Painting - Pencil of Ratios (Monge)

The history of projective geometry begins with the work of the French mathematician Gerard Desargues (1591–1661). During his lifetime his work was well known in some mathematical circles, but after his death, his contributions to the field were largely forgotten. When Gaspard Monge (1746–1818) and his student, Jean-Victor Poncelet (1788–1867) began their studies of projective geometry, they were largely unaware of the work of Desargues. This may be why Crockett Johnson included Monge's name as opposed to Desargues' in this painting's title.

One of the fundamental concepts of projective geometry, which was touched upon, but not fully understood, by the Greeks, is that of a cross-ratio, or "ratio of ratios." It is the topic of Johnson's painting. If points A, B, C, and D on line l are projected from point O, and if the line l’ crosses the four projected line segments, then the ratio of ratios (A’C’/C’B’)/(A’D’/ D’B’) of the corresponding points A’,B’,C’, and D’ is the same as the ratio of ratios (AC/CB)/(AD/DB). Thus, a cross-ratio is a projective invariant for all line segments l’.

The artist may have received inspiration for this painting from his copy of James R. Newman's The World of Mathematics (1956), p. 632. The figure is found there in an article by Morris Kilne entitled "Projective Geometry." This figure is not annotated, and the painting flips Kline's image.

Crockett Johnson chose purple, white, black, and brown to color this work. He executed the projection in three tints of purple and one shade of white. The background, which is divided by line l’, was executed in black and brown.

Pencil of Ratios, an oil painting on masonite, is #18 in the series. It was completed in 1966 and is signed: CJ66. It is signed on the back: Crockett Johnson 1966 (/) PENCIL OF RATIOS (MONGE). The painting is unframed.

Currently not on view