308 10. EUCLIDEAN GEOMETRY
FIGURE 21. Focal properties of an ellipse.
4.4. Foci and the three- and four-line locus. We are nowadays accustomed to
constructing the conic sections using the focus-directrix property, so that it comes
as a surprise that the original expert on the subject does not seem to recognize
the importance of the foci. He never mentions the focus of a parabola, and for
the ellipse and hyperbola he refers to these points only as "the points arising out
of the application." The "application" he has in mind is explained in Book 3.
Propositions 48 and 52 read as follows:
(Proposition 48) // in an ellipse a rectangle equal to the fourth part of the figure is
applied from both sides to the major axis and deficient by a square figure, and from
the points resulting from the application straight lines are drawn to the ellipse, the
lines will make equal angles with the tangent at that point.
(Proposition 52) // in an ellipse a rectangle equal to the fourth part of the figure is
applied from both sides to the major axis and deficient by a square figure, and from
the points resulting from the application straight lines are drawn to the ellipse, the
two lines will be equal to the axis.
The "figure" referred to is the rectangle whose sides are the major axis of the
ellipse and the latus rectum. In Fig. 21 the points Fi and F 2 must be chosen on the
major axis AB so that AF\ • F\B and AF2 • BF2 both equal one-fourth of the area of
the rectangle formed by the axis AB and the latus rectum. Proposition 48 expresses
the focal property of these two points: Any ray of light emanating from one will
be reflected to the other. Proposition 52 is the string property that characterizes
the ellipse as the locus of points such that the sum of the distances to the foci
is constant. These are just two of the theorems Apollonius called "strange and
beautiful." Apollonius makes little use of these properties, however, and does not
discuss the use of the string property to draw an ellipse.
A very influential part of the Conies consists of Propositions 54-56 of Book 3,
which contain the theorems that Apollonius claimed (in his cover letter) would pro-
vide a solution to the three- and four-line locus problems. Both in their own time
and because of their subsequent influence, the three- and four-line locus problems
have been of great importance for the development of mathematics. These proposi-
tions involve the proportions in pieces of chords inscribed in a conic section. Three
propositions are needed because the hyperbola requires two separate statements to
cover the cases when the points involved lie on different branches of the hyperbola.
Proposition 54 asserts that given a chord AT such that the tangents at the
endpoints meet at Ä, and the line from Ä to the midpoint Ε of the chord meets
the conic at B, any point è on the conic has the following property (Fig. 22). The
