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Science & Mathematics

The Museum's collections hold thousands of objects related to chemistry, biology, physics, astronomy, and other sciences. Instruments range from early American telescopes to lasers. Rare glassware and other artifacts from the laboratory of Joseph Priestley, the discoverer of oxygen, are among the scientific treasures here. A Gilbert chemistry set of about 1937 and other objects testify to the pleasures of amateur science. Artifacts also help illuminate the social and political history of biology and the roles of women and minorities in science.

The mathematics collection holds artifacts from slide rules and flash cards to code-breaking equipment. More than 1,000 models demonstrate some of the problems and principles of mathematics, and 80 abstract paintings by illustrator and cartoonist Crockett Johnson show his visual interpretations of mathematical theorems.

"Science & Mathematics - Overview" showing 2182 items.

Page 8 of 219

## Painting -

*Area and Perimeter of a Squared Circle*- Description
- To "square” a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle. It was not until the nineteenth century that the impossibility of squaring a circle was demonstrated.

- This painting is an original construction by Crockett Johnson. It begins with the assumption that the circle has been squared, the area of the larger square equals that of the circle. Crockett Johnson then constructed a smaller square so that it has perimeter equal to the circumference of the circle. His diagram for the painting is shown, with the large square having side AB and the small one side of length AC.

- The painting is #95 in the series. It has a black background. There is a rose circle superimposed on two gray squares. The painting is unsigned and has a metal frame.

- Reference: Carl B. Boyer and Uta C. Merzbach,
*A History of Mathematics*(1991), pp. 65-7, pp. 71–2.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.63

- catalog number
- 1979.1093.63

- accession number
- 1979.1093

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

## Painting -

*Velocities and Right Triangles (Galileo)*- Description
- This is the third painting by Crockett Johnson to represent the motion of bodies released from rest from a common point and moving along different inclined planes. In the
*Dialogues Concerning Two New Sciences*(1638), Galileo argued that the points reached by the balls at a given time would lie on a circle. Two such circles and three inclined planes, as well as a vertical line of direct fall, are indicated in the painting. One circle has half the diameter of the other. Crockett Johnson also joins the base of points on the inclined planes to the base of the diameters of the circles, forming two sets of right triangles.

- This oil painting on masonite is #96 in the series. It has a black background and a wooden and metal frame. It is signed on the back: VELOCITIES AND RIGHT TRIANGLES (GALILEO) (/) Crockett Johnson 1972. Compare to paintings #42 (1979.1093.30) and #71 (1979.1093.46), as well as the figure from Valens,
*The Attractive Universe: Gravity and the Shape of Space*(1969), p. 135.

- Location
- Currently not on view

- date made
- 1972

- referenced
- Galilei, Galileo

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.64

- catalog number
- 1979.1093.64

- accession number
- 1979.1093

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

## Painting -

*Squares of 1, 2, 3, 4 and Square Roots to 8*- Description
- This painting reflects Crockett Johnson's enduring fascination with square roots and squaring. As the title suggests, it includes four squares whose areas are 1, 2, 3, and 4 square units, and seven line segments whose lengths are the square roots of 2, 3, 4, 5, 6, 7, and 8.

- One may construct these squares and square roots by alternate applications of the Pythagorean theorem to squares running along the diagonal of the painting, and to rectangles running across the top (not all the rectangles are shown). More specifically, assume that the light-colored square in the upper left corner of the painting has side of length 1 (which equals the square root of 1). Then the diagonal is the square root of two, and a quarter circle with this radius centered at upper left corner cuts the sides of the square extended to determine two sides of a second, larger square. The area of this square (shown in the painting) is the square of the square root of 2, or two.

- One can then consider the rectangle with side one and base square root of two that is in the upper left of the painting. It will have sides one and the square root of 2, and hence diagonal of length equal to the square root of three. The diagonal is not shown, but an circular arc with this radius forms the second arc in the painting. It determines the sides of a square with side equal to the square root of three and area 3. It also forms a rectangle with sides of length one and the square root of 4 (or two). This gives the third arc and the largest square in the painting.

- By continuing the construction (further squares and rectangles are not shown), Crockett Johnson arrived at portions of circular arcs that cut the diameter at distances of the square roots of 5, 6, 7, and 8. Only one point on the last arc is shown. It is at the lower right corner of the painting.

- Crockett Johnson executed the work in various shades and tints from his starting point at the white and pale-blue triangle to darker blues at the opposite corner.

- This oil painting on masonite is not signed and its date of completion is unknown. It is #97 in the series.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.65

- catalog number
- 1979.1093.65

- accession number
- 1979.1093

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

## Painting -

*Law of Orbiting Velocities*- Description
- This creation, similar to works #22 (1979.1093.16) and #76 (1979.1093.50), is a further example of Crockett Johnson's work relating to Kepler's first two laws of planetary motion. The ellipse represents the path of a planet and the white sections represent equal areas swept out in equal times. This work is a silk screen on paper. It is number 99 in the series, and is signed in the right corner: Crockett Johnson (/) 67. It draws on a figure from
*The World of Mathematics*by James R. Newman.

- Location
- Currently not on view

- date made
- 1967

- referenced
- Kepler, Johannes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.66

- catalog number
- 1979.1093.66

- accession number
- 1979.1093

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

## Painting -

*Square Root of Pi*- Description
- This oil painting is an original construction of Crockett Johnson, and proceeds from the assumption that the circle has been "squared." If the circle has radius one, and if square with the same center has the same area, Crockett Johnson argued that the inscribed rectangle shown, which has a diagonal that meets opposite points of intersection of the square and circle, has an area equal to the square root of pi.

- The verification of Crockett Johnson's construction is straightforward. The circle has radius one so that its area is pi. Because it is assumed that the circle has been "squared," the area of the square is also pi, and the length of one of its sides equals the square root of pi. The area of the rectangle is equal to the sum of the area of the two triangles formed by the diagonal. These triangles have bases equal to the diameter of the circle (2) and height equal to half the length of the side of the square (half of the square root of two). Hence each triangle has area half of the square root of pi, and the entire rectangle has area equal to the square root of pi. There is a second rectangle in the painting of the same area.

- There are two paintings in the collection with this title. The geometry of the two is identical; only the dimensions and colors are different. For this painting, #100 in the series, Johnson illustrates the subject vividly through the electric blue color of the rectangle with area equal to the square root of pi. Its partner, #89 in the series (1979.1093.58), displays the same rectangle in white, which contrasts brilliantly with its black and purple surroundings.

- This painting is unsigned and its precise date is unknown. It has a plain wooden frame.

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.67

- catalog number
- 1979.1093.67

- accession number
- 1979.1093

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

## Painting -

*Euclidian Values of a Squared Circle*- Description
- To "square" a figure, according to the classical Greek tradition, means to construct, with the aid of only straightedge and compass, a square equal in area to that of the figure. The Greeks could square numerous figures, but were unsuccessful in efforts to square a circle. It was not until the 19th century that the impossibility of squaring a circle was demonstrated.

- This painting is an original construction by Crockett Johnson. It begins with the assumprion that the circle has been squared. In this case, Crockett Johnson performed a sequence of constructions that produce several additional squares, rectangles, and circles whose areas are geometrically related to that of the original circle. These figures are produced using traditional Euclidean geometry, and require only straightedge and compass.

- The painting on masonite is #102 in the series. It has a blue-black background and a metal frame. It shows various superimposed sections of circles, squares, and rectangles in shades of light blue, dark blue, purple, white and blue-black. It is unsigned. See 1979.3083.02.13.

- References: Carl B. Boyer and Uta C. Merzbach,
*A History of Mathematics*(1991), Chapter 5.

- Crockett Johnson, "A Geometrical Look at the Square Root of Pi,"
*Mathematical Gazette*54 (February, 1970): pp. 59–60.

- Location
- Currently not on view

- date made
- ca 1970

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.69

- catalog number
- 1979.1093.69

- accession number
- 1979.1093

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

## Painting -

*Archimedes Transversal*- Description
- The construction of regular polygons using straightedge and compass alone is a problem that has intrigued mathematicians from ancient times. Crockett Johnson was particularly interested in the construction of regular seven-sided figures or heptagons, which require not only a compass but a marked straight edge. The mathematician Archimedes reportedly proposed such a construction, which was included in a treatise now lost. Relying heavily on Thomas Heath's
*Manual of Greek Mathematics*, Crockett Johnson prepared this painting.

- Archimedes had reduced the problem of finding a regular hexagon to that of finding two points that divided a line segment into two mean proportionals. He then used a construction somewhat like that of the painting to find a line segment divided as desired. Crockett Johnson's papers include not only photocopies of the relevant portion of Heath, but his own diagrams.

- The painting is #104 in the series. It is in acrylic or oil on masonite., and has purple, yellow, green and blue sections. There is a black wooden frame. The painting is unsigned and undated. Relevant correspondence in the Crockett Johnson papers dates from 1974.

- References: Heath, Thomas L.,
*A Manual of Greek Mathematics*(1963 edition), pp. 340–2.

- Crockett Johnson, "A construction for a regular heptagon,"
*Mathematical Gazette*, 59 (March 1975): pp. 17–18.

- Location
- Currently not on view

- date made
- ca 1974

- referenced
- Archimedes

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.71

- catalog number
- 1979.1093.71

- accession number
- 1979.1093

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

## Painting -

*Heptagon 1:3:3 Triangle*- Description
- This painting is part of Crockett Johnson's exploration of the properties of the heptagon, extended to include a 14-sided regular polygon. The design of the painting is shown in his figure, which includes many of the line segments in the painting. Here Crockett Johnson argues that the triangle ABF in the figure is the one he sought, with angle FAB being one seventh of pi. Segment CD in the figure, which appears in the painting, is the length of the edge of a regular 14-sided figure inscribed in a portion of the larger circle shown.

- The painting, of oil or acrylic on masonite, is number 105 in the series. It is drawn in shades of cream, blue, and purple on a light purple background. It has a metal frame and is unsigned.

- Location
- Currently not on view

- date made
- ca 1973

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.72

- catalog number
- 1979.1093.72

- accession number
- 1979.1093

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

## Painting -

*Heptagon Stated By Seven Toothpicks (Between Parallels)*- Description
- This whimsical painting is part of Crockett Johnson's exploration of ways to represent the sides and angles of a regular heptagon using line segments of equal length. In its mathematics, it follows closely the construction from isosceles triangles within a rhombus used in the painting
*Heptagon from Ten Equal Lines*(#104 in the series - 1979.1093.71). However, both the line segments shown and the appearance of the paintings are quite different.

- Here three pairs of carefully selected equal lines at appropriate equal angles combine with a seventh line of equal length to give a construction of three sides and two angles of a regular heptagon. All but one of the endpoints of the lines lie on a parallelogram (the rhombus mentioned previously), hence the title. The segment of the heptagon is on the right side of the painting. In Crockett Johnson's figure for the work, the segment is lettered BCPE.

- The painting, in oil or acrylic on masonite, is #106 in the series. It has a dark purple background. The pairs of line segments are in turquoise, green, and lavender, with the vertical one in white. This increases the drama of the painting, but obscures the heptagon. There is a wooden frame. The painting is signed on the back: HEPTAGON STATED BY (/) SEVEN TOOTHPICKS (/) (BETWEEN PARALLELS) (/) Crockett Johnson 1973.

- Location
- Currently not on view

- date made
- 1973

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.73

- catalog number
- 1979.1093.73

- accession number
- 1979.1093

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

## Painting -

*Heptagon from Ten Equal Lines*- Description
- This is one of a series of paintings in which Crockett Johnson explored ways of constructing the regular heptagon. The construction is his own, and a drawing for it is attached to the back of the painting. By an arrangement of ten equal line segments, he produced three sides and two angles of a regular heptagon. Two sides and one angle are actually shown in the painting.

- Crockett Johnson supposed that four equal isosceles triangles, constructed with six equal line segments, were arranged as shown in his figure to form sides of a rhombus and of a parallelogram within it. Two adjacent sides of the rhombus also served as the long sides of equal triangles oriented in the opposite direction. Finally, a line parallel to one of these sides passed through points of intersection of the sides of triangles.

- More specifically, in the drawing triangles BAF, DAR, DKE, and HBE are arranged within rhombus ABED, and around a central parallelogram. Two other equal triangles DES and BAG are also included. AFand EJ intersect at a point C and EK and BH at a point P. The tenth line, UL parallel to BE, passes through points C and P. Crockett Johnson claimed that BCPE represents three sides of a regular heptagon. His argument appears in his papers. The painting shows only the ten equal lines described in the title.

- The sections of the rhombus are in black, white, and rose, with a purple background. There is a wooden frame painted purple. This oil painting on masonite is #109 in the series. It is marked on the back: HEPTAGON FROM TEN EQUAL LINES (/) Crockett Johnson 1973. Taped to the back is a sheet of paper with an explanation that is entitled: HEPTAGON FROM TEN EQUAL LINES.

- Location
- Currently not on view

- date made
- 1973

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.75

- catalog number
- 1979.1093.75

- accession number
- 1979.1093

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