history of mathematics - National STEM Centre

The Greeks
Activity 4.8 The cone
These sections were f
described in Activity 5.6
of the Plane curves unit
in Book 5,
Activity 4.9 Naming the conies
M
Figure 4.9
Apollonius gave names to
the line segments FL and
LP, which are usually
translated as 'abscissa'
and 'ordinate'
respectively.
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Greece and beyond, and as a result of it Apollonius became known as The Great
Geometer'
Apollonius was responsible for naming the conic sections: parabola, hyperbola and
ellipse. The names refer to area properties of the curves.
In the definitions in Book I of Conies, Apollonius defines his cone.
If a straight line infinite in length and passing through a fixed point be
made to move around the circumference of a circle which is not in the
same plane with the point so as to pass successively through every point
of that circumference, the moving straight line will trace out the surface of
a double cone.
1 Define in your own words the vertex of the cone and the axis of the cone.
2 With the help of diagrams, describe briefly how different sections of the shape
described above produce the parabola, the hyperbola, and the ellipse.
Apollonius's definition of the cone was more general than earlier definitions in that
the axis is not required to be perpendicular to the base, and the cone extends in both
directions. He went on to investigate the different plane sections of the cone.
Questions 2 and 3 are more difficult than the other question and are optional.
Consider a cone with vertex A, and a circular base with diameter BC (Figure 4.9).
ABC is a vertical plane section of the cone through its axis. Let DE be a chord on
the circular base, perpendicular to the diameter BC, intersecting BC at G, and let F
be a point on the line AB. Then DFE defines a section of the cone, and FG is called
a diameter of the section. Let the line MN be the diameter of a section of the cone
parallel to the base, cutting FG at L. P is the point on the circumference of this
section where it cuts DFE.
Finally, let FH be the diameter of the section of the cone parallel to the base.
Apollonius considered three cases, depending on the angle that the line FG makes
with AC.
1 In the first case FG is parallel to AC.
a Explain why LN = FH.
b Explain why LP 2 = LM x LN. You will find it helpful to refer back to Activity
3.3, questions le and If.
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c Explain why LM = FL x ——.
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d Hence show that for any section MNP parallel to the base, LP = k x FL, where
k is a constant for the section DFE.