52 The Division of Angles and Arcs of Circles. 



The idea of the instrument is based on this difference : the 

 pointing radii are worked at different speeds in such way that 

 when they have brought the end of the bar that starts at 30® 

 at its proper speed and the end of the bar that starts at 45* 

 at its proper and greater speed to exactly opposite points on the 

 diameter, the fraction required is marked. Like as in the tri- 

 sector, I ascertain the exact opposite points by the parallelism 

 to the diameter by the use of a spring face ; but the machine is 

 worked by screws and gearing, and this method, though 

 theoretically exact, is subject to mechanical errors, which deprive 

 its actual results of anything like the correctness that I have 

 succeeded in obtaining with the trisector. For example, to find 

 one-fifth part of angle 45°, or any other angle, set the radius 

 pointing to the angle, and set the other radius at 30°. Five- 

 sixths of the angle proposed to be divided, that is, itself minus 

 its sixth part, is the distance this radius must traverse ; and 

 30°, minus five-sixths of the proposed angle, is the distance 

 that the other radius must traverse. Place pinions in the axles 

 that respectively work the radii, such as will give these speeds. 

 (Here the lecturer went into some mechanical details of the 

 instrument, and recommended that in no event should the 

 screws ever be taken out.) 



In this plan of dividing angles we divide the arc rather than 

 the angle. But in the rough model of another instrument, which 

 I now produce, there is no circle employed. I he model is rather 

 rough, having been made in some haste last week by a cabinet- 

 maker who has some knowledge of mathematics. It is not for 

 the division of angles in general, it is only for the division of 

 the circumference of the circle into seven equal arcs ; and other, 

 instruments, based on the same method as it is, could be con- 

 si ructed for the division of the circle into any odd number of 

 equal arcs. Some of course are easier to make and simpler than 

 others, which must have far more parts or bars than this. We 

 can draw with theoretical correctness a square, an equilateral 

 triangle, and a pentagon, with their integral multiples, regard- 

 ing, as we may, the hexagon as a multiple of the equilateral 



