Painting - Cross Ratio in an Ellipse (Poncelet)

From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle. Nineteenth-century mathematicians were much interested in the properties of conics that were preserved under projection. They knew from the work of the ancient mathematician Pappus that the cross ratio of line segments created by two straight lines cutting the same pencil of lines was a constant (a pencil of lines is a set of lines emanating from a common point). In the drawing, which is Figure 5 from an article by Morris Kline in James R. Newman's The World of Mathematics (1956), if line segment l’ crosses lines emanating from the point O at points A’, B’, C’, and D’; and line segment l croses the same lines at points A, B, C, and D, the cross ratio:

(A’C’/C’B’) / (A’D’/D’B’) = (AC/BC) / (AD/DB), in other words it is independent of the cutting line. (see Crockett Johnson's painting Pencil of Ratios (Monge)).

The French mathematician Michel Chasles introduced a related result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio. This opened up an entirely different way of describing conic sections. Crockett Johnson associated this painting with both Chasles and another French advocate of projective geometry, Victor Poncelet.

The painting, in oil or acrylic on masonite, is #69 in the series. It has a dark gray or blue background and a black wooden frame. It shows a white ellipse, two points on the ellipse (on the left side of the painting), and two pencils of lines that produce the same cross ratio. The painting is not signed. It is inscribed on the back, in Crockett Johnson’s hand: CROSS RATIO IN AN ELLIPSE (PONCELET) (/) Crockett Johnson 1968. Compare #21 (1979.1093.15).

Reference: Morris Kline in James R. Newman, The World of Mathematics (1956), p. 634. This volume was in Crockett Johnson's library. The figure on this page is annotated.

Currently not on view