PART X.

THE CONIC SECTIONS.

HISTORICAL INTRODUCTION.

IF a solid be cut into two parts by a plane passing through it, the surface made in the solid by the cutting plane, is called A SECTION.

If a fixed point be taken above a plane, and one of the extremities of a straight line passing through it be made to describe a circle on the plane, then will the segments of this line by their revolution, describe two solids (one on each side of the fixed point) which are called

OPPOSITE CONES.

A plane may be made to cut a cone five ways; first, by passing through the vertex and the base; secondly, by passing through the cone parallel to the base; thirdly, by passing through it parallel to its sides; fourthly, by passing through the side of the cone and the base, so as likewise to cut the opposite cone; and fifthly, so as to cut its opposite sides in unequal angles ", or in a position not parallel to the base.

a If the segment of the generating line between the fixed point and the base be of a given length, the cone described by its motion will be a right CONE, having its axis perpendicular to the base; but if the length of the segment be variable in any given ratio, so as to become in one revolution a maximum and a minimum, the cone produced will be AN OBLIQUE CONE, and its axis will make an oblique angle with the base.

Of course a right cone is here understood; for if the cone be oblique, the base, which is a circle, will cut the opposite sides in unequal angles, and the segment made by cutting them in equal angles will evidently be an ellipse.

If the plane pass through the vertex and the base, the section is a triangle; if it be parallel to the base, the section is a circle; if parallel to the side of the cone, the section is called A PARABOLA; if the plane pass through the side and cut the opposite cone, the section is called AN HYPERBOLA; and if it cut the opposite sides of the cone at unequal angles, the section is called

AN ELLIPSE.

The triangle and circle pertain to common elementary Geometry, and are treated of in the Elements of Euclid; the parabola, the ellipse, and the hyperbola, are the three figures which are denominated THE CONIC SECTIONS.

There are three ways in which these curves may be conceived to arise, from each of which their properties may be satisfactorily determined; first, by the section of a cone by a plane, as above described, which is the genuine method of the ancients; secondly, by algebraic equations, wherein their chief proporties are exhibited, and from whence their other properties are easily deduced, according to the methods of Fermat, Des Cartes, Roberval, Schooten, Sir Isaac Newton, and others of the moderns; thirdly, these curves may be described on a plane by local motion, and their properties determined as in other plane figures from their definition, and the principles of their construction. This method is employed in the following pages.

WHEN, or from whom the ancient Greek geometricians first acquired a knowledge of the nature and properties of the cone and its sections, we are not fully informed, although there is every reason to suppose that the discovery owes its origin to that inventive genius, and indefatigable application to science, which distinguished that learned people above all the other nations of antiquity. Some

of the most remarkable properties of these curves were in all probability known to the Greeks as early as the fifth century before Christ, as the study of them appears to have been cultivated (perhaps not as a new subject) in the time of Plato, A. C. 390. We are indeed told, that until his time the conic sections were not introduced into Geometry, and to him the honour of incorporating them with that science is usually ascribed. We have nothing remaining of his expressly on the subject, the early history of which, in common with that of almost every other branch of science, is involved in impenetrable obscurity.

The first writer on this branch of Geometry, of whom we have any certain account, was Aristæus, the disciple and friend of Plato, A. C. 380. He wrote, a treatise consisting of five books, on the Conic Sections; but unfor tunately this work, which is said to have been much valued by the ancients, has not descended to us. Menechmus, by means of the intersections of these curves (which appears to have been the earliest instance of the kind) shewed the method of finding two mean proportionals, and thence the duplication of the cube; others applied the same theory, with equal success, to the trisection of an angle; these curious and difficult problems were attempted by almost every geometrician of this period, but the solution (as we have remarked in another place) has never yet been effected by pure elementary Geometry. Archytas, Eudoxus, Philolaus, Denostratus, and many others, chiefly of the Platonic school, penetrated deeply into this branch, and carried it to an amazing extent; succeeding geometers enriched it by the addition of several other curves as the cycloid, cissoid, conchoid, quadratris, spiral, &c. the whole forming a branch of science justly considered by the ancients

as possessing a more elevated nature than common Geometry, and on this account they distinguished it by the name of THE HIGHER or SUBLIME GEOMETRY.

Euclid of Alexandria, the celebrated author of the Elements, A. C. 280, wrote four books on the Conic Sections, as we learn from Pappus and Proclus; but the work has not descended to modern times. Archimedes was profoundly skilled in every part of science, especially Geometry, which he valued above every other pursuit; it appears that he wrote a work which is lost, expressly on the subject we are considering, and his writings which remain respecting spiral lines, conoids, and spheroids, the quadrature of the parabola, &c. are sufficient proofs that he was deeply skilled in the theory of the Conic Sections. In his tract on the parabola he has proved by two ingenious methods, that the area of the parabola is two-thirds that of its circumscribing rectangle; which is said to be the earliest instance on record of the absolute and rigorous quadrature of a space included between right lines and a curve. But the most perfect work of the kind among the ancients is a treatise originally consisting of eight books by Apollonius Pergæus of the Alexandrian School, A. C. 230. The first four only of these, have descended to us in their original Greek, the fifth, sixth, and seventh, in an Arabic version; the eighth has not been found, but Dr. Halley has supplied an eighth book in his edition, printed at Oxford, in 1710.

This excellent treatise is the most ancient work in our possession, on the subject; it supplied a model for the earliest writers among the moderns, and still maintains its classical authority: the improvements on the system of Apollonius by modern geometricians are comparatively few, except such as depend on the application of Algebra

and the Newtonian Analysis. Hitherto the ancients had admitted the right cone only (of which the axis is perpendicular to the base) into their Geometry; they supposed all the three sections to be made by a plane cutting the cone at right angles to its side. According to this method, if the cone be right angled (def. 18. 11.), the section will be a parabola; if acute angled, the section will be an ellipse; and if obtuse angled, an hyperbola; hence they named the parabola, The section of a right angled cone; the ellipse, The section of an acute angled cone; and the hyperbola, The section of an obtuse angled cone. But Apollonius first shewed that the three sections depend only on the different inclinations of the cutting plane, and may all be obtained from the same cone, whether it be right or oblique, and whether the angle of its vertex be right, acute, or obtuse. Pappus of Alexandria, who flourished in the fourth century after Christ, wrote valuable lemmata and observations on the writings of Apollonius, particularly on the conics, which are to be found in the seventh book of his Mathematical Collections and Eutocius, who lived about a century later, composed an elaborate commentary on several of the propositions.

In 1522 John Werner, published, at Nuremberg, some tracts on the subject; and about the same time Franciscus Maurolycus, Abbot of St. Maria del Porta, in Sicily, published a treatise on the Conic Sections, which has been highly spoken of by some of our best geometers for its perspicuity and elegance. The application of Algebra to Geometry, first generally introduced by Vieta, and afterwards improved and extended by Des Cartes, Fermat, Torricellius, and others, furnished means for the further developement of the nature and properties of curves. The indivisibles of Roberval and Cavalerius;