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.