Cutting a cone by a plane yields several different curves of second degree, which are known as conic sections. This white plaster model of a circular cone has three parts. One is fixed to the base. A second and the largest is cut by a plane on one side to give a hyperbola and on the other side to give an ellipse. The third piece, on the other side of the piece attached to the base, reveals a parabola when cut away. A fourth piece, which attaches to the ellipse, is missing. This example has no labels.
Compare 1985.0112.012, 1985.0112.013, and 1979.3002.21.
In the nineteenth and early twentieth century, students studying technical subjects often learned about the representation of surfaces by equations in courses in solid analytic geometry. Schools in Europe, the United States, and Japan sometimes purchased models to illustrate such surfaces. This object is part of series of models of quadric surfaces (surfaces of degree two) designed in 1878 by Rudolf Diesel, then a student at the technical high school in Munich. It was published by the firm of Ludwig Brill in Darmstadt. This example was exhibited at the German Educational Exhibit at the Columbian Exposition held in Chicago in 1893, where it was purchased by Wesleyan University.
The saddle-shaped plaster model shows a hyperbolic paraboloid. The surface is represented by the equation: + y2/ b2 - x2/a2 = - 2z. Sections by any plane where x = c or y=c (c being an arbitrary constant) are parabolas. Sections parallel to the plane z = 0 are hyperbolas. One hyperbola and one parabola, both of which pass through the center of the surface, are shown on the model. Also shown are two sets of perpendicular straight lines. Such lines can be used to represent the hyperbolic paraboloid as a ruled surface (see objects 1985.0112.022 and 1985.0122.023).
A paper tag near the base of the model reads: 25. Another paper tag reads: Hyperbolisches Paraboloid. (/) Verl. v. L. Brill. 3. Ser. Nr. 15.
References:
Ludwig Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill, 1892, p. 7, 59.
Henry Burchard Fine and Henry Dallas Thompson, Coordinate Geometry, New York: Macmillan Company, 1931, p. 243-244, 251-254.
In the nineteenth and early twentieth century, students studying technical subjects often learned about the representation of surfaces by equations in courses in solid analytic geometry. Schools in Europe, the United States, and Japan sometimes purchased models to illustrate such surfaces. This object is part of series of models of quadric surfaces (surfaces of degree two) designed in 1878 by Rudolf Diesel, then a student at the technical high school in Munich. It was published by the firm of Ludwig Brill in Darmstadt. This example was exhibited at the German Educational Exhibit at the Columbian Exposition held in Chicago in 1893, where it was purchased by Wesleyan University.
The plaster model shows an elliptic cone. The surface can be represented by the equation x2/a2 + y2/ b2 - z2/c2 = 0. Here x and y are axes in the horizontal plane and z is the vertical axis. Assuming that z=0 at the apex of the cone, a plane cutting the surface parallel to the plane z = 0 produces an ellipse. Sections by the planes x = 0 and y = 0 give the four straight line segments shown on the model. A tag on the model reads: Kegel. (/) Verl. v. L. Brill 3. Ser. Nr. 17
References:
Ludwig Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill, 1892, p. 7, 59.
Henry Burchard Fine and Henry Dallas Thompson, Coordinate Geometry, New York: Macmillan Company, 1931.
In 1888, Ludwig Brill of Darmstadt published a series of nine models for teaching about surfaces of the second degree that have the same foci (confocal surfaces). The plaster models were patterned after some designed by the Finnish mathematician Edvard Rudolf Neovius (1851-1917) of Helsinki. This, the fourth in the series, represents a hyperbola of one sheet. It has the same foci as the ellipsoid that first model in the series and the hyperbola of two sheets that is the fifth model (for it, see 1985.0112.149).
A paper tag on the model reads: 191. Another tag reads: Einschaliges Hyperboloid. (/) [. . .] L. Brill. 16. Ser. I. Nr. IV.
This example of the model was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893, where it was purchased by Wesleyan University.
Reference:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p. 35, 58.
This white plaster model of a third order surface has three symmetrically arranged peaks that meet at a point. Two intersecting straight lines are indicated.
A paper tag on the model reads: 45. Another paper tag reads: [. . .] pl. Knotenpunkt U6 (/) [. . .] 7. Ser. Nr. 16.
This model, along with all the models of Series 7, is on the design of Carl Rodenberg of the technical high school in Munich.. It was first published by Brill in 1881.
The object was exhibited at the German Educational Exhibit at the Columbian Exposition, a World’s Fair held in Chicago in 1893. It there was purchased by Wesleyan University in Connecticut, and subsequently was donated to the Smithsonian.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p.14, 62.
Accession file.
G. Fischer, Mathematical Models, Braunschweig: Vieweg, 1986, vol. 1, p. 28, vol. 2, pp. 12-14.
This white plaster model of a third order surface has a square base. There are two peaks with an overarching peak joining them. This model, along with all the models of Series 7, is on the design of Carl Rodenberg of the technical high school in Munich.. It was first published by Brill in 1881. A paper tag at the base reads: 35. Another tag reads: Fl. 3. Ord. mit 4 reelen con. Knpktn. (/) Verl. v. L. Brill. . . .7 Ser. Nr. 6.
The object was exhibited at the German Educational Exhibit at the Columbian Exposition, a World’s Fair held in Chicago in 1893. It there was purchased by Wesleyan University in Connecticut, and subsequently was donated to the Smithsonian.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill,1892, p.14, 61.
Accession file.
G. Fischer, Mathematical Models, Braunschweig: Vieweg, 1986, vol. 1, p. 17, vol. 2, pp. 12-14.
The black metal frame of this string model has two opposite oval sides, with a loop in each oval. The other opposite sides are I-shaped (one I is very squat). The top and bottom are T-shaped. The bits of thread that remain (most are missing) are red and green. There is no maker's tag on this example.
This is one of a series of models of ruled surfaces representing space curves of fourth order designed by Karl Rohn, professor of mathematics at the technical high school in Dresden. The series was first published by Brill in 1892. This model is a special case of model 5 in the series.
This example of the model was exhibited at the Columbian Exposition, a World’s Fair held in Chicago in 1893.
Compare models 1985.0112.181, 1985.0112.182, 1985.0112.183, 1985.0112.184, and 1985.0112.186.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill, 1892, pp. 40-42, 75
An example of this model is #44 in the collection of mathematical models at the University of Illinois. An image of it has a tag with the the Brill series 21, number 6 designation. http://www.mathmodels.illinois.edu/cgi-bin/cview?SITEID=4&ID=355.
This tan paper model of polyhedra for 120-cell consists of about 113 pentagonal faces. Each face measures 2 cm. w. x 2 cm. deep - the model is not assembled, incomplete, and in fragments. A mark reads: 15. Ser. (/) 6, 1.
This model, is one of a series of models of projections of four-dimensional surfaces into three space that was designed by Victor Schlagel in Hagen and first published by Brill in 1886. It is supposed to have 119 dodecahedra lying with in a dodecahedron. The complete model would have 720 faces, 1200 edges, and 600 vertices. Brill offered the model for 120 marks. By 1911, Schilling sold itl for 200 marks.
This example of the model was exhibited at the Columbian Exposition, a world’s fair held in Chicago in 1893.
This white plaster model has numerous flat faces. It is one of a series of models designed by A. Schoenflies in Göttingen to illustrate the regular partition of space. Schoenflies designed “stones” which could be arranged into larger blocks (sometimes with congruent stones, as in this case). The series was first published by Brill in 1891. Plaster stones that comprise the object with museum number 1985.0112.175 could be arranged to be part of the block with museum number 1985.0112.160. A somewhat similar model of a block and stones at the University of Göttingen (#331 in their collection) has two differently shaped blocks and three stones. The Smithsonian collections contain one block and three stones with these shapes.
This example of the model was exhibited at the Columbian Exposition, a World’s Fair held in Chicago in 1893.
References:
L. Brill, Catalog mathematischer Modelle. . ., Darmstadt: L. Brill, 1892, pp. 46-47, 90-91.
A Schoenflies, “Uber Reguläre Gebietstheilungen des Raumes,” Nachrichten von der Königl. Gesellschaft der Wissenschaften, #9, June 27, 1888, pp. 223-237.
Göttingen Collection of Mathematical Models, presently online at http://modellsammlung.uni-goettingen.de/, accessed September 6, 2019.