263 



Let a series of radii vectores be drawn from the point F to the 

 Logocyclic curve in geometrical progression, and let them be 

 (sec0 + tan0), (sec0 + tan0) 2 , (sec0-f-tan0)3 ____ (sec0 + tan0) w , 



meeting the vertical tangent to the parabola in the points T /} T ;/ , 

 T /y/ ---- T M , and let the tangents drawn from the points T /} T /p &c. 

 touch the parabola in the points Q /} Q y ,, Q^, ..... Q ; let the dif- 

 ference between the first parabolic arc and its protangent be , then 

 we shall have OQ 



Or while numbers increase in geometrical progression, their loga- 

 rithms increase in arithmetical progression. 



As every number whose logarithm is to be exhibited must be put 

 under the form sec + tan 0, which is of the form 1 + x, since the 

 limiting value of sec is 1, we discover the reason why in developing 

 the logarithm of a number the number itself must be put under the 

 form l+#, or some derivative from it, and not simply under that 

 of*. 



If we equate sec -f tan with 1 + #, we shall find 



x= 





 2 tan o 



Let u tan 



^ 



then n= sec + tan 0= 1 +x=^, which is another familiar form 



1 u 



under which a number is put, whose logarithm is to be developed in 

 a series. 



Let be the angle which the line (sec + tan 0) makes with the 

 axis, and let 0, 6 lt 6 ni be the angles which the lines (sec + tan 0) 2 , 

 (sec + tan 0) 3 . . . . (sec + tan 0) n make with the same axes, then 



=0 

 0, =0-^-0 



to n terms. 



The definitions of the symbols - 1 - , -r , which I call logarithmic or 

 parabolic plus and minus, are given in the paper referred to. 



