A history of Greek mathematics - Wilbourhall.org

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.