A history of Greek mathematics - Wilbourhall.org

THE COLLECTION. BOOK III 365
Pappus does not seem to have seen this, for he observes
that the geometer in question, though saying that DF is
a harmonic mean, does not say how it is a harmonic mean
or between what straight lines.
In the next chapters (pp. 84-104) Pappus, following Nicomachus
and others, defines seven more means, three of which
were ancient and the last four more modern, and shows how
we can form all ten means as linear functions of oc, ft, y, where
a, ft, y are in geometrical progression. The expositiop has
already been described (vol. i, pp. 86-9).
Section (3).
The 'Paradoxes' of Erycinus.
The third section of Book III (pp. 104-30) contains a series
of the same sort, which are curious rather
of propositions, all
than geometrically important. They appear to have been
taken direct from a collection of Paradoxes by one Erycinus. 1
The first set of these propositions (Props. 28-34) are connected
with Eucl. I. 21, which says that, if from the extremities
of the base of any triangle two straight lines be drawn meeting
at any point within the triangle, the straight lines are together
less than the two sides of the triangle other than the base,
but contain a greater angle. It is pointed out that, if the
straight lines are allowed to be drawn from points in the base
other than the extremities, their sum may be greater than the
other two sides of the triangle.
The first case taken is that of a right-angled triangle ABC
right-angled at B. Draw AD to any point D on BC. Measure
on it BE equal to AB, bisect AE
in F, and join FC. Then shall A
DF+FC be > BA + AC.
For EF+FC=AF + FC> AC.
Add BE and AB respectively,
and we have
BF+FC> BA + AC.
More elaborate propositions are next proved, such as the
following.
1 . In
any triangle, except an equilateral triangle or an isosceles
1
Pappus, iii, p. 106. 5-9.