Painting - <I>Cross-Ratio in a Conic (Poncelet)</I>
date: label: date made
content: 1966
name: label: referenced
content: Poncelet, Jean-Victor
label: painter
content: Johnson, Crockett
notes: label: Description
content: 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.
label: Description
content: 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 <I>The World of Mathematics</I>, 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:
label: Description
content: (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 <I>Pencil of Ratios (Monge) </I>).
label: Description
content: 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.
label: Description
content: 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).
label: Description
content: Reference: This painting is based on a figure in James R. Newman, <I>The World of Mathematics</I> (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.
label: Location
content: Currently not on view
setName: label: See more items in
content: Medicine and Science: Mathematics
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content: Science & Mathematics
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content: Crockett Johnson
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content: Art
creditLine: label: Credit Line
content: Ruth Krauss in memory of Crockett Johnson
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: 1979.1093.15
label: catalog number
content: 1979.1093.15
label: accession number
content: 1979.1093
objectType: label: Object Name
content: painting
physicalDescription: label: Physical Description
content: masonite (substrate material)
label: Physical Description
content: wood (frame material)
label: Physical Description
content: metal (frame material)
label: Measurements
content: overall: 84.5 cm x 124 cm x 3.8 cm; 33 1/4 in x 48 13/16 in x 1 1/2 in