Cast of Characters

a.k.a. Who's Who in the History of Models of Surfaces

Richard P. Baker, 1866-1937

Born in England, graduate from Oxford, PhD from Chicago, taught at Iowa State University. Produced and sold models in the 1930's. [Kidwell, p. 204.]

Robert S. Ball, 1840-1913

A popular astronomer who studied mathematics at Trinity College Dublin [website on Ball]. Constructed a model of the Cylindroid, which related both to Plücker's work on complexes and Ball's own work, "Theory of Screws." See [SKensington1, p. 35].

Professor Björling

According to the 9th edition of the Encyclopaedia Britannica, Björling (of Lund) made thread models of devopable surfaces, p. 628.

Carl Borchardt, 1817-1880

Borchardt studied under a variety of mathematicians including Dirichlet and Jacobi and was privately tutored by Plücker and Steiner [MacTutor]. He was editor or Crelle's Journal from 1856 to 1880. There are five models by Borchardt in the catalog of the Kensington collection, but the note at the end of the list indicates that the models were executed by the late Ferdinand Engel, "known from the drawings, which he has furnished to Prof. Schellbach's `Darstellende Optik'" [SKensington1, p. 38]. Perhaps Borchardt copied the models from these drawings?

Anton von Braunmühl, 1853-1908

Braunmuehl was the son of a famous architecht by the same name and studied under Klein, Brill, and Johann Bischoff at TU Munich. He became a popular lecturer in the history of mathematics [Gillispie, p. 430 Vol I]. Built models [Kidwell, p. 203].

Alexander Willhelmvon Brill, 1842-1935

Brill was a student of Clebsch at the Politechnikum in Karlsruhe and at the University of Giessen, where he graduated in 1864, and did his Habilitation in 1867. From 1869-75 he was a proffessor at the Politechnikum in Darmstadt, and then from 1975-84 he was at the Politechnikum (later TU) Munich with Klein (who left in 1880), who was a big influence. From 1884 until his retirement in 1918 he was at the University of Tübingen. [Gillispie, p. 465 Vol I]. According to a number of sources, Brill and Klein both believed in the use of models in teaching and they had their students build models. Also of interest is the fact that C. Wiener, another builder of models, was Brill's uncle, and Brill's brother, Ludwig, was a manufacturer of models.
According to Dyck's catalog, Henrici exhibited some models composed of semi-circles at a meeting of mathematicians in Göttingen and these inspired Brill to expand and complete the collection to represent all second order surfaces [Dyck1, p. 258??]. There are other models by Brill (or completed under his direction) in Dyck's catalog, including plaster models of second order surfaces and models of cyclides. There are also cardboard models of Brill's listed in the catalog of the Kensington collection, including a series of second order surfaces that was deformable, and another series that was fixed, as well as a large cardboard model of an ellipsoid [SKensington1, p. 36].

Ludwig Brill

Manufacturer of mathematical models in Darmstadt. Brother of Alexander von Brill.

Alexander Crum Brown, 1838-1922

A chemist from Edinburgh who initially worked under Kolbe, Brown had a great interest in the application of mathematics to chemistry [Gillispie, p. 514, Vol I]. There is a list of models by Brown in the Napier Catalogue, including descriptions and discussion. There are surface models in plaster, models of polyhedra, pictures of "interlacing surfaces" (eg borromean rings), and a number of other interesting models, including on of "half-twist" surface which not modeled in the "solid" was of most surfaces (with plaster filling in one part of space), but is made out of some thin, unidentified material. [Napier Catalogue, pp 302-313]. There is a museum of Brown's work in the chemistry department at the University of Edinburg.

Max Bruckner

Check out models in Uber die gleicheckig-gleichflachigen discontinuerlichen und nichtconvexen Polyeder and Vielecke und Vielflache.

Arthur Cayley, 1821-1895

Cayley was something of a prodigy, being elected in 1842 as a fellow at his college in Oxford at the earlies age of anyone in the 1800s. He became a lawer and during his 14 years at the bar authored 300 mathematical papers. He eventually left the law and took a professorship in Cambridge which he kept until his death. While working at the bar he met and became friends with J.J. Sylvester. He corresponded with and influenced many mathematicians, both in England and elsewhere. [Gillispie, p. 162-170, Vol 2]. Cayley made many contributions across a wide spectrum of mathematics, including the discovery that there are exactly 27 lines on any nonsingular cubic surface (with George Salmon). Cayley was president of the London Mathematical Society from 1868 to 1870.In the catalog of the South Kensington exhibition, there is a "Rough Model of Steiner's Surface," by Cayley [SKensington1, p. 34].

George Chrystal, p. 1851-1911

Chrystal studied mathematics at oxford as an undergraduate and was influenced by Maxwell beginning in 1872. He held a chair of mathematics at Edinburgh University for 32 used and was also a dean there. He greatly improved the mathematics program and Edinburgh. [Gillispie, p. 264-5, Vol II]. Crum Brown says that Chrystal created plaster models of a couple of algebraic surfaces ( z=3a(x^2-y^2)-(x^3+y^3) and z=a^2(x^2+3y^2)-(x^4+6x^2y^2+y^4) ) "to illustrate his lectures on equations." [Napier Catalogue, p. 303].

George W. Cussons

George Cussons founded the scientific apparatus manufacturing company G Cussons Ltd in 1876 (according to the company web site). He aslo authored the article on mathematical models in the 14th edition of the Encyclopaedia Britannica.

Walter von Dyck

Dyck studied under F. Klein. He wrote Katalog Mathematischer und Mathematisch Physikalischer Modelle, Apparate, und Instrumente. Advertisement for Brill models includes him as one of the people under whose direction the models were created. Dyck was appointed director of the Polytechnikum Munich and helped it to rise to university standing and become Technical Univeristy of Munich. He also helped to found Deutsches Museum of Natural Science and Technology, the first science museum in the world, and he was one of the founders of the Encyclopaedie der Mathematischen Wissenshaflen. [Gillispie, p. 268-9, Vol. II].

Christian Eberback and son Ottmar

Firm in Ann Arbor started as a pharmacy ini 1843, in 1874 began to import and manufacture scientific instruments. Sold some models similar to Brill's.Kidwell, p. 204.

Arnold Emch, PhD 1895 (University of Kansas)

Emch recieved his PhD in 1895 from the University of Kansas under the direction of Henry Newson [Math. Genealogy Project]. Emch wrote a number of article on mathematical models and published a series of small catalogs of his own models (see references).

Christian Gottlieb Ferdinand Engel, 1805-1866.

[Except where noted, from Engel's obituary, Gibbs1, p. 282-284] Engel studied carpentry at his father's insistance as a youth, but studied mathematical drawing secretly and won a prize from the Academy of Arts in Berlin. At that point his father relented and Engel entered the Royal School of Arts in Magdeburg. After graduation he taught drawing in Magdeburg and later descriptive geometry in Hamburg. Later he lectured on geometry and optics in Berlin, and his lectures became very popular. He illustrated Wolff's Descriptive Geometry, and in 1856 co-authored Darstellende Optik with Karl Schellbach (Schellbach provided the illustrations and Engel providing the drawings). I have not yet seen the book, but according to the obituary it was a work "of extraordinary beauty and value."
Engel's studies of optics led him to draw and then execute models in wood of "the wave surface in biaxial crystals." One model was sent to Plücker and the other "was sent to the great Exhibition in London, where Mr. Engel received a prize medal." I believe this "great exhitbition was the 1851 world's fair in london (Crystal Palace Exhibition). In 1855 he published a book which included a catalog of 38 models for sale in wood and plaster, including quadric surfaces, helicoids, screws, compounds of cubes, optical surfaces. [Kidwell, p. ?]. "Engel's book contained testimonials from Wolcott Gibbs of New York and the German mathematician G.P. Lejeune Dirichlet. Some of Engel's models survive in the Physics Collection of the University of Mississippi." Around 1848, Engel emmigrated to the United States, but as he found no means of supporting himself, he left for England. There "he made the acquaintance of a few men of scientific tastes and pursuits who appreciated the beauty of his work...He now found occupation in making optical models of various kinds in wood and plaster. Of these perhaps the most beautful are the model of wave-surfaces in biaxial crystals and a wooden model exhibiting the passage of a plane through all the varieties of elliptic to circular vibrations and back again to a plane vibration at right angles to the first." Engels models and drawings caught the attention of Professor Bache and Engel took a job for the US Coast and Geodetic Survey, where he "also made plaster models of variations in the earth's magnetic field...One of these models survives at the American Philosophical Society and another at the Smithsonian" [Kidwell , p. 201].
According the the catalog of the Kensington collection, five of the models of their collection were constructed at some point by Engel, evidenced by drawings he supplied in `Darstellende Optik' by Prof. Schellbach (p. 38).
Note: Ferdinand Engel is not to be confused with Friedrich Engel, and assistant of Lie.

Fabre de Lagrange

This Paris firm contructed a number of models. A collection of models of ruled surfaces were constructed for the South Kensington Museum [SKensington1, p. 22) in 1872. A collection of about 20 models of surfaces of revolution were constructed for the Escola Politécnica de Lisboa (Polytechnical School of Lisbon) around 1861, apparently copied from models already existing in the Conservatoire des Arts et Métiers in Paris [CMAF website]. Fabre also produced and sold models designed by Olivier (see below).

Percival Frost

Constructed a model of a nonsingular surface with 27 real lines around 1882. In his article "On the Twenty Seven Lines..." he gives "hints" at the construction of his model, including a number of equations. He notes, "My model is anything but perfect, two or three of the lines are too far off to appear, and with them all their ten points of intersection are out of sight...perhaps, if I should ever have time for another model, I might succeed in bringing them all into view." (Frost, p. 96)

Edward Heis

Münster, Wesphalia. There are two models in the Kensington collection by Heis, both called "Podoids" whose equation represents a particular elliptic function.

Prof. Hennessy, Dublin

Listed in the catalog of the Kensington collection as having built a series of models "illustrative of Plücker's Researches in Geometry of Three Dimensions" (p. 35). I have another vague reference to a Hennessy who discussed a model in "Phil. Mag (1895)" illustrating "Prince Rupert's Problem."

Olaus Henrici, 1840-1918

According to the MacTutor archive, Clebsch saw mathematical talent in Henrici, who was at the time an engineering apprentice, and convinced Hesse to take Henrici on as a student. Henrici later studied under Weierstrass and Kronecker. Henrici then went to England. He built the model of the elliptic paraboloid which inspired A. Brill. Also built a model of two moveable hyperboloids which always remain confocal (in Dyck's catalog, number 161, page 261), which was presented to the Lonton Math Society in January of 1874 (Dyck, p. 261-262). There is a model of a cubic surface by Henrici with the 27 lines coinciding in 3 real lines in the catalog of the Kensington exhibition. The equation is given as xyz=k^3(x+y+z-1)^3. There is also a listing for a surface of the ninth order by Henrici called "Sylvester's Amphigenous Surface."

Josiah Holbrook and his sons Alfred and Dwight.

Josiah wrote a book on devices (including geometric models) for use in education. Around 1826, Holbrook started an exchange for such objects. In the 1840's, Alfred and Dwight established Lyceum Village in Berea, Ohio and sold apparatus. Around 1854, Dwight moved to Hartford, Conn and started the Holbrook School Apparatus Manufacturing Company. Kidwell, p. 199.

Elling Holst

Has a "Model representing the Right Lines on a Surface of the Third Class" in the catalog of the Kensington collection. Seems to be connected with Klein and Lie (refs I've found so far are not in english).

Abraham Gotthelf Kästner, 1719-1800

Professor of mathematics at Göttengin from 1956, used models of polyhedra for his teaching. His are the oldest models in the collection at Göttengin. From Mühlhausen, p. 50.

Felix Klein

A major builder and advocate of the construction and use of models. See my 1999 talk. Klien's "Erlangen program" address in 1872 contained mention of models! Kidwell, p. 203. Klien joined the faculty of Technische Hochschule Munchen in 1875, the same year as A. Brill. He left in 1880, four years before Brill.

Eduard Kummer, 1810-1893

During the 60's and 70's, Kummer presented (at meetings of the Preussische Akademie der Wissenschaften) models of a series surfaces with tetrahedral symmetry (see Barth and Knorrer, p. 17). These were copied and sold by Brill as Series 9 (Kidwell, p. 200). In the 1960's, he constructed models of surfaces of the fourth order and certain focal surfaces (Fink, p. 278). See also the notes under L. Lohde below.

M. Langley

Reproduced models of Max Bruckner exhibited at Napier Catalogue p. 327.

Ludwig Lohde

Berlin. Has five models in the catalog of the Kensington collection, including two cyclides, a "Curvilinear central surface of the Ellipsoid," a model of "Maximum of Attraction of the Earth's Surface," and a minimal surface which the catalog notes was "submitted to the Berlin Academy of Sciences by Pressor Kummer, on the 6th April 1865" (p. 37). I do not know if the surface from the catalog was copied by Lohde or if it is Kummer's original surface, as the note is unclear. One of the cyclides, the Dupin's Cyclide, is "according to the calculation of Professor Kummer, at Berlin" and the reader is referred to "Monatsbericht der Akademie der Wissenschaften zu Berlin, 1863, pp. 328 and 336."

Magnus

Built a model of a wave surface in Berlin in 1840 (perhaps with Soleil?). (Fink, p. 277)

James Clerk Maxwell, 1831-1879

Made a model of a surface surface gives the relationship between volume, energy, and entropy of a certain mass of substance. The model was given to Professor Chrystal, who gave it to Crum Brown. Rumour has it that he gave a lecture on thermodynamics at the exhibition of scientific apparatus in London that was illustrated by his own model of the thermodynamic surface. Has a series of stereograms of various curves and surfaces and a stereoscope in the catalog of the Kensington exhibition (p. 40).

Rudolf Mehmke, 1857 - 1944

Helped Dyck to organize an exhibition of models and instruments in 1893. See notes.

Gaspard Monge, 1746-1818

Monge developed descriptive geometry, which involves representing three dimensional objects in two dimensions. His ideas were developed at a school for military engineering and were a military secret until after the French Revolution. During the 1790's, Monge became an influential figure in technical education and descriptive geometry became a widely taught subject (in 1801 students at the 'Ecole polytechnique spent 20% of their first two years studying it). He also studied ruled surfaces and made at least two models of surfaces which were at the Conservatoire National des Arts et M'etiers in Paris. Kidwell, p. 200. According to the 9th edition of the Encyclopaedia Britannica, "As regards tridimensional figuring, the oldest known models for instruction in the higher geometry are the thread models of skew surfaces constructed about the year 1800 under the direction of G. Monge for the École Polytechnique in Paris" (p. 628).

Muret

According to the 9th edition of the Encyclopaedia Britannica, Muret was a publisher in Paris (later in Delagrave) that published a series of thread and plaster models "intended for instruction in descriptive geometry" (p. 628).

Prof. Neovius

Listed in Brill's Advertisement as having a series of models in the catalog. From Helsingfors according to ad.I believe this is E. Neovius, a pubil of H.A. Schwarz. He has a triply periodic surface named after him. (See this website describing a mobile which contains the Neovius surface.)

Otto Neugebauer, 1899-1990

Neugebauer was a mathematician and historian, in charge of the collection of mathematical objects at Gootingen, from Muhlhausen, p. 50.

Théodore Olivier, 1793-1853

Olivier was a graduate of the
Ecole Polytechnique and taught descriptive geometry at the /Ecole Centrale des Artes et Manufactures and at the Conservatoire National des Arts et Méetiers. Olivier designed models of ruled surfaces, including some where were movable, to suggest how the surfaces were generated. His models also showed the curves resulting from the instersection of surfaces -- these were marked by special grommets. Olivier's models were manufactured by the firm of Pixii, Père et Fils. Their sucessor, Fabre de Lagrange, continued to manufacture the models. Kidwell, p. 200. See more in notes. The 9th edition of the Encyclopaedia Britannica reports that Olivier created models similar to those of G. Monge, but that were moveable.

Professor Peddie

Mentioned in the Napier Catalogue in the section from the University of Edinburgh.

Saul Pollock, 1904-?

Pollock got his doctorate from University of California and taught at Indiana State Teacher's College in Terre Haute, Indiana. Pollock made moveable models of ruled surfaces with aluminum frames, models of ruled surfaces showing intersections of surfaces. Exhibited at the 1934 Century of Progress Exhibition in Chicago Kidwell, p. 204.

Julius Plücker, 1801-1868

According to the 9th edition of the Encyclopaedia Britannica, Plücker constructed a series of 27 complex surfaces. After his death "not very satisfactory" copies were made "from zinc casts" (p. 628).Plücker attended the universities of Heidelberg, Berlin, and Paris. According to the catalog of the South Kensington exhibition (p. 30), in 1866, Plücker read a paper "Complexes of the Second Order" to the British Association at Nottingham." With this paper he showed a series of models constructed by Epkens of Bonn, who was probably a scientific instruments manufacturer. Copies of these models were made "for Dr. Hirst," and presented to the London Mathematical Society. The source goes on to note that "The Society possesses a series of 14 wooden models of surfaces, constructed under the direction of the late Prof. Plücker, in illustration of the teory developed in his posthumous work `Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raum-element,' Leipzig, 1869." I am not sure if these are the same as the 1866 models or different. I believe Klein worked to finish Plücker's postumous work -- perhaps this is important. It is later noted that the models were remounted under the direction of Henrici (p. 34).

Carl Rodenberg

A student of Klein's who died shortly after completing his thesis. Along with his thesis he published a series of models of singular cubics.

Karl Rohn

Mentioned in Fink, p. 278 as a builder of models. Student of Klein and Brill at Munich.

W. W. Ross

Sold a set of models for teaching students about surface area and volume. Ross was a school superintendant in Fremont Ohio. (Kidwell, p 197.

Martin Schilling

Schilling's firm produced models for sale after Brill's. First in Halle, then Leipzig.

Ludwig Schliermacher

Student of Klein and Brill at Munich. Built models. Kidwell, p. 203.

Arthur Schoenflies, 1853-1928

Held a professorship for applied mathematics at Gõttengin starting in 1892. Made models of crystal structures, in plaster. From Mühlhausen, p. 51.

J. Schroeder

Schroeder and his students constructed models at the Polytechnisches Arbeits-Insitut in Darmstadt, Prussia. The models they built included some movable string models and "models of intersecting lines and solids." They also made models showing the projection of various solids onto planes (to help student learn drawing) and wooden models of polyhedra. His models were sold by J.W. Queen and Company from at least 1881. Kidwell, p. 201.

Herman Schwarz, 1843-1921

Schwarz was a student of Kummer and built a series of models, including some minimal surfaces and "the surfaces of centers of the ellipsoid" (Fink, p. 278).

Henry J.S. Smith, 1826-1883

Henry Smith was a number theorist and geometer who was president of the London Mathematical Society from 1874-1876. He wrote an article "Geometrical Instruments and Models" covering that portion of the 1876 exhibition of scientific apparatus at the South Kensington Museum.

Walter Smith, 1836-1886

Originally from England, Smith was hired by the city of Boston to set up courses in industrial drawing for the city. He founded the Massachusetts Art Normal School (later Mass College of Art). He believed students should begin to learn to draw by drawing plane and solid geometrical figures and that they should use the actual objects as models. "Smith designed four sets of what he called `American Drawing Models.' The first set had 30 pieces and included not only common geometrical solids, but also a flight of four steps, a cross, and three vases. Several shapes were shown in skeleton forms...The models were made in Worcester, Massachusetts, at the Washburn Machine Shop of the Friee Institute of Industrial Science (later Worcester Polytechnic Institute)." They were manufactured as late as 1897. Dealers included Devoe, Andrews, and Frost & Adams of Boston. After Smith returned to England John B. Clark, Walter Perry and Mary Dana Hicks contined to develop textbooks with Prang, Smith's publisher. The Use of Models emphasised models in art education.Kidwell, p. 202.

Soleil

Built a model of a wave surface in Paris in 1840 (perhaps with Magnus?). (Fink, p. 278).

Duncan MacLaren Young Sommerville, 1879-1934

Built some polyhedral models exhibited at the Napier exhibition, Napier Catalogue p 320-321.

J.E.A. Steggall

There is a list of four groups of models by Steggall in the Napier Catalogue, p.319-320. These include four deformable models of quadratic surfaces, some polyhedra, and some other models.

William E. Story, PhD, University of Leipzig, 1875

Story used models at Johns Hopkins, where he taught. Story was a Harvard grad.

B.G. Teubner

Teubner's firm in Leipzig sold models designed by Hermann Wiener. From Cajori's History of Math, p.328-329.

James Thompson

Constructed model of surface that shows relation of temperature, pressure, and volume of a constant mass of carbonic anhydride. Napier Catalogue p 302, 311.

A. Harold Wheeler, 1873-1950

Graduate of Worcester Polytechnic institute.Taught math in high schools. Built polyhedra models and had his students build them as well using string, balsa wood, and paper from about 1910 to 1950.

Dr. Wiecke

Cassel. Has two models in the catalog of the Kennsington Collection, one a string model of a hyperboloid of one sheet, the other a plaster model of the "eighth part" of the hyperboloid of one sheet (p. 38-9).

Christian Wiener

Karlsruhe. The MacTutor archive has this to say: "At Clebsch's suggestion Wiener constructed plaster of Paris models of mathematical surfaces which were exhibited in London, Munich and Chicago." According to Fink, Christian Wiener was the first to build a model of the surface of a cubic with 27 lines in 1868. Dyck's calog lists stereosopic photos available of a model of a cubic surface with all 27 lines real contructed by Weiner, and published by Teubner. The date is listed as 1869 (see p. 263, catalog number 162 in Dyck's catalog). The Kensington collection also lists models of the cubic surfaces with 27 real lines, one made of plaster of paris, the other of "discs of card-board" (p. 37). According to the catalogue, the construction of the plaster model "is described on a placard fixed to the model."
Weiner studied engineering and archtechture and was a teacher of physics, mechanics, hydraulics and descriptive geometry. His nephew A. Brill comes into our story as well. He wrote a couple of texts on descriptive geometry which can be found on the Cornell digital library page. If he is the "Professor Wiener of Carlsruhe, then according to the 9th edition of the Encyclopaedia Britannica, he made wire models of tortuous curves.

Hermann Wiener

Designed models that were sold through the firm of B.G. Teubner.