Art

Panorama of the Army of the Cumberland

The National Museum of American History is not an art museum. But works of art fill its collections and testify to the vital place of art in everyday American life. The ceramics collections hold hundreds of examples of American and European art glass and pottery. Fashion sketches, illustrations, and prints are part of the costume collections. Donations from ethnic and cultural communities include many homemade religious ornaments, paintings, and figures. The Harry T Peters "America on Stone" collection alone comprises some 1,700 color prints of scenes from the 1800s. The National Quilt Collection is art on fabric. And the tools of artists and artisans are part of the Museum's collections, too, in the form of printing plates, woodblock tools, photographic equipment, and potters' stamps, kilns, and wheels.

Name: Johnson, Crockett

Painting - Cross-Ratio in a Conic (Poncelet)

From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle.
Description
From ancient times, mathematicians have studied conic sections, curves generated by the intersection of a cone and a plane. Such curves include the parabola, hyperbola, ellipse, and circle. Each of these curves may be considered as a projection of the circle.
Nineteenth-century mathematicians were much interested in the properties of conics that were preserved under projection. They knew from the work of the ancient mathematician Pappus that the cross ratio of line segments created by two straight lines cutting the same pencil of lines was a constant. In Figure 5, which is from an article by Morris Kline in James R. Newman's The World of Mathematics, if line segment l’ crosses lines emanating from the point O at points A’, B’, C’ and D’; and line segment l croses the same lines at points A, B, C, and D, the cross ratio:
(A’C’/C’B’) / (A’D’/D’B’) = (AC/BC) / (AD/DB). In other words, it is independent of the cutting line. (see the Crockett Johnson painting Pencil of Ratios (Monge) ).
The French mathematician Michel Chasles introduced a related result, which is the subject of this painting. He considered two points on a conic section (such as an ellipse) that were both linked to the same four other points on the conic. He found that lines crossing both pencils of rays had the same cross ratio. Moreover, a conic section could be characterized by its cross ratio.This opened up an entirely different way of describing conic sections. Crockett Johnson associated this particular painting with another French advocate of projective geometry, Victor Poncelet.
This oil painting on masonite is #21 in the series. It has a dark gray background and a wood and metal frame. It shows a large black ellipse with two pencils of lines linked to the same four lines of the ellipse. The painting is signed: CJ66. It is inscribed on the back: Crockett Johnson 1966 ( /) CROSS-RATIO IN A CONIC (/) (PONCELET). Compare painting #69 (1979.1093.44).
Reference: This painting is based on a figure in James R. Newman, The World of Mathematics (1956), p. 634. This volume was in the Crockett Johnson library. The figure on this page is annotated. For a figure on cross-ratios, see p. 632.
Location
Currently not on view
date made
1966
referenced
Poncelet, Jean-Victor
painter
Johnson, Crockett
ID Number
1979.1093.15
catalog number
1979.1093.15
accession number
1979.1093

Painting - Measurement of the Earth (Eratosthenes)

The determination of the size and shape of the Earth has occupied philosophers from antiquity. Eratosthenes, a mathematician in the city of Alexandria in Egypt who lived from about 275 through 194 BC, proposed an ingenious way to measure the circumference of the Earth.
Description
The determination of the size and shape of the Earth has occupied philosophers from antiquity. Eratosthenes, a mathematician in the city of Alexandria in Egypt who lived from about 275 through 194 BC, proposed an ingenious way to measure the circumference of the Earth. It is illustrated by this painting. Eratosthenes claimed that the town of Syene (now Aswan) was directly south of Alexandria, and that the distance between the cities was known. Moreover, he reported that on a day when the vertical rod of a sundial cast no shadow at noon in Syene, the shadow cast by a similar rod at Alexandria formed an angle of 1/50 of a complete circle.
In the Crockett Johnson painting, the circle represents the Earth and the two line segments drawn from the center display the direction of the two rods. The two parallel lines represent rays of sunlight striking the Earth, the dark-purple region the shadowed area. The angle of the shadow equals the angle subtended at the center of the Earth, hence the circumference of the entire Earth can be computed when the angle and the distance of the cities is known.
Crockett Johnson's painting may be after a diagram from the book by James R. Newman entitled The World of Mathematics (p. 206), although the figure is not annotated. Newman published a brief extract describing ideas of Eratosthenes, based on a first century BC account by Cleomedes.
The Crockett Johnson painting is #15 in the series. It is marked on the back : Crockett Johnson 1966 (/) MEASUREMENT OF THE EARTH (/) (ERATOSTHENES).
Reference: O. Pederson and M. Phil, Early Physics and Astronomy (1974), p. 53.
Location
Currently not on view
date made
1966
referenced
Eratosthenes
painter
Johnson, Crockett
ID Number
1979.1093.10
catalog number
1979.1093.10
accession number
1979.1093

Painting - Locus of Point on Chord (Plato)

The locus of the midpoints of the chords of a given circle that pass through a fixed point is a circle when the point lies inside of or on the circle.
Description
The locus of the midpoints of the chords of a given circle that pass through a fixed point is a circle when the point lies inside of or on the circle. The small circle painted white is the locus of the midpoints of chords drawn in the large circle that pass through a point toward the top left of the inside of the circle. Three chords of the large circle are suggested. These are the diameter, whose midpoint is the center of the circle, a vertical chord through the point, and a horizontal chord through the point (only a small part of this chord is indicated). The painting is based on a diagram from College Geometry by Nathan Court. It is unclear why Crockett Johnson associated this painting with Plato.
The oil painting on masonite is #41 in the series. It has a background of two purple and gray rectangles. It has a metal and wooden frame. It shows a circle with a smaller circle inside it. The smaller circle is in two shades of white, the larger one in orange, black, gray and light purple. The painting is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) LOCUS OF POINT ON CHORD (PLATO).
Reference: Nathan Court, College Geometry, (1964 printing), p. 13. This figure is annotated in Crockett Johnson's copy of this volume.
date made
1966
referenced
Plato
painter
Johnson, Crockett
ID Number
1979.1093.29
catalog number
1979.1093.29
accession number
1979.1093

Painting - Doubled Cube (Newton)

Two paintings in the Crockett Johnson collection concern the ancient problem of doubling the volume of a given cube, or the Problem of Delos.
Description
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.
Location
Currently not on view
date made
ca 1970
referenced
Newton, Isaac
painter
Johnson, Crockett
ID Number
1979.1093.55
catalog number
1979.1093.55
accession number
1979.1093

Painting - Duality (Pascal-Brianchon)

As a 21-year-old student, the Frenchman Charles Jules Brianchon (1785–1864) discovered that in any hexagon circumscribed about a conic section (such as a circle), the three lines that join opposite diagonals meet in a single point.
Description
As a 21-year-old student, the Frenchman Charles Jules Brianchon (1785–1864) discovered that in any hexagon circumscribed about a conic section (such as a circle), the three lines that join opposite diagonals meet in a single point. He also pointed out connections between his result and Pascal's theorem concerning the points of intersection of opposite sides of a hexagon inscribed in a conic section.
In the painting, a hexagon (only the vertices are shown) is inscribed in a circle. Three diagonal lines (edges of the gray and black polygon) are collinear. The line in question is the line joining the points of intersection, white on one side and purple on the other. Crockett Johnson's painting closely resembles a diagram of A. S. Smogorzhevskii in which Brianchon's theorem is applied to a proof of Pascal's theorem.
The painting on masonite is #81 in the series. It has a purple background and a black wooden frame. It is signed: CJ66.
References: A. S. Smogorzhevskii, The Ruler in Geometrical Constructions (1961), p. 37. This volume was in Crockett Johnson's library. The figure is not annotated.
Carl Boyer and Uta Merzbach, A History of Mathematics (1991), p. 534.
Location
Currently not on view
date made
1966
referenced
Pascal, Blaise
Brianchon, Charles Julien
painter
Johnson, Crockett
ID Number
1979.1093.53
catalog number
1979.1093.53
accession number
1979.1093

Painting - Parabolic Triangles (Archimedes)

According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. Finding the area bounded by curved surfaces was not an easy task.
Description
According to the classical Greek tradition, the quadrature or squaring of a figure is the construction, with the aid of only straight edge and compass, of a square equal in area to that of the figure. Finding the area bounded by curved surfaces was not an easy task. The parabola and other conic sections had been known for almost a century before Archimedes wrote a short treatise called Quadrature of the Parabola in about 240 BC. This was the first demonstration of the area bounded by a conic section.
In his proof, Archimedes first constructed a triangle whose sides consisted of two tangents of a parabola and the chord connecting the points of tangency. He then showed that the area under the parabola (shown in white and light green in the painting) is two thirds of the area of the triangle that circumscribes it. Once the area bounded by the tangent could be expressed in terms of the area of a triangle, it was easy to construct the corresponding square. Crockett Johnson’s painting is based on diagrams illustrating a discussion of Archimedes’s proof given by H. Dorrie (Figure 54) or J. R. Newman (Figure 9).
This oil painting is #43 in the series, and is signed: CJ69. It has a gray background and a gray frame. It shows a triangle that circumscribes a portion of a parabola. The large triangle is divided into a triangle in shades of light green, which touches a triangle in shades of dark green. The region between the triangles is divided into black and white areas. A second painting in the series, #78 (1979.1093.52) illustrates the same theorem.
References: Heinrich Dorrie, trans. David Antin, 100 Great Problems of Elementary Mathematics: Their History and Solution (1965), p. 239. This volume was in Crockett Johnson’s library and his copy is annotated.
James R. Newman, The World of Mathematics (1956), p. 105. This volume was in Crockett Johnson's library. The figure on this page is annotated.
Location
Currently not on view
date made
1969
referenced
Archimedes
painter
Johnson, Crockett
ID Number
1979.1093.31
catalog number
1979.1093.31
accession number
1979.1093

Painting -Hippias' Curve

This painting is a construction of Crockett Johnson, relating to a curve attributed to the ancient Greek mathematician Hippias. This was one of the first curves, other than the straight line and the circle, to be studied by mathematicians.
Description
This painting is a construction of Crockett Johnson, relating to a curve attributed to the ancient Greek mathematician Hippias. This was one of the first curves, other than the straight line and the circle, to be studied by mathematicians. None of Hippias's original writings survive, and the curve is relatively little known today. Crockett Johnson may well have followed the description of the curve given by Petr Beckmann in his book The History of Pi (1970). Crockett Johnson's copy of Beckmann’s book has some light pencil marks on his illustration of the theorem on page 39 (see figure).
Hippias envisioned a curve generated by two motions. In Crockett Johnson's own drawing, a line segment equal to OB is supposed to move uniformly leftward across the page, generating a series of equally spaced vertical line segments. OB also rotates uniformly about the point O, forming the circular arc BQA. The points of intersection of the vertical lines and the arc are points on Hippias's curve. Assuming that the radius OK has a length equal to the square root of pi, the square AOB (the surface of the painting) has area equal to pi. Moreover, the height of triangle ASO, OS, is √(4 / pi), so that the area of triangle ASO is 1.
The painting has a gray border and a wood and metal frame. The sections of the square and of the regions under Hippias's curve are painted in various pastel shades, ordered after the order of a color wheel.
This oil painting is #114 in the series. It is signed on the back: HIPPIAS' CURVE (/) SQUARE AREA = (/) TRIANGLE " = 1 = [ . .] (/) Crockett Johnson 1973.
Location
Currently not on view
date made
1973
referenced
Hippias
painter
Johnson, Crockett
ID Number
1979.1093.76
accession number
1979.1093
catalog number
1979.1093.76

Documentation, Letters and Drawings of Crockett Johnson Sent to Milton Rosenau

Between 1968 and 1975, the artist and author Crockett Johnson (1906-1975) sent a variety of mathematical inquiries to his Connecticut acquaintance Milton (Mickey) Rosenau. The questions relate to Crockett Johnson's mathematical paintings and drawings for them.
Description
Between 1968 and 1975, the artist and author Crockett Johnson (1906-1975) sent a variety of mathematical inquiries to his Connecticut acquaintance Milton (Mickey) Rosenau. The questions relate to Crockett Johnson's mathematical paintings and drawings for them. Rosenau's replies apparently do not survive.
For related transactions, see the collection of Crockett Johnson's mathematical paintings (transaction 1979.1093), related nonaccessioned correspondence and drawings of Crockett Johnson (transaction 1979.3083), and correspondence received from Harley Flanders (transaction 2009.3005).
Location
Currently not on view
date made
1968-1975
maker
Johnson, Crockett
ID Number
2012.3120.01
nonaccession number
2012.3120
catalog number
2012.3120.01

Correspondence and Drawings of Crockett Johnson

These materials are Intermixed correspondence and drawings of Crockett Johnson relating to his mathematical paintings from the period 1973-1974. . Correspondents include G.
Description
These materials are Intermixed correspondence and drawings of Crockett Johnson relating to his mathematical paintings from the period 1973-1974. . Correspondents include G. Stanley Smith, Douglas Quadling, and Harley Flanders.
For related transaction, see 1979.1093.
Location
Currently not on view
date made
1973-1974
maker
Johnson, Crockett
ID Number
1979.3083.05
catalog number
1979.3083.05
nonaccession number
1979.3083

Documents Relating to Crockett Johnson's Painting Heptagon 1:3:3 Triangle

These documents and diagrams relate to Crockett Johnson's painting Heptagon 1:3:3 Triangle (1979.1093.72).Currently not on view
Description
These documents and diagrams relate to Crockett Johnson's painting Heptagon 1:3:3 Triangle (1979.1093.72).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.16
catalog number
1979.3083.02.16
nonaccession number
1979.3083

Documentation on Crockett Johnson's Painting Equal Areas, Their Triangular Square Root and Pi

These materials document Crockett Johnson's painting Equal Areas, Their Triangular Square Root and Pi (1979.1093.59).Currently not on view
Description
These materials document Crockett Johnson's painting Equal Areas, Their Triangular Square Root and Pi (1979.1093.59).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.07
catalog number
1979.3083.02.07
nonaccession number
1979.3083

Documentation Relating to A Circle and Square are Finitely Equal in Area

Currently not on view
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.04.04
catalog number
1979.3083.04.04
nonaccession number
1979.3083

Documents Relating to Crockett Johnson's painting Hippias's Curve

These documents and diagrams relate to Crockett Johnson's painting Hippias's Curve (1979.1093.76).Currently not on view
Description
These documents and diagrams relate to Crockett Johnson's painting Hippias's Curve (1979.1093.76).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.15
catalog number
1979.3083.02.15
nonaccession number
1979.3083

Documentation on Crockett Johnson's Painting Square Root of Pi - 0.00001

These materials document Crockett Johnson's painting Square Root of Pi - 0.00001(1979.1093.60).Currently not on view
Description
These materials document Crockett Johnson's painting Square Root of Pi - 0.00001(1979.1093.60).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.08
catalog number
1979.3083.02.08
nonaccession number
1979.3083

Documentation Relating to Crockett-Johnson's Painting Area and Perimeter of a Squared Circle

These materials relate to Crockett Johnson's painting Area and Perimeter of a Squared Circle (1979.1093.63).Currently not on view
Description
These materials relate to Crockett Johnson's painting Area and Perimeter of a Squared Circle (1979.1093.63).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.11
catalog number
1979.3083.02.11
nonaccession number
1979.3083

Documents Relating to Crockett Johnson's Painting Construction of Heptagon

These documents and diagrams relate to Crockett Johnson's painting Construction of Heptagon (1979.1093.78).Currently not on view
Description
These documents and diagrams relate to Crockett Johnson's painting Construction of Heptagon (1979.1093.78).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.18
catalog number
1979.3083.02.18
nonaccession number
1979.3083

Documentation Relating to Cube Doubled in Volume

Currently not on view
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.04.05
catalog number
1979.3083.04.05
nonaccession number
1979.3083

Documentation Relating to Crockett Johnson's Painting Squared Rectangle and Euler Line

These materials document Crockett Johnson's painting Squared Rectangle and Euler Line (1979.1093.62).Currently not on view
Description
These materials document Crockett Johnson's painting Squared Rectangle and Euler Line (1979.1093.62).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.10
catalog number
1979.3083.02.10
nonaccession number
1979.3083

Documentation on Crockett Johnson's Painting Squares of 2, 4, 16 from Square Root of x

These materials document Crockett Johnson's painting Squares of 2, 4, 16 from Square Root of x (1979.1093.57).Currently not on view
Description
These materials document Crockett Johnson's painting Squares of 2, 4, 16 from Square Root of x (1979.1093.57).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.06
catalog number
1979.3083.02.06
nonaccession number
1979.3083

Documentation Relating to Crockett Johnson's Painting Approximation of Pi to .0001

Currently not on view
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.12
nonaccession number
1979.3083
catalog number
1979.3083.02.12

Documentation on Crockett Johnson's Painting Division of a Square by Conic Rectangles

These sheets provide documentation for Crockett Johnson's painting Division of a Square by Conic Rectangles (1979.1093.37).Currently not on view
Description
These sheets provide documentation for Crockett Johnson's painting Division of a Square by Conic Rectangles (1979.1093.37).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.05
catalog number
1979.3083.02.05
nonaccession number
1979.3083

Documentation Relating to David Crockett Johnson's Painting Euclidian Values of a Squared Circle

These materials document David Crockett Johnson's painting Euclidian Values of a Squared Circle (1979.1093.69).Currently not on view
Description
These materials document David Crockett Johnson's painting Euclidian Values of a Squared Circle (1979.1093.69).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.13
catalog number
1979.3083.02.13
nonaccession number
1979.3083

Painting - Morley Triangle

In the late 1960s and early 1970s, the American cartoonist Crockett Johnson created a series of paintings on mathematical subjects.
Description
In the late 1960s and early 1970s, the American cartoonist Crockett Johnson created a series of paintings on mathematical subjects. This oil painting, #74 in the series, dates from 1969 and is signed "CJ69." It is based on a theorem in plane geometry proved by the English-born mathematician Frank Morley (1860–1937). Morley emigrated to the United States and taught at Haverford College and Johns Hopkins University.
The painting illustrates his best-known result. It shows lines that divide the three angles of the large triangle into three equal parts. Lines coming from different vertices of the triangle meet in points. The triangle formed by joining the intersections of the trisectors, which lie nearest to the three sides of the triangle, is shown in white in the painting. According to Morley's theorem, this is an equilateral triangle.
Date made
1969
painter
Johnson, Crockett
ID Number
1979.1093.48
catalog number
1979.1093.48
accession number
1979.1093

Documentation Relating to Crockett Johnson's Painting Biblical Squared Circles

These diagrams and documents relate to David Crockett Johnson's painting Biblical Squared Circles (1979.1093.61).Currently not on view
Description
These diagrams and documents relate to David Crockett Johnson's painting Biblical Squared Circles (1979.1093.61).
Location
Currently not on view
maker
Johnson, Crockett
ID Number
1979.3083.02.09
catalog number
1979.3083.02.09
nonaccession number
1979.3083

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