Villalpando's proportionatrix secunda
Construction of the proportionatrix secunda in a semicircle of diameter AG. Steps to find a point K of the curve from a point F of the semicircle:
- Semicircle of diameter AG and center D(diagram).
- Semicircle of diameter DG.
- A point F is selected on the large semicircle, FG is drawn cutting the small semicircle in H.
- Perpendicular HI from H to AG.
- Semicircle of center I and radius IG.
- K is the intersection of this semicircle with FG.
- L is the intersection of this semicircle with AG. FL is drawn.
Once the curve is drawn, it is easy to prove that, for each K, AG/GF = GF/GL = GL/GK: it is enough to show that AGF, FGL and GKL are similar triangles. This statement is proved in Villalpando's proposition XI (http://www.e-rara.ch/zut/content/pageview/3800973). Villalpando's proportionatrix secunda can be used to find two mean proportionals between lines a,b, with a>b. If we set AG = a and the proportionatrix secunda is met by a circle of center G and radius b, we obtain K, being GK = b. F is obtained prolongating GK and cuttig it with the biggest semicircle; L is obtained tracing a perpendicular line from F to AG. An Archytas' solution is the figure ALGKF (Villalpando's actual use of this curve to find the two mean proportionals between two assigned straight lines is a bit different, still is equivalent to the one just expounded).