Painting - Geometric Mean (Archytas)

This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as diameter. Erect a perpendicular at the point where the line segments meet (d in the figure), and consider this to be the altitude of a right triangle inscribed in the semicircle. By similar triangles, the length of the perpendicular of a triangle inscribed in a semicircle is the geometric mean of the two lengths into which it divides the diameter of the circle. Hence the length of d is the mean of the segments e and f.

This painting an orange-red background, and shows a triangle inscribed in an orange semicircle. The perpendicular from the right angle of the triangle divides the triangle into triangles similar to it, painted in black and white.

The painting, and the attribution of the theorem to Archytas, are based on a passage from Evans G. Valens, The Number of Things: Pythagoras, Geometry and Humming Strings (1964), p. 118. The figure on this page of this book from Crockett Johnson's library is annotated.

This oil painting on masonite is #65 in the series. It is inscribed on the back: GEOMETRIC MEAN (ARCHYTAS) (/) Crockett Johnson 1968. It has a wooden frame.

Currently not on view