259' 



III. O being the vertex of the parabola, in which the two branches 

 of the curve cut at right angles, the angle ROR' is always a right 

 angle. 



IV. Through the points R and R! let normals be drawn to the 

 Logocyclic curve, they will meet the parabola in the same point Q, 

 and the parabolic arc OQ diminished by the protangent QT is the 

 logarithm of the number FR or of its reciprocal FR,. 



Q, the intersection on the parabola of the normals, drawn to the 

 Logocyclic curve at the reciprocal points R, R a may be called the 

 Logarithmic point of the line FR. 



V. The normals RQ and R,Q are equal. This is also a property 

 of the circle. 



VI. At the points R and R u the reciprocal points, let tangents 

 be drawn to the curve and meet in V. The tangents RV and RjV 

 are equal, and they are equally inclined to the chord RR r This is 

 another property of the circle. 



VII. The locus of V, the intersection of tangents drawn to the 

 curve through the reciprocal points, is the Cissoid of Diocles, a curve 

 discovered in the Schools of Alexandria. 



VIII. Since the line VQ is at right angles to the line VO, we 

 have this new property of that old curve, that a line drawn through 

 any point of a Cissoid at right angles to the radius vector drawn to 

 the cusp of the curve envelopes a parabola. 



IX. The directrix of the parabola is the asymptote of the Cissoid, 

 and its cusp is at O the vertex of the parabola. 



X. The sum of the ordinates of the reciprocal points R and R x is 

 equal to the ordinate of the logarithmic point Q on the parabola, 

 and the sum of the distances of the points R and RI from the 

 asymptote DS is constant and equal to 2a. 



XI. The sum of the products of the abscissae and ordinates of 

 the reciprocal points on the curve is constant, or 



XII. The distances of any point Q on a parabola from its focus 

 and directrix are equal. We may generalize this well-known theo- 

 rem, and say the distances are equal of any point on a parabola 

 from its Logocyclic curve, measured along the two normals to this 

 curve drawn through the point Q. 



