1915-16.] The “ Geometria Organica” of Colin Maclaurin. 135 
Hence 
(x 2 + y 2 - dx) 2 = r 2 (x 2 + y 2 ) . . . . (3) 
So that the pedal is a quartic curve. 
[The curve is now called the Linton o£ Pascal. When d = r, so that 
O is on the circumference of the given circle, it is called a cardioid.] 
Cor. 1. The origin is a double point on the curve (as are also the 
circular points at infinity, making in all three double points). The 
branches through O are real when 0 is outside the circle, and imaginary 
when O is inside the circle, while O is a cusp if d = r. 
Cor. 2. The lima 9 on touches the circle at A and B, and is a finite 
closed curve. 
Cor. 3. The chords of the quartic that pass through O are of constant 
length = 2 r. 
Cor. 4. Let T' be a point on the circle infinitely near to T, and find 
the corresponding points P' Q', q'. 
Then ATPP'~ AQgg'; so that on integrating we find that the area of 
the lima 9 on between AB and the semi-circle ATB is equal to a quadrant 
of the circle of radius d, and the area ABPA = 7 r(d 2 /4<-\-r 2 /2). 
Cor. 5. The curve can be rectified only when it is a cardioid. In this 
latter case the arc BP of the cardioid = 2BT, i.e. double the chord of the 
corresponding arc of the original circle (proof later), so that the whole 
length of the curve is Sr. 
Cor. 6. The curve is the epicycloid generated by the rolling of a circle 
of radius r/2 upon an equal circle, the centre of the fixed circle being 
midway between 0 and C. 
Let 0, bisect OC : 0 X K = K0 8 = T0 2 = r/2 : TR = TC. 
Let R be carried into position R r , and let R" be the image of R' in 
