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THE 



BRITISH ENCYCLOPEDIA, 



ELLIPSIS. 



ELLIPSIS, in geometry, a curve line 

 returning into itself, and produced 

 from the section of a cone by a plane cut- 

 ting- both its sides, but not parallel to the 

 base. See CONIC SECTIONS. 



The easiest way of describing this 

 curve, in piano, when the transverse 

 and conjugate axis A B, E D, (Plate V. 

 MiscelK fig. 1.) are given, is this: first 

 take the points F, /, in the transverse 

 axis A B, so that the distances C F, C/, 

 from the centre C, be each equal to 

 v/ A C C D ; or that the lines F D,/D, 

 be each equal to A C : then, having fixed 

 two pins ; n the points F,,/J which are 

 culled the foci of the ellipsis, take a 

 thread equal in length to the transverse 

 axis A B; and fastening its two ends, 

 one to the pin F, and the other to/, with 

 another pin M stretch the thread tight ; 

 then if this pin M be moved round till it 

 returns to the place from whence it first 

 s'et out, keeping 'the thread always ex- 

 ter.aed so as to form the triangle F M/, 

 it will describe an ellipsis, whose axes 

 are A B, D E 



The greater axis, A B, passing through 

 the two foci F/, is called the transverse 

 axis ; and the lesser one D E, is called 

 the conjugate, or second axis : these two 

 always bisect each other at right angles, 

 and the centre of the ellipsis is the point 

 C, where they intersect. Any right line 

 passing through the centre, and terminat- 

 ed by the curve of the ellipsis on each 

 side, is called a diameter ; and two dia- 

 meters, which naturally bisect all the 

 parallels to each other, bounded by the 



ellipsis, are called conjugate diameters, 

 Any right line, not passing through the 

 centre, but terminated by the ellipsis, 

 and bisected by a diameter, is called the 

 ordinate, or ordinate-applicate, to that 

 diameter; and a third proportional to two 

 conjugate diameters is called the latus 

 rectum, or parameter of that diameter, 

 which is the first of the three propor- 

 tionals. 



The reason of the name is this : let 

 B A, E D, he any two conjugate diame- 

 ters of an ellipsis (fig. 2, where they are 

 the two axes) at the end A, of the dia- 

 meter A B, raise the perpendicular A F, 

 equal to the latus rectum, or parameter, 

 being a third proportional to A B, E D, 

 and draw the right line B F ; then, if any 

 point P betaken in B A, and an ordinate 

 P M be drawn, cutting B F in N, the 

 rectangle under the absciss A P, and the 

 line P N will be equal to the square of the 

 ordinate P M. Hence drawing N O pa- 

 rallel to A B, it appears that this rec- 

 tangle, or the square of the ordinate, 

 is less than that under the absciss A P, 

 and the parameter A F, by the rectan- 

 gle under A P and O F," or N O and 

 OF; on account of which deficiency, 

 Appollonius first gave this curve the 

 name of an ellipsis, from eAA5/9re*v, to be 

 deficient. 



In every ellipsis, as A E B D, (fig. 2), 

 the squares of the semi-ordinates M P, 

 m p. are as the rectangles under the seg- 

 ments of the transverse axis A P X P B, 

 Ap X p B, made b) these ordinates re- 

 spectively ; which holds equally true of 



