THE 
BRITISH ENCYCLOPEDIA. 
ELLIPSIS. 
ELLIPSIS, in geometry, a curve line re- 
turning into itself, and produced from 
Ihe section of a cone by a plane cutting 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 conju- 
axes AB, ED, (Plate V. Miscell. fig. 1.) 
are given, is this : first take the points F ,f, 
in tiie transverse axis A B, so that tlie dis- 
tances C F, C/j from the centre C, be each 
equal to y/ A C — C D ; or, that the lines 
FD,/D, be each equal to AC; then, hav- 
ing fixed two pins in the points F, which 
are called 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 ano- 
ther pin M stretch the thread tight ; then 
if this pin M be moved round till it returns 
to the place from whence it first set out, 
keeping the thread always extended so as 
to form the triangle F M R 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 eacli 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 terminated by the 
curve of the ellipsis on each side, is called 
a diameter ; and two diameters, which na- 
turally bisect all the parallels to each other, 
bounded by the ellipsis, are called conju- 
gate diameters. Any right line, not pass- 
ing through the centre, but terminated by 
called the ordinate, dr ordinate-applicate’ 
to that diameter ; and a third proportional 
to two conjugate diameters; is called the la- 
tus 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, be any two conjugate diameters of an 
ellipsis (fig. 2, where they are the two 
axes) at the end A, of the diameter A B, 
raise the perpendicular A F, equal to the 
latus rectum, or parameter, being a third 
proportional to AB, E D, and draw the 
right line B F ; then if any point P be 
taken in B A, and an ordinate P M be 
drawn, cutting B F in N, the rectangle un- 
der the absciss A P, and the line P N will 
be equal to the square of the ordinate P M. 
Hence drawing N O parallel to AB, it ap- 
pears that this rectangle, or the square of 
the ordinate, is less than that under the ab- 
sciss A P, and the parameter A F, by the 
rectangle under A P and O F, or N O and 
O F ; on account of which deficiency, Apol- 
lonius first gave this curve the name of an 
ellipsis, from A'kMim, to be deficient. 
In every ellipsis, as A E B D, (fig. 2), the 
squares of the semi-ordinates MP, nip, are 
as the rectangles under the segments of the 
transverse axis AP x P B, A p x p B, made 
by these ordinates respectively ; which holds 
equally true of the circle, where the squares 
of the ordinates are equal to such rectan- 
gles, as being mean proportionals between 
the segments of the diameter. In the same 
manner, the ordinates to any diameter 
whatever, are as the rectangles under the 
segments of that diameter, 
the ellipsis, and bisected by a diameter, is As to the other principal properties of 
VOL. III. B 
