CON

189

CON

by o ® n< Hhe solidity of a cone, multiply the area of the bases °lidW a ^'*- u de, and one-third of the product will give theVb; c v^.‘ ® r > multiply one-third of the area of the base,p f0( j 18 ^e mean area, by the altitude of the cone, and theI ji U ®t w hl give the solidity. See Circle .

8i(j e the curved surface of a cone, multiply the slanting

the

cone by the semi-circumference of the base, and

If 9^ u ct will be the area of the curved surface.trnj S t, ae diameter of the base be given, the circumferencejp , e found as directed under the article Circle .the eo 6 P er P en dicular altitude be given, the slanting side ofBut ip 16 , be ascertained by the 47th prop., Book i., Euclid.^ a, itin • COne S iven > w ih he much easier to take thetu dfi „ g SK J e ar| d the circumference of the base, than its alti-by (a ,. diameter; the operation will also be much shorterSe C T, the former dimensions than the latter. See Conic8 0u N ’ Ellipsis, Envelope, Hyperbola , and Para-

Pkce° NPESSI ONAL, or Confessionary, in churches, ade C p ’ foully under the main altar, wherein the bones ofC 0 v 8a,n ts, martyrs, and confessors were deposited.desig na t Essi ® NAL is also used in the Romish church, topfi es j e a little box, or desk, in the church, in which theEe w eCeiVe8 the confessions of the penitents.a ^thouo-i c ® n E‘essi° n als, if any, are to be found in England,1 is a common practice to set down all niches,this ti t j e n ° °ther use can be immediately discovered, under

^Perti^j^^ERATION, (from the French ) the exterior®gure es °f a body, from which it receives its particular

">ce,

i a column, is part of the interior surface of a cylin-c < n<; t g ’ a . n< ^ 'ts section is generally a quarter of a circle.

! fl '' tlU ig derivprl frnm flip l-'ivni.li t'dti n pnwp ■ 1 1 1

in

, ^ a concavity at the extremity of a vertical sur-ttttii it 6re ‘t bends off in a tangent, and projects forward\le. ^ et a fillet, or other vertical surface, at an external^Ppep a _ , lUs the shaft of a column bends forward at the^.Pl>ed to ° We , r en ds, until it meet the fillet. The conge, when

1'hi_ .

a ln 18 derived from the French , conge, a curve ; then'cli tij^pNation is apophyge ; and the Latin, scapus , frompONfiv'n'^'^h word scape is derived. See Apophyge.'ted j n hiES, a collection or heap of several bodies,mass or aggregate.

Of Es > fir, hflTY, in geometry, a term applied to lines,gt ?°'ticid e lreS ’ an< ^ s °hds, which exactly cover each other,g hity. ■ Figures that are equal and similar have a con-tj Us ■v P iti' aVe s °hds, the figures of whose sides are con-o 'iclin Phines of the corresponding side at the

o 0 ^fC t,On '

Ci aUng ^ a cone. See Cone .lfc, il;ir C0 CTI °NS. the curves formed by the intersection of^,'' e a nd a plane, the former being either obliqueis „ lr * tv I.; i. e Wor ks of Apollonius and Archimedes are the>O ch these

1 till

sections were treated of; and their history’ ll U tv, ~ °f the addition of a few remarkable pro-to^ u np es ; ® discovery that the path of a projected bodythe r S ln » space is a parabola, and that of a planetl he s C ^t ion Un an ellipse. Though the name, therefore, ofc Hrvp 0 3 remains, the interest which attaches to

the method of treating them, has no longerl W‘ Thp p, to the accident from which they derive their

VNv es 6 , Gl -eek

» ft, • ’ ^VUUIWV/IUJ AAA (/UAL' OpVOUIUUUlJ j V<jVU^flV/U

V 8 'ttip] v v t ^ le different methods in which a cone may*H C ^) c£ because the conical (with the cylindrical and"d pl acp j w ithin the restrictive definitions under which6 a dud rl geometl T >—hut since the discovery to whiched > we might as well attempt to write the his-3 c

geometers, in pure speculation, occupied

K2>

tory of mathematics and physics, as that of conic sections intheir results and consequences.

Some sections of a cone are considered in elementary geo-metry, for a plane may meet a cone in a point, or in a singlestraight line, in two intersecting straight lines, or in a circle.But the curves, which are peculiarly conic sections, are theoval made by a plane which cuts the cone entirely on oneside of the vertex, called the Ellipse ; the indefinitelyextended modification of this when the plane becomes parallelto any one slant side of the cone, called the Parabola ; andthe curve, which is partly on one side, and partly on theother 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 thesections upon a plane, without any reference to the cone.

If each end of a string of greater length than the distanceE p, Plate 1, Figure 1, to be tied to the points E and f, andany intermediate point, b, be taken in the string, then thepoint B being carried round the line e f, so as to keepthe parts, eb,bf, always stretched till it come to the pointwhence it began to move, the point b will trace out a curve,abcb, which will be an ellipsis.

If the end of a straight inflexible line, or rod, of a greaterlength than the distance e f, Figure 2, be fixed to oneextremity, e, of the line, and one end of a string of greaterlength than the difference between e f and the length of therod, be fixed to f, and the other end to the other end ofthe rod at n ; then, if any point, B, be taken in the string,and the rod moved round the point E, so as to keep the partsN 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 atf, and one end of the string moved from f, and fixed at e,the curve described after the same manner, is called an oppo-site hyperbola.

In the ellipsis and hyperbola, the points e and f are calledthe foci; the line, AC, passing through the foci, joining theopposite parts of the curve, or curves, is called the transverseaxis ; and the point, c, in the middle of the transverse axis,is called the centre.

In the ellipsis, any line - drawn through the centre, andterminated by the opposite parts of the curve, is calleda diameter ; if another right line, terminated by the curve,be drawn parallel to a tangent at one extremity of the otherdiameter, such line is called a double ordinate ; and if it passthrough the centre, it becomes a diameter ; then the two dia-meters, thus situated, are called conjugate diameters.

When the conjugate diameters are at right angles to eachother, they are called the axis of the curve.

If there be a diameter, and a double ordinate to thatdiameter, the two segments of the diameter are called theabscissce.

Concentric ellipses are such as are similar, and have thesame centre with the greater axis of the one upon the greateraxis of the other, and the less upon the less.

Most of the above definitions apply also to the hyperbola.

If the side, a b, Figure 3, of a right angle or square, abc,be applied to the straight-edge, a d, of a rule, and a thread,equal in length to bc, be fastened to the end, C, of the rightangle, with the other end to the fixed point, F ; and if anypoint, e, be taken in the line, then if the edge, A b, of thesquare be moved along the straight-edge, ad, keeping thevariable point, e, upon the side, b c, of the square, andthe two portions c e and e f stretched, the point E will traceout a curve, which is a parabola.

The point F is called the focus.

The line a d is the directrix.