The Division of Angles and Arcs of Circles. 49 



that, although the compasses have been known from time 

 immemorial as instruments which draw a circle with absolute 

 theoretical accuracy, there has not been till within less than 

 thirty years an instrument enabling us to draw a straight line 

 with absolute theoretical accuracy. I believe it was in 1867 or 

 1868 that M. Paucellier invented what he calls his cell, which per- 

 forms that operation, and which for the first lime in the history 

 of mechanics produces a perfect parallel motion, deriving in fact 

 the straight line from the motion of two circles of different radii. 



Not to dwell on this, however, and to return to the trisection 

 of angles. While plane geometry proper admits only of those 

 instruments which have for their correlatives the two lines of 

 the second degtee, there are thousands if not millions of problems 

 solvable with the aid of other instruments, with the same 

 theoretic accuracy as the problems of plane geometry and its 

 trigonometrical branch. One of these is the trisection of an 

 angle. Greek geometers — whether before or subsequent to 

 Euclid I do not know, but after him I believe — had constructed 

 a curve, which they called a conchoid, by the aid of which they 

 were successful in trisecting arcs with as absolute accuracy as 

 the imperfection of the drawing of the curve would allow. 

 My instrument does the same. There is a solution of the 

 problem given by Professor Casey in the appendix to his Euclid 

 which, so well as I remember, is virtually the same in slightly 

 different form. These processes divide the arc first, and the 

 angle by the result ; the equation derived from them is the 

 same. Thus, let ^ -5 C be a right-angled triangle, B C or a 

 the hypotenuse, A C ox b the base, and x the segment of b 

 towards C, which subtends two-thirds of the angle at B on that 

 side ; then the equation is x^—Z a^ x-{-2 a^ b=0 ; which is a 

 cubic. Likewise, the value of the sine of three times an angle 

 is 8 sin— 4 sin^. 



In addition to these processes, which are based on making the 

 external segment of a secant (not a trigonometrical but an 

 ordinary secant) meeting a diameter produced, equal to the 

 radius, as on the board, there is another which is considered to 



