Draw Pp and Qq touching both catenaries, Pp and Qq will intersect at T, a point in the directrix; for since any catenary with its directrix is a similar figure to any other catenary with its directrix, if the directrix of the one coincides with that of the other the centre of similitude must lie on the common directrix.
He is shown the " holy church " under the similitude of a tower in building, and the great and final tribulation (already alluded to as near at hand) under that of a devouring beast, which yet is innocuous to undoubting faith.
Hence T, the point of intersection of Pp and Qq, must be the centre of similitude and must be on the common directrix.
That returns from the bottomless pit, "that was, and is not, and yet is"; the head "as it were wounded to death" that lives again; the gruesome similitude of the Lamb that was slain, and his adversary in the final struggle.
- The " centres of similitude " of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the " external centre," the transverse tangents to the " internal centre."