MaricJiesier Memoirs, Vol. lix. (191 5), No. 13. 19 



is the intersection of a radius vector OR and an ordinate 

 NP, each of which lines moves at a uniform rate such 

 that N describes the diameter BC in the same time as R 

 describes the semi-circumference BC. In other words, 

 BN is the same fraction of the radius OB as the arc BR 

 is of the quadrant BD. 



If RS be drawn parallel to OB to meet the ordinate 

 NP at S, the locus of 5 is a sine (or cosine) curve. This 

 curve was used for trisection purposes by Dinostratus, a 

 disciple of Plato (see Pappus), and by Tschirnhausen 

 (165 1 -1708), to whom Saxony owed its porcelain 

 manufactory. 



The Spiral of Archimedes is also shown in Fig. 12. 

 It is the locus of the point X on the radius vector OR 

 such that OX increases uniformly as the angle BOR 

 increases uniformly. In the particular curve shown, OX 

 is equal to BN. 



The use of these three curves to obtain an angle 

 BOQ equal to one-third of the angle AOB will be 

 obvious from the figure. BM is made equal to one-third 

 of BN, and OZ or Y equal to one-third of OX. 



It is interesting to notice the connection of the 

 Quadratrix curve with the quadrature of the circle. 



Let the angle DOR be equal to radians, and let 

 OP = r. Also let OD = a and OE = b. 



Then 



r sin 0= 0N= OB x. — -„- =« • 



quadrant Db 7172 



Therefore 



At the point I) 



Therefore 



a 6 



r — 



sin tf 



7r/2 sin 6 

 = 1 and r = b. 



6 = JL 



77 / 2 



