Construction of Viviani's curve inspired by Villalpando's proportionatrix secunda (as Viviani claims) in a semicircle of diameter AC. Steps to find a point K of the curve from a point F of the semicircle:
- Semicircle of diameter AC (see diagram).
- A point F on the large semicircle is selected and FC drawn.
- FC is drawn.
- FL is perpendicular to AC.
- LK is perpendicular to CF.
- Unite AF.
Once the curve is drawn, it is easy to prove that AC/CF = CF/CL = CL/CK because the figure ALCKF is precisely an Archytas' triangle. This statement is proved in Viviani's Diporto Geometrico, pp.279-280, http://www.e-rara.ch/zut/content/pageview/3800973. The two-dimensional curve described by Knorr is almost this same definition.