A C—C D ; or that the lines F D,f D, be each equal to A C : then, having fixed two pins in the points Fj, which are called the foci of the ellipsis, take a thread equalin length to the transverse axis A B; and fastening its two ends, one to the pin F, and the other to f, 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 set out, keeping—the thread always ex tended so as to form the triangle F Mf, it will describe an ellipsis, whose axes are A B, D E.
The greater axis, A B, passing through the two foci Ff, 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, be 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 laths rectum, or parameter, being a third proportional to A B, 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 under the absciss A P, and the line P N will be equal to the square of the ordinate P M. Hence drawing N 0 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 0 F, or N 0 and 0 F; on account of which deficiency, Appollonius first gave this curve the name of an ellipsis, from EAJtEivretv, 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 AP xP B, Ap x p B, made by these ordinates re spectively ; which holds equally true of