Mathematical Paintings of Crockett Johnson - Introduction
"Mathematical Paintings of Crockett Johnson - Introduction" showing 3 items.
Painting - Parabolic Triangles (Archimedes)
- 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. 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 white and light green in the painting) is two thirds of the area of the triangle that 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 is based on diagrams illustrating a discussion of Archimedes’s proof given by H. Dorrie (Figure 54) or J. R. Newman (Figure 9).
- This oil painting is #43 in the series, and is signed: CJ69. It has a gray background and a gray frame. It shows a triangle that circumscribes a portion of a parabola. The large triangle is divided into a triangle in shades of light green, which touches a triangle in shades of dark green. The region between the triangles is divided into black and white areas. A second painting in the series, #78 (1979.1093.52) illustrates the same theorem.
- 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 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 is annotated.
- Location
- Currently not on view
- date made
- 1969
- referenced
- Archimedes
- painter
- Johnson, Crockett
- ID Number
- 1979.1093.31
- catalog number
- 1979.1093.31
- accession number
- 1979.1093
- Data Source
- National Museum of American History, Kenneth E. Behring Center
Painting - Parabolic Triangles (Archimedes)
- 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
- date made
- 1967
- referenced
- Archimedes
- painter
- Johnson, Crockett
- ID Number
- 1979.1093.52
- catalog number
- 1979.1093.52
- accession number
- 1979.1093
- Data Source
- National Museum of American History, Kenneth E. Behring Center
Painting - Archimedes Transversal
- Description
- The construction of regular polygons using straightedge and compass alone is a problem that has intrigued mathematicians from ancient times. Crockett Johnson was particularly interested in the construction of regular seven-sided figures or heptagons, which require not only a compass but a marked straight edge. The mathematician Archimedes reportedly proposed such a construction, which was included in a treatise now lost. Relying heavily on Thomas Heath's Manual of Greek Mathematics, Crockett Johnson prepared this painting.
- Archimedes had reduced the problem of finding a regular hexagon to that of finding two points that divided a line segment into two mean proportionals. He then used a construction somewhat like that of the painting to find a line segment divided as desired. Crockett Johnson's papers include not only photocopies of the relevant portion of Heath, but his own diagrams.
- The painting is #104 in the series. It is in acrylic or oil on masonite., and has purple, yellow, green and blue sections. There is a black wooden frame. The painting is unsigned and undated. Relevant correspondence in the Crockett Johnson papers dates from 1974.
- References: Heath, Thomas L., A Manual of Greek Mathematics (1963 edition), pp. 340–2.
- Crockett Johnson, "A construction for a regular heptagon," Mathematical Gazette, 59 (March 1975): pp. 17–18.
- Location
- Currently not on view
- date made
- ca 1974
- referenced
- Archimedes
- painter
- Johnson, Crockett
- ID Number
- 1979.1093.71
- catalog number
- 1979.1093.71
- accession number
- 1979.1093
- Data Source
- National Museum of American History, Kenneth E. Behring Center