Can You Really Derive Conic Formulae from a Cone? - Menaechmus' Constructions

Author(s):
Gary S. Stoudt

Let us consider how Menaechmus constructed the two mean proportionals. [14, pp. 278-283]. Let the given lengths be a and b. Let a straight line be given to serve as the axis, with a point D on the line to serve as the origin. Construct a parabola with the given axis, with vertex at D, and latus rectum a. This is equivalent to constructing a square on ordinate y equal to the rectangle on the latus rectum with side x. Note that this makes y the geometric mean of the latus rectum and the side x. Consider the diagram below to see why.

The height of the rectangle is x, the abscissa.
Click the red X in the lower right of the sketch to clear the trace.

From the above sketch you can convince yourself that this does indeed generate a parabola.

Next, using the same axis and origin as above, construct a hyperbola such that the rectangle contained by the abscissa x and the ordinate y makes an area equal to the area of the rectangle contained by the lengths a and b.

The base of the rectangle at lower left is a; the height is b. Click the red X in the lower right of the sketch to clear the trace. (Download the Geometer's Sketchpad 4.0 file.)

From the above sketch you can convince yourself that this does indeed generate a hyperbola.

Now, what about the mean proportionals of a and b? The abscissa x and ordinate y of the point of intersection of the hyperbola and parabola so generated give the mean proportionals of a and b .