5© The Division of Angles and Arcs of Circles. 



be much more ingenious. If in the hyperbola the transverse 

 axis of which is to the conjugate axis as i : >/ 3, there be drawn 

 lines from the focus, and from the extremity of the transverse 

 axis beyond the centre from the focus, meeting at any point in 

 the hyperbola, then the angle of the triangle so formed, which 

 is at the focus, is double of the angle at the extremity of the 

 axis ; when, by drawing a line from the vertex parallel to the 

 base, that line evidently trisects the external angle. I need not 

 pause to give the demonstration of this, which may be either 

 by the ordinary analysis by ordinates or, more simply, trig- 

 onometrically. The history of this solution of the problem is 

 interesting. I was for the first time informed of the method 

 while I was engaged in the construction of the instrnments on 

 the table ; and told that the author of it was a Danish astron- 

 omer, who published it in the astronomical transactions of 1868. 

 But a short time afterwards I found the solution in a mathe- 

 matical book of the early years of the century, which I found 

 at a bookstall. The interest belonging to this method is chiefly 

 that the angle is trisected without using a circle, and thus the 

 division of the arc follows the division of the angle, instead of 

 preceding it as in the method of any instrument. In the division 

 of angles into other fractional parts, one-seventh, &c., no circle 

 need be used ; but in my instrument (that on the table), for the 

 division of angles into fractional parts, it is the arc that I divide. 

 I now proceed to the demonstration of the proposition on 

 which the working of my trisecting instrument depends. On 

 examining the instrument, it will be perceived that the manner 

 in which the parallelism between the two bars is ascertained is 

 mechanical purely ; and in this model the strength of the parts 

 is not properly proportioned. The leverage, owing chiefly to 

 the fineness of the screws (35 per in.), is so great that it over- 

 comes the resistance which the passing of the pressing point 

 from the end of the spring face that contains the spring to the 

 other end, ought to give to working the machine any longer ; 

 the machine, in fact, ought to lock itself when the angle is 

 trisected. The spring accordingly is plainly not strong enough 



