#
Art

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.

"Art - Overview" showing 3219 items.

Page 5 of 322

## Painting -

*Squared Circle*- Description
- This oil painting on pressed wood, #52 in the series, shows an original construction of Crockett Johnson. He executed this work in 1968, three years after he began creating mathematical paintings. It is evident that the artist was very proud of this construction because he drew four paintings dealing with the problem of squaring the circle. The construction was part of Crockett Johnson's first original mathematical work, published in
*The Mathematical Gazette*in early 1970. A diagram relating to the painting was published there.

- To "square a circle," mathematically speaking, is to construct a square whose area is equal to that of a given circle using only a straightedge (an unmarked ruler) and a compass. It is an ancient problem dating from the time of Euclid and is one of three problems that eluded Greek geometers and continued to elude mathematicians for 2,000 years. In 1880, the German mathematician Ferdinand von Lindermann showed that squaring a circle in this way is impossible - pi is a transcendental number. Because this proof is complicated and difficult to understand, the problem of squaring a circle continues to attract amateur mathematicians like Crockett Johnson. Although he ultimately understood that the circle cannot be squared with a straightedge and compass, he managed to construct an approximate squaring.

- Crockett Johnson began his construction with a circle of radius one. In this circle he inscribed a square. Therefore, in the figure, AO=OB=1 and OC=BC=√(2) / 2. AC=AO+OC=1 + √(2) / 2 and AB=√(AC² + BC²) = &#*&#);(2+√(2)). Crockett Johnson let N be the midpoint of OT and constructed KN parallel to AC. K is thus the midpoint of AB, and KN=AO - (AC)/2=(2-&#*&#);(2)) / 4. Next, he let P be the midpoint of OG, and he drew KP, which intersects AO at X. Crockett Johnson then computed NP=NO+OP=(√(2))/4+(1/2). Triangle POX is similar to triangle PNK, so XO/OP=KN/NP. From this equality it follows that XO=(3-2√(2))/2.

- Also, AX=AO-XO=(2√(2)-1)/2 and XC=XO+OC=(3-√(2))/2. Crockett Johnson continued his approximation by constructing XY parallel to AB. It is evident that triangle XYC is similar to triangle ABC, and so XY/XC=AB/AC. This implies that XY=[√((2+√(2)) × (8-5√(2))]/2. Finally he constructed XZ=XY and computed AZ=AX+XZ=[2√(2)-1+(√(2+√(2)) × (8-5√(2))]/2 which approximately equals 1.772435. Crockett Johnson knew that the square root of pi approximately equals 1.772454, and thus AZ is approximately equal to √(Π) - 0.000019. Knowing this value, he constructed a square with each side equal to AZ. The area of this square is (AZ)² = 3.1415258. This differs from the area of the circle by less than 0.0001. Thus, Crockett Johnson approximately squared the circle.

- The painting is signed: CJ68. It is marked on the back: SQUARED CIRCLE* (/) Crockett Johnson 1968 (/) FLAT OIL ON PRESSED WOOD) (/) MATHEMATICALLY (/) DEMONSTRATED (/) TO √π + 0.000000001. It has a white wooden frame. Compare to painting #91 (1979.1093.60).

- References: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,”
*Leonardo*5 (1972): p. 98.

- C. Johnson, “A Geometrical look at √π,"
*Mathematical Gazette*, 54 (1970): p. 59–60. the figure is from p. 59.

- Location
- Currently not on view

- date made
- 1968

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.35

- catalog number
- 1979.1093.35

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Problem of Delos Constructed from a Solution by Isaac Newton (Arithmetica Universalis)*- 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. . . ."(p. 99). Hence the reference to the problem of Delos in the title of the painting.

- Isaac Newton suggested a solution to the problem in his book
*Arithmetica Universalis*, first published in 1707. His construction served as the basis of the painting. Newton’s figure, as redrawn by Crockett Johnson, begins with a base (OA), bisected at a point (B), with an equilateral triangle (OCB) constructed on one of the halves of the base. Newton then extended the sides of this triangle through one vertex. Placing a marked straightedge at one end of the base (O), he rotated the rule so that the distance between the two lines extended equaled the sides of the triangle (in the figure, DE = OB = BA = OC = BC). If these line segments are of length one, one can show that the line segment OD is of length equal to the cube root of two, as desired.

- In Crockett Johnson’s painting, the line OA slants across the bottom and the line ODE is vertical on the left. The four squares drawn from the upper left corner (point E) have sides of length 1, the cube root of 2, the cube root of 4, and two. The distance DE (1) represents the edge of the side and the volume of a unit cube, while the sides of three larger squares represent the edge (the cube root of 2), the side (the square of the cube root of 2) and the volume (the cube of the cube root of two) of the doubled cube.

- This oil painting on masonite is #56 in the series and dates from 1970. The work is signed: CJ70. It is inscribed on the back: PROBLEM OF DELOS (/) CONSTRUCTED FROM A SOLUTION BY (/) ISAAC NEWTON (ARITHMETICA UNIVERSALIS) (/) Crockett Johnson 1970. The painting has a wood and metal frame. For related documentation see 1979.3083.04.06. See also painting number 85 (1979.1093.55), with the references given there.

- Reference: Crockett Johnson, “On the Mathematics of Geometry in My Abstract Paintings,”
*Leonardo*5 (1972): pp. 98–9.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Newton, Isaac

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.36

- catalog number
- 1979.1093.36

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Division of the Square by Conic Rectangles*- Description
- This painting shows three rectangles of equal area, one in shades of blue, one in shades of purple, and one in shades of pink. The height of the middle rectangle equals the height of the first rectangle less its own width, while the height of the third rectangle equals the height of the first triangle less the width of the first triangle. Crockett Johnson associated these properties with conic curves. The construction is that of the artist. The coloring was suggested by a recently discovered French cave painting. The narrow rectangle on the left side and the dark, thin triangle at the base were also added to correspond to the cave painting.

- The oil painting on masonite is #60 in the series. It is signed: CJ70, and inscribed on the back: DIVISION OF THE SQUARE BY CONIC RECTANGLES (/) (GNOMON ADDED AT THE SUGGESTION OF A CRO-MAGNON (/) ARTIST OF LASCAUX (/) Crockett Johnson 1970. The painting is in a black wooden frame. For related documentation see 1979.3083.02.05.

- Reference: Crockett Johnson, "On the Mathematics of Geometry in My Abstract Paintings,"
*Leonardo*5 (1972): pp. 98–101.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.37

- catalog number
- 1979.1093.37

- accession number
- 1979.1093

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Aligned Triangles (Desargues)*- Description
- In the 17th century, the French engineer and architect Girard Desargues (1591–1661) explored interconnections between extensions of the lines within a pencil of three line segments (a pencil of line segments consists of several line segments originating at a common point). His theorems, as published in his own extremely obscure work and also by his contemporary, Abraham Bosse, were extended in the 19th century, and proved of fundamental importance to projective geometry.

- Crockett Johnson's library contains discussions of Desargues' theorem by H. S. M. Coxeter, N. A. Court, Heinrich Dorrie, and William M. Ivins. This painting most resembles a figure from Coxeter, although the diagram is not annotated. Suppose that the vertices of two triangles (PQR and P'Q'R' in Figure 1.5B from Coxeter) lie on a pencil of three line segments emanating from the point O. Suppose that similarly situated sides of the two triangles can be extended to meet in the three points denoted by A, C and B in the figure. According to Desargues' theorem, A, C, and B are collinear.

- In the painting, the two concurrent triangles are shown in shades of gray and black, while the top of the pencil of three lines is in shades of gold. Extensions of the sides and their points of intersection are clearly shown. Both the figure and the background of the painting are divided by the line joining the points of intersection

- The painting is #63 in the series. It is painted in oil or acrylic on masonite, and has a brown wooden frame. The painting is signed: CJ70.

- References:

- Newman, J. R.,
*The World of Mathematics*, p. 133. Figure annotated.

- Court, N. A.,
*College Geometry*(1952), pp. 163–5. The figure is not annotated.

- Coxeter, H. S. M.,
*The Real Projective Plane*, (1955 edition), p. 7. The figure resembles the painting but is not annotated.

- Dorrie, Heinrich,
*100 Great Problems of Elementary Mathematics: Their History and Solution*(1965), p. 267. There is an annotated figure here for another theorem of Desargues, the theorem of involution.

- Field, J. V.,
*The Invention of Infinity: Mathematics and Art in the Renaissance*(1997), pp. 190–206.

- Ivins, William M. Jr.,
*Art & Geometry: A Study in Space Intuitions*(1946), pp. 87–94.

- Location
- Currently not on view

- date made
- 1970

- referenced
- Desargues, Girard

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.38

- accession number
- 1979.1093

- catalog number
- 1979.1093.38

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Golden Rectangle*- Description
- Crockett Johnson annotated several diagrams in his copy of Valens’s book
*The Number of Things*, and used a few of them as the basis of paintings. This is one example. It shows three golden rectangles, the curves from a compass used to construct the rectangles, and a section of a five-pointed Pythagorean star.

- Euclid showed in his
*Elements*that it is possible to divide a line segment into two smaller segments wherein the ratio of the whole length to the longer part equals the ratio of the longer part to the smaller. He used this theorem in his construction of a regular pentagon. This ratio came to be called the “golden ratio.”

- A golden rectangle is a rectangle whose sides adhere to the golden ratio (in modern terms, the ratio of its length to its width equals (1 + √(5) ) /2, or about 1.62). The golden rectangle is described as the rectangle whose proportions are most pleasing to the eye.

- This painting shows the relationship between a golden rectangle and a five-pointed Pythagorean star by constructing the star from the rectangle. It follows a diagram on the top of page 131 in Evans G. Valens,
*The Number of Things*. This diagram is annotated. Valens describes a geometrical solution to the two expressions f x f = e x c and f = e - c, and associates it with the Pythagoreans. The right triangle on the upper part of Valens's drawing, with the short side and part of the hypotenuse equal to f, is shown facing to the left in the painting. It can be constructed from a square with side equal to the shorter side of the rectangle. Two of the smaller rectangles in the painting are also golden rectangles. Crockett Johnson also includes in the background the star shown by Valens and related lines.

- This painting on masonite, #64 in the series, dates from 1970 and is signed: CJ70. It also is marked on the back: ”GOLDEN RECTANGLE (/) Crockett Johnson 1970. It is executed in two hues of gold to emphasize individual sections. While this method creates a detailed and organized contrast, it disguises the three rectangles and the star. Compare paintings 1979.1093.33 (#46) and 1979.1093.70 (#103).

- Reference: Evans G. Valens,
*The Number of Things*(1964), p. 131.

- Location
- Currently not on view

- date made
- 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.39

- accession number
- 1979.1093

- catalog number
- 1979.1093.39

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Geometric Mean (Archytas)*- Description
- This painting demonstrates a construction for finding the geometric mean of two line segments credited to the Greek mathematician Archytas (flourished 400–350 BC), an admirer of Pythagoras. Place the line segments end to end, and draw a circle with this length as radius. Erect a perpendicular at the point where the line segments meet (d in the figure), and consider this to be the altitude of a right triangle inscribed in the semicircle. By similar triangles, the length of the perpendicular of a triangle inscribed in a semicircle is the geometric mean of the two lengths into which it divides the diameter of the circle. Hence the length of d is the mean of the segments e and f.

- This painting an orange-red background, and shows a triangle inscribed in an orange semicircle. The perpendicular from the right angle of the triangle divides the triangle into triangles similar to it, painted in black and white.

- The painting, and the attribution of the theorem to Archytas, are based on a passage from Evans G. Valens,
*The Number of Things: Pythagoras, Geometry and Humming Strings*(1964), p. 118. The figure on this page of this book from Crockett Johnson's library is annotated.

- This oil painting on masonite is #65 in the series. It is inscribed on the back: GEOMETRIC MEAN (ARCHYTAS) (/) Crockett Johnson 1968. It has a wooden frame.

- Location
- Currently not on view

- date made
- 1968

- referenced
- Archytas

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.40

- accession number
- 1979.1093

- catalog number
- 1979.1093.40

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Square Roots of One, Two and Three*- Description
- Crockett Johnson much enjoyed constructing square roots of numbers geometrically. He offered the following account of this painting, as well as the figure shown: "Let AN and BN be 1. Then the diagonal AB is the square root of 2, because it is the hypotenuse of a right triangle with sides of length √1 and √1. The large right triangle √1 plus √2 adds up to a hypotenuse of √3. The compass traces pronounce a statement and also declare its proof. The square root of 2 is 1.4142 . . . and the square root of 3 is 1.7321 . . . Their decimals run on and on but as produced by the compass and blind straightedge both numbers are quite as finite as 1. The triangle embodies three dimensions of the cube. CB is any edge, AB is a face diagonal, and AC is an internal diagonal." Crockett-Johnson described the source of the painting as "Artist's Construction, or Anybody's."

- The triangle with three sides equal to the lengths of interest is painted white. Remaining segments of the construction are in dark gray and purple, with a black background. The painting has a brown wooden frame.

- The painting is #66 in the series and is signed: CJ69. For a related painting, see #45 (1979.1093.32).

- Reference: "Geometric Geometric [sic] Paintings by Crockett Johnson" NMAH Collections.

- Location
- Currently not on view

- date made
- 1969

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.41

- accession number
- 1979.1093

- catalog number
- 1979.1093.41

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Squared Lunes (Hippocrates Of Chios)*- Description
- Classical Greek mathematicians were able to square all convex polygons. That is, given any polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straight edge. Figures with curved sides proved more difficult. However, as this painting suggests, the mathematician Hippocrates of Chios (5th century BC) squared a lune, a figure bounded by arcs of two circle with different radii (lunes resemble quarter moons, hence the name). Finding the area of a lune in terms of a square might seem more difficult than squaring a circle, but the latter problem would prove intractable.

- The painting follows annotated figures in Evans G. Valens's
*The Number of Things*(1964), p.103, which was part of Crockett Johnson's mathematical library. It corresponds to an early diagram in Valens's discussion of squaring the circle. According to Valens, Hippocrates began by arguing that the areas of similar segments of different circles are in the same ratio as the squares of their bases. Suppose an isosceles right triangle is inscribed in a semicircle of diameter c. Construct smaller semicircles of diameter a and b on the sides of the inscribed triangle. As the square of a plus the square of b equals the square of c, the area of the two smaller semicircles equals that of the large one. The proof goes on to consider the area of the two crescents and the triangle.

- In this version of
*Squared Lunes*Crockett Johnson uses brown, black, red, and white against a gray background. This oil painting is #67 in the series, and the first in the series with the title "Squared Lunes." It was completed in 1968 and is signed: CJ68. It is inscribed on the back: SQUARED LUNES (/) (HIIPPOCRATES OF CHIOS) (/) Crockett Johnson 1968. A related painting is #68 (1979.1093.43).

- Location
- Currently not on view

- date made
- 1968

- referenced
- Hippocrates of Chios

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.42

- accession number
- 1979.1093

- catalog number
- 1979.1093.42

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Squared Lunes (Hippocrates Of Chios)*- Description
- The title of this painting refers to Hippocrates of Chios (5th century BC), one of the greatest geometers of antiquity. Classical Greek mathematicians were able to square convex polygons. That is, given a polygon, they could produce a square of equal area in a finite number of steps using only a compass and a straightedge. They were unable to square a circle. This painting is based on the earliest known squaring of a figure bounded by curves rather than straight lines. The mathematician Hippocrates squared a lune, a figure bounded by arcs of two circles with different radii. This achievement might seem more difficult than squaring a circle.

- Crockett Johnson's painting follows two annotated figures in Evans G. Valens's
*The Number of Things*(1964), pp. 103–104, a book in the artist’s mathematical library. The finished piece shows isosceles triangles T, and a second congruent triangle connected to it base to base to form a square. Also present in the painting are three lunes, two small and one large. The area of triangle T is equal to the sum of the areas of lunes A and B (see figures). The area of triangle T is also equal to the area of a lune composed of X, Y, and the area T-C. Furthermore, because triangle T is congruent to the triangle below it, triangle T is equal to the area of this lune. Thus, the area of the square is equal to the sum of the areas of the three lunes. In summary, Johnson pictorially represented a "squared" curvilinear region; that is, he successfully constructed a square with the same area as that of the region of three lunes bounded by curves.

- Crockett Johnson executed this painting in 4 tints and darker shades of purple upon a black background. The center triangle is the darkest shade of purple. As one moves outward, the colors grow lighter. This allows a dramatic distinction to be seen between the figure and the background, and thus puts a greater emphasis on the lunes.

- This oil painting on masonite is #68 in Crockett Johnson's series. Its date of completion is unknown and the work is unsigned. It is closely related to painting #67 (1979.1093.42).

- Location
- Currently not on view

- date made
- ca 1965

- referenced
- Hippocrates of Chios

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.43

- accession number
- 1979.1093

- catalog number
- 1979.1093.43

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Painting -

*Cross Ratio in an Ellipse (Poncelet)*- 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 (a pencil of lines is a set of lines emanating from a common point). In the drawing, which is Figure 5 from an article by Morris Kline in James R. Newman's
*The World of Mathematics*(1956), 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 Crockett Johnson's 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 painting with both Chasles and another French advocate of projective geometry, Victor Poncelet.

- The painting, in oil or acrylic on masonite, is #69 in the series. It has a dark gray or blue background and a black wooden frame. It shows a white ellipse, two points on the ellipse (on the left side of the painting), and two pencils of lines that produce the same cross ratio. The painting is not signed. It is inscribed on the back, in Crockett Johnson’s hand: CROSS RATIO IN AN ELLIPSE (PONCELET) (/) Crockett Johnson 1968. Compare #21 (1979.1093.15).

- Reference: Morris Kline in James R. Newman,
*The World of Mathematics*(1956), p. 634. This volume was in Crockett Johnson's library. The figure on this page is annotated.

- Location
- Currently not on view

- date made
- 1968

- referenced
- Poncelet, Jean-Victor

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.44

- accession number
- 1979.1093

- catalog number
- 1979.1093.44

- Data Source
- National Museum of American History, Kenneth E. Behring Center

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