#
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 3 of 219

## Painting -

*Star Construction*- Description
- Crockett Johnson's interest in regular polygons included the pentagram, or five-pointed star. The relation between the pentagon and the star is simple. If each side of a regular pentagon is extended, a regular five-pointed star results. Similarly, connecting each diagonal of a regular pentagon creates a regular five-pointed star. The star will have a pentagon in it, so the method is self-perpetuating.

- A method for a pentagram's construction in described in Book IV, Proposition II of Euclid's
*Elements*, but the construction illustrated in this painting is the artist's own creation. It builds on the relationship between the sides of a regular five-pointed star and the golden ratio. As Crockett Johnson may have recalled from his earlier paintings, the five rectangles that surround the central pentagon of the star are golden, that is to say the ratio of the length of the two equal sides of the triangle to the side of the enclosed pentagon is (1 + √5) / 2. Hence one can construct the star by finding a line segment divided in this ratio. No figure by Crockett Johnson showing his construction has been found.

- The pentagram, executed appropriately enough in hues of gold, contrasts vividly with the purple background in
*Star Construction*.

- The painting is #103 in the series. It is in oil or acrylic on pressed wood and has a gold-colored metal frame. The painting is unsigned and undated. Compare #46 (1979.1093.33) and #64 (1979.1093.39).

- Location
- Currently not on view

- date made
- 1970-1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.70

- catalog number
- 1979.1093.70

- accession number
- 1979.1093

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

## Painting -

*Heptagon from Its Seven Sides*- Description
- Toward the end of his life, Crockett Johnson took up the problem of constructing a regular seven-sided polygon or heptagon. This construction, as Gauss had demonstrated, requires more than a straight edge and compass. Crockett Johnson used compass and a straight edge with a unit length marked on it. Archimedes and Newton had suggested that constructions of this sort could be used to trisect the angle and to find a cube with twice the volume of a given cube, and Crockett Johnson followed their example.

- One may construct a heptagon given an angle of pi divided by seven. If an isosceles triangle with this vertex angle is inscribed in a circle, the base of the triangle will have the length of one side of a regular heptagon inscribed in that circle. According to Crockett Johnson's later account, in the fall of 1973, while having lunch in the city of Syracuse on Sicily during a tour of the Mediterranean, he toyed with seven toothpicks, arranging them in various patterns. Eventually he created an angle with his menu and wine list and arranged the seven toothpicks within the angle in crisscross patterns until his arrangement appeared as is shown in the painting.

- Crockett Johnson realized that the vertex angle of the large isosceles triangle shown is exactly π/7 radians, as desired. The argument suggested by his diagram is more complex than what he later published. The numerical results shown in the figure suggest his willingness to carry out detailed calculations.

*Heptagon from its Seven Sides*, painted in 1973 and #107 in the series, shows a triangle with purple and white sections on a navy blue background. This oil or acrylic painting on masonite is signed on its back : HEPTAGON FROM (/) ITS SEVEN SIDES (/) (Color sketch for larger painting) (/) Crockett Johnson 1973. No larger painting on this pattern is at the Smithsonian.

- Reference: Crockett Johnson, "A Construction for a Regular Heptagon," Mathematical Gazette, 1975, vol. 59, pp. 17–21.

- Location
- Currently not on view

- date made
- 1973

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.74

- catalog number
- 1979.1093.74

- accession number
- 1979.1093

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

## Painting -

*Hippias' Curve*- 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

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

## Painting -

*Construction of a Heptagon*- Description
- This is one of three very similar Crockett Johnson paintings closely related to the construction of a side of an inscribed regular heptagon which the artist published in
*The Mathematical Gazette*in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π / 7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon.

- Crockett Johnson described the construction of his isosceles triangle in the diagram shown. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the largest triangle in the painting is triangle is ABF in the figure, with vertex angle equal to one seventh of pi. This angle is inscribed in the large circular arc KDC. The side of the heptagon is the chord KC.

- This version of Crockett Johnson's construction of a heptagon is #115 in the series. It has a dark blue background and a wood and metal frame. The painting is an oil or acrylic on masonite. The work is unsigned. See also #108 (335571) and #117 (1979.1093.79).

- References: Crockett Johnson, “A Construction for a Regular Heptagon,”
*Mathematical Gazette*, 1975, vol. 59, pp. 17–21.

- Location
- Currently not on view

- date made
- ca 1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.77

- accession number
- 1979.1093

- catalog number
- 1979.1093.77

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

## Painting -

*Construction of Heptagon*- Description
- This painting represents one of Crockett Johnson's early constructions of a heptagon. It shows a large purple circle, a pink triangle superimposed, and two smaller circles. Crockett Johnson's diagram for the painting is shown. Two equal circles are constructed, with the center of the first on the second and conversely (circles with centers C and D in the diagram), and a line segment drawn that includes their points of intersection. Then, in Crockett Johnson's words, "Against a straight edge controlling their alignment the sought points B, U, and E, are determined by the adjustment of compass arcs BC from U and EC from B. Angles FBC, CBD, DBE, and BAF are π/ 7." Detailed examination of the triangles in the drawing shows that this is indeed the case.

- The colors of the painting highlight the circles, lines, and arcs central to the construction, and the largest of the resulting isosceles triangles with vertex angle π/7 is shown in bold shades of pink. The short line called CF in the drawing (as well as line segments CD and DE, which are not shown), is the length of the side of a heptagon inscribed in a circle centered at B with radius BF.

- The oil on masonite work is #116 in the series. It has a gray background and a wood and metal frame. It is inscribed on the back: CONSTRUCTION OF HEPTAGON (/) . . .(8) (/) Crockett Johnson 1973.

- Location
- Currently not on view

- date made
- 1973

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.78

- accession number
- 1979.1093

- catalog number
- 1979.1093.78

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

## Painting -

*Construction of Heptagon*- Description
- Three very similar paintings in the Crockett Johnson collection are closely related to the the construction of a side of an inscribed regular a heptagon which he published in
*The Mathematical Gazette*in 1975. The paper presents a way of producing an isosceles triangle with angles in the ratio 3:3:1, so that the smallest angle in the triangle is π/ 7. This angle is then inscribed in a large circle, and intercepts an arc length of π/7. A central angle of the same circle intercepts twice the angle, that is to say 2π/7, and the corresponding chord the side of an inscribed heptagon. Crockett Johnson described the construction of his isosceles triangle in the diagram reproduced. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the triangle is ABF. Three of the four light-colored sections of the painting highlight important points in the construction. The critical steps are drawing a perpendicular bisector to the line segment BF, marking off an arc of radius equal to the √(2) with center F, and measuring the unit length AO along a marked straightedge that passes through B and intersects the perpendicular bisector at A. Finally, one finds the side of the regular inscribed heptagon.

*Construction of Heptagon*is #117 in the series. The oil painting on masonite is in shades of purple, cream, turquoise, and black. It has a black wood and metal frame. The work is unsigned. The surface appears damaged, perhaps from water. See also #115 (1979.1093.77) and #108 (335571).

- Reference: Crockett Johnson, “A Construction for a Regular Heptagon,”
*Mathematical Gazette*, 1975, vol. 59, pp. 17–21.

- Location
- Currently not on view

- date made
- ca 1975

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.79

- accession number
- 1979.1093

- catalog number
- 1979.1093.79

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

## Dollond Refracting Telescope with Divided Glass Micrometer

- Description
- This brass telescope has an achromatic objective of 2.75 inches aperture, a finder, several eyepieces, two telescoping braces, two control rods, a split objective micrometer, a brass equatorial mount, and a wooden tripod. The tube is 43.5 inches long. The faceplate at the eye end is marked “DOLLOND * LONDON.” For storage, the telescope fits into a mahogany box with a hinged lid.

- This seems to be an example of the brass telescope “of 3½ feet focal length, with an aperture of 2¾ inches, two eye tubes for Land Objects, and two tubes for Astronomical purposes” that George Dollond was offering in the 1830s. That instrument could be had with either a brass stand for use on a table, or “a mahogany folding stand, to be used on the Floor.” It could, moreover, be “supported in the centre of Gravity, and applied to a socket that may be turned to any latitude, so that the Telescope may have an Equatorial Motion” The complete outfit cost £50. The micrometer would be extra.

- The Dollond family began working as opticians in London in 1750, and gained fame in 1758 when John Dollond introduced his design for achromatic lenses. John Dollond was also responsible for the split objective micrometer.

- Ref: “A Description of a Contrivance for Measuring Small Angles, by Mr. John Dollond; Communicated by Mr. J. Short, F.R.S.,”
*Philosophical Transactions of the Royal Society of London*48 (1753): 178-181.

- “An Explanation of an Instrument for measuring small Angles, the first Account of which was read before the Royal Society May 10, 1753. By Mr. John Dollond,”
*Philosophical Transactions of the Royal Society of London*48 (1754): 551-564.

- George Dollond,
*A Catalogue of Optical, Mathematical, Philosophical Instruments*(London, ca. 1830).

- Gloria Clifton, “Dollond Family,” in
*Oxford Dictionary of National Biography*.

- Location
- Currently not on view

- date made
- probably 1760s

- maker
- Dollond

- ID Number
- 1979.1110.01

- accession number
- 1979.1110

- catalog number
- 79.1110.1

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

## Semicircular Protractor Retailed by Post

- Description
- This semicircular brass protractor is divided by single degrees and marked by tens from 0° to 180° in both the clockwise and counterclockwise directions. The bottom edge is indented so that a pencil or pricker may be placed at the origin point. The protractor bears three marks: POSTS; MADE IN GERMANY; and a fleur-de-lis pointing to the origin. The Frederick W. Post Company, a Chicago mathematical instruments dealer established in 1893, used the type style found on this protractor in the 1920s and 1930s. However, the brass protractor depicted in Post's 1936 catalog is not indented on its lower edge, and it shows an eagle under the maker's mark.

- William J. Ellenberger (1908–2008) donated this object. He studied electrical and mechanical engineering at the George Washington University between 1925 and 1934. He then worked for the Potomac Electric Power Company and the National Bureau of Standards. During World War II, Ellenburger served in the U.S. Army Signal Corps. He was a civilian construction management engineer for the army from 1954 to 1968, when he became a private consultant.

- References: Frederick W. Post Company,
*Dependable Drawing Materials*, 18th ed. (Chicago, 1936), 195; "The G[eorge] W[ashington] Engineering Hall of Fame 2006 Inductees," http://www.seas.gwu.edu/ifaf/hall_of_fame_inductees_2006.php.

- Location
- Currently not on view

- date made
- 1920-1940

- maker
- Frederick Post Co.

- ID Number
- 1981.0933.18

- accession number
- 1981.0933

- catalog number
- 1981.0933.18

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

## Semicircular Protractor

- Description
- This semicircular brass protractor is divided by single degrees and marked by tens from 10° to 170° in both the clockwise and counterclockwise directions. The origin point is marked with an arrow. A 2-1/2 inch ruler, divided by sixteenths of an inch, runs along the lower edge of the protractor. The letters U.S.A. are engraved above the right end of the ruler. No maker's mark is present.

- Location
- Currently not on view

- date made
- ca 1930

- ID Number
- 1981.0933.19

- accession number
- 1981.0933

- catalog number
- 1981.0933.19

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

## Keuffel & Esser Model 1868 Semicircular Protractor

- Description
- By the 20th century, the makers of drawing instruments transitioned to new and inexpensive materials, particularly for objects intended for student use. This semicircular xylonite protractor is divided by half-degrees and marked by tens from 0° to 180° in both the clockwise and counter-clockwise directions. An upside-down T marks the origin point. The maker's mark forms a circle to the left of the origin point: KEUFFEL & ESSER Co (/) N.Y. Above the origin point is the K&E product number (1868) and diameter in inches (6). The company logo, the top half of an eagle, is to the right of the origin point. Underneath the logo are the words TRADE MARK. There is a hole just above the maker's mark to the left of the origin point. Perhaps the hole enabled the owner to store the protractor in a notebook.

- K&E marketed this protractor at least as early as 1909. It was intended to replace protractors made of horn. By 1936, K&E gave the instrument a new product number, 1276-6. This example of the protractor is yellowed and slightly warped.

- Alfred John Betcher (1887–1971) used this protractor, perhaps at the University of Minnesota (1906) or at West Point (1907–1911). He was commissioned as a captain, served at posts in New York, Vermont, and Kentucky, and retired in 1939 at the rank of lieutenant-colonel. In 1940, he was elected mayor of Canajoharie, N.Y.

- Reference:
*Catalogue of Keuffel & Esser*(New York, 1909), 214. Biographical information in accession file.

- Location
- Currently not on view

- date made
- 1909-1936

- maker
- Keuffel & Esser Co.

- ID Number
- 1982.0386.05

- accession number
- 1982.0386

- catalog number
- 1982.0386.05

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