306 10. EUCLIDEAN GEOMETRY
asymptotes, things necessary for analyzing problems to see what data permit a
solution, and the three- and four-line locus. He continues:
The third book contains many remarkable theorems of use for the
construction of solid loci and for distinguishing when problems
have a solution, of which the greatest part and the most beauti-
ful are new. And when we had grasped these, we knew that the
three-line and four-line locus had not been constructed by Euclid,
but only a chance part of it and that not very happily. For it
was not possible for this construction to be completed without the
additional things found by us.
We have space to discuss only the definition and construction of the conic
sections and the four-line locus problem, which Apollonius mentions in the passage
just quoted.
4.3. Apollonius' definition of the conic sections. The earlier use of conic
sections had been restricted to cutting cones with a plane perpendicular to a gen-
erator. As we saw in our earlier discussion, this kind of section is easy to analyze
and convenient in the applications for which it was intended. In fact, only one kind
of hyperbola, the rectangular, is needed for duplicating the cube and trisecting the
angle. The properties of a general section of a general cone were not discussed.
Also, it was considered a demerit that the properties of these planar curves had to
be derived from three-dimensional figures. Apollonius set out to remove these gaps
in the theory.
First it was necessary to define a cone as the figure generated by moving a line
around a circle while one of its points, called the apex and lying outside the plane of
the circle, remains fixed. Next, it was necessary to classify all the sections of a cone
that happen to be circles. Obviously, those sections include all sections by planes
parallel to the plane of the generating circle (Book 1, Proposition 4). Surprisingly,
there is another class of sections that are also circles, called subcontrary sections.
Once the circles are excluded, the remaining sections must be parabolas, hyperbolas,
and ellipses. We have space only to consider Apollonius' construction of the ellipse.
His construction of the other conies is very similar. Consider the planar section of
a cone in Fig. 20, which cuts all the generators of the cone on the same side of its
apex. This condition is equivalent to saying that the cutting intersects both sides
of the axial triangle. Apollonius proved that there is a certain line, which he called
the [up]right side, now known by its Latin name latus rectum, such that the square
on the ordinate from any point of the section to its axis equals the rectangle applied
to the portion of the axis cut off by this ordinate (the abscissa) and whose defect
on the axis is similar to the rectangle formed by the axis and the latus rectum.
He gave a rule, too complicated to go into here, for constructing the latus rectum.
This line characterized the shape of the curve. Because of its connection with the
problem of application with defect, he called the resulting conic section an ellipse.
Similar connections with the problems of application and application with excess
respectively arise in Apollonius' construction of the parabola and hyperbola. These
connections motivated the names he gave to these curves.
In Fig. 20 the latus rectum is the line EH, and the locus condition is that the
square on LM equal the rectangle on EO and EM; that is, LM^2 = EO • EM.
