THE COLLECTION. BOOK III 
363 
Section (2). The theory of means. 
Next follows a section (pp. 69-105) on the theory of the 
different kinds of means. The discussion takes its origin 
from the statement of the ‘ second problem which was that 
of ‘ exhibiting the three means ’ (i. e. the arithmetic, geometric 
and harmonic) ‘in a semicircle’. Pappus first gives a con 
struction by which another geometer (aAAoy ns) claimed to 
have solved this problem, but he does not seem to have under 
stood it, and returns to the same problem later (pp. 80-2). 
In the meantime he begins with the definitions of the 
three means and then shows how, given any two of three 
terms a, b, c in arithmetical, geometrical or harmonical pro 
gression, the third can be found. The definition of the mean 
(b) of three terms a, b, c in harmonic progression being that it 
satisfies the relation a:c = a — b:b — c, Pappus gives alternative 
definitions for the arithmetic and geometric means in corre 
sponding form, namely for the arithmetic mean a: a=a — b:b—c 
and for the geometric a:b = a — b:b — c. 
The construction for the harmonic mean is perhaps worth 
giving. Let AB, BG be two given straight lines. At A draw 
DAE perpendicular to AB, and make DA, AE equal. Join 
DB, BE. From G draw GF at right 
angles to A B meeting DB in F. 
Join EF meeting AB in C. Then 
BC is the required harmonic mean. 
For 
AB: BG — DA : FG 
= EA: FG 
= AG: CG 
= {AB-BG):{BG-BG). 
Similarly, by means of a like figure, we can find BG when 
AB, BC are given, and AB when BC, BG are given (in 
the latter case the perpendicular DE is drawn through G 
instead of A). 
Then follows a proposition that, if the three means and the 
several extremes are represented in one set of lines, there must 
be five of them at least, and, after a set of five such lines have 
been found in the smallest possible integers, Pappus passes to