CONIC SECTIONS
A CONIC section is the curve in which a plane cuts a cone, which is defined in Euclid's Elements as "a solid ligure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed." Though the properties of conic sections can be investigated from this point of view, we consider it more advantageous to start from the following definition, which is derived from one of the properties which all conic sections possess in common.
Definition.—If a point move in such a way in a plane that its distance from a fixed point in the plane always bears a fixed ratio to its distance from a fixed straight line in the plane, the point will trace out a conic section.
The curve is called an ellipse if the distance from the fixed point is less than, a parabola if it is equal to, and a hyperbola, if it is greater than, the distance from the fixed straight line.
The fixed point is called a focus, and the fixed straight line a directrix of the curve.
The fixed ratio of the distance from the focus to the distance from the directrix is called the eccentricity of the curve.
The discovery of the conic sections seems to have originated in the school of Plato. It is probable that the followers of that philosopher were led to the discovery of these curves, and to the investigation of many of their properties, in seeking to resolve the two famous problems of the duplication of the cube and the trisection of an angle, for which the artifices of the ordinary or plane geometry were insufficient. Two solutions of the former problem, by the help of the conic sections, are preserved by Eutocius, and are attributed by him to Menaschmus, the scholar of Eudoxus, who lived a little after the time of Plato.
The writings of Archimedes that have reached us show that the geometers before his time had advanced a great length in investigating the properties of the conic sections. This author expressly mentions numerous de monstrations of preceding writers, and often refers to properties as known to mathematicians. His own discoveries are worthy of the most profound and inventive genius of antiquity. In the quadrature of the parabola he gave the first and the most remarkable instance that has yet been discovered of the exact equality of a curvilinear to a rectilinear space. He determined the proportion of the elliptic spaces to the circle ; and he invented many pro positions respecting the mensuration of the solids formed by the revolution of the conic sections about their axes.
It is chiefly from the writings of Apollonius of Perga that we know how far the ancient mathematicians carried their speculations concerning these curves. (See Apollonius.) His work on the conic sections, written in eight books, was held in such high estimation by the ancients as to procure for him the name of the Great Geometer. The first four books of this treatise only have come down to us in the original Greek ; in these the author claims no further merit than that of having collected, amplified, and arranged the discoveries of preceding mathematicians. One improvement he introduced deserves particular notice. The geometers who preceded him derived each curve from a right cone, which they conceived to be cut by a plane perpendicular to its slant side; and Apollonius was the first to show that all the curves are produced from any sort of cone, whether right or oblique, according to the different inclinations of the cutting plane. An Arabic MS. dis covered in 1658, and two others brought from the East a few years later, contain the first seven books of the treatise of Apollonius ; the eighth book appears to be irrecoverably lost. Dr Halley, who in 1710 put forth a correct edition of the Conics of Apollonius, guided by the account of the different books preserved by Pappus, has given a very able restoration of the eighth book. The last four books of the Conics of Apollonius, containing the higher or more recondite parts of the science, are generally supposed to be the fruit of the author s own researches, and do much honour to his geometrical skill and invention. Even in our times the whole treatise must be regarded as a very extensive work on the conic sections, modern mathematicians having made few discoveries of which there are not some traces to be found in the work of Apollonius.
The geometers who followed Apollonius seem to have contented themselves with commenting on his treatise. It was only about the middle of the 16th century that the study of this branch of mathematical science was revived ; since that time no part of mathematics has been more cultivated, or has w been illustrated by a greater variety of ingenious writings. The applications of the properties of these curves in natural philosophy have, in modern times, given them a degree of importance which they did not formerly possess ; and a knowledge of them is now indispensable to any one who seeks to acquaint himself with the remarkable physical discoveries of the present age.
Apollonius and all the earlier writers on conic sections derived the elementary properties of the curves from the nature of the cone; but in 1665 Dr Wallis, in his De Sectionibus Conicis, laid aside the consideration of the cone, deriving the properties of the curves from a description in plano. In the following treatise, as has been already stated, the properties of the conic sections are deduced from their description in a plane.
An assemblage of points, all of which satisfy some condition, whether or not they form a continuous curve, is called a locus; as, for example, we could define a circle as the locus of a point whose distance from a fixed point is constant, or a conic section as the locus of a point whose distance from a fixed point always bears a constant ratio, to its distance from a fixed straight line.
The following is a proposition which is very useful in the discussion of the properties of conic sections.
Lemma.—The locus of a point in a plane whose distances from two fixed points in the plane always bear a constant ratio to one another is a circle.
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Let A, B (fig. 1) be the two fixed points, in the common ratio, and P any point on the locus.
Divide BA internally and exter- in the given ratio, so that
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CA_ DA CB ~ DB
Join PC, PD.
Then, because PA : PB//CA : CB, PC is the internal bisector of the angle APE (Eucl. vi. 3); and because PA : PB // DA : DB , PD is the external bisector of the angle APB (Eucl. vi. A).
Therefore the angle CPD is a right angle, and the locus of P a circle described 011 the line CD as diameter.
PART I.—THE PARABOLA.
Definitions.
A straight line perpendicular to the directrix, terminated at one extremity by the parabola, and produced indefinitely within it, is called a diameter.
The point in which the diameter meets the parabola is called its vertex.
