#
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 42 items.

Page 1 of 5

## 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 -

*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 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 the Heptagon*- Description
- Three very similar paintings in the Crockett Johnson collection are closely related to 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 shown in the image. The horizontal line segment below the circle on the painting corresponds to unit length BF in the figure, and the triangle is ABF. The light colors of the painting highlight important points in the construction - marking off an arc of radius equal to the square root of 2 with center F, measuring the unit length AO along a marked straight edge that passes through B and ends at point A on the perpendicular bisector, and finding the side of the regular inscribed heptagon.

- This version of the construction of a heptagon is #108 in the series. The oil painting on masonite with chrome frame was completed in 1975 and is unsigned. It is marked on the back: Construction of the Heptagon (/) Crockett Johnson 1975. See also paintings #115 (1979.1093.77) and #117 (1979.1093.79) in the series.

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

- Location
- Currently not on view

- date made
- 1975

- painter
- Johnson, Crockett

- ID Number
- MA*335571

- accession number
- 322732

- catalog number
- 335571

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

## Lasico L30-A Electronic Polar Compensating Planimeter Prototype

- Description
- This metal prototype for an electronic polar planimeter has an adjustable 12" tracer arm with lens. The top of the arm is divided to millimeters and numbered from 10 to 24 centimeters. The bottom is marked with a serial number: 45254. The arm slides into a painted metal holder for an electronic measuring unit with a plug. The holder has a vernier for the scale on the tracer arm and is marked: LASICO. The plug attaches to a Series 40 processor with a digital screen for displaying the measurement and a knob for setting the instrument to OFF, A, ACCU, or B. An AC adapter by Calrad, a Taiwanese company, powers the processor.

- An adjustable 10" pole arm fits into the holder at one end and a rectangular painted metal pole weight at the other end. The weight is marked: LASICO (/) U.S.A. The arm is divided to millimeters and numbered by tens from 30 to 60 millimeters. The adjusting part of the arm is marked: LASICO. An additional tracer arm with a point instead of a lens has serial number: 45275. A business card for the designer, who also donated the instrument, an extra lens, and two plastic washers for the lens are inside a black plastic case lined with foam.

- Maximilian Berktold (b. 1929) immigrated from Kempten-Allgäu, West Germany, in 1950 and almost immediately began working for the Los Angeles Scientific Instrument Company. He oversaw design and production of the firm's planimeters, integrators, pantographs, and various optical instruments until Lasico closed in 2008. He developed this prototype around 1970 from the company's model L30 mechanical planimeter, but the final version was sold as model series 40 and 50. These devices cost several hundred dollars.

- An 18-page booklet, "LASICO Instruction Manual [for] Digital Compensating Polar Planimeters," was received with the instrument. It contains the calibration settings for a model L50-E, serial number 65879. For company history, see 2011.0043.01.

- Reference: Accession file.

- Location
- Currently not on view

- date made
- 1970

- maker
- Los Angeles Scientific Instrument Company

- ID Number
- 2011.0043.03

- accession number
- 2011.0043

- catalog number
- 2011.0043.03

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

## Sterling Metric Converter Slide Rule

- Description
- This ten-inch, one-sided plastic rule has a yellow base, a white slide, and a transparent indicator. Identical logarithmic scales are on the top and the bottom of the base. Both sides of the slide are marked with pairs of metric and conventional units. On one side, the user can read off conversions between: inches and centimeters; meters and feet; meters and yards; miles and kilometers; square inches and square centimeters; square meters and square feet (times ten); square meters and square yards; and square miles and square kilometers. The other side of the slide permits readings of cubic inches and cubic centimeters (times ten); cubic meters and cubic feet (times ten); cubic meters and cubic yards; liters and quarts; ounces and grams (times ten); kilograms and pounds; metric tons and short tons; and gallons and liters.

- The top left of the base is marked with the letters SP in a circle and the word STERLING. The top middle of the base is marked: METRIC CONVERTER. The bottom left of the base is marked: MADE IN U.S.A. The rule was received with its original packaging, a clear plastic cover on a blue paper backing. The packaging is marked at the top: SP STERLING #651 (/) metric (/) converter. At the bottom, it is marked: BORDEN ® (/) © 1972 STERLING PLASTICS (/) DIVISION OF BORDEN CHEMICAL, BORDEN INC. (/) MOUNTAINSIDE, N.J. 07092 (/) MADE IN U.S.A.

- Sterling Plastics, a 20th-century manufacturer of drawing instruments for schools, was purchased by Borden Chemical in 1970. Since Sterling stopped making slide rules in 1972, this example of model number 651 was probably one of the last rules produced by the company. The five braces holding together the base of the instrument are also consistent with this date; early Sterling slide rules had only two braces. For instructions, see 1990.0689.03. For a Sterling slide rule with standard scales, see 1988.0807.01.

- Reference: Mike Konshak, "Sterling Plastics," http://sliderulemuseum.com/Sterling.htm.

- Location
- Currently not on view

- date made
- 1972

- maker
- Sterling Plastics

- ID Number
- 1990.0689.01

- accession number
- 1990.0689

- catalog number
- 1990.0689.01

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

## Impact Metric Converter Slide Rule for Regal Beloit

- Description
- Around 1970 many American companies and government agencies encouraged Americans to adopt the metric system. Regal Beloit of Wisconsin and other manufacturers of cutting tools and gear boxes adopted the units of measure and distributed devices like this one to assist in their use.

- The one-sided white cardboard rule is printed in orange and black and has eight windows. Two logarithmic scales on the slide are viewed through four of the windows so that the user can convert between yards or feet and meters; centimeters and inches; pounds and kilograms; and tons and metric tons. Two more logarithmic scales on the slide permit conversions between square yards and square meters; square centimeters and square inches; cubic yards and cubic meters; and liters and imperial gallons or U.S. gallons. Below the windows is a scale for converting between Celsius and Fahrenheit temperatures. The rule is marked: REGAL BELOIT. It is also marked metric/inch (/) CONVERTER. It is also marked SWANI PUBLISHING COMPANY (/) P.O. Box 284 • Roscoe, Illinois 61073 (/) 815 / 389-3065.

- The back of the rule has small windows for reading conversions between fractional inches, decimal inches, and millimeters from columns of numbers printed on the slide. Tables of equivalents appear above more windows for reading conversions between inches and centimeters and miles and kilometers. After another table of prefixes and equivalents, instructions for using this side of the rule are provided. More small windows permit conversions between U.S. gallons and liters and cubic feet and cubic meters. At the bottom, the rule is marked: DISTRIBUTED BY (/) C-6862. The back of the slide is marked ©1971, IMPACT, Culver City, Callf. (/) Printed in U.S.A.

- Impact was presumably a printing company. Swani was a division of Regal Beloit that published a few elementary textbooks on the metric system. Compare this rule to 1990.0689.01.

- Location
- Currently not on view

- date made
- 1971

- maker
- Impact

- ID Number
- 1990.3231.01

- nonaccession number
- 1990.3231

- catalog number
- 1990.3231.01

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

## Datalizer Universal English/SI Converter Slide Rule

- Description
- This white rectangular cardboard rule is held together with four metal rivets. The front has logarithmic scales for making measurement conversions for length, mass, area, and volume. A chart for converting temperatures runs along the bottom edge. The top is marked: THE NEWS (/) NEW YORK'S PICTURE NEWSPAPER Universal ENGLISH/SI (metric) Datalizer. The bottom is marked: © 1976
*datalizer*by DATALIZER SLIDE CHARTS, INC., Addison, IL 60101 PRINTED IN U.S.A. FORM NO. EM2.

- The back has charts for converting between cubic meters and cubic feet; gallons and liters; miles and kilometers; inches and centimeters; and fractional inches and millimeters. A table of miscellaneous conversions appears in the center of the back.

- Around 1960 a former employee of the Perry Graf Corporation (see 1979.3074.03) established Datalizer in Addison, a Chicago suburb. The company made this promotional rule during a time of considerable interest in adopting the metric system in the United States. The
*New York Daily News*used the "picture newspaper" slogan between 1920 and 1991.

- References: "Slide Chart Specialists,"
*Datalizer Slide Charts*, http://www.datalizer.com/about-us/; Lance Gould, "The Lenses And Legacy Of New York's Picture Newspaper,"*New York Daily News*, January 25, 2002, http://articles.nydailynews.com/2002-01-25/entertainment/18195567_1_photographers-gallery-exhibit-exposed.

- Location
- Currently not on view

- Currently not on view

- date made
- 1976

- maker
- Datalizer Slide Charts, Incorporated

- ID Number
- 1981.0922.15

- catalog number
- 1981.0922.15

- accession number
- 1981.0922

- 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