# Painting - Aligned Triangles (Desargues)

Description
In the 17th century, the French engineer and architect Girard Desargues (1591–1661) explored interconnections between extensions of the lines within a pencil of three line segments (a pencil of line segments consists of several line segments originating at a common point). His theorems, as published in his own extremely obscure work and also by his contemporary, Abraham Bosse, were extended in the 19th century, and proved of fundamental importance to projective geometry.
Crockett Johnson's library contains discussions of Desargues' theorem by H. S. M. Coxeter, N. A. Court, Heinrich Dorrie, and William M. Ivins. This painting most resembles a figure from Coxeter, although the diagram is not annotated. Suppose that the vertices of two triangles (PQR and P'Q'R' in Figure 1.5B from Coxeter) lie on a pencil of three line segments emanating from the point O. Suppose that similarly situated sides of the two triangles can be extended to meet in the three points denoted by A, C and B in the figure. According to Desargues' theorem, A, C, and B are collinear.
In the painting, the two concurrent triangles are shown in shades of gray and black, while the top of the pencil of three lines is in shades of gold. Extensions of the sides and their points of intersection are clearly shown. Both the figure and the background of the painting are divided by the line joining the points of intersection
The painting is #63 in the series. It is painted in oil or acrylic on masonite, and has a brown wooden frame. The painting is signed: CJ70.
References:
Newman, J. R., The World of Mathematics, p. 133. Figure annotated.
Court, N. A., College Geometry (1952), pp. 163–5. The figure is not annotated.
Coxeter, H. S. M., The Real Projective Plane, (1955 edition), p. 7. The figure resembles the painting but is not annotated.
Dorrie, Heinrich, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965), p. 267. There is an annotated figure here for another theorem of Desargues, the theorem of involution.
Field, J. V., The Invention of Infinity: Mathematics and Art in the Renaissance (1997), pp. 190–206.
Ivins, William M. Jr., Art & Geometry: A Study in Space Intuitions (1946), pp. 87–94.
Location
Currently not on view
1970
referenced
Desargues, Girard
painter
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
Measurements
overall: 64 cm x 123.8 cm x 4.5 cm; 25 3/16 in x 48 3/4 in x 1 3/4 in
ID Number
1979.1093.38
accession number
1979.1093
catalog number
1979.1093.38
Credit Line
Ruth Krauss in memory of Crockett Johnson
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Medicine and Science: Mathematics
Science & Mathematics
Art
Crockett Johnson
Data Source
National Museum of American History