# Geometrical Models for Arithmetic Teaching

• Contents

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.

### Holbrook's Geometrical Forms and Arithmetical Solids

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.
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.
1859
maker
Holbrook School Apparatus Manufacturing Company
ID Number
1986.1025.01
accession number
1986.1025
catalog number
1986.1025.01

### Stereometry Made Easy, A Set of Geometric Models

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.
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
ca 1860
maker
A. N. Myers & Company
ID Number
1990.0539.41
catalog number
1990.0539.41
accession number
1990.0539
catalog number
323474

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

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.190
accession number
1985.0112
catalog number
1985.0112.190

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

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.191
accession number
1985.0112
catalog number
1985.0112.191

### Dissected Polygon, Probably a Ross Surface Form

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.194
accession number
1985.0112
catalog number
1985.0112.194

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

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.
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 1985.0112.190.
Location
Currently not on view
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.192
accession number
1985.0112
catalog number
1985.0112.192

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

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.193
accession number
1985.0112
catalog number
1985.0112.193

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

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.
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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.195
accession number
1985.0112
catalog number
1985.0112.195

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

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.196
accession number
1985.0112
catalog number
1985.0112.196

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

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.
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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.197
catalog number
1985.0112.197
accession number
1985.0112

### Dissection of a Parallelogram into Triangles, Ross Surface Form #10

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.198
catalog number
1985.0112.198
accession number
1985.0112

### Trapezium or Quadrilateral, Ross Surface Form

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.199
catalog number
1985.0112.199
accession number
1985.0112

### Dissected Square, Ross Surface Form #11

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.200
catalog number
1985.0112.200
accession number
1985.0112

### Regular Octagon, Ross Surface Form

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.201
catalog number
1985.0112.201
accession number
1985.0112

### Sixteen-Sided Regular Polygon, Ross Surface Form

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.
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 16-sided 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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.202
catalog number
1985.0112.202
accession number
1985.0112

### Model Illustrating Finding the Area of a Circle, Ross Surface Form #14

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.203
catalog number
1985.0112.203
accession number
1985.0112

### Pyramid and Frustrum of Pyramid, Ross Solid #11

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.205
catalog number
1985.0112.205
accession number
1985.0112

### Square Prism, Ross Solid

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.207
catalog number
1985.0112.207
accession number
1985.0112

### Cylinder, Ross Solid #8

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.209
catalog number
1985.0112.209
accession number
1985.0112

### Dissected Triangular Prism, Ross Solid #9

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.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.211
catalog number
1985.0112.211
accession number
1985.0112

### Dissected Frustum of a Triangular Prism, Ross Solid

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.
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. The unpainted wooden model of the frustum of a triangular pyramid has three trapezoidal sides and a triangular top and base. It is dissected into three pieces. A paper label attached to one side reads: Triangular Frustum.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.215
catalog number
1985.0112.215
accession number
1985.0112

### Cone Dissected into Two Pieces, Ross Solid #12

This is the twelfth 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 of a cone is divided into two pieces by a plane parallel to the base.
Description
This is the twelfth 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 of a cone is divided into two pieces by a plane parallel to the base. A label on the side of the frustum reads: CONE. Ross compared the volume of a cone to the volume of a pyramid with a regular polygon for its base.
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
ca 1895
maker
Ross, W. W.
ID Number
1985.0112.216
catalog number
1985.0112.216
accession number
1985.0112

### Model for the "Devil's Coffin" Diagram Relating to Computing the Volume of a Parallelepiped, Ross Solid

This wooden model is one in a series illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The incomplete unpainted wooden model has two pieces.
Description
This wooden model is one in a series illustrating the volume of solids designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The incomplete unpainted wooden model has two pieces. One is a cube, the second is part of a parallelepiped with one square face the same size as the cube. A paper label pasted to a square side of both pieces of the model reads: DEVIL’S COFFIN (/) Phillips & Fisher, p. 305 Van Velzer & Shutts, p. 300 (/) Wentworth, p. 303 Wells, p. 278. This is a reference to four American geometry textbooks published between 1894 and 1899.
In the course of the 19th century, American geometry textbooks came to be more than reproductions of British works. By the 1890s, several texts discussing solid geometry used a figure demonstrating the volume of a parallelepiped that apparently arose in the United States.
In this construction, the volume of an arbitrary parallelepiped is first compared to one constructed having the same altitude and rectangular bases equal in area to those of the original solid. This figure is then compared to a third parallelepiped, this with the same altitude and six rectangular sides. John Farrar, following A.-M. Legendre, proposed such a construction in his Elements of Geometry . By the 1890s, the figure had taken a rather different form. Perhaps because it was difficult imagine from a two dimensional drawing, it was known as “the devil’s coffin.”
Ross’s model of the construction had three parts, a parallelepiped with six sides in the shape of equilateral parallelograms, a parallelepiped with two square sides and four rhombic sides, and a cube. The parallelepipeds are dissected. The two models in the Smithsonian collections are the cube and one piece of one of the parallelepipeds.
This model is not mentioned in Ross’s original manual for his surface forms and solids. The texts referred were published several times, but show the devil’s coffin construction on the pages indicated on the model on editions published between 1894 and 1899. Hence the date of about 1900 assigned to the model.
References:
A.-M. Legendre, Éléments de géométrie, avec des notes, Paris: Didot, 1794, pp. 178–184, Plate 8.
John Farrar, Elements of geometry, by A. M. Legrendre. Translated from the French for the use of the students at the University at Cambridge, New England, Boston : Hilliard and Metcalf printers, 1819, pp. 134–139, plates IX and X.
Thomas Heath, ed., The Thirteen Books of Euclid’s Elements, vol. 3, Book XI, propositions 29 and 30, especially the commentary on Proposition 30, New York: Dover, 1956, esp. pp. 333–336.
Andrew Wheeler Phillips and Irving Fisher, Elements of Geometry New York: American Book Company, 1896, p. 305–306.
C. A. Van Velzer and George C. Shutts, Plane and Solid Geometry Suggestive Method Madison, WI: Tracy Gibbs, 1894, p. 300.
Webster Wells, The Elements of Geometry, rev. ed., Boston: Leach, Shewell and Sanborn, 1894, p. 278.
George A. Wentworth, Plane and Solid Geometry, rev. ed., Boston: Ginn, 1899, p. 303.
Location
Currently not on view
ca 1900
maker
Ross, W. W.
ID Number
1985.0112.217
catalog number
1985.0112.217
accession number
1985.0112

### Dissected Circle Transformable into Parallelogram

In the years following the Civil War, a handful of American educators designed and sold wooden solids or flat shapes hinged or doweled so that they could be transposed into other shapes that had areas known to students. One of them was Albert H.
Description
In the years following the Civil War, a handful of American educators designed and sold wooden solids or flat shapes hinged or doweled so that they could be transposed into other shapes that had areas known to students. One of them was Albert H. Kennedy (1848–1940), Superintendent of Schools in Rockport, Indiana. He sold this business to the Rockport School Desk Company. Modified forms of the solids would be sold by the Western School Supply House of Des Moines, Iowa, A. Cowles and Company of Chicago, Illinois, and the American School Furniture Company of Chicago.
From ancient times, mathematicians sought to find a polygon with straight sides equal in area to the circle. This model represents Kennedy’s attempt to demonstrate that the area of a circle equaled half of the product of its circumference and its radius. It consists of a dissected circle, transformable into a parallelogram. The circle has of two semicircular portions. Each portion is divided into eight equal wooden segments, which are held together by cloth tape that is nailed to each segment around the circumference. Rearranging the pieces gives a rough parallelogram that has one side equal to half the circumference of the circle and a height equal to the radius. Multiplying the two factors together gives the desired area.
In 1882, the German mathematician Lindemann demonstrated that no exact geometric squaring of the circle is possible. His work undoubtedly was unknown to Kennedy.
The object has no maker's marks.
Compare 2005.0054.01, 2005.0054.02, 2005.0054.03 and 2005.0054.04.
References:
Arithmetic of Practical Measurements for Teachers' Instruction and Class Work in Mensuration. Published by Western School Supply House, Des Moines: Iowa Printing Co., 1893. This reportedly was ”To accompany Kennedy’s improved dissecting mathematical blocks. 20th ed.” A copy of the sixteenth edition, which has the same date, is 2005.3099.01.
C. L. F. von Lindemann, “Über die Zahl π,” Mathematische Annalen, 20 (1882), 215.
“Paintings Presented to Local Schools,” Rockport Journal May 15, 1964.
P. A. Kidwell, "American Mathematics Viewed Objectively: The Case of Geometric Models," in Vita Mathematica: Historical Research and Integration with Teaching, ed. Ronald Calinger, Washington, D.C.: Mathematical Association of America, 1996, pp. 197–207.
Location
Currently not on view