150 Proceedings of the Royal Society of Edinburgh. [Sess. 
Page 89. 
The Cubic Duplicatrix of G. de Longchamps. 
A is a fixed point, P any point on the ^/-axis. PQ ± r AP meets the cc-axis 
in Q. If QR is drawn parallel to the y-axis and cuts AP in R, R traces the 
curve in question. 
Page 90. 
The Parabolic Leaf of De Longchamps, 1890. 
A is a fixed point, P any point on the y- axis. PQRS is a rectangle, Q 
being on OX, R on OY, and S on AP between A and P. The locus of S is 
the parabolic leaf. 
Page 223. 
Given a circle of centre 0 and radius R, the locus of the vertices of the 
parabolas which touch the circle and have a fixed point on the circumference 
as focus is a curve whose equation is given by Barisien (. Intermediate des 
Math., 1896), which Retali ( J . de Math. Spec., 1897) observed to be the pedal 
of the cardioid, when the pole is at the cusp. 
Page 498. 
The cardioid as a special epicycloid is ascribed to Cramer, and not to 
Maclaurin. 
These extracts may serve to show the importance of Maclaurin’s methods 
in the invention of curves. 
The Geometria Organica is, in fact, remarkable for the great number and 
variety of the curves invented by the young Maclaurin, and had he never 
written another page of mathematics, Maclaurin’s name would have been 
entitled to a conspicuous place in the annals of mathematicians. 
If I have succeeded in pointing this out in the foregoing summary of 
his work, my object in writing it has been attained. 
{Issued separately October 20, 1916.) 
