CONIC SECTIONS, the curves formed by the intersection of a circular cone and a plane, the former being either oblique or right. The works of Apollonius and Archimedes are the first in which these sections were treated of ; and their history is nothing hut that of the addition of a few remarkable pro perties, till the discovery that the path of a projected body in an unresisting space is a parabola, and that of a planet round the sun an ellipse. Though the name, therefore, of conic sections still remains, the interest which attaches to these curves, and the method of treating them, has no longer any reference to the accident from which they derive their name. The Greek geometers, in pure speculation, occupied themselves with the different methods in which a cone may be cut, simply because the conical (with the cylindrical and spherical) came within the restrictive definitions under which they had placed geometry ;—but since the discovery to which we have alluded, we might as well attempt to write the his tory of mathematics and physics, as that of conic sections in their results and ennsequences.
Some sections of a cone arc considered in elementary geo metry, for a plane may meet a cone in a point, or in a single straight line, in two intersecting straight lines, or in a circle. But the curves, which are peculiarly conic sections, are the oval made by a plane which cuts the cone entirely on one side of the vertex, called the ELLIPSE; the indefinitely extended modification of this when the plane beounes parallel to any one slant side of the cone, called the PARABOLA ; and the curve, which is partly on one side, and partly on the other of the vertex, formed by a plane which cuts both sur faces of the cone, called the HYPERBOLA.
Below is appended some convenient methods of forming the sections upon a plane, without any reference to the cone.
If each end of a string of greater length than the distance E F, Plate 1, Figure 1, to be tied to the points E and F, and any intermediate point a, be taken in the string, then the point B being carried round the line E F, so as to keep the parts, E B, 13 F, always stretched till it come to the point whence it began to move, the point B will trace out a curve, A B C B, which will be an ellipsis.
If the end of a straight inflexible line, or roil, of a greater length than the distance E F, Figure 2, be fixed to one extremity, E, of the line, and one end of a string of greater length than the difThrence between E r and the length of the rod, be fixed to F, and the other end to the other end of the rod at N ; then, if any point, n, be taken in the string, and the rod moved round the point E, so as to keep the parts N B, B F always stretched, the point B will trace out a curve, which will be an hyperbola.
And if the end of the rod be moved from E, and fixed at F, and one end of the string moved from F, and fixed at E, the curve described after the same manner, is called an appo site hyperbola.
In the ellipsis and hyperbola, the points E and F are called the foci ; the line, A c, passing through the tbci, joining the opposite parts of the curve, or curves, is called the transverse axis ; and the point, c, in the middle of the transverse axis, is called the centre.
In the ellipsis, any line drawn through the centre, and terminated by the opposite parts of the curve, is called a diameter ; if another right line, terminated by the curve, be drawn parallel to a tangent at one extremity of the other diameter, such line is called a double ordinate ; and if it pass through the centre, it becomos a diameter ; then the two dia meters, thus situated, are called conjugate diameters.
When the conjugate diameters are at right angles to each other, they are called the axis of the curve.
It' there lie a diameter, and a double ordinate to that diameter, the two segments of the diameter are called the abscissa.
Concentric ellipses are such as are similar, and have the same centre with the greater axis or the one upon the greater axis of the other, and the less upon the less.