THE

IMPERIAL

ENCYCLOPAEDIA.

/

CONIC SECTIONS.

ONIC SECTIONS are the figures formed by cutting acone by a plane. They are five in number corresponding' to the different positions of the cutting plane ; viz. a Tri-angle, a Circle, an Ellipse, a Parabola, and an Hyperbola.'rK * ast dlr<;e of these only are peculiarly called conic sections.1 be more ancient mathematicians, before the lime of Apolloniusf ersaais, admitted only the right cone into their geometry, andi e T sl !PP°scd a section made of it by a plane perpendicular to oneSld cs: and as the vertical angle of a right cone may be eitherrn> it, acute, or obtuse, this method of cutting these several conesproduced all the three conic sections. The parabola was calledtne section of a right-angled cone, the ellipse the section of theat -r'gUa, Cone> an( ] t i )e hyperbola the section of the obtuse-fh^ S ?°" e - But Apollonius, who on account of his writings ontms su iject, obtained the title of the Great Geometrician, observ-ed 6 S< r clions might be obtained in every cone, hoth oblique

fi Jr ’ ,'that they depended on the different inclinations of1 v; . , l . he tone itself. There have been two methods em-

F.rlered -is of tlle tonic sections; by the one they are con-

^nr-ients and ^ ut of l| ie Holid cone, which is the method of the‘ 1 v tile „ti° SOnie of the most elegant writers of the moderns;

ime nroDcrtv i n,eth °d certain curves are delined, either from

them, or ehe' 1)y J w 11u'I k a, ‘>' 1,u ? b " of !’, 01! ! tS ma - V be iol,nd in

,p ( »ne; r ,h4; , “ | ;Kv.~ 5 .

trop'ittai'S SS'eM?, 0 "''''* m ,be "" “ ta " f' T" r

Ind cone Earh ^ , cl ‘ “re formed by the intersections of a planeand cone. > of lhese methodg ,/ as u# advantages; although

geometrically by the K»° ns , of ' vliters who havt ‘ treatwl the L s "bj ectothers upon which denim'’ be sllort and po‘ s P lcllou3 > TV 1 lh * r< i aretedious and difficult? ‘mlT of the i ,nlR 'T ai properties, that are

nr UDV mimoer oi ptrnub iuav Ut KHUJU III

nlane • or thev- W u<; ,b the V mav be described mechanically uponml in either AAr lltfl ned by'means of an algebraical equation,

;ry samef a plane

tome of the demo.^t‘“ ese methods has its advantages; althoughSO F,.r,r,,,ii‘ .. 0 .'‘ strdll ons of writers who have treated the subject

perspicuous, yet there are

tedious *..u umicult T , - r . ineipal properties, that are

pursued the first method (1 « mon sTations ol writers who havenerallv plain and concil d '/ e free h ' om Bus objection, being ge-duce so* many " I but they have been obliged to intro-

touching and cutti.T ° pr>Sltl0lls concerning tlie properties otirincipal pronertie* ,?., conical surfaces, in order to arrive at

th^P’incipa' propei-ties of t) COI t V Ca ' s

ciderable portion of time V1 U ’ rec sections > thatit requires a con-matical studies to go throu! /i^ 0 ’ 01 for a beg 11111 -’ 1 ', 1,1 mall,e -.,,1 the subject algebraical! v l®' 1 ' - fa ome writers, who have treat-rovver compass; but in t)' : lave reduced the whole into a nar-i,, ve fallen into another ea S c rness to avoid prolixity, theyJ? which they have dVducm? f!xc '-‘Pfio,uble fault/ The methodt)ie relations of the abscissa/ S °i nie of tllc l )1 ' 0 P crties > particularly„ n d inelegant; each step in thJ* 1 ' 11 ordinates, is extremely operoseyou n.— no. 56. 1 - e Process is so little connected w ith the

j! preceding one, that it is scarcely possible to retain them in theI memory. The conic sections are of great use in physical andj geometrical astronomy, as well as in the physico-mathcmalicalsciences, and therefore they have been much cultivated ever sincetheir great importance in these sciences was known.

Definitions of the Cone.

If an indefinite straight line Z X, fig. 1, Plate XLVUI. bemade to revolve in the circumference of the circle B C D E so asalways to pass through a fixed point A w ithout the circle, the sur-faces so generated are called conical superficies, or conic surfaces,and when mentioned together, opposite conical superficies. Alsothe solid contained by the conical superficies and a circle is calleda cone, and the solicls contained by the opposite conical superfi-cies, and two circles in parallel planes are called opposite cones.The terminating circle is called the base of the cone. The fixedpoint A is the vertex, and the line A F joining the vertex andcentre of the base is the axis of the cone. When the axis is per-pendicular to the plane of the base the cone is called a right cone,and when the axis is inclined to the plane of the base it is a sca-lene cone. See Cone.

Corollary 1. lienee it is evident, that a straight line drawn fromthe vertex to any point in the circumference base lies wholly in theconical superficies ; for it coincides with the generating, line.

Cor. 2. A section passing through the vertex will be a triangle.For let V O B, fig. 4, be a section passing through the vertex V,and meeting the circumference of the base in B and O, then V O,V B, are straight lines by cor. 1. and O B is a straight line, sinceit is the common section of two planes.

Cor. 3. A section parallel to the base is a circle. For letF G II, fig. 5, be a section parallel to the base meeting the axisin X, through the axis draw the plane V A D, and let its commonsections with the given plane and with the base be FXHand ACD,also pass any other plane through the axis, as V C 13, cutting thegiven plane and the base in XG and CB; then because the planesEGIJ, ABD, are parallel, Eli is parallel to Al>, and XG to CB,hence by similar triangles VC : YX : : Ci5: XG :: CD : XII: :CA : XF, but CB, CD, CA, are equal; therefore XG, XII, andXF, are 1 equal, hence FGI1 is a circle.

Cor. 4: If a scalene cone be cut through the axis by a planeperpendicular to the base, as YI'C, fig. 6,"and in this plane a line| EM he drawn, making the angle VLM — VBC, and the cone hrcut by a plane MPLQ, passing through l.M, perpendicular to theplane VBC, the section will be a circle. Through any point II inML draw a plane G PLQ parallel to the base; J’Qits commonB section