Painting - <I>Simple Equation (Descartes)</I>
date: label: date made
content: 1966
name: label: referenced
content: Descartes, Rene
label: painter
content: Johnson, Crockett
notes: label: Description
content: In a pathbreaking book <I>La Géométrie</I>, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting. More specifically, Descartes described geometrical methods for finding the roots of simple polynomials. He wrote (as translated from the original French): "Finally, if I have z² = az -b², I make NL equal to (1/2)a and LM equal to b as before: then, instead of joining the points M and N, I draw MQR parallel to LN, and with N as center describe a circle through L cutting MQR in the points Q and R; then z, the line sought, is either MQ or MR, for in this way it can be expressed in two ways, namely: z = (1/2)a + √((1/4)a² - b²) and z = (1/2)a - √((1/4)a² - b²)."
label: Description
content: To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.
label: Description
content: Crockett Johnson's painting directly imitates Descartes's figure found in Book I of <I>La Géométrie</I>. A translation of part of Book I is found in the artist’s copy of James R. Newman's <I> The World of Mathematics</I>. The figure on page 250 is annotated.
label: Description
content: This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.
label: Location
content: Currently not on view
setName: label: See more items in
content: Medicine and Science: Mathematics
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content: Science & Mathematics
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content: Crockett Johnson
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content: Art
creditLine: label: Credit Line
content: Ruth Krauss in memory of Crockett Johnson
dataSource: label: Data Source
content: National Museum of American History
identifier: label: ID Number
content: 1979.1093.24
label: catalog number
content: 1979.1093.24
label: accession number
content: 1979.1093
objectType: label: Object Name
content: painting
physicalDescription: label: Physical Description
content: masonite (substrate material)
label: Physical Description
content: wood (frame material)
label: Measurements
content: overall: 94.5 cm x 65 cm x 2.5 cm; 37 3/16 in x 25 9/16 in x in