Painting - Square Root of Two (Descartes)

La Géométrie, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots. This construction is the subject of Crockett Johnson's painting. Descartes explained: "If the square root of GH is desired, I add, along the same straight line, FG equal to unity, then bisecting FH at K, I describe the circle FIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root." (this is a translation of portion of La Géométrie, as published by J. R. Newman, The World of Mathematics (1956), p. 241)
To understand Descartes' description and the title of this painting, consider the diagram. An angle inscribed in a semicircle is a right angle, thus triangle FGI is similar to triangle IGH. Because this two triangles are similar, their corresponding sides are proportional. Thus, G/IFG = GH/GI. But FG is equal to one, so GH is the square of GI, and GI the square root of GH desired.
In his painting, Crockett Johnson has flipped the image from La Géométrie found in his copy of The World of Mathematics. This figure is not annotated. The artist divided his painting into squares of area one, suggesting what came to be called Cartesian coordinates. The division indicates that the GH chosen has length two.
Johnson chose white for the section of the semicircle that contains the edge of length equal to the square root of GH. This section provides a vivid contrast against the dull, surrounding colors. Crockett Johnson purposefully creates this area of interest to draw focus to the result of Descartes' construction.
Square Root of Two is painting #19 in the series. It was painted in oil or acrylic on masonite, completed in 1965, and is signed: CJ65. The wooden frame is painted black.
Currently not on view
Object Name
date made
Descartes, Rene
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 54.6 cm x 85.2 cm x 3.8 cm; 21 1/2 in x 33 9/16 in x 1 1/2 in
ID Number
catalog number
accession number
Science & Mathematics
Crockett Johnson
See more items in
Medicine and Science: Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center
Credit Line
Ruth Krauss in memory of Crockett Johnson
Additional Media

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