262 



To represent Numbers and their Logarithms by the Logocyclic 

 Curve and its Conjugate Parabola. 



XXVII. A parabola whose focal vertical distance is a or 1 being 

 drawn, and also its Logocyclic curve, let a radius vector be drawn 

 to the latter equal to the given number n. Then w=sec0+tan0. 



Let this line meet the vertical tangent in T, the parabolic arc 

 OQ QT is the logarithm of n. 



It is clear that the infinite branch of the curve from + GO to O will 

 give radii vectores of every magnitude from oo to 1, and parabolic 

 arcs from oc to ; hence, while the numbers range from oo to 1, the 

 parabolic arcs range from oo to 0. When the number lies between 

 1 and 0, the radius vector representing it is drawn below the axis ; its 

 extremity will be found on the loop, and the corresponding arc of the 

 parabola will be negative, hence the logarithm of a positive number is 

 equal to the logarithm of its reciprocal, with the sign changed ; for 

 the magnitude of the parabolic arc depends on 0, and is the same 

 in sec + tan 0, as in its reciprocal sec tan 0. 



Hence while the infinite branch of the Logocyclic curve from + x> 

 through R, O, p, to F, may by its radii vectores represent all positive 

 numbers from +00 to + 0, the two infinite branches of the parabola 

 will be used in representing the logarithms of positive numbers from 

 -f oo to + ; that is, the upper or positive branch of the parabola will 

 be "used up" in representing the logarithms of positive numbers from 

 + oo to + 1, and the lower or negative branch of the parabola in re- 

 presenting the logarithms of positive fractional numbers from -j- 1 

 to +0. Hence there is no construction by which we can represent 

 negative numbers or their logarithms, therefore such numbers can 

 have no logarithms. 



Let radii vectores be drawn from F to the Logocyclic curve equal 

 to e, e 2 , e 3 , e 4 . . . . e n , then these lines will meet the tangent to 



the vertex of the parabola in the points T, T p T^ T n ; and 



tangents being drawn from these points, touching the parabola 

 in Q, Q,, Q /<} Q //p Q n , the logarithms of these numbers will be 

 OQ-QT=1, OQ,-Q,T,= 2, OQ M -Q W T W =3, . . . 



OQ w -Q n T re =0 + 1); 

 hence the logarithms of e t e 2 e 3 , e n are 1, 2, 3, ... n. l 



In like manner we should find the logs of e, e&> e* ei . . . . e 11 to be 



111 i 



'2'3'4 



