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.
- 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."
The circle on the line joining the internal and external centres of similitude as diameter is named the " circle of similitude."
With a system of three circles it is readily seen that there are six centres of similitude, viz.
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.
Hence T, the point of intersection of Pp and Qq, must be the centre of similitude and must be on the common directrix.
Again, Moses differs from all other prophets in that Yahweh speaks to him face to face, and he sees the similitude of Yahweh.
The theory of centres of similitude and coaxal circles affords elegant demonstrations of the famous problem: To describe a circle to touch three given circles.
John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his Sequel to Euclid; an analytical solution by Gergonne is given in Salmon's Conic Sections.
In this paper Vieta made use of the centre of similitude of two circles.