Arithmetic Teaching Apparatus  Geometrical Models for Arithmetic Teaching
Geometrical 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.
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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.
 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

Stereometry Made Easy, A Set of Geometric Models
 Description
 From the 16th through the 19th centuries, Englishspeaking mathematicians referred to the measurement of solid bodies as stereometry. This set of fortyodd models, made in London in the mid19th 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

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 191718, 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

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 1985.0112.190.
 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

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 1985.0112.190.
 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

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 rightangled 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 1985.0112.190.
 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

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 1985.0112.190.
 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

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 obtuseangled triangle.
 Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.190.
 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

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 1985.0112.190.
 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

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 obtuseangled triangles result.
 Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.190.
 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

Dissection of a Parallelogram into Triangles, Ross Surface Form #10
 Description
 This is the tenth 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 rhomboid (parallelogram) is bisected along a diagonal into two scalene triangles. Two adjacent sides on the left are equal, as are two adjacent sides on the right. A paper label attached to the model reads: Rhomboid B 4x6 Bisected. According to Ross, the model shows that if a rhomboid is cut diagonally through the obtuse angles, two equal scalene triangles result.
 Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.198
 catalog number
 1985.0112.198
 accession number
 1985.0112
 Data Source
 National Museum of American History

Trapezium or Quadrilateral, Ross Surface Form
 Description
 This is one of a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. This example, what Ross called a “trapezium,” is a quadrilateral with four unequal sides, none of them parallel. A diagonal groove joining two opposite vertices, dividing the quadrilateral into two triangles. Ross recommended finding the area of these triangles from the length of their sides.
 A paper sticker attached to the model reads: Trapezium. Another sticker reads: SCALENE TRIANGLE. A second mark on this sticker reads: It is the only operation for which the Ross Blocks have no objective proof or illustration, such objective proof is probably impossible.
 This model is not listed in Ross’s 1891 manual. Here he had written: “The trapezium is measured by dividing it up into triangles. This disposes of all the quadrilaterals.” He apparently revised this view.
 If none of the angles of an arbitrary convex quadrilateral is known, knowing the length of the sides does not suffice to determine the area of the figure.
 Compare models 1985.0112.190 through 1985.0112.202. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.199
 catalog number
 1985.0112.199
 accession number
 1985.0112
 Data Source
 National Museum of American History

Dissected Square, Ross Surface Form #11
 Description
 This is the eleventh 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 square is bisected along one diagonal, with wooden dowels to hold the pieces together. Along the other diagonal, one of the triangles is bisected. On the back, an inscribed circle is indicated as well as the radius of the circle and a square inscribed inside it. A paper label glued to the object reads: SQUARE 6x6.
 This is the first in a series of models in which Ross considered the area of a regular polygon to be made up of isosceles triangles, with base equal to the length of the side of the polygon and height equal to the radius of the inscribed circle. Summing the area of the triangles, he found that the total area equaled half the perimeter of the polygon times the radius. In this case, each of the four triangles had base 6, height 3 (the radius of the inscribed circle), and area 9. The perimeter is 4x6 or 24, half the perimeter is 12, and the area is 36. This was the same area found by multiplying the length of the sides of the square.
 Although they are not listed in his 1891 Manual, Ross also would make models of regular polygons with 8 and with 16 sides, similarly divided into triangular sectors. Examples of these have catalog numbers 1985.0112.201 and 1985.0112.202. He then, like A. H. Kennedy before him, generalized the dissection to represent the area of a circle (see 1985.0112.203).
 For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.200
 catalog number
 1985.0112.200
 accession number
 1985.0112
 Data Source
 National Museum of American History

Regular Octagon, Ross Surface Form
 Description
 This is one of the models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The flat wooden object is in the shape of a regular octagon. On the side of the model opposite from the label, an inscribed circle is indicated, as well as four lines joining opposite vertices of the octagon and meeting at the center of the circle. A paper tag attached to the model reads: OCTAGON.
 In constructing his visual demonstration of the area of a circle, Ross built several regular polygons, and showed that they had areas equal to the sum of the area of triangles with height equal to the radius of an inscribed circle and sides equal to the sides of the polygons. In other words, the area of the regular polygon equaled half the perimeter of the polygon times the radius of the inscribed circle.
 This is the example for an octagon. Compare 1985.0112.200 and 1985.0112.202. For the circle, see 1985.0112.203. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.201
 catalog number
 1985.0112.201
 accession number
 1985.0112
 Data Source
 National Museum of American History

SixteenSided Regular Polygon, Ross Surface Form
 Description
 This is one of the models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The flat unpainted wooden object is in the shape of a regular polygon with sixteen sides. On the opposite side from the paper label, it has eight straight lines drawn joining opposite vertices, dividing the polygon into 16 equal triangles. The lines meet at a point. The label reads: POLYGON OF 16 SIDES.
 In constructing his visual demonstration of the area of a circle, Ross built several regular polygons, and showed that they had areas equal to the sum of the area of triangles with height equal to the radius of an inscribed circle and sides equal to the sides of the polygons. In other words, the area of the regular polygon equaled half the perimeter of the polygon times the radius of the inscribed circle.
 This is the example for a 16sided figure. Compare 1985.0112.200 and 1985.0112.201. For the circle, see 1985.0112.203. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.202
 catalog number
 1985.0112.202
 accession number
 1985.0112
 Data Source
 National Museum of American History

Model Illustrating Finding the Area of a Circle, Ross Surface Form #14
 Description
 This is one of the models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The flat wooden disc can be arranged as a circle which is divided into six wedges that are hinged together along the perimeter. These may be rearranged to form what the model calls a “rhomboid.”
 One side of the model has four paper stickers and the other has six. One of them reads: AREA OF CIRCLE.
 Ross, like A. H. Kennedy before him, argued that a circle could be considered as the most general case of a polygon with area equal to the sum of the area of triangles, with height equal to the radius of an inscribed circle, and with sides equal to the sides of the polygons. In other words, the area of the regular polygon equaled half the perimeter of the polygon times the radius of the inscribed circle, and the area of a circle half the circumference of the circle times the radius.
 For further information about Ross models, including references, see 1985.0112.190. Closely related models are 1985.0112.200, 1985.0112.201, and 1985.0112.202. Kennedy’s version of this model is 2005.0054.01.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.203
 catalog number
 1985.0112.203
 accession number
 1985.0112
 Data Source
 National Museum of American History

Pyramid and Frustrum of Pyramid, Ross Solid #11
 Description
 This is the eleventh in a series of models illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The unpainted wooden model has a square base and four equal triangles for sides. A plane parallel to the base divides it into a square pyramid and the frustum of a square pyramid. A paper label on the model reads: Frustum of a Pyramid. Another mark on this label reads: (See Metallic Frustum). A mark on another paper label reads: Pyramid.
 Compare models 1985.0112.205 through 2012.0112.217. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.205
 catalog number
 1985.0112.205
 accession number
 1985.0112
 Data Source
 National Museum of American History

Square Prism, Ross Solid
 Description
 This is one of a series of models illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. It is a wooden square prism with a base of 1 inch by 1 inch and a height of 3 inches. The object has no maker’s label.
 Ross took the fundamental unit of measure of rectangles to be one square inch, and the fundamental unit of measure for solids to be one cubic inch. He argued from there that a 1 inch x 6 inch rectangle had an area of 6 square inches (see 1985.0112.191). Similarly, this solid model consisted of 3 cubic inches. He would go on to consider several square prisms lined up end to end, and may have intended for this to be one of them. See 1985.0112.206 for two closely related models. These are also shown in the photograph.
 Compare models 1985.0112.205 through 2012.0112.217.
 For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.207
 catalog number
 1985.0112.207
 accession number
 1985.0112
 Data Source
 National Museum of American History

Cylinder, Ross Solid #8
 Description
 This is the eighth in a series of models illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio.
 The unpainted wooden model is in the shape of a cylinder. Inscribed on the top of the cylinder is a square, with its diagonals indicated. An incomplete paper tag reads: C [. . .] R 3x6 [. . .] (/) When the [. . .] of the prism become infinite, it becomes a cylinder, the perimeter of a prism with an infinite number of sides being termed the circumference.
 In the series of plane figures, Ross compared the area of a circle to the area of circumscribing polygons of increasing numbers of sides. To demonstrate the volume of a cylinder, he compared it to various regular prisms inscribed in it. This model suggests how a square pyramid might be inscribed in a cylinder.
 Compare 1985.0112.208 and 1985.0112.210. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.209
 catalog number
 1985.0112.209
 accession number
 1985.0112
 Data Source
 National Museum of American History

Dissected Triangular Prism, Ross Solid #9
 Description
 This is the ninth in a series of models illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The unpainted wooden model is a triangular prism with three rectangular sides and a triangular base and top. It separates into three pyramids of equal volume; two of these are identical. A diagram of the dissection appears on one of two paper stickers glued to the model. A mark on one label reads: Triangular Pris [. . .].
 Finding the volume of pyramids was not only important for practical reasons but was central to Ross’s demonstrations for the volume of a cone and of a sphere.
 For Ross solids, see 1985.0112.205 through 2012.0112.217. For further information about Ross models, including references, see 1985.0112.190.
 Location
 Currently not on view
 date made
 ca 1895
 maker
 Ross, W. W.
 ID Number
 1985.0112.211
 catalog number
 1985.0112.211
 accession number
 1985.0112
 Data Source
 National Museum of American History