INTRODUCTION.

In treating of so considerable a branch of the mathematical sciences as the Conic Sections, it would be improper to pass over in total silence the history of those remarkable curves. But this topic will not require any long detail. None of the works of the more early Greek geometers have reached our time; nor have we any work of antiquity professedly written on the subject of our inquiry. Our curiosity must therefore rest satisfied with the knowledge of a few incidental notices and facts, gleaned from different authors.

The discovery of the conic sections seems to have originated in the school of Plato, in which geometry was highly respected, and much cultivated. 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 Menaechmus, the scholar of Euclid, who lived not much posterior to the time of Plato: and this circumstance, added to a few words in an epigram of Eratosthenes, has been thought sufficient authority, by some authors, to ascribe the honour of the discovery of the conic sections to Menaechmus. We may at least infer that, at this epoch, geometers had made some progress in developing the properties of these curves.

The writings of Archimedes that have reached us explicitly shew, that the geometers before his time had advanced a great length in investigating the properties of the conic sections. This author expressly mentions many principal propositions to have been demonstrated by preceding writers; and he often refers to properties of the conic sections, as truths commonly divulged, and known to mathematicians. His own discoveries in this branch of science 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 rectilineal space. He determined the proportion of the elliptic spaces to the circle; and he invented many propositions 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, a town in Pamphylia, on the subject of the conic sections, that we know how far the ancient mathematicians carried their speculations concerning these curves. Apollonius flourished under Ptolemy Philopator, about forty years later than Archimedes. He formed his taste for geometry, and acquired that superior skill in the science to which he is indebted for his fame, in the school of Alexandria, under the successors of Euclid. Besides his great work on the conic sections, he was the author of many smaller treatises, relating chiefly to the geometrical analysis, the originals of which have all perished, and are only known to modern mathematicians by the account given of them by Pappus of Alexandria, in the seventh book of his Mathematical Collections.

The work of Apollonius 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. It is the purpose of these four books, as we are informed in the prefatory epistle to Eudemus, to deliver the elements of the science; and in this part of his labour, the author claims no farther merit than that of having collected, amplified, and reduced to order, the discoveries of preceding mathematicians. One improvement introduced by Apollonius is too remarkable to be passed over without 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. It will readily be perceived, from what is shewn in the first section of the fourth part of the following treatise, that the section would be a parabola when the vertical angle of the cone was a right angle; an ellipse when it was acute; and a hyperbola when it was obtuse. Thus each curve was derived from a different sort of cone. Apollonius was the first to shew that all the curves are produced from any sort of cone, whether right or oblique, according to the different inclinations of the cutting plane. This fact is one remarkable instance of the adherence of the mind to its first conceptions, and of the slowness and difficulty with which it generalizes.

The original of the first four books of the treatise of Apollonius is lost; nor is it easy to ascertain in what age it disappeared. In the year 1658 Borelli discovered at Florence an Arabic manuscript, entitled Apollonii Pergei Conicorum Libri Octo. By the liberality of the Duke of Tuscany he was permitted to carry the manuscript to Rome, and, with the aid of an Arabic scholar, Abraham Ecchellensis, he published in 1661 a Latin translation of it. The manuscript, although from its title it was expected to be a complete translation of all the eight books, was yet found to contain only the first seven books: and it is remarkable, that another manuscript, brought from the east by Golius, the learned professor of Leyden, so early as 1664, as well as a third, of which Rarius published a translation in 1669, have the same defect: all the three manuscripts agreeing in the want of the eighth book, we may now consider that part of the work of Apollonius as irrecoverably lost. Fortunately, in the Collectiones Mathematicae of Pappus, in whose time the entire treatise of Apollonius was extant, there is preserved served some account of the subjects treated in each book, and all the Lemmata required in the investigations of the propositions they contain. Dr Halley, who in 1710 gave a correct edition of the Conics of Apollonius, guided in his researches by the lights derived from Pappus, has restored the eighth book with so much ability as to leave little room to regret the original.

The four last 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 they do much honour to the geometrical skill and invention of the Great Geometer. Even in our times the whole treatise must be regarded as a very extensive, if not a complete work on the conic sections. Modern mathematicians make important applications of these curves, with which the ancients were unacquainted; and they have been thus led to consider the subject in particular points of view, suited to their purposes; but they have made few discoveries, of which there are not some traces to be found in the work of the illustrious ancient.

The geometers who followed Apollonius seem to have contented themselves with the humble task of commenting on his treatise, and of rendering it of more easy access to the bulk of mathematicians. Till about the middle of the 16th century, the history of this branch of mathematical science presents nothing remarkable. The study of it was then revived; and since that time this part of mathematics has been more cultivated, or has been illustrated by a greater variety of ingenious writings.

Among the ancients, the study of the conic sections was a subject of pure intellectual speculation. The applications of the properties of these curves in natural philosophy have, in modern times, given to this part of the mathematics a degree of importance that it did not formerly possess. That which, in former times, might be considered as interesting only to the learned theorist and profound mathematician, is now a necessary attainment to him who would not be ignorant of those discoveries in nature, that do the greatest honour to the present age.

It is curious to remark the progress of discovery, and the connexion that subsists between the different branches of human knowledge; and it excites some degree of admiration to reflect, that the astronomical discoveries of Kepler, and the sublime theory of Newton, depend on the seemingly barren speculations of Greek geometers concerning the sections of the cone.

Apollonius, and all the writers on conic sections before Dr Wallis, derived the elementary properties of the curves from the nature of the cone. In the second part of his treatise De Sectionibus Conicis, published in 1665, Dr Wallis laid aside the consideration of the cone, deriving the properties of the curves from a description in plano. Since his time authors have been much divided as to the best method of defining those curves, and demonstrating their elementary properties; many of them preferring that of the ancient geometers, while others, and some of great note, have followed the example of Dr Wallis.

In support of the innovation made by Dr Wallis, it is urged, that in the ancient manner of treating the conic sections, young students are perplexed, and discouraged by the previous matter to be learnt respecting the generation and properties of the cone; and that they find it no easy matter to conceive steadily, and to understand diagrams rendered confused by lines drawn in different planes: all which difficulties are avoided by defining the curves in plano from one of their essential properties. It is not our intention particularly to discuss this point; and we have only to add, that, in the following treatise, we have chosen to deduce the properties of the conic sections from their description in plano, as better adapted to the nature of a work designed for general readers.

A geometrical treatise on the conic sections must necessarily be founded upon the elements of geometry. As Euclid's Elements of Geometry are generally studied, and in every one's hands, we have chosen to refer to it in the demonstrations. The edition we have used is that published by Professor Playfair of Edinburgh. Although the references are made to Euclid's Elements, yet they will also apply to the treatise on GEOMETRY given in this Work; for a table is there given, indicating the particular proposition of our treatise that corresponds to each of the most material propositions in Euclid's Elements.

The references are to be thus understood: (20. i. E.) means the 20th prop. of the 1st book of Euclid's Elements; (2 cor. 20. 6. E.) means the 2d corollary to the 20th prop. of the sixth book of the same work; and so of others. Again, (7.) means the seventh proposition of that PART of the following treatise in which such reference happens to occur: (cor. 1.) means the corollary to the first proposition: (2 cor. 3.) means the 2d corollary to the third proposition, &c.—such references being all made to the propositions in the division of the treatise in which they are found.

PART I. OF THE PARABOLA.

Definitions,

I. If a straight line BC, and a point without it F, be given by position in a plane, and a point D be supposed to move in such a manner that DF, its distance from the given point, is equal to DB, its distance from the given line, the point D will describe a line DAD, called a Parabola.

Corollary. The line DF, DB, may become greater than any given line; therefore the parabola extends to a greater distance from the point F, and the line BC, than any that can be assigned.

II. The straight line BC, which is given by position, is called the Directrix of the Parabola.

III. The given point F is called the Focus.

IV. A straight line perpendicular to the directrix, terminated at one extremity by the parabola, and produced indefinitely within it, is called a Diameter.

V. The point in which a diameter meets the parabola is called its Vertex.

VI. The VI. The diameter which passes through the focus of the parabola is called the Axis of the parabola; and the vertex of the axis is called the Principal Vertex.

Cor. A perpendicular drawn from the focus to the directrix is bisected at the vertex of the axis.

VII. A straight line terminated both ways by the parabola, and bisected by a diameter, is called an Ordinate to that diameter.

VIII. The segment of a diameter between its vertex and an ordinate, is called an Abscissa.

IX. A straight line quadruple the distance between the vertex of a diameter and the directrix, is called the Parameter, also the Lotus Rectum of that diameter.

X. A straight line meeting the parabola only in one point, and which everywhere else falls without it, is said to touch the parabola at that point, and is called a Tangent to the parabola.

PROPOSITION I.

The distance of any point without the parabola from the focus is greater than its distance from the directrix; and the distance of any point within the parabola from the focus is less than its distance from the directrix.

Let D be a parabola, of which F is the focus, GC the directrix, and P a point without the curve, that is, on the same side of the curve with the directrix; PF, a line drawn to the focus, will be greater than PG, a perpendicular to the directrix. For, as PF must necessarily cut the curve, let D be the point of intersection; draw DB perpendicular to the directrix, and join PB. Because D is a point in the parabola, DB = DF (Definition 1.), therefore PF = PD + DB; but PD + DB is greater than PB (29. 1. E.), and therefore still greater than PG (19. 1. E.), therefore PF is greater than PG.

Again, let Q be a point within the parabola, QF, a line drawn to the focus, is less than QB, a perpendicular to the directrix. The perpendicular QB necessarily cuts the curve; let D be the point of intersection; join DF. Then DF = DB (Def. 1.), and QD + DF = QB; but QF is less than QD + DF, therefore QF is less than QB.

COROLLARY. A point is without or within the parabola, according as its distance from the focus is greater or less than its distance from the directrix.

PROPOSITION II.

Every straight line perpendicular to the directrix meets the parabola, and every diameter falls wholly within it.

Let the straight line BQ be perpendicular to the directrix at B, BQ shall meet the parabola. Draw BF to the focus, and make the angle BFP equal to FBO; then, because QBC is a right angle, QBF and PFB are each less than a right angle, therefore QB and PF intersect each other; let D be the point of intersection, then DB = DF (5. 1. E.); therefore, D is a point in the parabola. Again, the diameter DQ falls wholly within the parabola; for take Q any point in the diameter, and draw FQ to the focus, then QB or QD + DF is greater than QF, therefore Q is within the parabola (Cor. 1.).

Cor. The parabola continually recedes from the axis, and a point may be found in the curve that shall be at a greater distance from the axis than any assigned line.

PROPOSITION III.

The straight line which bisects the angle contained by two straight lines drawn from any point in the parabola, the one to the focus, and the other perpendicular to the directrix, is a tangent to the curve in that point.

Let D be any point in the curve; let DF be drawn to the focus, and DB perpendicular to the directrix; the straight line DE, which bisects the angle FDB, is a tangent to the curve. Join BF meeting DE in I, take H any other point in DE, join HF, HB, and draw HG perpendicular to the directrix. Because DF = DB, and DI is common to the triangles DFI, DBI, and the angles FDI, BDI, are equal, these triangles are equal, and FI = IB, and hence FH = HB (4. 1. E.) but HB is greater than HG (19. 1. E.); therefore the distance of the point H from the focus is greater than its distance from the directrix, hence that point is without the parabola (Cor. 1.), and therefore HDI is a tangent to the curve at D (Def. 10.).

Cor. 1. There cannot be more than one tangent to the parabola at the same point. For let any other line DK, except a diameter, be drawn through D; draw FK perpendicular to DK; on D for a centre, with a radius equal to DB, or DF, describe a circle, cutting FK in N; draw NL parallel to the axis, meeting DK in L, and join FL. Then FK = KN (3. 3. E.) and therefore FL = LN. Now BD being perpendicular to the directrix, the circle FBN touches the directrix at B (16. 3. E.); and hence N, any other point in the circumference, is without the directrix, and on the same side of it as the parabola, therefore the point L is nearer to the focus than to the directrix, and consequently is within the parabola.

Cor. 2. A perpendicular to the axis at its vertex is a tangent to the curve. Let AM be perpendicular to the axis at the vertex A, then RS, the distance of any point in AM from the directrix, is equal to CA, that is, to AF, and therefore is less than RF, the distance of the same point from the focus.

Cor. 3. A straight line drawn from the focus of a parabola perpendicular to a tangent, and produced to meet the directrix, is bisected by the tangent. For it has been shown that FB, which is perpendicular to the tangent DI, is bisected at I.

Cor. 4. A tangent to the parabola makes equal angles with the diameter which passes through the point of contact, and a straight line drawn from that point to the focus. For BD being produced to Q, DQ is a diameter, and the angle HDQ is equal to BDE, that is, to EDF.

Cor. 5. The axis is the only diameter which is perpendicular to a tangent at its vertex. For the angle HDQ, or BDE, is the half of BDF, and therefore less than a right angle, except when BD and DF lie...