Trisectrix of Maclaurin

The curve pictured to the left is given by the equation: Point O is the node of the curve, located at the origin. Point R is the other point of intersection with the x-axis.  The distance OR is 3a units.  Point P is located along the x-axis such that the distance OP is 2a units.   Using only Euclidean tools (a compass and a straightedge), the trisection of an angle is an impossibility.  But if one adds another tool in the form of a plane curve, then trisection may become a possibility.  The Trisectrix of Maclaurin is one of those curves which make angle trisections possible.

To trisect an angle, place a copy of the angle in such a way that the vertex is at point P, and one side of the angle lies along the positive x-axis.  Then the other side of the angle will intersect the loop of the trisectrix at point A above the x-axis (or point B below the x-axis).  In other words, the original angle is ∠AOR (or ∠BOR).  Draw line segment OA (or OB).  Then:

Colin Maclaurin (1698-1746) became a professor of mathematics at the University of Aberdeen in 1717, and at the University of Edinburgh in 1725.  Among his many areas of study were higher plane curves.