The Division of Angles and Arcs of Circles. 55 



circles, in other words, by the use of a ruler and pair of com- 

 passes. For example, the trisection of an angle and the finding 

 two mean proportionals are impossible problems in this sense. 

 These and other like problems can be effected by the descrip- 

 tion of ellipses and by hyperbolas. Of course these are plane 

 curves, but they were studied by the old geometers as the 

 sections of a cone, and were considered by them therefore 

 rather as belonging to solid geometry, and their use was 

 regarded by them as inadmissible in what they called Plane 

 Geometry. It is curious that the great mathematician Gauss 

 proved that though we cannot divide the circumference of a 

 circle into seven equal parts by a rule and compass, we can 

 divide it into seventeen. I would only add that I have listened 

 with great interest to Mr. Murphy's description of his beautiful 

 instrument. 



Mr. Murphy thanked those present for the flattering way 

 in which they had received his communication and expressed 

 the pleasure it had given him to deliver his lecture. 



