Cavalieri’s parabolograph

A beau­tiful tool for drawing a parabola was invented by an Italian math­e­mati­cian Bonaven­tura Cava­lieri (it. Bonaven­tura Francesco Cava­lieri, lat. Cava­lerius, , 1598—1657) back in the XVII century.

The Cava­lieri’s parabolo­graph consists of three parts: a ruler and two rigid right angles with slots on their sides.

One right angle slides on a ruler, which is fixed rela­tive to a sheet, in such a way that its hori­zontal side constantly touches the ruler.

The second right angle forms a right triangle with the ruler. The right angle vertex is equipped with a stylus and slides along the slot of the first right angle vertical side. Two other sides of the second right angle slide their slots on the guide pins, one of which is fixed to the ruler, and the other — to the hori­zontal side of the movable angle.

When the Cava­lieri’s parabolo­graph is moving, the stylus is drawing a parabola. The distance from the pin to the vertex of the right angle, which is adja­cent to the ruler, is a para­meter of the parabola. The curve changes then the pin is moved, but it remains being the parabola.

The proof that the curve being drawn is the parabola is based on the theorem from the modern school math­e­matics course. Consider the triangle formed by the sides of the second right angle and the ruler. In this triangle, the square its alti­tude length (the distance from the stylus to the ruler), dropped on its hypotenuse is equal to a product of catheti projec­tions on the hypotenuse. The projec­tion of the “right” сathetus is constant by the parabolo­graph design and it is the para­meter that defines the parabola.