BRITISH ENCYCLOPEDIA, OR DICTIONARY OF ARTS AND SCIENCES; COMPRISING AN ACCURATE AND POPULAR VIEW OF THE PRESENT IMPROVED STATE OF HUMAN KNOWLEDGE. BY WILLIAM NICHOLSON, Mathematical Works. ILLUSTRATED WITH UPWARDS OF 150 ELEGANT ENGRAVINGS, BY MESSRS. LOWRY AND SCOTT. VOL. III. E....I. LONDON: Goswell Street ; AND LETTER MAN ; CUTHELL AND MARTIN ; R. LEA; LACKINGTON AND CO.; VERNOR, HOOD, AND 1809. THE BRITISH ENCYCLOPEDIA. ELLIPSIS. ELLIPSIS, in geometry, a curve line re called the ordinate, or ordinate-applicate' turning into itself, and produced from to that diameter; and a third proportional te section of a cone by a plane cutting both to two conjugate diameters, is called the laits sides, but not parallel to the base. See tus rectum, or parameter of that diameter Coxic SECTIONS. which is the first of the three propor. The easiest way of describing this curve, tionals. in plano, when the transverse and conju- The reason of the name is this : let B A, 26 AB, ED, (Plate V. Miscell. fig. 1.) ED, be any two conjugate diameters of an are given, is this : first take the points F,f, ellipsis (fig. 2, where they are the two in the transverse axis A B, so that the dis axes) at the end A, of the diameter A B, tances CF, Cf, from the centre C, be each raise the perpendicular A F, equal to the equal to VĀC-CD; or, that the lines latus rectum, or parameter, being a third FD, FD, be each equal to AC; then, hav. proportional to A B, ED, and draw the ing fixed two pins in the points F,f, which right line BF; then if any point P be are called the foci of the ellipsis, take a taken in BA, and an ordinate PM be thread equal in length to the transverse drawn, cutting B F in N, the rectangle unaxis AB; and fastening its two ends, one der the absciss A P, and the line PN will to the pin F, and the other to f, with ano- be equal to the square of the ordinate PM. ther pin M stretch the thread' tight; then Hence drawing N O parallel to AB, it apif this pin M be moved round till it returns pears that this rectangle, or the square of to the place from whence it first set out, the ordinate, is less than that under the abkeeping the thread always extended so as sciss AP, and the parameter AF, by the to form the triangle FMf, it will describe rectangle under AP and O F, or NO and an ellipsis, whose axes are A B, D E. OF; on account of which deficiency, ApolThe greater axis, AB, passing through lonius first gave this curve the name of an the two foci Ff, is called the transverse ellipsis, from a 2617!19, to be deficient. axis; and the lesser one D Е, is called the In every ellipsis, as A E B D, (fig. 2), the conjugate, or second axis : these two always squares of the senii-ordinates MP, mp, are bisect each other at right angles, and the as the rectangles under the segments of the eentre of the ellipsis is the point C, where transverse axis A P X PB, Ap X p B, made they intersect. Any right line passing by these ordinates respectively; which holds through the centre, and terminated by the equally true of the circle, where the squares curve of the ellipsis on each side, is called of the ordinates are equal to such rectana diameter; and two dianieters, which na. gles, as being mean proportionals between turally bisect all the parallels to each other, the segments of the diameter. In the same bounded by the ellipsis, are called conju- manner, the ordinates to any diameter gate diameters. Any right line, not pass- whatever, are as the rectangles under the ing through the centre, but terminated by segments of that diameter. the ellipsis, and bisected by a diameter, is As to the other principal properties of VOL. III, B : a + 2.2 the ellipsis, they may be reduced to the fol- this series will be more simple : for if lowing propositions. 1. If from any point 3 as c=2r, then MB=at M in an ellipsis, two right lines, MF, Mf, + 96 m2 2048 m (fig. 1.) be drawn to the foci F,f, the sum 113 a? 3419 a' of these two lines will be equal to the trans &c. This se+ 458752 " 75497472 pol? verse axis A B. This is evident from the ries will serve for an hyperbola, by making manner of describing an ellipsis. 2. The the even parts of all the terms affirmative, square of half the lesser axis is equal to the and the third, fifth, seventh, &c. terms nerectangle under the segments of the greater gative. axis, contained between the foci and its The periphery of an ellipsis, according to vertices; that is, DC=AF XF B=Af Mr. Simpson, is to that of a circle, whose xf B. 3. Every diameter is bisected in diameter is equal to the transverse axis the centre C. 4. The transverse axis is the d 3 d greatest, and the conjugate axis the least, of the ellipsis, as 1 2.2.4. of all diameters. ' 5. Two diameters, one 3.3.5d 2.3.5.5.7d" of which is parallel to the tangent in the 2.2.4.4.6.6 2.2.4.4.6.6.8.8, vertex of the other, are conjugate diameters; &c. is to 1, where d is equal to the differand vice versa, a right line drawn through ence of the squares of the axis applied to the vertex of any diameter parallel to its the square of the transverse axis. conjugate diameter, touches the ellipsis in Ellipsis, in grammar, a figure of synthat vertex. 6. If four tangents be drawn tax, wherein one or more words are not exthrough the vertices of two conjugate diameters, the parallelogram contained under pressed; and from this deficiency it has got the name ellipsis. them will be equal to the parallelogram ELIIPsis, in rhetoric, a figure nearly alcontained under tangents drawn through the lied to pretérition, when the orator, through vertices of any other two conjngate dia- transport of passion, passes over many meters. 7. If a right line, touching an el- things : which, had be been cool, ought to lipsis, meet two conjugate diameters pro- have been mentioned. In preterition, the duced, the rectangle under the segments of omission is designed ; which, in the ellipsis, the tangent, between the point of contact is owing to the vehemence of the speaker's and these diameters, will be equal to the passion, and his tongue not being able to square of the semi-diameter, which is conjugate to that passing through the point of keep pace with the emotion of his mind. ELLIPTIC, or ELLIPTICAL, something 8. In every ellipsis, the sum of belonging to an ellipsis. Thus we meet the squares of any two conjógate diameters with elliptical compasses, elliptic conoid, is equal to the sum of the squares of the two elliptic space, elliptic stairs, &c. The ellipaxes. 9. In every ellipsis, the angles FGI, tic space is the area contained within the fGH, (fig. 1), made by the tangent HI, curve of the ellipsis, which is to that of a and the lines FG, G, drawn from the foci circle described on the transverse axis, as to the point of contact, are equal to each the conjugate diameter is to the transverse other. 10. The area of an ellipsis is to the axis ; or it is a mean proportional between area of a circumscribed circle, as the lesser two circles, described on the conjugate and axis is to the greater, and vice versa with transverse axis. respect to an inscribed circle ; so that it is ELLIPTOIDES, in geometry, a name a mean proportional between two circles, used by some to denote infinite ellipses, having the transverse and conjugate axes mtn for their diameters. This holds equally defined by the equation ay true of all the other corresponding parts (a— x)". belonging to an ellipsis. Of these there are several sorts: thus, The curve of any ellipsis may be obtain- if ay'=bx? (a — 2) it is a cubical elliped by the following series. Suppose the toid; and if a y' = b x(a — «)', it desemi-transverse axis CB=r, the semi-con notes a biquadratic elliptoid, which is an jugate axis CD=c, and the semi-ordi- ellipsis of the third order in respect of the nate sa; then the length of the curve appollonian ellipsis. pod ad 4 m2 2 a -ras MB = at + + ELLISIA, in botany, so called in me6€* 40c mory of John Ellis, F. R. S. a genus of 8 c* r* a' tro a? – 48** a?, &c. And, if the Pentandria Monogynia class and order. 4 ? 112 c2 Natural order of Luridæ. Borragineæ, the species of the ellipsis be determined, Jussieu. Essential character: corolla fun. contact. m = bx n |