Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the Problem of Delos. Crockett Johnson wrote of this problem: "Plutarch mentions it, crediting as his source a now lost version of the legend written by the third century BC Alexandrian Greek astronomer Eratosthenes, who first measured the size of the Earth. Suffering from plague, Athens sent a delegation to Delos, Apollo’s birthplace, to consult its oracle. The oracle’s instruction to the Athenians, to double the size of their cubical altar stone, presented an impossible problem . . . . It could not be done with the compass and an unmarked straightedge."

(p. 99).

Crockett Johnson's paintings follow a construction proposed by the eminent English mathematician Isaac Newton. As Lucasian professor of mathematics at Cambridge University, Newton was required to deposit copies of his lectures in the university library. In 1683, after he had taught a course in algebra for 11 years, he finally deposited the notes for it. After Newton left Cambridge in 1696, his successor, William Whiston, arranged to have the lectures published in a book with the short title *Arithmetica Universalis*. Latin editions of the book appeared in 1707, 1722, 1732, and 1761; and English translations in 1720, 1728, and 1769.

In an appendix to this book, Newton discussed ways of finding the roots of numbers through geometric constructions. One problem was that of finding two mean proportions between given numbers. One case of this problem gives the cube root of a number. [Suppose the numbers are a and b and the proportionals x and y. Then a / x = x / y = y /b). Squaring the first and last term, a² / x² = y² / b². But, from the first equation, one also has x = y² / b. By substitution, a² / x² = x / b, or x³ = a² b. If a is 1, x is the cube root of b, as desired.]

Newton and Crockett Johnson represented the quantities involved as lengths of the sides of triangles. Newton’s figure is #99 in his *Arithmetica Universalis*. Crockett Johnson's figure is differently lettered, and the mirror image of that of Newton.

Following the artist's notation (figure 1979.3083.04.05), suppose AB = 1, bisect it at M, and construct an equilateral triangle MBX on MB. Draw AX and MX extended. Using a marked straightedge, construct line segment BZY, intersecting AX at Z and MX at Y in such a way that XY = AM = MB = 1/2. Then the distance BZ will have a length of one half the cube root of 2, that is to say the length of the side of a cube of side 1/2.

A proof of Newton’s construction is given in Dorrie. Crockett Johnson's copy of a drawing in this volume is annotated. The duplication of the cube also was discussed in at least two other books in Crockett Johnson's library. One is a copy of the 1764 edition of an English translation of the *Arithmetica Universalis*, which Crockett Johnson purchased in January of 1972. The second is W. W. Rouse Ball’s *Mathematical Recreations and Essays*, which also discusses Newton's solution.

Crockett Johnson's painting emphasized doubled lines in the construction, building on the theme of the painting. His diagram for the painting is oriented differently from the painting itself.

This oil painting on masonite is #85 in the series. It depicts overlapping blue, pink and gray circular segments in two adjacent rectangles. These rectangles are divided by various lines into gray and black sections. A lighter gray border goes around the edge. There is a metal and wooden frame. The painting is unsigned. For a mathematically related painting, see #56 (1979.1093.36).

References: Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings," *Leonardo* 5 (1972): pp. 98–100. This specific painting is not discussed in the article.

Heinrich Dorrie, trans. David Antin, *100 Great Problems of Elementary Mathematics: Their History and Solution* (1965) p. 171. The figure on this page, figure 27, is annotated.

Isaac Newton, *Universal Arithmetick*, (1769), esp. pp. 486–87, figure 99. This volume was in Crockett Johnson's library. It is not annotated.

W. W. Rouse Ball, rev. H. S. M. Coxeter, *Mathematical Essays and Recreations*, (1962 printing), pp. 327–33. This is a slightly different construction. The volume was in Crockett Johnson's library.

Isaac Newton, *The Mathematical Works of Isaac Newton*, assembled by Derek T. Whiteside, vol. 2, (1967). This includes a reprint of the 1728 English translation of the *Arithmetica Universalis*.

Our collection database is a work in progress. We may update this record based on further research and review. Learn more about our approach to sharing our collection online.

If you would like to know how you can use content on this page, see the Smithsonian's Terms of Use. If you need to request an image for publication or other use, please visit Rights and Reproductions.