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 Nico- 
machus 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 a, /3, y, where 
a, (3, y are in geometrical progression. The exposition 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 propositions, all of the same sort, which are curious rather 
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 AT) to any point P on BC. Measure 
on it DE equal to AB, bisect AE 
in F, and join FG. Then shall 
DF+FG be > BA + AG. 
For EF+FC=AF+FG>AG. 
Add DE and AB respectively, 
and we have 
DF+FG > 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.