#
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 278 items.

Page 1 of 28

## Flexible Polyhedron

- Description
- The mathematician Leonard Euler once wrote,"A closed spatial figure allows no changes, as long as it is not ripped apart." Proving the "rigidity" of polyhedra was another matter. In 1813, Augustin-Louis Cauchy showed that a convex polyhedral surface is rigid if its flat polygonal faces are held rigid. In 1974, Herman Gluck proved that almost all triangulated spherical surfaces were rigid. However, in 1977 Robert Connelly of Cornell University found a counterexample, that is to say a flexible polyhedron. He built this model of such a surface some years later. It is made of cardboard and held together with duct tape. Two cutout plastic windows allow the viewer to observe changes when the polyhedron is flexed. The top section has 12 large faces and a six-faced appendage. The bottom section has 12 corresponding faces but no appendage.

- Location
- Currently not on view

- date made
- 1985

- maker
- Connelly, Robert

- ID Number
- 1990.0492.01

- accession number
- 1990.0492

- catalog number
- 1990.0492.01

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

## Geometric Model of a Regular Icosahedron by A. Harry Wheeler or One of His Students

- Description
- Greek mathematicians knew in ancient times that there are only five polyhedra that have identical faces with equal sides and angles. These five regular surfaces, called the Platonic solids, are the regular tetrahedron (four equilateral triangles as sides), the cube (six square sides), the regular octahedron (eight equilateral triangles as sides), the regular dodecahedron (twelve regular pentagons as sides) and the regular icosahedron (twenty equilateral triangles as sides). This is an early 20th-century model of a regular icosahedron. The sides are covered with sateen and brocade fabrics of various designs and colors, in the style of late 19th-century piece work. Catch stitches are along the edges.

- The model is unsigned, but associated with the Worcester, Massachusetts, schoolteacher A. Harry Wheeler. Wheeler taught undergraduates at Wellesley College, a Massachusetts women’s school, from 1926 until 1928. It is possible that one of his students there made the model.

- Reference:

- Judy Green and Jeanne LaDuke,
*Pioneering Women in American Mathematics: The Pre-1940 PhD’s*, Providence: American Mathematical Society, 2009, p. 21.

- Location
- Currently not on view

- date made
- ca 1926

- ID Number
- 1979.0102.188

- accession number
- 1979.0102

- catalog number
- 1979.0102.188

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

## Geometric Model of a Deltahedron (also a Form of Stellated Icosahedron) by A. Harry Wheeler and His Students

- Description
- Polyhedra in which all faces are equilateral triangles are called deltahedra. The regular tetrahedron, octahedron, and icosahedron are the simplest deltahedra. It also is possible to replace each face of a regular dodecahedron with a “dimple” having five equilateral triangles as sides. This is a model of such a surface. It also may be considered as one of the polyhedra formed by extending the sides of—or stellating—a regular icosahedron.

- This deltahedron is folded from paper and held together entirely by hinged folds along the edges. Fifteen of the sixty faces have photographs of students of A. Harry Wheeler at North High School in Worcester, Massachusetts. All are boys. Another face reads: 1927 (/) Stanley H. Olson. A seventeenth face reads: Royal Cooper. Cooper is also shown on one of the sides with a photograph. There is a photograph of Lanley S. Olson, but not Stanley H. Olson. Yet another face of the model has a pencil mark that reads: June – 1927.

- Reference:

- Magnus J. Wenninger,
*Polyhedron Models*, Cambridge: The University Press, 1971, p. 48.

- Location
- Currently not on view

- date made
- 1927

- ID Number
- 1979.0102.308

- accession number
- 1979.0102

- catalog number
- 1979.0102.308

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

## Plaster Model for Function Theory by L. Brill, No. 175, Ser. 14 No. 3

- Description
- This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 3 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886.

- The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later.

- Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on the front of model 3, indicating that the vertical axis can represent either u or v.

- On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1/2 cm. on model 3. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 3 is related to the level curves on that model.

- The equation involving complex variables on which model 3 is based is w
^{4}= 1 - z^{2}. That equation defines a surface in four dimensions. Model 3 is a three-dimensional model and is defined by two almost identical equations, each using three real variables. Those, very complicated, equations are found by replacing w by u + vi and z by x + yi, and then eliminating u or v. Normally this process produces two very different equations and three-dimensional models, but in this case, the two equations are identical except that one includes u’s and the other v’s and they define the same model. The polynomial equations that define model 3 start with a term with a coefficient 256 and exponent of the variable 16.

- Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. Computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program
*Mathematica*show scales to indicate the direction of at least two of the variables. Each has an R and an I superimposed approximately where they appear on model 3. While model 3 does not show it, the computer generated plot positioned to look directly at the front shows a hole with a complicated boundary.

- References:

- L. Brill,
*Catalog mathematischer Modelle*, Darmstadt, 1892, pp. 29-30, 70-71.

- G. Fischer, ed.
*Mathematical Models: From the Collections of Universities and Museums*, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 125, p.122, and vol. 2 (*Commentary*), pp. 71-72, 75-76.

- Location
- Currently not on view

- date made
- 1892

- maker
- L. Brill

- ID Number
- 1985.0112.137

- catalog number
- 1985.0112.137

- accession number
- 1985.0112

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

## Plaster Model for Function Theory by L. Brill, No. 176, Ser. 14 No. 4

- Description
- This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. It was manufactured by the Darmstadt publishing company of Ludwig Brill and is model number 4 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886.

- The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later.

- Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on model 4, but on adjacent vertical faces, with the I on what looks like the back of the model.

- On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm. on model 4. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 4 is related to the level curves.

- The equation involving complex variables on which model 4 is based is w = 1/z. That equation defines a surface in 4 dimensions. Model 4 is a three-dimensional model and is defined by an equation using three real variables. That equation is found by replacing w by u + vi and z by x + yi, to get u + vi = (x – yi)/(x
^{2}+ y^{2}), which is equivalent to the two equations u = x/(x^{2}+ y^{2}) and v = –y/(x^{2}+ y^{2}). . Normally this process produces two very different equations and three-dimensional models, but in this case, the two equations are identical except that switching from u to v, changes x to –y and y to –x. Therefore, these equations define the same model but with the x and y axes rotated 90 degrees.

- Only points on the curved surface of the solid model satisfy the equation that defines it; points in the solid plaster that supports that surface do not satisfy that equation. Computer generated versions show only the surface so are able to show details of the model lying below the complex z-plane. Plots of the surfaces produced using the program
*Mathematica*show scales to indicate the direction of at least two of the variables. Each has an R and an I superimposed to show the faces where they appear on model 4. The computer plots looking directly at the faces with the inscribed R and I show a hollow cropped spire and the same spire rotated 180 degrees around the origin to face downward.

- References:

- L. Brill,
*Catalog mathematischer Modelle*, Darmstadt, 1892, pp. 29-30, 70-71.

- G. Fischer, ed.
*Mathematical Models: From the Collections of Universities and Museums*, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 126, p. 123, and vol. 2 (*Commentary*), pp. 71-73.

- Location
- Currently not on view

- date made
- 1892

- maker
- L. Brill

- ID Number
- 1985.0112.138

- catalog number
- 1985.0112.138

- accession number
- 1985.0112

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

## Group of Two Plaster Models for Function Theory by L. Brill, No. 179, Ser. 14 No. 7a and 7b

- Description
- This group of two plaster models was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The models were manufactured by the Darmstadt publishing company of Ludwig Brill and this group contains model numbers 7a and 7b of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886.

- The mathematician Felix Klein came to Chicago as a representative of the Prussian Ministry of Culture and presented several lecture–demonstrations about the mathematical models on display there. After the World’s Fair, the models displayed in Chicago were purchased by Wesleyan University; they were donated to the museum about ninety years later.

- Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. In this group, there is an R inscribed on the front of model 7a and an I inscribed on the front of model 7b.

- On each model in series 14 there are two sets of curves that act much like the lines on two-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm. on models 7a and 7b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 7a is related to the level curves on model 7b. Similarly, the placement of the gradient curves model 7b is related to the level curves on the model 7a.

- Models 7a and 7b are based on a Weierstrass P-function. These complex valued functions are named after the nineteenth century German mathematician, Karl Weierstrass and each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. The tiling associated with models 7a and 7b is made up of squares with sides parallel to the x and y axes. There are four such squares in each of the models so models 7a and 7b are both made up of four congruent sections each of which has a square base and has at its center a pair of cropped spires and a pair of narrowing holes.

- Only points on the curved surfaces of each model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surfaces produced using the program
*Mathematica*show scales to indicate the direction of at least two of the variables. Each has an R or an I superimposed approximately where it appears on the corresponding model. For models 7a and 7b, the computer generated versions show the four congruent sections, each of which includes two spires that are hollow and two holes that are downward pointing versions of the hollow spires.

- References:

- L. Brill,
*Catalog mathematischer Modelle*, Darmstadt, 1892, pp. 29-30, 70-72.

- G. Fischer, ed.
*Mathematical Models: From the Collections of Universities and Museums*, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photos 129 (model 7a) and 130 (model 7b), pp. 126-127. and vol. 2 (*Commentary*), pp. 71-72, 75-76.

- Location
- Currently not on view

- date made
- 1892

- maker
- L. Brill

- ID Number
- 1985.0112.139

- catalog number
- 1985.0112.139

- accession number
- 1985.0112

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

## Plaster Model for Function Theory by L. Brill, No. 180, Ser. 14 No. 8

- Description
- This plaster model was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 8 of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886.

- Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is both an R and an I inscribed on model 8. On adjacent faces of model 9 there is an R and an I inscribed, with the R on the face with the labels.

- On each model in series 14 there are two sets of curves that act much like the lines on 2-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm on model 8. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 8 is related to the level curves.

- Model 8 is based on the derivative of the Weierstrass P-function on which Brill models 7a and b (1985.0112.139) are based. Complex valued Weierstrass P-functions are named after the nineteenth century German mathematician, Karl Weierstrass and the derivative of each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the derivative of the Weierstrass P-function is the same for corresponding points of the parallelograms of the tiling. The tiling associated with models 7a, 7b, and 8 is made up of squares with sides parallel to the x and y axes and there are four such squares in each of the models. Model 8 is made up of four congruent sections each of which has a square base and three cropped spires alternating with three narrowing holes equally spaced around the center of each square.

- Only points on the curved surfaces of model 8 satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces so are able to show details that would be difficult to portray on a plaster model. Plots of the surface produced using the program
*Mathematica*show scales to indicate the direction of at least two of the variables. Although each plot has both an R and an I superimposed approximately where it appears on the model, if there is an R or an I facing the front, the x axis is parallel to that face and the vertical axis is labeled u if R is on that face and is labeled v if I is on it. For model 8, as well as models 7a and 7b, the computer generated versions show four congruent sections. In model 8 each section includes three hollow spires alternating with three downward pointing versions of the hollow spires.

- References

- L. Brill,
*Catalog mathematischer Modelle*, Darmstadt, 1892, pp. 29-30, 70-72.

- G. Fischer, ed.
*Mathematical Models: From the Collections of Universities and Museums*, Braunschweig/Wiesbaden: F. Vieweg & Sohn, vol. 1, photo 131, p. 128, and vol. 2 (*Commentary*), pp. 71-72. 75-76.

- Location
- Currently not on view

- date made
- 1892

- maker
- L. Brill

- ID Number
- 1985.0112.140

- catalog number
- 1985.0112.140

- accession number
- 1985.0112

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

## Plaster Model for Function Theory by L. Brill, No. 182, Ser. 14 No. 10b

- Description
- This model is one of a group of two plaster models that was shown at the German Universities Exhibit in Chicago at the 1893 World’s Fair: Columbian Exposition. The model was manufactured by the Darmstadt publishing company of Ludwig Brill and is number 10b of the group that also contains model 10a of Brill’s series 14. The series was designed under the direction of the German mathematician Walther Dyck, on the pattern of originals in the mathematical institute at the technical high school in Munich. Brill first sold them in 1886.

- Each model in Brill’s series 14 represents a surface related to an equation involving pairs of complex numbers, (z, w), where z = x + yi, w = u + vi, x, y, u, and v are real numbers, and i is the square root of –1. The horizontal plane passing through the center of each of those models represents the complex z-plane, which is the real Cartesian plane with axes x and y. Each model in series 14 has an R and/or an I inscribed on a vertical face to indicate that the face is the front of the model, i.e., it is parallel to the x axis with positive real x values on the right. R is inscribed if the vertical axis represents u, the real part of w, while I is inscribed if the vertical axis represents v, the imaginary part of w. There is an I inscribed on a vertical face of model 10b but, since there is no vertical face of the model that is parallel to the x axis, the I is inscribed on a face that meets the x axis in what appears to be a 30 degree angle.

- On each model in series 14 there are two sets of curves that act much like the lines on 2-dimensional graph paper. One set of curves, called the level curves of the surface, lies on horizontal planes that are spaced at a fixed distance between them, which is 1 cm on model 10b. The other set, called the gradient curves of the surface, are perpendicular to the level curves. The placement of the gradient curves on model 10ba is related to the level curves on model 10a, which is not in the museum collections.

- Model 10b is based on the derivative of the Weierstrass P-function on which Brill models 9a and b (1985.0112.141) are based. Complex valued Weierstrass P-functions are named after the nineteenth century German mathematician, Karl Weierstrass and the derivative of each of these functions can be associated with tilings of the horizontal complex z-plane by congruent parallelograms (like graph paper) so that the complex value of the derivative of the Weierstrass P-function is the same for corresponding points of the One can see one such rhombus in model 10b by joining the four points where the tops of three cropped spires meet. At each vertex of the rhombus the sides of the rhombus parallel to the x axis pass in front of a cropped spire. parallelograms of the tiling. The tiling associated with models 9a, 9b, 10a, and 10b is made up of rhombuses whose angles are 60 and 120 degrees and with one pair of sides parallel to the x axis. One can see one such rhombus in model 10b by joining the four points where the tops of three cropped spires meet. At each vertex of the rhombus the sides of the rhombus parallel to the x axis pass in front of a cropped spire.

- The tiling by rhombuses leads to an alternate tiling by regular hexagons with one pair of sides parallel to the y axis and with the structures that were centered at each vertex of the tiling by rhombuses sitting at the center of each hexagon of the tiling. Model 10b is made up of four such hexagons that have been slightly trimmed. This tiling is difficult to see directly on the model but can be easily seen in a two dimensional plot mimicking a bird’s-eye view of the surface that were produced using the program
*Mathematica*. This plot has been superimposed with an outline in red of four of the tiling hexagons and thicker black lines that show the model’s footprint. In addition, +’s and –‘s have been placed to indicate the location of the cropped spires (+’s) and the tapering holes (-‘s). These two-dimensional plots also show why the models do not have rectangular footprints and why the R and I could not be placed on faces parallel to the x axis.

- Only points on the curved surfaces of the model satisfy the equation that defines the model; points in the solid plaster that connects those surfaces do not satisfy that equation. The computer generated versions show only the surfaces, and so are able to show details that would be difficult to portray on a plaster model. Plots of the surface produced using the program
*Mathematica*show scales to indicate the direction of at least two of the variables and each has an I superimposed approximately where it appears on the model. For model 10b, as well as models 9a and 9b, the computer generated versions show four congruent sections. In model 10b each section includes three hollow spires alternating with three downward pointing versions of the hollow spires. The plots are produced by taking x and y values from a rectangle grid so extra portions of the surface are seen, including parts of additional spires that are outside the footprint of the model. A version of this plot has been overlaid with the two sides of the rhombus that are parallel to the x axis and the long diagonal of the rhombus, which is parallel to the base of the vertical face with the inscribed I.

- References:

- L. Brill,
*Catalog mathematischer Modelle*, Darmstadt, 1892, pp. 29-30, 70-73.

- “Mathematische Modelle angefertigt im mathematischen Institut der k. technischen Hochschule in München unter Leitung von Professor Dr. Walther Dyck. Modelle zur Functionentheorie (Zu Serie XIV)” [“Mathematical Models Made in the Mathematical Institute of the Royal Technical College in Munich under the Direction of Professor Dr. Walther Dyck. Models for Function Theory (For Series 14)”], pp. 1-3, 7-8, 11-13 and fig. 7 plate III.

- Location
- Currently not on view

- date made
- 1892

- maker
- L. Brill

- ID Number
- 1985.0112.142

- catalog number
- 1985.0112.142

- accession number
- 1985.0112

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

## Model of An Oblong or Rectangle, Ross Surface Form #3

- Description
- In 1891, William Wallace Ross (1834–1906), the superintendent of schools in Fremont, Ohio, published a set of “dissected surface forms and geometrical solids” for teaching practical geometry and measurement in schools and colleges. He also prepared a manual that describes their use. Ross extended earlier work of Albert H. Kennedy, including a much larger number of surfaces. His models would be distributed at least as late as 1917, when they were listed in the catalog of the Atlas School Supply Company of Chicago, Illinois.

- In his manual, Ross listed eighteen “surface forms”, eighteen solids or volumes, and the five Platonic or regular solids. By the time of the 1917–1918 catalog, a set of the model reportedly contained fifty pieces. The Smithsonian collections include thirteen of the surface forms, ten of which correspond to objects in the 1891 list. They also contain all or part of twelve of the solid forms, at least five of which correspond to the 1891 list.

- This is the second of Ross’s surface forms, a rectangle (or, in Ross’s language, an oblong) that measures 6 inches by 1 inch. The first surface form was a square one inch on a side. Taking the area of this square to be one square inch, students were to observe that the area of the rectangle was six square inches. A paper label attached to the model reads: Oblong 1x6.

- Compare models 1985.0112.190 through 1985.0112.202.

- References:

- W. W. Ross,
*Mensuration Taught Objectively with Lessons on Form . . . Manual for the Use of the Author’s Dissected Surface Forms and Geometrical Solids*, Fremont, Ohio, 1891.

- Atlas School Supply Company,
*Catalog No. 39 1917-18*, Chicago, Illinois, 1917, p. 86.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.190

- accession number
- 1985.0112

- catalog number
- 1985.0112.190

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

## Model of a Rectangle or Oblong, Ross Surface Form #2

- Description
- This is the third in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The model is a 6 inch by 4 inch rectangle, divided into 24 one inch by one inch squares. A paper label attached to the model reads: Oblong 4x6.

- Comparing its area to that of a 6 inch by 1 inch rectangle (1985.0112.191), Ross noted that the area was four times 6 square inches, or 24 square inches. He generalized to argue that the area of a rectangle equaled the number of square units corresponding to the product of the length times the breadth.

- Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.191.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.191

- accession number
- 1985.0112

- catalog number
- 1985.0112.191

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

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