#
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 5 of 219

## Model of a Riemann Surface by Richard P. Baker, Baker #405wn

- Description
- This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

- The mark "405 w" is carved into one edge of the wooden base of this model and the typed part of a paper label on the base reads: No. 405 zn (/) Riemann surface : w
^{2}= z^{3}- z. The label should have read "405 wn" and someone added a handwritten question mark after the "zn." Model 405wn is listed on page 17 of Baker’s 1931 catalog of models as “w^{2}= z^{3}- z” under the heading*Riemann Surfaces*. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w^{2}= z^{3}- z. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. The spheres of the model are called complex, or Riemann, spheres. They are formed with north and south poles representing ∞ and 0, respectively. The equator represents the points x + yi with x^{2}+ y^{2}= 1. Thus the points ±1 and ±i are equally spaced along the equator. Riemann surfaces and spheres are named after the 19th-century German mathematician Bernhard Riemann.

- Baker explains in his catalog that the wn after the number of the model indicates that the model is made up of spheres representing w-values. These spheres are called the sheets of the model. There is no part of this model in which values of z or pairs (z,w) are represented. However, it is possible that the coloring on this model is related to the painted part of the wooden base of one of three other Baker models of Riemann surfaces that are associated with the equation of this model.

- If w = ±
^{4}√ (4/27) or w = ±i^{4}√ (4/27), the equation w^{2}= z^{3}- z is satisfied by two distinct values of z. For the two real values of w, ±^{4}√ (4/27), z takes the values –1/√3 and 2/√3, and for the two imaginary values of w, ±i^{4}√ (4/27), z takes the values 1/√3 and –2/√3. These four points together with the point z = ∞ are called branch points of the model and for all other points on the w-sphere the equation w^{2}= z^{3}- z is satisfied by three distinct values of z, each of which produces a different pair on the Riemann surface (if w = 0, the three distinct pairs on the Riemann surface are (0,0) and (±1,0)). Thus there are three sheets representing the complex w-sphere and together they represent what is called a branched cover of the complex w-sphere.

- On each of the sheets the equator is a thin circle and there are two great circles through the poles. On one of the great circles the values of w are purely imaginary while on the other they are real. Baker’s usual use of colors implies that the great circles facing the front and back represent imaginary numbers, while those facing the sides represent real numbers. Normally w = ∞ is at the north pole and w = 0 is at the south pole. However, as the four finite branch points of this model lie in the northern hemisphere, it appears that this model has that assignment of values reversed. The great circle facing the front and back has a thick white segment that connects the two imaginary branch points by way of w = ∞ at the south pole, while the other has a thick black segment connecting the two real branch points by way of w = 0 at the north pole. The parts of the great circles that connect two branch points are called branch cuts. This model has three, one is the black arc mentioned above and the others are the two halves of the white arc with ends at an imaginary branch point and infinity. These branch cuts are curves on a sheet that produce movement to another sheet. This movement occurs when meeting a branch cut while following a path of the inputs of w values into the equation.

- There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, a model with Baker’s no. 405w (MA*211257.068), has “405w” carved on the base Two others, Baker’s no. 405z (MA*211257.070) and Baker’s no. 405zn (MA*211257.071), both have the mark “405z“ carved on them. Baker carved a "w" or a "z" to indicate which variable is represented on the sheets of the model and added an "n" after the "w" or "z" to indicate that the sheets of the model are spheres.

- Location
- Currently not on view

- date made
- ca 1915-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.069

- accession number
- 211257

- catalog number
- 211257.069

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

## Model of a Riemann Surface by Richard P. Baker, Baker #405z

- Description
- This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

- The mark 405 z is carved into one edge of the wooden base of this model and the typed part of a paper label on the base reads: No. 405 w (/) Riemann surface : w
^{2}= z^{3}- z. Someone corrected the error on the label by hand, crossing out the w and inserting a z. Model 405z is listed on page 17 of Baker’s 1931 catalogue of models as “w^{2}= z^{3}- z” under the heading*Riemann Surfaces*. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w^{2}= z^{3}- z. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. A complex plane is like the usual real Cartesian plane but with the horizontal axis representing the real part of the number and the vertical axis representing the imaginary part of the number. Riemann surfaces are named after the 19th-century German mathematician Bernhard Riemann.

- Baker explains in his catalogue that the z after the number of the model indicates that the metal disks above the wooden base represent copies of a disk in the complex z-plane. These disks are called the sheets of the model. The painted part of the wooden base of the model represents a square in the complex w-plane with the point w = 0 at its center and the real axis along the line between the yellow and dark green stripes.

- If z = 0 or z = ±1, the equation w
^{2}= z^{3}- z is satisfied by only one value of w, i.e., w = 0. These three points on the z-plane are called branch points of the model and for all other points on the z-plane the equation w^{2}= z^{3}- z is satisfied by two distinct values of w, each of which produces a different pair on the Riemann surface (if z = 2, the two distinct pairs on the Riemann surface are (2, ±√6)). Thus there are two sheets representing the same disk in the complex z-plane and together they represent part of what is called a branched cover of the complex z-plane. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base in which the w coordinate lies.

- For each sheet, the center of the disc is the point z = 0 and the solid black line through that point is the real axis. The branch points of this model all lie on the real axis. The point z = –1 is the point inside the green and yellow oval where the real axis meets the small red circle representing the unit circle with center z = 0. The point z = 1 is the other point where the real axis meets the small red circle; it is inside the oval that includes all eight colors used in the model.

- The vertical surfaces between the two sheets are not part of the Riemann surface but call attention to what are called branch cuts of the model, i.e., curves on a sheet that produce movement to another sheet. This movement occurs when meeting a branch cut while following a path of the inputs of z values into the equation. While the defining equation determines the branch points, the branch cuts are not fixed by the equation but, normally, each branch cut goes through two of the surface’s branch points or runs out to infinity. In this model, one of the branch cuts connects z = 0 to z = 1 and the other runs from z = –1 to infinity; they are represented by the horizontal edges of the vertical surfaces.

- There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, a model with Baker's number 405zn (MA*211257.071), has "405z" carved on the base. Two others, one with Baker's number 405w (MA*211257.068) and the other with Baker's number 405wn (MA*211257.069) have "405w" carved on the edge of the base. Baker carved a "z" or a "w" to indicate which variable is represented on the sheets of the model and added an "n" after the "z" or "w" to indicate that the sheets of the model are spheres.

- Location
- Currently not on view

- date made
- ca 1915-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.070

- accession number
- 211257

- catalog number
- 211257.070

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

## Model of a Riemann Surface by Richard P. Baker, Baker #405zn

- Description
- This geometric model was constructed by Richard P. Baker in about 1930 when he was Associate Professor of Mathematics at the University of Iowa. Baker believed that models were essential for the teaching of many parts of mathematics and physics, and over one hundred of his models are in the museum collections.

- The mark "405 z" is carved into one edge of the wooden base of this model and the typed part of a paper label on the base reads: No. 405 wn (/) Riemann surface : w
^{2}= z^{3}- z. The label is incorrect and should read "405 zn". Model 405zn is listed on page 17 of Baker’s 1931 catalogue of models as “w^{2}= z^{3}- z” under the heading*Riemann Surfaces*. This means that the model represents a Riemann surface consisting of pairs of complex numbers, (z,w), for which w^{2}= z^{3}- z. Complex numbers are of the form x + yi for x and y real numbers and i the square root of –1. The spheres of the model are called complex, or Riemann, spheres. They are formed with north and south poles representing ∞ and 0, respectively. The equator represents the points x + yi with x^{2 + y}2 = 1. Thus the points ±1 and ±i are equally spaced along the equator. Riemann surfaces and spheres are named after the 19th-century German mathematician Bernhard Riemann.

- Baker explains in his catalog that the zn after the number of the model indicates that the model is made up of spheres representing z-values. These spheres are called the sheets of the model. It appears as if painted part of the wooden base of the model represents the Riemann surface as a torus, i.e., a donut, formed by pasting together the ends of the stripes to form a cylinder and then joining the ends of the cylinder.

- If z = 0 or z = ±1, the equation w
^{2}= z^{3}- z is satisfied by only one value of w, i.e., w = 0. These three points together with the point z = ∞ are called branch points of the model and for all other points on the z-sphere the equation w^{2}= z^{3}- z is satisfied by two distinct values of w, each of which produces a different pair on the Riemann surface (if z = 2, the two distinct pairs on the Riemann surface are (2, ±√6)). Thus there are two sheets representing the complex z-sphere and together they represent what is called a branched cover of the complex z-sphere. The color of a region on a sheet is chosen with the aim of indicating the stripe on the base into which it is mapped.

- On each of the sheets the equator is colored red and there are great circles through the poles that are colored yellow and black. The points on the yellow great circle are purely imaginary while those on the black great circle are real. Thus the real non-zero branch points, z = ±1, lie on the equator and on the black great circle, while the other two branch points are at the north and south poles. The darkened parts of the black great circle are called branch cuts. Assuming the pair (1,0) lies on the Riemann surface along edge shared by the center (yellow and green) stripes on the base and that the pair (–1,0) lies along the edges of the outer stripes on the base, one of the branch cuts runs between join z = 0 and z = 1 and other between z = –1 and z = ∞. These branch cuts are curves on a sheet that produce movement to another sheet. This movement occurs when meeting a branch cut while following a path of the inputs of z values into the equation. Thus one can construct the Riemann surface as a torus by cutting the spheres along the branch cut and sewing the two spheres together along those cuts while matching the four branch points.

- There are three other models of Riemann surfaces in the collections that are associated with the equation of this model. One, with Baker's number 405z (MA*211257.070) has "405z" carved on the base. Two others, Baker's number 405 w (MA*211257.068) and Baker's number 405wn (MA*211257.069) have the mark "405w" on the base. Baker carved a "z" or a "w" to indicate which variable is represented on the sheets of the model and added an "n" after the "z" or "w" to indicate that the sheets of the model are spheres.

- Location
- Currently not on view

- date made
- ca 1915-1935

- maker
- Baker, Richard P.

- ID Number
- MA*211257.071

- accession number
- 211257

- catalog number
- 211257.071

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

## Geometric Model by A. Harry Wheeler, One-Sided Polyhedron

- Description
- In the late 1930s and early 1940s, A. Harry Wheeler took great interest in polyhedra with interpenetrating sides, such as had been discussed by the German mathematician August F. Moebius. In this example, each of the two like-colored quadrilaterals (e.g. the two yellow sides) on the top pass through the model and appear as a white quadrilateral on the bottom. These three figures thus contribute only one side to the polygon.

- A mark on the model reads: 695. This was Wheeler’s number for the model. Models MA*304723.413, MA*304723.397, and MA*304723.398 fit together. Model MA*304723.409 is a compound of four models like MA*304723.413.

- Reference:

- Kurt Reinhardt, “Zu Moebius’ Polyhedertheorie,”
*Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe*, 37, pp. 106-125. Wheeler referred to this article.

- Location
- Currently not on view

- date made
- ca 1940

- maker
- Wheeler, Albert Harry

- ID Number
- MA*304723.413

- accession number
- 304723

- catalog number
- 304723.413

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

## Protractor Signed by Cox

- Description
- This semicircular brass protractor is graduated to half-degrees. It is marked by tens from 10° to 170° in both directions, from left to right and from right to left. A brass rectangle with a curved notch has been soldered on at the origin point. The rectangle contains a small hole for locating the vertex of the angle being measured. The base of the protractor bears the maker's mark: W. C. Cox, Devonport. The letters DB are scratched near the maker's mark.

- William Charles Cox, a British instrument maker who worked in Plymouth and Devonport, had his shop in Devonport from 1830 to 1851. He presumably made this protractor during that period. The Smithsonian purchased this instrument in 1959 from the estate of Henry Russell Wray via an auction conducted by Maggs Bros. Ltd. of London.

- Reference: Gloria Clifton,
*Directory of British Scientific Instrument Makers 1550-1851*(London: National Maritime Museum, 1995), 69–70.

- Location
- Currently not on view

- date made
- ca 1840

- maker
- Cox, William Charles

- ID Number
- MA*316927

- accession number
- 228694

- catalog number
- 316927

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

## Protractor Signed by Delure

- Description
- This brass semicircular protractor is graduated by single degrees and marked by tens, from left to right and from right to left, from 10° to 180°. The degree lines are probably stamped rather than engraved by hand and thus represent a notable increase in workmanship over MA*326978 and MA*316861. Higher levels of accuracy were not reached until machine division was achieved by instrument makers such as Jesse Ramsden, who worked approximately fifty years after this protractor was manufactured.

- The inner edge of the base of the protractor is slanted, and there is a notch at the origin point. The base carries a maker's mark: Delure À Paris. The protractor is very similar to one depicted in the famous manual on mathematical instruments by Jean-Baptiste Delure's son-in-law, Nicolas Bion. The protractor dates to about 1720. It was purchased in 1959 from the estate of Henry Russell Wray via an auction conducted by Maggs Bros. Ltd. of London.

- References: Nicolas Bion,
*Traité de la construction et des principaux usages des instruments de mathematique*(Paris, 1709), 25–27; Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts,*Tools of American Mathematics Teaching, 1800–2000*(Baltimore: Johns Hopkins University Press, 2008), 166–168; Maya Hambly,*Drawing Instruments: 1580–1980*(London: Sotheby's Publications, 1988), 120–121; Jean-Dominique Augarde, "La fabrication des instruments scientifiques du XVIIIe siècle et la corporation des fondeurs," in Christine Blondel et al., eds.,*Studies in the History of Scientific Instruments*(London, 1989), 62–63.

- Location
- Currently not on view

- date made
- ca 1720

- maker
- Delure, Jean-Baptiste-Nicolas

- ID Number
- MA*316930

- accession number
- 228694

- catalog number
- 316930

- 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

## Lutz Refracting Telescope

- Description
- This is a brass instrument on a pillar-and-tripod stand. Its achromatic objective has an aperture of slightly more than 2 inches. The tube is 29 inches long and, with eyepieces, extends to 32½ inches. The “Ed LUTZ / Paris” signature on the eyepiece refers to Édouard Lutz, an optical instrument maker who showed his wares at the international exhibitions held in Paris in 1878 and 1889.

- The federal Bureau of Education was formed in 1867 and charged with providing educational information to the states and territories. To this end it collected apparatus and text-books from around the world, and recommended that funds be provided for the organization of an educational museum and the exchange of educational “appliances.” It mounted an extensive display at the World’s Industrial and Cotton Centennial Exposition in New Orleans in 1885, but folded soon thereafter. This telescope was among its instruments. It came to the Smithsonian in 1910, a transfer from the U.S. Department of the Interior.

- Ref: “Catalogue of the Educational Museum. U.S. Bureau of Education. Sketch of the Origin, Growth, and Objects of the Museum,” in accession file, NMAH.

- date made
- late 19th century

- maker
- Lutz, Edouard

- ID Number
- PH*261258

- catalog number
- 261258

- accession number
- 51116

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

## Zeiss Refracting Telescope

- Description
- The American military began using telescopes with stable mounts and tubes of fixed length in the 1890s, to supplement the spyglasses it had used since the Revolution and the field glasses (binoculars) it had used since the Civil War. This example is of that sort. It has an achromatic objective of 110 mm (about 4¼ inches) aperture, a turret with three terrestrial eyepieces, a sunshade, an alt-azimuth mount, and a wooden tripod. The tube is 51 inches long. The “Zeiss” signature refers to the important German optical firm that began in business in Jena in 1846. The “6366” on the prism housing, if it is a serial number, suggests a date of manufacture in the 1890s. The eyepiece turret is marked: “Orthoskop. Okular f = 40”.”

- This instrument came to the Smithsonian from the U.S. Weather Bureau which had installed it on Tatoosh Island, off the Olympic Peninsula in Washington State, and used it for spotting ships at sea. It may earlier have been used by the Army Corps of Engineers.

- Ref: D. J. Warner, “Optical Elements of Fire Control, 1890-1921,”
*Rittenhouse*18 (2004): 21-59.

- date made
- ca 1900

- maker
- Zeiss, Carl

- ID Number
- PH*314578

- accession number
- 204611

- catalog number
- 314578

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

## Projector

- Description
- Instruments of this sort, which projected the image of objects placed on them, came into use in the second half of the nineteenth century. The “J. Duboscq / à Paris / No. 54” inscription on this example refers to Jules Duboscq, an important optical instrument maker in Paris.

- This form came to be known, in the twentieth century, as a viewgraph or an overhead projector.

- J. Duboscq, “Appareil pour la projection des corps placés horizontalement,”
*Journal de Physique Theorique et Appliquee*5 (1876): 216-218.

- Debbie Griggs, “Projection Apparatus for Science in Late Nineteenth Century America,”
*Rittenhouse*7 (1992): 9-15.

- Location
- Currently not on view

- date made
- after 1876

- 1834-1844

- 1870s

- maker
- Duboscq, Jules

- Duboscq, Jules

- Soleil, Jean Baptiste Francois

- ID Number
- PH*315416

- catalog number
- 315416

- accession number
- 217544

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