The Division of Angles and Arcs of Circles. 53 



triangle. The pentagon is derived from an isosceles triangle, 

 the angles at the base of which are double the angle at the 

 vertex. That triangle contains, by drawing a line across it 

 bisecting one of the base angles, two isosceles triangles, the 

 smaller of which is similar to the whole ; it is composed of 

 them in fact. The triangle which the instrument on the table 

 is constructed to give, that is an isosceles triangle, each of the 

 base angles of which is treble the vertical angle, consists also of 

 two isosceles triangles. But, unlike the triangle of the pentagon, 

 one of the triangles of which is similar to itself, the heptagonal 

 4:riangle consists of two isosceles triangles, not similar to one 

 another, and neither of them similar to the greater triangle. 

 The nonagonal triangle consists of three isocles triangles, one 

 of them similar to it. The undecagonal triangle consists 

 of three isoceles triangles, part of one of whichs overlaps 

 the another ; trimdecagonal triangle likewise. The non- 

 adecagonal triangle consists of four, one of them overlapping 

 another. The eistrincontial triangle, that the base angles of 

 which are 15 times the vertical, consists of four exactly ; and 

 all of the form the base angles of which are 2"— i times the 

 vertical, consist of sets of isocles triangles without any over- 

 lapping. It is to this that we owe the power of forming by a 

 system of linkage, the idea of which is indicated by the model 

 on the table, triangles of the order, angles at base n times angle 

 at vertex, and by their aid we are accordingly able to divide 

 the circle into 2«+ 1 equal arcs. It is worth while for me to 

 mention that when we draw these triangles with the lines 

 dividing one of the base angles meeting the opposite side, the 

 analogies of the sides, the dividing lines, and the segments of 

 the side to which they are drawn are most interesting and 

 extremely numerous. 



Mr. Grkenhill. — I did not know I was going to be called 

 upon to make any remarks, but merely to express the thanks 

 of the Council to Mr. Murphy for his very curious instruments. 

 I think this building contains some extraordinary examples of 

 skill I but I venture to say that there are none more extra- 



