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# Painting - Parabolic Triangles (Archimedes)

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Description
According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. But finding the area bounded by curved surfaces was not an easy task. The parabola and other conic sections had been known for almost a century before Archimedes wrote a short treatise called Quadrature of the Parabola in about 240 BC. This was the first demonstration of the area bounded by a conic section. In his proof, Archimedes first constructed a triangle whose sides consisted of two tangents of a parabola and the chord connecting the points of tangency. He then showed that the area under the parabola (shown in gray and black in the painting) is two thirds of the area of the triangle which circumscribes it. Once the area bounded by the tangent could be expressed in terms of the area of a triangle, it was easy to construct the corresponding square. Crockett Johnson’s painting follows two diagrams illustrating a discussion of Archimedes’s proof given by Heinrich Dorrie (Figure 54).
This oil or acrylic painting on masonite is #78 in the series and is signed “CJ67” in the bottom left corner. It has a gray wooden frame. For a related painting, see #43 (1979.1093.31).
References: Heinrich Dorrie, trans. David Antin, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965), p. 239. This volume was in Crockett Johnson's library and the diagram in his copy is annotated.
James R. Newman, The World of Mathematics (1956), p. 105. This volume was in Crockett Johnson's library. The figure on this page (Figure 9) is annotated.
Location
Currently not on view
Object Name
painting
1967
referenced
Archimedes
painter
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
Measurements
overall: 76 cm x 61 cm x 3.2 cm; 29 15/16 in x 24 in x 1 1/4 in
ID Number
1979.1093.52
catalog number
1979.1093.52
accession number
1979.1093
Credit Line
Ruth Krauss in memory of Crockett Johnson
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Art
Crockett Johnson
Data Source
National Museum of American History
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