66 



Colorado College Studies 



either of these circles, we will call the origin of arcs, for that 

 circle. 



Suppose now, that a polygon of 2" sides, say a quadrilateral, 

 is inscribed in the circle of the amplitude, having one vertex at 

 the origin; and at the same time a regular polygon of the same 

 number of sides — a square — is inscribed in the circle of the 

 argument, this also having a vertex at the origin. All polygons 

 inscribed in this circle are to be regular. In order to avoid con- 

 fusing the diagram, the vertices only of either set of polygons 



are noted in the figure. 

 (Fig. 4.) The trapezium 

 inscribed in the circle of 

 the amplitude is Ah^ Fho, 

 and the square in the 

 circle of the argument is 

 XB.EB,. Next let an 

 ^ octagon be inscribed in 

 each circle. A new ver- 

 tex is now inserted be- 

 tween each of the former 

 vertices and the succeed- 

 ing. In the circle of the 

 amplitude these new ver- 

 tices are 5/, 6o', 63', h^ ; and the corresponding new vertices in 

 the other circle are B^', B/, Bi, B^. Next, the polygon of 

 sixteen sides may be constructed, adding the new vertices 

 6/', 60",... 6s" in one circle and B,", B.:',...B^' in the other. 

 And so the process may be continued as far as desired, with 

 the result of fixing — in a perfectly definite way in each case — 

 as many points as we please on one circle, and the correspond- 

 ing points on the other. 



We are now able to define, for any point on one circle, its 

 corresponding point on the other. Let a point -be given at 

 pleasure on the circle of the argument, and also let an arc of 

 the circle of the amplitude be stated in magnitude but not ia 



