Conic Sections, the name given to the sections formed by cutting a right cone by a plane. The term is also constantly used to denote the curves formed by the intersection of the cutting plane with the surface of the cone. If the plane be parallel to the base of the cone, the section is a circle, or through the vertex a point. If the angle between the cutting plane and the plane of the base is less than the angle between the side of the cone and the base, the section is an ellipse, or through the vertex a point. If the angle between the cutting plane and the plane of the base is equal to that between the side of the cone and the base, the section is a parabola, or through the vertex a straight line. If the angle between the cutting plane and the plane of the base is greater than that between the side of the cone and the base, the section is a hyperbola, or through the vertex a triangle. If we suppose two similar cones to be so placed that they touch each other only at their vertices, and their axes form one straight line, then in the case of the hyperbola the cutting plane will cut both cones, giving two curves, which however are generally regarded as two branches of one curve.
The properties of the conic sections were investigated with great thoroughness by the ancient Greek mathematicians of the school of Plato. Four books by Apollonius of Perga on conic sections have come down to us in the original Greek, and three more in Arabic translations. They are wonderfully full and accurate, and have left comparatively little for modern geometers to do in the investigation of the properties of these curves. Conic sections were in his day merely speculative theories; but after the lapse of 18 centuries it was discovered by Kepler that the orbits of the planets are ellipses, and from that time nearly all the most brilliant applications' of mathematics to natural science and to the practical arts have been possible only through the use of conic sections. What was pure geometrical speculation among the Greeks, has proved of much practical advantage to us, the inheritors of their knowledge. The curves are now generally treated by the methods of analytical geometry. Every conic section may be represented by an equation of the second degree, and conversely every equation of the second degree may be represented by a conic section.
One of the best purely geometrical treatises on the subject is the "Conic Sections" of Prof. Jackson of Union college; and the most elaborate and at the same time clear and practical analytical treatise is the "Conic Sections " of Prof. Salmon of Dublin.