Volume 2

The imperial encyclopaedia; or, dictionary of the sciences and arts / [William Moore Johnson].

  • Johnson, William Moore, 1772?-1849.
Date:
[1812]
Catalogue details

Licence: Public Domain Mark

Credit: The imperial encyclopaedia; or, dictionary of the sciences and arts / [William Moore Johnson]. Source: Wellcome Collection.

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    section with MPLQ is perpendicular to the plane VBC, because the planes GPFQ PQ is bisected in MHG and FHL are equal, and also the vertical angles at H equal, hence ME: HG:: FH: HL; therefore MH x HL= HG x f H = PH?, hence MPLQ isa circle whose diameter is LM. The circle MPLQ is called a sub-contrary section. If a cone be cut by a plane which does not pass through the vertex, and which is neither parallel to the base, mor to the plane of a subcontrary section 5 the common section of the plane and the surface of the cone will be an ellipse, a parabola, or an hyperbola, according as a plane passmg through the vertex parallel to the cutting plane, falls without the cone, touches it, or falls within the cone; as explained in the following definitions : _ DEFINITIONS. 1. A conic section is the figure’ formed by the intersection. of a cone and a plane. Scholium, ‘Vhe triangle and eirele, though conic sections, are not treated under that denomination, but their proper- ties are considered together in plane geometry. : 2. A line drawn from thé vertex_to a point in the circumference of the base of the cone is called a side of the cone. 3. A plane passing through the vertex of the cone, and parallel to the plane of the section, is called the vertical plane. : 4, A plane not cutting the cone, and passing through a side of the cone, is called a tangent plane. ; 5. A straight line drawn in the tangent plane is called a tangent to the cone. 6. When the vertical plane as VBN, fig. 7, is wholly without the cone, and not parallel to the base, the section PQ is called an ellipse. Cor. It is evident that because the vertical plane, meets all the sides of the cone, the plane of the section which is parallel to it will meet all the sides of the cone, and cannot meet the opposite superficies. Hence the ellipse surrounds the cone, 7. When the vertical plane is a tangent plane as VBN, fig. 8, the section PQ is called a parabola. » Cor. it is evident that the cutting plane can never meet VB, the side of the cone through which the tangent plane passes ; hence the curve of the section will nct return into itself. . - 8. When the vertical plane VEN cuts the cone, as in MN, fig. 9, the section is called a Ayperdola, Cor. Since the vertical plane meets all the sides of the cone, the cutting plane will likewise meet them, and it is evident that it will meet those which are on one side of the vertical plane in one of the conical superficies, and those on the other side of the vertical plane in the opposite conical superficies, forminga hyperbola in each cone. - 9. ‘The two hyperbolas formed in the opposite cones are called opposiie hyperbolas. 10. A \me which meets a conic section but does not cut it, is called a tangent; any other line meeting a conic section is called a secant, Cor. It is evident, that a tangent to a conic section is in the tan- gent plane, and is a tangent to the cene. il, Ifthe triangular section VAD, fig. 7,8, 9, be perpendicular to ihe cutting plane, the commen section of these planes, as PQ, terminated by the conical superficies, is called the transverse axis, and the extremities of the axis are its vertices. Cor. 1. It is evident, that the transverse axis in the ellipse and opposite hyperbolas has two vertices, and that of the parabola only one vertex. , Cor. 2, It is also evident, that the transverse axis in the ellipse ~ and parabola, is entirely within the section; and the transverse axis of the opposite hyperbolas entirely without the section. 12. The middle of the axis, as C, is the centre of the section. _ Cor. The parabola has no centre, since PQ is unlimited. 13. Ifa plane ELFPH be drawn parallel to the base of the cone, and cutting the conic section, the common section IH of this plane and the conic section, is called a double ordinate, and its half an ordinate to the transverse axis. Cor. }. ‘The ordinate is at right angles to the axis, for the planes EIFH and PiQH are at right angles to VAD. right angles te the diameter of the circle EIH. ts an ordinate. Hence IG or GH A 14, The segment ef an axis contained between the vertex and ordinate, is called an absciss. 15. The double ordinate passing through the centre of the el- lipse is called the conjugate axis, and its extremities are called the vertices of the conjugate axis. _ 16. If through C, the centre of the opposite hyperbolas, a plane be drawn forming in the cone the circle RS, and if from C a tangent CL be drawn to the circle, the tangent is called the semiz- conjugate axis of the opposite hyperbolas. : : 17, The straight line which is a third proportional to the axis, tional to an absciss and ofdinate in the parabola, is the parameter of the axis of the parabola. Cor. The semi-transyerse is to the semi-conjugate, as the semi- 18. [he point where that double ordinate which is equal to the parameter cuts the axis, is called the focus of the section. Cor.. The ellipse and opposite hyperbolas will have two foci, the parabola only one. ; 19. If F, fig. 17 and 19, be a focus of an ellipse, or hyperbola, or opposite hyperbolas, and AG be taken in the transverse axis (on the opposite side of the vertex to the focus F), such, that AF is to AG as CF isto CA; then a line, as HK, drawn through G per- pendicular to the transverse axis, iscalled a directrix of the ellipse, hyperbola, or opposite hyperbolas. Fig. 18. But the directrix of a parabola isa line, as HK, per- pendicular to the axis, drawn through a point G as far distant from the vertex of the axis on the one side as the focus is on the other side. Cor. An ellipse, hyperbola, or opposite hyperbolas, Have two directrices ; one corresponding to each focus. For the same con- struction that is made for one focus may be made for the other focus. 20. The tangents DLQ, DT 4q, fig. 23, 24, which are drawn through the extremities of the ordinate passing through the focus, are called focal tangents, 21, Any line passing through the centre of an ellipse or hyper- bola, and terminated both ways by the curve in the former, but by the opposite curves in thelatter, is called a diameter. 22. A line drawn through any point in the parabola parallel to the axis, is called a déameter of the parabela. 23, Any point where the diameter meets the curve is called a ” vertex to that diameter. “ 24, If from the centre C, fig. 27, at the distance CA, half the transyerse axis, a circle be described, cutting the directrix of the hyperbola in the points H, A, and lines drawn from the centre through the points of intersection, these lines are called asymp- totes. 25, If AM be the transverse axis, and B }, the conjugate axis of two opposite hyperbolas, and two other hyperbolas be described, ef which the transverse axis is B 6, and conjugate axis AM, these hyperbolas are-said to be conjugate to the former. 26. When the two axes are equal; the hyperbolas are said to be equilateral. 27. The straight line PR, fig. 33, 34, which is drawn from the point of contact perpendicular to the tangent, intercepted between the tangent and the axis of a conic section, is called a normal. 28. ‘The segment of the axis NR, which is intercepted between the ordinate and thenormal, is called a subnormal, 29. Ifa straight line be drawn through any point in the diame- ter of a conic section, parallel to a tangent at the vertex, and ter- minated both ways by the curve, it is called an ordinate to that dianeter. 30, Thesegment of any diameter of aconic section; which is in- tercepted between an ordinate and the vertex, is called an ab- — SCISS. 31. A diameter which is parallel to the tangent at the vertex of- any diameter of the ellipse or hyperbola, is called a conjugate diameter, ; 32. A line which is a third proportional to any diameter of the ellipse or hyperbola, and its conjugate, is. called a parameter of that diameter. re 7 33. Ifa line be drawn through the focus of 4 parabola parallel to the ordinates of any diameter, terminated both ways by the curye, it ig called a parameter to that diameter, ;
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