Also the auxiliarly circle is the locus of the feet of the perpendiculars from the foci on any tangent.
Since the area of the triangle FPP' is one half the product of FP into the perpendicular p from P on FP', it follows that if these perpendiculars were equal all round the orbit, the areas described during the infinitesimal time would be smallest at the pericentre and continually increase during the passage of the body to B.
Inside an equilateral triangle, for instance, of height h, - 2Ra/3y/h, (8) where a, 13, y are the perpendiculars on the sides of the triangle.
The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.
From every point of the curve of intersection, perpendiculars are drawn to the axis.