Painting - Harmonic Series from a Quadrilateral (Pappus)

The concept of a harmonic set of points can be traced back through Girard Desargues (1591–1661) and Pappus of Alexandria (3rd century AD) to Apollonius of Perga (240–190 BC). Crockett Johnson's painting seems to be based upon a figure associated with Pappus. It is likely that Crockett Johnson was inspired by a figure found in H. W. Turnbull's article "The Great Mathematicians" found in his copy of James R. Newman's The World of Mathematics, p. 111. This figure is annotated.
The construction begins with a given set of collinear points (A, B, and Y). An additional point (W) is sought such that AW, AB, and AY are in harmonic progression. That is, the terms AW, AB, and AY represent a progression of terms whose reciprocals form an arithmetic progression. To do this, any point Z, not on line AB, is chosen, and line segments ZA and ZB are constructed. Next, any point D, on ZA, is chosen, and DY, which will intersect ZB at C, is constructed. AC and DB intersect each other at X, and ZX will intersect AB at W. The location of point W is entirely independent of the choice of points Z and D. It follows that AW, AB, and AY form a harmonic progression, and thus the points A, W, B, and Y form a harmonic set.
Crockett Johnson flipped the annotated image for his painting. The boldest portion of his painting, and thus the area with greatest interest, is the quadrilateral ABCD. In addition, the background of his painting is divided into three differently colored sections to illustrate the harmonic series constructed from the quadrilateral. This careful color choice reinforces the painting's title.
This painting was executed in oil on masonite and is painting #24 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) HARMONIC SERIES FROM A QUADRILATERAL (/) (PAPPUS). It has a gray wooden frame.
Currently not on view
Object Name
date made
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 84 cm x 124 cm x 3.8 cm; 33 1/16 in x 48 13/16 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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