#
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 2381 items.

Page 3 of 239

## Blunt Reflecting Telescope

- Description
- This is a brass Gregorian telescope with a speculum metal mirror 3¾ inches diameter. The rod that runs along the tube serves to adjust the smaller secondary mirror. The “T. Blunt, London” signature refers to Thomas Blunt, an instrument maker who worked on his own from 1794 to 1823, or to his son of the same name, who worked for a year or two later.

- Location
- Currently on loan

- date made
- 1794-1824

- maker
- Blunt, Thomas

- ID Number
- PH*329777

- accession number
- 283516

- catalog number
- 329777

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

## Rutherfurd Diffraction Grating

- Description
- This is one of the earliest gratings made by Lewis M. Rutherfurd, and one of three that the pioneer astrophysicist, Henry Draper, acquired in the fall of 1872. The glass plate measures 1⅜ inches square and is marked "Nov. 19, 1872, 6480 per inch L. M. Rutherfurd."

- Ref. D. J. Warner, "Lewis M. Rutherfurd: Pioneer Astronomical Photographer and Spectroscopist,"
*Technology and Culture*12 (1971): 190-216.

- Location
- Currently not on view

- date made
- 1872

- maker
- Rutherfurd, Lewis Morris

- ID Number
- PH*334273

- accession number
- 304826

- catalog number
- 334273

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

## Felsenthal A-10 Protractor and Map Coordinator

- Description
- This clear plastic semicircular protractor is divided by ten mils and marked by hundreds from 100 to 3,100 in both the clockwise and counterclockwise directions. It is also divided by single degrees and marked by tens from 0° to 180° in both directions. Diagonal lines extend some of the measurement markings out to the edges of the rectangle surrounding the protractor. Pinholes are at the origin point and in the upper left and right corners. The interior of the protractor has cutout stencils for a circle, triangle, square, and two oblong shapes. The middle also contains scales placed at right angles to each other. They are divided and marked by hundreds from 1,000 to 2[00]. The scales are labeled: 1:21120.

- The left edge of the rectangular plastic piece is divided by tenths of an inch and marked by ones from 1 to 3. Inside the 3-inch ruler is a scale for mils divided by hundreds and marked by thousands from 5,000 to 1,000. The scale continues on the top of the rectangle, again divided by hundreds and marked by thousands from 5,000 to 1,000. The scale is labeled: 1:62500. On the right side of the top is a scale labeled: 1:20,000. It is divided and marked by hundreds from 1,000 to 2[00]. This scale also repeats on the right side of the rectangle. On the right edge of the rectangle, there is a scale divided by millimeters and marked by ones from 1 to 7. It is labeled: METRIC.

- The bottom of the protractor bears a scale divided by hundreds and marked by thousands from 1,000 to 8,000. It is labeled: 1:62500. The bottom edge has a second scale, divided by hundreds and marked by five hundreds from 500 to 3,000. It is labeled: 1:21120. The name of the instrument is printed on the very bottom edge: MAP COORDINATOR AND PROTRACTOR - A-10. Donor Ben Rau dated the object to 1942.

- See also 1977.1141.01, 1977.1141.02, 1977.1141.03, 1977.1141.05, 1977.1141.08, 1977.1141.09, 1977.1141.10, 1977.1141.11, 1977.1141.12, 1977.1141.18, 1977.1141.19, 1977.1141.20, 1977.1141.22, 1977.1141.23, 1977.1141.24, 1977.1141.30, and 1977.1141.39.

- Location
- Currently not on view

- date attributed by donor
- 1942

- maker
- Felsenthal

- ID Number
- 1977.1141.21

- accession number
- 1977.1141

- catalog number
- 336405

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

## Clark Refracting Telescope

- Description
- Alvan Clark & Sons were the leading telescope opticians in the United States in the second half of the 19th century. The firm came to prominence in 1865 when their 18½-inch refractor, then the largest in the world, was installed in the Dearborn Observatory in Chicago. Other notable achievements included the 26-inch telescope installed in the U.S. Naval Observatory in Washington, D.C., in 1873; the 30-inch objective lens installed in the Imperial Russian Observatory at Pulkowa in 1883; the 36-inch objective lens installed in the Lick Observatory in California in 1887; and the 40-inch objective lens installed in the Yerkes Observatory in Williams Bay, Wisconsin, in 1897.

- The Clarks also made many smaller instruments for investigation, education, and recreation. This example is marked "Alvan Clark & Sons 1894 Cambridgeport, Mass." It has a nickel-plated brass tube assembly, an objective lens of 5 inches aperture, an equatorial mount, and a wooden tripod.

- Ref: Deborah Jean Warner and Robert B. Ariail,
*Alvan Clark & Sons. Artists in Optics*(Washington, D.C., 1996).

- Location
- Currently not on view

- date made
- 1894

- maker
- Alvan Clark & Sons

- ID Number
- 1979.1017.01

- accession number
- 1979.1017

- catalog number
- 79.1017.1

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

## Painting -

*Proof of the Pythagorean Theorem (Euclid)*- Description
- The Pythagorean theorem states that in any right triangle, the square of the side opposite the right angle (the hypotenuse), is equal to the sum of the squares of the other two sides. This painting depicts the “windmill” figure found in Proposition 47 of Book I of Euclid’s
*Elements*. Although the method of the proof depicted was written about 300 BC and is credited to Euclid, the theorem is named for Pythagoras, who lived 250 years earlier. It was known to the Babylonians centuries before then. However, knowing a theorem is different from demonstrating it, and the first surviving demonstration of this theorem is found in Euclid’s*Elements*.

- Crockett Johnson based his painting on a diagram in Ivor Thomas’s article on Greek mathematics in
*The World of Mathematics*, edited by James R. Newman (1956), p. 191. The proof is based on a comparison of areas. Euclid constructed a square on the hypotenuse BΓ of the right triangle ABΓ. The altitude of this triangle originating at right angle A is extended across this square. Euclid also constructed squares on the two shorter sides of the right triangle. He showed that the square on side AB was of equal area to the rectangle of sides BΔ and Δ;Λ. Similarly, the area of the square on side AΓ was of equal area to the rectangle of sides EΓ and EΛ. But then the square of the hypotenuse of the right triangle equals the sum of the squares of the shorter sides, as desired.

- Crockett Johnson executed the right triangle in the neutral, yet highly contrasting, hues of white and black. Each square area that rests on the sides of the triangle is painted with a combination of one primary color and black. This draws the viewer’s attention to the areas that complete Euclid’s proof of the Pythagorean theorem.

*Proof of the Pythagorean Theorem*, painting #2 in the series, is one of Crockett Johnson’s earliest geometric paintings. It was completed in 1965 and is marked: CJ65. It also is signed on the back: Crockett Johnson 1965 (/) PROOF OF THE PYTHAGOREAN THEOREM (/) (EUCLID).

- Location
- Currently not on view

- date made
- 1965

- referenced
- Euclid

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.01

- catalog number
- 1979.1093.01

- accession number
- 1979.1093

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

## Painting -

*Squares of a 3-4-5 Triangle in Scalene Perspective (Dürer)*- Description
- This painting, while similar in subject to the painting entitled
*Perspective (Alberti)*, depicts three planes perpendicular to the canvas. These three planes provide a detailed, three-dimensional view of space through the use of perspective. Three vanishing points are implied (though not shown) in the painting, one in each of the three planes.

- The painting shows a 3-4-5 triangle surrounded by squares proportional in number to the square of the side. That is, the horizontal plane contains nine squares, the vertical plane contains sixteen squares, and the oblique plane, which represents the hypotenuse of the 3-4-5 triangle, contains twenty-five squares. This explains the extension of the vertical and oblique planes and reminds the viewer of the Pythagorean theorem. Thus, Crockett Johnson has cleverly shown the illustration of two of his other paintings;
*Squares of a 3-4-5 Triangle (Pythagoras)*and*Proof of the Pythagorean Theorem (Euclid)*, in perspective; hence the title of the painting.

- The title of this painting points to the role of the German artist Albrecht Dürer (1471–1528) in creating ways of representing three-dimensional figures in a plane. Dürer is particularly remembered for a posthumously published treatise on human proportion. In his book entitled
*The Life and Art of Albrecht Dürer*, art historian Erwin Panofsky explains that the work of Dürer with perspective demonstrated that the field was not just an element of painting and architecture, but an important branch of mathematics.

- This construction may well have originated with Crockett Johnson. The oil painting was completed in 1965 and is signed: CJ65. It is #8 in his series of mathematical paintings.

- Location
- Currently not on view

- date made
- 1965

- referenced
- Duerer, Albrecht

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.04

- catalog number
- 1979.1093.04

- accession number
- 1979.1093

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

## Painting -

*Conic Curve (Apollonius)*- Description
- In ancient times, the Greek mathematician Apollonius of Perga (about 240–190 BC) made extensive studies of conic sections, the curves formed when a plane slices a cone. Many centuries later, the French mathematician and philosopher René Descartes (1596–1650) showed how the curves studied by Apollonius might be related to points on a straight line. In particular, he introduced an equation in two variables expressing points on the curve in terms of points on the line. An article by H. W. Turnbull entitled "The Great Mathematicians" found in
*The World of Mathematics*by James R. Newman discussed the interconnections between Apollonius and Descartes, and apparently was the basis of this painting. The copy of this book in Crockett Johnson's library is very faintly annotated on this page. Turnbull shows variable length ON, with corresponding points P on the curve.

- The analytic approach to geometry taken by Descartes would be greatly refined and extended in the course of the seventeenth century.

- Johnson executed his painting in white, purple, and gray. Each section is painted its own shade. This not only dramatizes the coordinate plane but highlights the curve that extends from the middle of the left edge to the top right corner of the painting.

*Conic Curve*, an oil or acrylic painting on masonite, is #11 in the series. It was completed in 1966 and is signed: CJ66. It is marked on the back: Crockett Johnson 1966 (/) CONIC CURVE (APOLLONIUS). It has a wooden frame.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Apollonius of Perga

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.06

- catalog number
- 1979.1093.06

- accession number
- 1979.1093

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

## Painting -

*Square Root of Two (Descartes)*- Description
*La Géométrie*, one of the most important works published by the mathematician and philosopher René Descartes (1596–1650), includes a discussion of methods for performing algebraic operations using a straight edge and compass. One of the first is a way to determine square roots. This construction is the subject of Crockett Johnson's painting. Descartes explained: "If the square root of GH is desired, I add, along the same straight line, FG equal to unity, then bisecting FH at K, I describe the circle FIH about K as a center, and draw from G a perpendicular and extend it to I, and GI is the required root." (this is a translation of portion of*La Géométrie*, as published by J. R. Newman,*The World of Mathematics*(1956), p. 241)

- To understand Descartes' description and the title of this painting, consider the diagram. An angle inscribed in a semicircle is a right angle, thus triangle FGI is similar to triangle IGH. Because this two triangles are similar, their corresponding sides are proportional. Thus, G/IFG = GH/GI. But FG is equal to one, so GH is the square of GI, and GI the square root of GH desired.

- In his painting, Crockett Johnson has flipped the image from
*La Géométrie*found in his copy of*The World of Mathematics*. This figure is not annotated. The artist divided his painting into squares of area one, suggesting what came to be called Cartesian coordinates. The division indicates that the GH chosen has length two.

- Johnson chose white for the section of the semicircle that contains the edge of length equal to the square root of GH. This section provides a vivid contrast against the dull, surrounding colors. Crockett Johnson purposefully creates this area of interest to draw focus to the result of Descartes' construction.

*Square Root of Two*is painting #19 in the series. It was painted in oil or acrylic on masonite, completed in 1965, and is signed: CJ65. The wooden frame is painted black.

- Location
- Currently not on view

- date made
- 1965

- referenced
- Descartes, Rene

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.13

- catalog number
- 1979.1093.13

- accession number
- 1979.1093

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

## Painting -

*One Surface and One Edge (Möbius)*- Description
- Most geometric surfaces have a distinct inside and outside. This painting shows one that doesn’t. Take a strip of material, give it a half-twist, and attach the ends together. The result is a band with only one surface and one edge. Mathematicians began to explore such surfaces in the nineteenth century. In 1858 German astronomer and mathematician August Ferdinand Möbius (1790–1868), who had studied theoretical astronomy under Carl Friedrich Gauss at the University of Goettingen, discovered the one-sided surface shown in the painting. It has come to be known by his name. As often happens in the history of mathematics, another scholar, Johann Benedict Listing, had found the same result a few months earlier. Listing did not publish his work until 1861.

- If one attaches the ends of a strip of paper without a half twist, the resulting figure is a cylinder. The cylinder has two sides such that one can paint the outside surface red and the inside surface green. If you try to paint the outside surface of a Möbius band red you will paint the entire band red without crossing an edge. Similarly, if you try to paint the inside surface of a Möbius band green you will paint the entire surface green. A cylinder has an upper edge and a lower edge. However, if you start at a point on the edge of a Möbius band you will trace out its entire edge and return to the point at which you began. Since Möbius's time, mathematicians have discovered and explored many other one-sided surfaces.

- This painting, #34 in the series, was executed in oil on masonite and is signed: CJ65. The strip is shown in three shades of gray based on the figure’s position. The shades of gray, especially the lightest shade, are striking against the rose-colored background, and this contrast allows the viewer to focus on the properties of the Möbius band. The painting has a wooden frame.

- Crockett Johnson's painting is similar to illustrations in James R. Newman's
*The World of Mathematics*(1956), p. 596. However, the figures are not annotated in the artist's copy of the book.

- Location
- Currently not on view

- date made
- 1965

- referenced
- Moebius, August Ferdinand

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.23

- catalog number
- 1979.1093.23

- accession number
- 1979.1093

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

## Painting -

*Simple Equation (Descartes)*- Description
- In a pathbreaking book
*La Géométrie*, René Descartes (1596–1650) described how to perform algebraic operations using geometric methods. One such explanation is the subject of this Crockett Johnson painting. More specifically, Descartes described geometrical methods for finding the roots of simple polynomials. He wrote (as translated from the original French): "Finally, if I have z² = az -b², I make NL equal to (1/2)a and LM equal to b as before: then, instead of joining the points M and N, I draw MQR parallel to LN, and with N as center describe a circle through L cutting MQR in the points Q and R; then z, the line sought, is either MQ or MR, for in this way it can be expressed in two ways, namely: z = (1/2)a + √((1/4)a² - b²) and z = (1/2)a - √((1/4)a² - b²)."

- To verify that z = MR is a solution to the equation z²= az - b², note that the square of the length of the tangent ML equals the product of the two line segments MQ and MR. As ML is defined to equal b, its square is b squared. The length of MR is z, and the length of MQ is the difference between the diameter of the circle (length a) and the segment MR, that is to say (a – z) . Hence b squared equals z (a – z) which, on rearrangement of terms, gives the result desired.

- Crockett Johnson's painting directly imitates Descartes's figure found in Book I of
*La Géométrie*. A translation of part of Book I is found in the artist’s copy of James R. Newman's*The World of Mathematics*. The figure on page 250 is annotated.

- This oil or acrylic painting on masonite is #36 in the series. It was completed in 1966 and is signed: CJ66. It has a wooden frame.

- Location
- Currently not on view

- date made
- 1966

- referenced
- Descartes, Rene

- painter
- Johnson, Crockett

- ID Number
- 1979.1093.24

- catalog number
- 1979.1093.24

- accession number
- 1979.1093

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