#
Arithmetic Teaching ApparatusGeometrical Models for Arithmetic Teaching

Standard topics in arithmetic teaching are calculations of the area of plane figures and the volume of solids. To make these processes clearer to students, educators introduced a variety of geometric models. Models were also used to teach about fractions and to explain the process of taking cube roots.

"Arithmetic Teaching Apparatus - Geometrical Models for Arithmetic Teaching" showing 29 items.

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## Holbrook's Geometrical Forms and Arithmetical Solids

- Description
- In the years before the Civil War, several Northern states opened free elementary or common schools. To communicate with large numbers of students, teachers used a wide range of objects, including these models of simple geometrical shapes. Connecticut school reformer and lecturer Josiah Holbrook developed a collection of apparatus for teaching by families and in schools. The models were part of this set. He designed them to help students learn the names of simple solids, basic rules for calculating the area of various flat surfaces, and elementary drawing. Holbrook advertised that his equipment was "Good enough for the best, and cheap enough for the poorest." It was used in thousands of schools. Even after Holbrook died in 1854, his family continued to manufacture school apparatus; these models date from about 1859.

- Location
- Currently not on view

- date made
- 1859

- maker
- Holbrook School Apparatus Manufacturing Company

- ID Number
- 1986.1025.01

- accession number
- 1986.1025

- catalog number
- 1986.1025.01

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

## Stereometry Made Easy, A Set of Geometric Models

- Description
- From the 16th through the 19th centuries, English-speaking mathematicians referred to the measurement of solid bodies as stereometry. This set of forty-odd models, made in London in the mid-19th century, assisted in teaching the subject. According to the maker, the solids also were well suited for use by art students.

- Included in the wooden box are a diagonal scale; three equal trapezoids, any two of which can be arranged to form a rectangle or a parallelogram; two equal triangles which together form a rectangle or a triangle; three equal quadrilaterals (with a fourth quadrilateral of the same size, they would form a square); and nine pieces that are lettered from a to i. Pieces a to c are equal oblique pyramids that can be arranged to form a cube. Pieces d to i are equal square pyramids which can be arranged to form a cube.

- The set also includes eight pieces of a cube root block. The smaller cube of the cube root block is not labeled, and three of the other pieces are mislabeled. Also included are six equal triangular prisms, one longer triangular prism, two additional cubes, a cylinder, a tetrahedron, an icosahedron, two rectangular parallelepipeds, one oblique parallelepiped, one taller square pyramid, two triangular pyramids, and an irregular tetrahedron.

- A discolored label on the lid of the box reads: STEREOMETRY (/) MADE EASY.

- An example of the set in the library of Princeton University also includes several lithographed cards and an instruction booklet, published in 1853. The
*Catalogue of the Educational Division of the South Kensington Museum*indicates that the set was made by Myers and Company of London. This example came to the Smithsonian from the Physics Department of Queens College of London University. An 1877 advertisement of A. N. Myers & Co. indicates that by that date, a set of 44 geometrical models sold in three sizes. This would correspond to the smallest size. As the advertised set contained 44 surfaces, it seems likely that one object in this example (perhaps the diagonal scale) was not part of the original.

- References:

*Catalogue of the Educational Division of the South Kensington Museum*, London: Eyre and Spottiswoode, 1876, p. 407.

*Stereometry Made Easy: A Short Compendium of the Facts and Principles of that Instructive and Amusing Science: Intended as a Companion to the Collection of Solids*, London: Thompson and Davidson, 1853.

- “Educational and Amusing Publications of A. N. Myers & Co.,”
*A Catalogue of Works of Natural Science, Art, General Literature, Medicine &c. Published by Hardwick & Bogue*, London, 1877, p. 1.

- Location
- Currently not on view

- date made
- ca 1860

- maker
- A. N. Myers & Company

- ID Number
- 1990.0539.41

- catalog number
- 1990.0539.41

- accession number
- 1990.0539

- catalog number
- 323474

- 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

## Dissected Polygon, Probably a Ross Surface Form

- Description
- This unpainted wooden model consists of two doweled pieces that can be arranged as a quadrilateral. The model is incomplete. It resembles other Ross surface forms.

- Compare models 1985.0112.190 through 1985.0112.202, especially 1985.0112.193. 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
- 1985.0112.194

- accession number
- 1985.0112

- catalog number
- 1985.0112.194

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

## Model of a Rectangle Bisected into Two Right Triangles, Ross Surface Form #8

- Description
- This is the eighth 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 unpainted rectangular wooden model is bisected along a diagonal. A paper label pasted to the model reads: Oblong 4x6 Bisected. According to Ross, this model demonstrates that a right-angled triangle with unequal sides adjacent to the right angle has half the area of a rectangle.

- 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
- 1985.0112.192

- accession number
- 1985.0112

- catalog number
- 1985.0112.192

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

## Model of a Dissected Trapezoid, Ross Surface Form #6

- Description
- This is the sixth 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 unpainted rectangular wooden model is cut into two pieces at one corner. It may be arranged so that the pieces form either a rectangle or a trapezoid. A paper label attached to the model reads: Dissected Trapezoid 5x7.

- Ross argued that the area of the trapezoid equaled half the sum of its parallel sides, multiplied by its 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
- 1985.0112.193

- accession number
- 1985.0112

- catalog number
- 1985.0112.193

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

## Rectangle Transformable Into an Obtuse Triangle, Probably a Ross Surface Form

- Description
- This is apparently is one 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 three doweled pieces of this unpainted wooden model can be arranged either as a rectangle or as an obtuse-angled triangle.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.195

- accession number
- 1985.0112

- catalog number
- 1985.0112.195

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

## Dissected Rhomboid, Ross Surface Form #5 (incomplete)

- Description
- This is the fifth 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 unpainted wooden model is divided into two pieces, with the smaller piece missing.

- With the smaller piece, the model could be arranged either as a parallelogram or a rectangle. A paper label attached to the model reads: Dissected Rhomboid 4x6.

- Ross argued that the parallelogram (or, in his terminology, rhomboid), like the rectangle, was the product of its length and its altitude.

- 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
- 1985.0112.196

- accession number
- 1985.0112

- catalog number
- 1985.0112.196

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

## Dissection of a Parallelogram into Triangles, Ross Surface Form #9

- Description
- This is the ninth 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 unpainted wooden parallelogram (rhomboid in Ross’s terminology) is bisected along a diagonal into two scalene triangles. A paper label attached to the model reads: Rhomboid A 4x6 Bisected. According to Ross, the model shows that if a rhomboid (parallelogram) is cut diagonally through the opposite acute angles, two equal obtuse-angled triangles result.

- Location
- Currently not on view

- date made
- ca 1895

- maker
- Ross, W. W.

- ID Number
- 1985.0112.197

- catalog number
- 1985.0112.197

- accession number
- 1985.0112

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

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