264 



Hence 



(sec0 + tan 6) n = sec(0-L0 --8 ---- n) + tan (0-^6-^ 0-^-0 ---- to ri). 

 Change sec0 into cos0, tan0 into VH~i S i n 0, and - 1 - into + , then 

 we shall have 



(cos0+\/ 1 sin0) n =cosw^ + V 1 sinrc^. 



Hence De Moivre's theorem, which represents an imaginary pro- 

 perty of the circle, has its counterpart in a real property of the para- 

 bola. 



The hase of the Neperian system e is that particular radius vector 

 drawn to the Logocyclic which gives the difference between the cor- 

 responding parabolic arc and iisprotangent equal to the focal vertical 

 distance of the parabola. Let e be the angle which this radius vector 

 makes with the axis. Then 



e=sece + tan e. 



e=2718281828. 



Hence also as the angles 0, + 0, + + 0, ..... nd give circular 

 arcs which increase in arithmetical progression, the angles 0, 0-^0, 

 0- J -0- J -0, &c. give parabolic arcs, whose excesses over their protan- 

 gents increase in arithmetical progression. 



If the lines (sec0 + tan0), (sec + tan 0) 2 , (sec + tan0) 3 , &c. 

 were drawn, making equal angles with each other, and therefore 

 multiples of with the axis, instead of the angles 0, 0-^-0, 0-^-0 --0, 

 the locus of the extremities would be the logarithmic spiral instead 

 of the Logocyclic curve. 



The spiral of Archimedes may also be used as a means of exhi- 

 biting logarithms or parabolic arcs. For the equation of the 

 spiral being r=0, the arc of the spiral is given by the equation 



=a I jj 



Let ad = a tan 0, then making the necessary substitutions, 

 - ; but this is the expression for an arc of a parabola 



measured from the vertex to a point whose ordinate is 2 tan (p=2r, 

 that is to twice the radius vector of the spiral. 



Hence, if a line be drawn along the vertical tangent of a parabola 

 equal to twice the radius vector of the spiral, and a line be drawn 

 parallel to the base, it will determine the parabolic arc correspond- 

 ing to the radius vector of the spiral. 



It is not difficult to construct a trammel which shall describe the 



