Agnesi did not discover the curve that bears her name. But it was through her book that the curve became known, because she used it as an example of how to pass from a geometric construction to an equation, which was exactly what Descartes’s new method promised. The construction Start with a circle of diameter $OM = a$, with $O$ at the bottom and $M$ at the top, and draw the line tangent to the circle at $M$. Now:
In the treatise On Spirals, also dedicated to Dositheus, Archimedes studies a new curve, defined by a motion. If a straight line in a plane revolves at a uniform rate about one of its extremities, which remains fixed, and returns to the position from which it started, and if, at the same time, a point moves at a uniform rate along the line, starting from the fixed extremity, the point will describe a spiral in the plane. — Archimedes, On Spirals, definition