CONIC Sections
, are the figures made by cutting a cone by a plane.
2. According to the different positions of the cutting plane, there arise five different figures or sections, viz, a triangle, a circle, an ellipse, a parabola, and an hyperbola: the last three of which only are peculiarly called conic sections.
3. If the cutting plane pass through the vertex of the cone, and any part of the base, the section will evidently be a triangle; as VAB.
4. If the plane cut the cone parallel to the base, or make no angle with it, the section will be a circle, as ABD.
5. The section DAB is an el- lipse, when the cone is cut obliquely through both sides, or when the plane is inclined to the base in a less angle than the side of the cone is.
6. The section is a parabola, when the cone is cut by a plane parallel to the side, or when the cutting plane and the side of the cone make equal angles with the base.
7. The section is an hyperbola, when the cutting plane makes a greater angle with the base than the side of the cone makes. And if the plane be continued to cut the opposite cone, this latter section is called the opposite hyperbola to the former; as dBe.
8. The vertices of any section, are the points where the cutting plane meets the opposite sides of the cone, or the sides of the vertical triangular section; as A and B. —Hence, the ellipse and the opposite hyperbolas have each two vertices; but the parabola only one; unless we consider the other as at an infinite distance.
9. The axis, or transverse diameter of a conic section, is the line or distance AB between the vertices.— Hence the axis of a parabola is infinite in length. Ellipse. Oppos. Hyperb. Parabola.
10. The centre C is the middle of the axis.—Hence the centre of a parabola is infinitely distant from the vertex. And of an ellipse, the axis and centre lie within the curve; but of an hyperbola, without. |
11. A Diameter is any right line, as AB or DE, drawn through the centre, and terminated on each side by the curve: and the extremities of the diameter, or its intersections with the curve, are its vertices.—Hence all the diameters of a parabola are parallel to the axis, and infinite in length; because drawn through the centre, a point at an infinite distance. And hence also every diameter of the ellipse and hyperbola have two vertices; but of the parabola, only one; unless we consider the other as at an infinite distance.
12. The conjugate to any diameter, is the line drawn through the centre, and parallel to the tangent of the curve at the vertex of the diameter. So FG, parallel to the tangent at D, is the conjugate to DE; and HI, parallel to the tangent at A, is the conjugate to AB. —Hence the conjugate HI, of the axis AB, is perpendicular to it; but the conjugates of other diameters are oblique to them.
13. An ordinate to any diameter, is a line parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter and curve. So DK and EL are ordinates to the axis AB; and MN and NO ordinates to the diameter DE.—Hence the ordinates of the axis are perpendicular to it; but of other diameters, the ordinates are oblique to them.
14. An absciss is a part of any diameter, contained between its vertex and an ordinate to it; as AK or BK, and DN or EN.—Hence, in the ellipse and hyperbola, every ordinate has two abscisses; but in the parabola, only one; the other vertex of the diameter being infinitely distant.
15. The parameter of any diameter, is a third proportional to that diameter and its conjugate.
16. The focus is the point in the axis where the ordinate is equal to half the parameter: as K and L, where DK or EL is equal to the semiparameter.—— Hence, the ellipse and hyperbola have each two foci; but the parabola only one.—The foci, or burning points, were so called, because all rays are united or reflected into one of them, which proceed from the other focus, and are reflected from the curve.
17. If DAE, FBG be two opposite hyperbolas, having AB for their first or transverse axis, and ab for their second or conjugate axis; and if dae, fbg be two other opposite hyperbolas, having the same axis, but in the contrary order, viz, ab their first axis, and AB their second; then these two latter curves dae, fbg, are called the conjugate hyperbolas to the two former DAE, FBG; and each pair of opposite curves mutually conjugate to the other.
18. And if tangents be drawn to the four vertices of the curves, or extremities of the axis, forming the inscribed rectangle HIKL; the diagonals HCK and ICL, of this rectangle, are called the asymptotes of the curves.
19. Scholium. The rectangle inscribed between the four conjugate hyperbolas, is similar to a rectangle circumscribed about an ellipse, by drawing tangents, in like manner, to the four extremities of the two axes; also the asymptotes or diagonals in the hyperbola, are analogous to those in the ellipse, cutting this curve in similar points, and making the pair of equal conjugate diameters. Moreover, the whole figure, formed by the four hyperbolas, is, as it were, an ellipse turned inside out, cut open at the extremities D, E, F, G, of the said equal conjugate diameters, and those four points drawn out to an infinite distance, the curvature being turned the contrary way, but the axes, and the rectangle passing through their extremities, remaining fixed, or unaltered.
From the foregoing definitions are easily derived the following general corollaries to the sections. Ellipse. Hyperbola. Parabola.
20. Corol. 1. In the ellipse, the semiconjugate axis, CD or CE, is a mean proportional between CO and CP, the parts of the diameter OP of a circular section of the cone, drawn through the centre C of the ellipse, and parallel to the base of the cone. For DE is a double ordinate in this circle, being perpendicular to OP as well as to AB.
21. In like manner, in the hyperbola, the length of the semiconjugate axis, CD or CE, is a mean proportional between CO and CP, drawn parallel to the base, and meeting the sides of the cone in O and P. Or, if AO′ be drawn parallel to the side VB, and meet PC produced in O′, making CO′=CO; and on this diameter O′P a circle be drawn parallel to the base: then the semiconjugate CD or CE will be an ordinate of this circle, being perpendicular to O′P as well as to AB.
Or, in both figures, the whole conjugate axis DE is a mean proportional between QA and BR, parallel to the base of the cone. See my Conic Sections, pa. 6.
In the parabola, both the transverse and conjugate are infinite; for AB and BR are both infinite.
22. Corol. 2. In all the sections, AG will be equal to the parameter of the axis, if QG be drawn making the angle AQG equal to the angle BAR. In like manner Bg will be equal to the same parameter, if Rg be drawn to make the angle BRg=the angle ABQ.
23. Corol. 3. Hence the upper hyperbolic section, or section of the opposite cone, is equal and similar to the lower one. For the two sections have the same transverse or first axis AB, and the same conjugate or second axis DE, which is the mean proportional between AQ and RB; and they have also equal parame- | ters AG, Bg. So that the two opposite sections make, as it were, but the two opposite ends of one entire section or hyperbola, the two being every where mutually equal and similar. Like the two halves of an ellipse, with their ends turned the contrary way.
24. Corol. 4. And hence, although both the transverse and conjugate axis in the parabola be infinite, yet the former is infinitely greater than the latter, or has an insinite ratio to it. For the transverse has the same ratio to the conjugate, as the conjugate has to the parameter, that is, as an infinite to a finite quantity, which is an infinite ratio.
The peculiar properties of each particular curve, will be best referred to the particular words Ellipse, HYPERBOLA, Parabola; and therefore it will only be proper here to lay down a few of the properties that are common to all the conic sections.
25. From the foregoing desinitions, &c, it appears, that the conic sections are in themselves a system of regular curves, naturally allied to each other; and that one is changed into another perpetually, when it is either increased, or diminished, in infinitum. Thus, the curvature of a circle being ever so little increased or diminished, passes into an ellipse; and again, the centre of the ellipse going off infinitely, and the curvature being thereby diminished, is changed into a parabola; and lastly, the curvature of a parabola being ever so little changed, there ariseth the first of the hyperbolas; the innumerable species of which will all of them arise orderly by a gradual diminution of the curvature; till this quite vanishing, the last hyperbola ends in a right line. From whence it is manifest, that every regular curvature, like that of a circle, from the circle itself to a right line, is a conical curvature, and is distinguished with its peculiar name, according to the divers degrees of that curvature.
26. That all diameters in a circle and ellipse intersect one another in the centre of the figure within the section: that in the parabola they are all parallel among themselves, and to the axis: but in the hyperbola, they intersect one another, without the figure, in the common centre of the opposite and conjugate sections.
27. In the circle, the latus rectum, or parameter, is double the distance from the vertex to the focus, which is also the centre. But in ellipses, the parameters are in all proportions to that distance, between the double and quadruple, according to their different species. While, in the parabola, the parameter is just quadruple that distance. And, lastly in hyperbolas, the parameters are in all proportions beyond the quadruple, according to their various kinds.
28. The first general property of the conic sections, with regard to the abscisses and ordinates of any diameter, is, that the rectangles of the abscisses are to each other, as the squares of their corresponding ordinates. Or, which is the same thing, that the square of any diameter is to the square of its conjugate, as the rectangle of two abscisses of that diameter, to the square of the ordinate which divides them. That is, in all the figures, the rect. AC. CB: rect. AE. EB :: CD^{2} : EF^{2} : But as, in the parabola the infinites CB and EB are in a ratio of equality, for this curve the same property becomes AC : AE :: CD^{2} : EF^{2}, that is, in the parabola, the abscisses are as the squares of their ordinates.
Or, when one of the ordinates is the semiconjugate GH, dividing the diameter equally in the centre, the same general property becomes, AG . GB or AG^{2} : AC . CB :: GH^{2} : CD^{2}, or AB^{2} : HI^{2} :: AC . CB : CD^{2}.
29. From hence is derived the equation of the curves of the conic sections; thus, putting the diameter AB =d, its conjugate HI=c, abscrss AC=x, and its ordinate CD=y; then is the other absciss CB=d-x in the ellipse, or d+x in the hyperbola, or d in the parabola; and hence the last analogy above, becomes y^{2}=((c^{2})/d) x or=px in the parabola, where the parameter p=((c^{2})/d) the third proportional to the diameter and its conjugate, by the definition of it.
And from this one general proposition alone, which is easily derived from the section in the solid cone itself, together with the definitions only, as laid down above, all the other properties of all the sections may easily be derived, without any farther reference to the cone, and without mechanical descriptions of the curves in plano; as is done in my Treatise on Conic Sections, for the use of the Royal Mil. Acad.; in which also all the similar propositions in the ellipse and hyperbola are carried on word for word in them both.
The more ancient mathematicians, before the time of Apollonius Pergæus, admitted only the right cone into their geometry, and they supposed the section of it to be made by a plane perpendicular to one of its sides; and as the vertical angle of a right cone may be either right, acute, or obtuse, the same method of cutting these several cones, viz, by a plane perpendicular to one side, produced all the three conic sections. The parabola was called the section of a right-angled cone; the ellipse, the section of the acute-angled cone; and the hyperbola, the section of the obtuse-angled cone. But Apollonius, who, on account of his writings on this | subject, obtained the appellation of Magnus Geometra, the Great Geometrician, observed, that these three sections might be obtained in every cone, both oblique and right, and that they depended on the different inclinations of the plane of the section to the cone itself. Apollon. Con. Halley's edit. lib. 1, p. 9.
Instead of considering these curves as sections cut from the solid cone, which is the true genuine way of all the ancients, and of the most elegant writers among the moderns, Descartes, and some others of the moderns, have given arbitrary constructions of curves on a plane, from which constructions they have demonstrated the properties of these, and have afterwards proved that some principal property of them belongs to such curves or sections as are cut from a cone; and hence it is inferred by them that those curves, so described on a plane, are the same with the conic sections.
The doctrine of the conic sections is of great use in physical and geometrical astronomy, as well as in the physico-mathematical sciences. The doctrine has been much cultivated by both ancient and modern geometricians, who have left many good treatises on the subject. The most ancient of these is that of Apollonius Pergæus, containing 8 books, the first 4 of which have often been published; but Dr. Halley's edition has all the eight. Pappus, in his Collect. Mathem. lib. 7, says that the first four of these were written by Euclid, though perfected by Appollonius, who added the other 4 to them. Among the moderns, the chief writers are Mydorgius de Sectionibus Conicis; Gregory St. Vincent's Quadratura Circuli & Sectionum Coni; De la Hire de Sectionibus Conicis; Trevigar Elem. Section. Con.; De Witt's Elementa Curvarum; Dr. Wallis's Conic Sections; De l'Hospital's Anal. Treat. of Conic Sections; Dr. Simson's Section. Con.; Milne's Elementa Section. Conicarum; Muller's Conic Sections; Steel's Conic Sections; Dr. Hamilton's elegant treatise; my own treatise, above cited; and at the writing of this, my friend Mr. Abram Robertson of Oxford is preparing a curious work on this subject, containing at the same time a treatise on the science, and a history of the writings relating to it.