A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE COLLECTION. BOOK III 363
Section (2). The thebry 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 construction
by which another geometer (dXXo? tl?) claimed to
have solved this problem, but he does not seem to have understood
it, and returns to the same problem later (pp. 8,0-2).
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 progression,
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 corresponding
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 A B, and make DA, AE equal. Join
DB, BE. From G draw GF at right
angles to AB meeting DB in F.
Join EF meeting AB in C. Then
BC is the required harmonic mean.
For
AB:BG=.DA:FG
= EA:FG
= AC-.CG
= (AB-BC):(BC-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
364 PAPPUS OF ALEXANDRIA
the problem of representing the three means with the respective
extremes by six lines drawn in a semicircle.
Given a semicircle on the diameter AC, and B any point on
the diameter, draw BD at right angles to A G. Let the tangent
H V
at D meet AG produced in G, and measure DH along the
tangent equal to DG. Join HB meeting the radius OD in K.
Let BF be perpendicular to OB.
Then, exactly as above, it is shown that OK is a harmonic
mean between OF and OD. Also BD is the geometric mean
between A B, BC, while OG (= OD) is the arithmetic mean
between A B, BG.
Therefore the six lines DO (= OC), OK, OF, AB, BG, BD
supply the three means with the respective extremes.
But Pappus seems to have failed to observe that the certain
'
other geometer ', who has the same figure excluding the dotted
lines, supplied the same in Jive lines. For he said that DF
is
'
a harmonic mean '. It is in fact the harmonic mean
between A B, BG, as is easily seen thus.
to
or
Since ODB is a right-angled triangle, and BF perpendicular
OD,
But
therefore DF .
Therefore AB . (DF-
DF:BD = BD:DO,
DF.DO = ED* = AB.BC.
DO =i(AB + BC);
(AB + BC) = 2AB. BG
BG) = BG . (AB-DF),
that is, AB:BC= (AB - DF) :
(DF-
BG),
and DF is the harmonic mean between AB, BG
Consequently the Jive lines DO (= OC), DF,
9
AB, BG, BD
exhibit all the three means with the extremes.