Painting - Every Positive Integer (Gauss)

This painting is loosely based on a theorem proven by the German mathematician Carl Friedrich Gauss (1777–1855) in 1776 when he was just nineteen years old. The proposition, one of Gauss’s many contributions to the branch of mathematics called number theory, states that every positive integer is the sum of three triangular numbers. The concept of triangular numbers dates to antiquity. Suppose one arranges dots in rows, with one in the first row, two in the second, three in the first and so forth. Three dots form a triangle, as do 6 dots, 10 dots, and 15 dots. The numbers 3, 6, 10, 15, and so forth are called "triangular numbers." The integers 0 and 1 are thought of as special cases of triangular numbers.
Crockett Johnson derived his painting from an entry in Gauss's diary published in an article by Eric Temple Bell included by James R. Newman in his book The World of Mathematics (1956), p. 304. The entry includes the phrase EUREKA in Greek, and indicates that any positive integer is the sum of three triangular numbers.
Crockett Johnson’s painting abstractly represents this theorem through the juxtaposition of three triangles. The triangles are equal, but each figure is painted a different color. It is possible that the artist chose to illustrate each triangle in its own color to demonstrate that each triangle generally represents its own triangular number when computing a positive integer. However, the triangles are congruent, which reminds the viewer that the triangles are related because they all represent a triangular number.
This work was painted in oil on masonite, completed in 1966, and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) EVERY POSITIVE INTEGER (/) (GAUSS). It is painting #29 in the series, and has a wooden frame.
Reference: J. R. Newman, The World of Mathematics, 1956, p. 304.
Currently not on view
date made
Gauss, Carl Friedrich
Johnson, Crockett
Physical Description
masonite (substrate material)
wood (frame material)
overall: 25.5 cm x 83 cm x 3.8 cm; 10 1/16 in x 32 11/16 in x 1 1/2 in
ID Number
catalog number
accession number
Credit Line
Ruth Krauss in memory of Crockett Johnson
See more items in
Medicine and Science: Mathematics
Science & Mathematics
Crockett Johnson
Data Source
National Museum of American History, Kenneth E. Behring Center


Add a comment about this object