# The Physics, Astronomy and Mathematics of the Solar System

## Discovering Conic Sections in the Motion of Heavenly Bodies

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## Discovering conics

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Figure 1.
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Wikipedia

However the fact that Apollonius used his theories of conic sections to create more accurate sundials suggests that the following scenario may be more likely. During its daily course above the horizon the Sun appears to describe a circular arc. Supplying in his mind's eye the missing portion of the daily circle, the Greek astronomer could imagine that his real eye was at the apex of a cone, the surface of which was defined by the Sun's rays at different times of the day and the base of which was defined by the Sun's apparent diurnal course. Our astronomer, using the pointer of a sundial, known as a gnomon, as his eye, would generate a second, shadow cone spreading downward. The intersection of this second cone with a horizontal surface, such as the face of a sundial, would give the trace of the Sun's image (or shadow) during the day as a plane section of a cone. (The possible intersections of a plane with a cone, known as the conic sections, are the circle, ellipse, point, straight line, parabola, and hyperbola.)

However, compilers of the ideas in the history of the philosophy of science (known as doxographers) ascribe the discovery of conic sections to Menaechmus (mid-4th century BC), a student of Eudoxus, who used them to solve the problem of duplicating the cube. His restricted approach to conics--he worked with only right circular cones and made his sections at right angles to one of the straight lines composing their surfaces--was standard down to Archimedes' era.

A right circular cone is defined as a cone with a circle as its base and the apex is centered directly over the center of the circle so that the height from the base to the apex is at a right angle from the center of the circle to the apex. Figure 2 shows both a right circular cone (on the left) and an oblique circular cone (on the right). A cone may also have a non-circular base.

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Figure 2.
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Wikipedia

Euclid adopted Menaechmus's approach in his lost book on conics, and Archimedes followed suit. Doubtless, however, both knew that all the conics can be obtained from the same right cone by allowing the section at any angle. Whereas his predecessors had used finite right circular cones, Apollonius considered arbitrary (oblique) double cones that extend indefinitely in both directions.

The reason that Euclid's treatise on conics perished is that Apollonius of Perga (c. 262-c. 190 BC) did to it what Euclid had done to the geometry of Plato's time. Apollonius reproduced known results much more generally and discovered many new properties of the figures. He first proved that all conics are sections of any circular cone, right or oblique. Apollonius introduced the terms
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ellipse
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hyperbola
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, and
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parabola
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for curves produced by intersecting a circular cone with a plane at an angle less than, greater than, and equal to, respectively, the opening angle of the cone.

For this section there will be a lesson with a hands on activity where students will be constructing and also slicing cones. A sundial will also be demonstrated.