Painting - Duality (Pascal-Brianchon)

As a 21-year-old student, the Frenchman Charles Jules Brianchon (1785–1864) discovered that in any hexagon circumscribed about a conic section (such as a circle), the three lines that join opposite diagonals meet in a single point. He also pointed out connections between his result and Pascal's theorem concerning the points of intersection of opposite sides of a hexagon inscribed in a conic section.
In the painting, a hexagon (only the vertices are shown) is inscribed in a circle. Three diagonal lines (edges of the gray and black polygon) are collinear. The line in question is the line joining the points of intersection, white on one side and purple on the other. Crockett Johnson's painting closely resembles a diagram of A. S. Smogorzhevskii in which Brianchon's theorem is applied to a proof of Pascal's theorem.
The painting on masonite is #81 in the series. It has a purple background and a black wooden frame. It is signed: CJ66.
References: A. S. Smogorzhevskii, The Ruler in Geometrical Constructions (1961), p. 37. This volume was in Crockett Johnson's library. The figure is not annotated.
Carl Boyer and Uta Merzbach, A History of Mathematics (1991), p. 534.
Currently not on view
Object Name
date made
Pascal, Blaise
Brianchon, Charles Julien
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 65 cm x 65 cm x 1.3 cm; 25 9/16 in x 25 9/16 in x 1/2 in
ID Number
catalog number
accession number
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center


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