If the circle be reduced to a point, as in Fig. 5, the resulting locus is a right line perpendicular to and bisecting D O. If on the other hand the diameter of the given circle be infinite, the circumference, as in Fig. 6, becomes a right line perpendicular to the axis at F, and the curve satisfies the familiar definition of the parabola, D E being equal to E H, D P equal to P L, and so on.

In Fig. 7, as in Fig. 1, DT is tangent at T to the given circle whose center is O, and at t to the circle about C whose diameter is AB, the major axis.  Since DTO is a right angle, T lies upon the circumference of the circle whose center is C, and diameter DO; this circle cuts the asymptote SCS at M and N. The semi-conjugate axis is a mean proportional between D A and AO; now drawing TM and TN, it is seen that Tt is that mean proportional; and a circle described about C with that radius will be tangent to TO.  DT, then, is the radius of the circle to be described about the focus of the conjugate hyperbola for its construction according to the enunciation first given:  and we observe that DT and TO are supplementary chords in the circle about C through D and O. The conjugate foci must therefore lie upon this circumference, at D’ and O’; and since D’O’ is perpendicular to DO, D’T will be perpendicular and T’O’ will be parallel to SCS.

[Illustration:  FIG 7.]

Now as TO increases, T’O’ will diminish, until, when TO equals DO, T’O’ will vanish and with it Ct’; and at this crisis, the case is the same as in Fig. 4; but the conjugate hyperbola logically reduces to two right lines, extending from C to infinity on the right and left.  As indeed it should from the familiar construction, since the distances from D’ and O’ to any point on the horizontal axis being equal, their difference is constant and equal to zero.

It appears, then, that a conic section may be defined as the locus of a point which is equally distant from a given point and from the circumference of a given circle.  Boscovich defines it as the locus of a point so moving that its distances from a given point and from a given right line shall have a constant ratio.

The latter definition involves the conceptions of a rectilinear directrix, and a varying ratio in the cases of the different curves, this ratio being unity for the parabola, less for the ellipse, and greater for the hyperbola.  The former involves the conception of a circular directrix with a ratio equal to unity in all cases; and the two definitions become identical in the construction of the parabola, which is in fact the only curve of which a clear idea is given by either of them.  That of Boscovich has been given a prominence far in excess of its merits, being made the foundation for the discussion of these important curves, and this in a textbook whose preface contains the following true and emphatic statement, viz.: 

    “The abstract nature of a ratio, and the fact that it is a
    compound concept, peculiarly unfit it for elementary
    purposes.”