + 2/3-fy 
a = 4 
a — b ~ c 
b = a + /5 + y 
b = 3 
c — (3 + y 
c = 2 
10 
a — c b 
b — c c 
a — a + (3 + y 
b = (3 + y 
a — 3 
b = 2 
c= y 
c = 1 
Pappus does not include a corresponding solution for his 
No. 1 and No. 7, and Tannery suggests as the reason for this 
that, the equations in these cases being already linear, there 
is no necessity to assume ay = (3 2 , and consequently there is 
one indeterminate too many. 1 Pappus does not so much prove 
(X /3 
as verify his results, by transforming the proportion ^ = 
in all sorts of ways, componendo, dividendo, &c. 
(y) Plato on geometric means between two squares 
or two cubes. 
It is well known that the mathematics in Plato’s Timaeus 
is essentially Pythagorean. It is therefore a priori probable 
that Plato nv6ayopi£eL in the passage 2 where he says that 
between two planes one mean suffices, but to connect two 
solids two means are necessary. By planes and solids he 
really means square and cube numbers, and his remark is 
equivalent to stating that, if p 2 , q 2 are two square numbers, 
p 2 : pq-= pq\ q 2 , 
while, if p 3 , q'- ] are two cube numbers, 
p 3 : p 2 q = p 2 q : pq 2 — pq 2 : q 3 , 
the means being of course means in continued geometric pro 
portion. Euclid proves the properties for square and cube 
1 Tannery, Joe. cit., pp. 97-8. 2 Plato, Tinmens, 82 A, B.