The Division of Angles and Arcs of Circles. 51 



by a great deal, and I am obliged to verify the parallelism by 

 the unsatisfactory way of mere observation. This defect is still 

 greater practically in the other model, as I shall show when I 

 arrive at it presently. But I need not point out that these 

 mechanical imperfections do not affect the theoretical perfection 

 of the instruments. The only angles which can be marked on 

 the circle with absolute theoretic accuracy, are 45°, 36°, and 

 30°, with their halves, quarters, &c., and their integral multiples. 

 45** and 30° are derived from the circle itself, 36° from the 

 triangle by which the pentagon is arrived at. There is no 

 geometrical way of pointing off 20^, 10°, &c. When I arranged 

 in my mind the details of my instrument for trisecting, I had 

 done so with the hope that to be able to point off 20° would 

 be of assistance in that extremely nice work, the division 

 of the circles of telescopes. I showed my method to Sir 

 Howard Grubb, who, while recommending me to patent the 

 instrument, at the same time assured me that the present 

 empirical division of the circle by the use of micrometers is so 

 practically accurate that it would be impossible for any instru- 

 ment, however theoretically exact, to enable that accuracy to be 

 surpassed. He asked me how many screws I intended to 

 connect the guilding bars with, and advised me to have five, 

 not three, as I had thought of, and to have the screws of 

 considerable fineness. 100 per inch — a screw of which pitch he 

 showed me — would not be too fine. In working the instrument 

 at first I found the strain greater than I had expected. Thinking 

 that this might be owing to the pressure of one of the bars upon 

 the outside of the groove of the circle — or perhaps I should say 

 the pressure of the pivot or swivel on the groove — I added a 

 carrying radius to keep the swivel always in the exact middle of 

 the groove without pressure, since when I find the machine 

 easier to work, and no increasing resistance, as there had been 

 when the bar began to near the trisecting point. 



Respecting the machine for dividing angles into fractional 

 parts, it will be observed that the two sides of the semi-circle 

 are set differently. I have one set at 45°, the other at 30°. 



