Properties of the limagon may be deduced from its mechanical construction; thus the length of a focal chord is constant and the normals at the extremities of a focal chord intersect on a fixed circle.
Apollonius' genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.
Now, if one of these latter children of the second brother marries a cousin - a child of the first brother, - their offspring, if large enough, will consist of some pure normals N, impure normals N(A), and of albinoes A.
Leonhard Euler, in his paper on curvature in the Berlin Memoirs for 1760, had considered, not the normals of the surface, but the normals of the plane sections through a particular normal, so that the question of the intersection of successive normals of the surface had never presented itself to him.
.and A, B, C, of the two funiculars draw normals to the plane of the diagram, to meet w and w respectively.