Ever since I first became acquainted with Wilkinson's method of establishing the existence of the nine-points circle of a triangle (see Mackay's “Euclid,” Appendix to Bk. IV., Prop. 2, the lettering of which I have followed in the first three sections of this paper), its simple and fundamental character has pleased me. I propose to point out first that this method yields probably the most elementary proof of the concurrence of the perpendiculars from the vertices, and then, after restating the investigation of the nine-points circle, to sketch some generalizations.

The following way of viewing triangles triply in perspective is of interest.

Two triangles ABC and PQR are said to be “in perspective” if the joins of corresponding vertices be concurrent.

Thus if U be the point of concurrency we have the following collineations:—

2. Triangles doubly in perspective are also triply in perspective.

Let the two triangles ABC and PQR be doubly in perspective, the two centres of perspective being U and V.

Not the least interesting portions of the wonderful “Mathematical Collections” of Pappus are those which reproduce parts of the νε⋯σεις, or the two lost books of Apollonius (247–205 B.C.). Pappus (c. 300 A.D.) writes:— “A line is said to verge (using Heath's translation) toward a point if, being produced, it reach the point,” and among other particular cases of the general problem he gives the following as treated by Apollonius:

Problem A: Between two lines, given in position, to place a straight line given in length and verging toward a given point.

A friend, Professor Smith of Sackville, New Brunswick, has called my attention to the fact that the term “Wallace's Line,” as defined in the Century Dictionary, is employed by other scientists than mathematicians. In these days, when increasing attention is paid to the nomenclature of lines and curves, it may be of interest to add here a brief quotation from the dictionary: “Wallace's Line [so named after Alfred R. Wallace, who defined it] in Zoögeography, a line assumed to separate the Indomalayan from the Austromalyan Zoological region or faunal area. It passes between Borneo and Celebro, through the Strait of Macassar. Southward between Bali and Lombok, northeastward between Mindanao and Gilolo. This line divides the shallow waters of the Indomalayan region from the much deeper Austromalayan seas; and the character of the fauna is quite different on the two sides of it.”