l894-] BAKER — CIRCULAR INVERSION. 257 



and the traced curve being circles, in which case the inverse curve is 

 also a circle, as is shown by substituting in the polar equation of the 

 circle 



j/ — 2fja cos d-\-ci' — r' = o, 



— for p, since oR=k", o being the radius vector of the circle to be 



inverted, and Ji the radius vector of the curve of inversion. 

 This substitution gives 



Ji'—2R ~\, cos ft -„^— .=0, 



a' — ;■"• a' — /' 



which is evidently the equation of a circle also. 



This will be better understood after a glance at fig. i, (plate 12) 

 where P is the centre of the circle of inversion, T the tracing point, 

 and / the inverse of the point T, r being the radius of the circle of 

 inversion. Here FT-FI = r". The curves traced hy F and / are 

 also shown, both being circles. Fig. 2 shows the effect of 

 enlarging the radius of the circle traced by T, the dexter vertex of 

 the horizontal diameter of 7^ being fixed. You will notice that as the 

 circle T enlarges, the circle /also enlarges, only much more rapidly, 

 and finally its arc becomes a straight line, and then begins to curve 

 the other way. / is a straight line when 7" passes through the centre 

 of inversion. This is shown by making r^a, in which case the 

 equation above becomes 



F cos ft = constant, 



the equation of a straight line. 



Now if we can only harness F, I and T together by some 

 mechanical contrivance, so that the product FJF T shoXl be constant, 

 it will only be necessary to make T move in a circle through F to get 

 / to trace a straight line. 



The importance of this will be appreciated when I state that 

 previous to the year 1864 there was no known method of tracing a 

 straight line primarily by the continuous sweep of a tracing point. 

 By primarily I mean without first having some other straight line as 

 a guide. I do not mean to say that there were no straight lines 

 previous to 1&64, but they were drawn by means of a straight edge 

 which had been previously constructed by trial, taking off " here a 

 little, there a little," until finally it satisfied the eye of its maker. 

 You will appreciate the significance of this if you try to construct a 

 circular ruler by cutting a circle as near as you can and then 



