## Painting - Parabolic Triangles (Archimedes)

- 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

- National Museum of American History

- referenced
- Archimedes
- painter
- Johnson, Crockett

- Credit Line
- Ruth Krauss in memory of Crockett Johnson

- 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

- See more items in
- Medicine and Science: Mathematics
- Science & Mathematics
- Art
- Crockett Johnson

- 1967

- Object Name
- painting

- ID Number
- 1979.1093.52
- catalog number
- 1979.1093.52
- accession number
- 1979.1093