Descartes’s Hyperbola Machine
A figure from Descartes’s Geometry (1637). For the modern reader, consider ray AK as the positive y axis and AG as the positive x axis (even though it points to the left). KNL is a rigid triangle whose side KL slides on the y axis. An infinitely long “ruler” is fixed at G on the x axis, and is attached to the moving point L of the triangle. The side KN of the triangle is extended so that it intersects the ruler at C. As the triangle moves on the y axis, point C traces a curve.
This curve is one nappe of a hyperbola. Why? You can read Descartes, or a paper by David Dennis: Rene Descartes' Curve-Drawing Devices: Experiments in the Relations Between Mechanical Motion and Symbolic Language. Mathematics Magazine, 1997, 70(3), 163-174.