1915-16.] The “ Geometria Organica ” of Colin Maclaurin. 137 
It is obtained from the circle in the same way as the rational circular 
cubics are obtained from the straight line (and is likewise a rational or 
unicursal curve). It appears to be the simplest of the curves of the fourth 
order, the conchoid excepted, just as the circular cubics with a double point 
are the simplest of the third order. 
[Teixeira has shown that just as the rational circular cubic is the 
cissoidal of a circle and a straight line, so these bicircular quartics are the 
cissoidais of a circle and a circle.] 
§ 50. Prop. XI. 
Pedals of the Conic Sections. 
(I.) For the Parabola. 
The pedal of the focus is the tangent at the vertex. Hence by Cor. 8 
of Prop. IX the pedal of any other point O' is the rational circular cubic 
already discussed in Lemma II of Part I. The curve has a double point at 
the pole with real, coincident, or imaginary tangents according as the pole 
is outside, on, or inside the parabola. It has a line of symmetry when the 
pole is on the axis of the parabola, and is the cissoid of Diodes when the 
pole is the vertex of the parabola. 
(II.) For the Ellipse. 
The pedal of the focus is the major auxiliary circle (Maclaurin’s 
theorem). Hence the pedal of any other point is the bicircular quartic of 
Cor. 10 of Prop. X. 
(III.) For the Hyperbola. 
The pedal of the focus is again a circle, and we have a conclusion 
similar to that of (II). 
Cor. If O is a fixed point, P any point on a circle, and OPT a constant 
angle, then PT envelops a conic section. 
§ 51. Prop. XII. 
When a curve rolls on a congruent curve , corresponding points being 
points of contact , the roulette of any carried point can be obtained easily 
as a pedal. 
The usual proof is given. 
Cor. 1. The curves described by this method coincide with the epi- 
cycloids of Nicole generated by a curve rolling on a congruent curve. 
Cor. 2. Thus the epicycloids generated by a parabola rolling upon a 
congruent parabola are (1) a straight line for the focus, (2) a cissoid of 
Diodes for the vertex, (3) a rational circular cubic for any other point. 
