Painting - Cross-Ratio in a Conic (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. In Figure 5, which is from an article by Morris Kline in James R. Newman's The World of Mathematics, 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 the Crockett Johnson 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 particular painting with another French advocate of projective geometry, Victor Poncelet.

This oil painting on masonite is #21 in the series. It has a dark gray background and a wood and metal frame. It shows a large black ellipse with two pencils of lines linked to the same four lines of the ellipse. The painting is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 ( /) CROSS-RATIO IN A CONIC (/) (PONCELET). Compare painting #69 (1979.1093.44).

Reference: This painting is based on a figure in James R. Newman, The World of Mathematics (1956), p. 634. This volume was in the Crockett Johnson library. The figure on this page is annotated. For a figure on cross-ratios, see p. 632.

Currently not on view