List of References

Art

See References Relating to Art/Artists (and relationships with models of surfaces)

General Mathematical History and Biography

F. Cajori. A History of Mathematics. Second Edition. Macmillan and Co., London, 1919.

General reference. Does contain a short article specifically about models.

F. Cajori. The Teaching and History of Mathematics in the United States. Government Printing Office, Washington, DC, 1890.

I have not looked very much at the first part of this book, but the last part contains results of a survey sent to high schools all the way up through universities. There were 168 respondents to the long list of questions, which included a question on the use of mathematical models.

J. L. Coolidge. A History of Geometric Methods.

J. Dieudonne. History of Algebraic Geometry.(Translation of Cours de Geometrie Algibrique I.) Wadsworth, Monterey, 1985.

K. Fink. A Brief History of Mathematics. Translated from German by W. Beman and D. Smith. Open Court Publishing Co, Chicago, 1910.

General history of mathematics. Contains a couple of pages on models.

W. Gibbs. "Obituary: Christian Gottlieb Ferdinand Engel" from The American Journal of Science and Arts, Vol. 45, pp. 282-284.

C. Gillispie, ed. Dictionary of Scientific Biography. Charles Scribner and Sons, NY, 1981.

Mathematical Models (and Scientific Instruments)

W. Barth, H. Knorrer. ``Algebraic Surfaces'' in Mathematical Models, Volume II: Commentary.

This is an excelent article that does a great job of discussing models of algebraic surfaces from a historical point of view, as well as relating the models to more modern results. The surfaces discussed are those of orders 3 and 4.

G.W. Cussons. "Mathematical Models" in Encyclopaedia Brittanica, 14th edition, 1929.

Cussons (of the firm Messrs. G. Cussons Ltd) was a maker of scientific and technical apparatus in Manchester. The article is excellent, containing many illustrations and photographs, descriptions of a wide range of models, and some ideas and tips concerning construction. Only a very few names are mentioned in connection with model building, unfortunately, and none of the models illustrated indicate who was involved in construction or design.

H. M. Cundy, A. P. Rollett. Mathematical Models. Second Edition. Oxford University Press, London, 1961.

A great "how to" book -- describes methods for building a wide variety of models and instruments. This has been reprinted and can be found on Amazon.com.

W. Dyck. Katalog Mathematischer und Mathematisch Physikalischer Modelle, Apparate, und Instrumente (Reprint). George Olms Verlag, 1994. Originally published by C. Wolf & Sohn in 1892.

A catalog of over 300 mathematical models, instruments, and other objects available from various publishers in the 1890s. There is a new introduction with photographs by Joachim Fischer.

A. Emch, Mathematical Models, Univ Illinois Bulletin 18 (1920) 12.

Series of catalogues of models including explanations and pictures.

A. Emch. "A Model for the Peano Surface" in American Mathematical Monthly, Vol 29, Issue 10 (Nov-Dec, 1922), p 388-391.

In this paper, Emch discusses the Peano surface, which is an example that Peano used to show certain accepted max/min criteria were bogus. The article include a picture of a plaster model built by Emch.

A. Emch. "On the Construction and Modelling of Algebraic Surfaces" in American Mathematical Monthly, Vol 28, Issue 2 (Feb, 1921), p 46-54.

In this paper, Emch discusses a "projective method" for modelling surfaces.

A. Emch. "The Value of Mathematical Models and Figures" in American Mathematical Monthly, Volume 34, Issue 2 (Feb, 1927), p. 76-79.

In this article, Emch gives some arguments about the usefulness of visualization (through models and drawings) in mathematics instruction. He discusses some specific examples.

G. Fischer. Mathematical Models: From the Collections of Universities and Museums. (Two volumes). Friedr. Vieweg and Sohn, Braunschweig, 1986.

This great two volume set got me started investigating the history of models. The first volume has photographs and the second has some excellent companion articles.

Frost, "Model of the 27 lines" in Quarterly Journal of Pure and Applied Math Vol XVIII 1882 p 89-96.

Have this, haven't yet read it.

J. Gray. Review of Mathematiche Modelle/Mathematical Models in Mathematical Intelligencer, Vol. 10, No. 3, pp. 64-69.

E.M. Horsburgh. Modern Instruments and Methods of Calculation: A Handbook of the Napier Tercentenary Exhibition. G. Bell and Sons, Ltd., London, 1914.

The Napier Tercentenary Exhibition was held in Edinburgh, . There is a section on mathematical models, presented by Crum Brown. There is also memoribilia related to John Napier (as well as a history of Napier's life). There are tables, calculating machines, laboratory instruments, and ruled papers.

P. A. Kidwell. ``American Mathematics Viewed Objectively: The Case of Geometric Models'' in Vita Mathematica, R. Calinger, ed. Mathematical Association of America, 1996.

H. Smith. "Geometrical Instruments and Models" from Handbook to the Special Loan Collection of Scientific Apparatus, 1876, South Kensington Museum. Chapman and Hall, London, 1876.

This article describes models and instruments from the South Kensington exhibition in 1876. Included are instruments used in drawing and a variety of models, mainly of algebraic surfaces, although there are also some models of surfaces of constant curvature. Unfortunately, there are no illustrations, but the models are well-described, and the descriptions often include information on the significance or interesting points of the model, as well as information on who built the model.

[SKensington1] South Kensington Museum. Catalogue of the Special Loan Collection of Scientific Apparatus at the South Kensington Museum, Third Edition. Eyre and Spottiswoode, London, 1877.

Contains a section describing in detail many of the models on display at the exhibition.

Advertisement for models published by L. Brill. Back matter from American Journal of Mathematics, Volume 12, Issue 3.

This advertisement lists the names of those involved with the construction of the models as well as numbers of different types of models (plaster, string, etc) in the collection.

"Mathematical Drawing and Modelling" in Volume XV of The Encylopaedia Brittanica: A Dictionary of Arts, Sciences, and General Literature, 9th Edition, p. 628-629.

This article discusses the uses of drawings and models in mathematics discusses some of the mathematicians involved in makings models for "higher geometry," including Monge, Plücker, Olivier, and others. The article also includes some notes about the construction of models. I would like to find out who wrote this article.

E. Mählhausen, "Riemann Surface -- Crocheted in Four Colors" in The Mathematical Intelligencer, Vol. 15, No. 3, p. 49-53.

An article about the collection of mathematical objects (plaster surfaces, string models, polyhedral models, and more) at Göttingen.

Mathematics of Surfaces

A. Henderson. The 27 Lines Upon the Cubic Surface. Hafner Publishing, New York, 1911.

Discusses the nonsingular cubic surface and its 27 lines in depth, including a discussion of techniques for modelling the 27 lines. Includes an historical overview and numerous plates illustrating the configuration of the lines.

R.W.H.T. Hudson. Kummer's Quartic Surface. (Reissue: Orig. 1905). Cambridge University Press, Briston, 1990.

Other Possible Sources

Cayley, "A memoir on cubic surfaces" in the collected papers of Cayley (vol 6 p 359-456).

Have this currently, haven't read yet.

H.F. Baker. ``On some recent advances in the Theory of Surfaces.'' {\sl Proc. London Math Society}. 1912 (2) 1-40. (Cajori cites as vol 12 not 2). Cajori says see also Proceed of the 5th Intern Congress, Vol I Cambridge 1913, p 49.

Have this currently, haven't read yet. Need to fix reference.

H. A. Schwarz. Gesammelte Mathematische Abhandlungen. Springer, Berlin, 1890.

Have this currently, haven't read yet.

Lucien Godeaux. "Sur la construction de modeles de surfaces algebrique content des involutions cycliques" in Bulletin de la Societe Royale des Sciences de Liege #17 1948 p 56-61.

Have this, haven't yet read it.

Visual Education. Journal published by the society for visual education -- have a 1920 ad for this pub.

Harvard, Gutman Education Per LB 1044.A2.v5.

Benno Artmann, "Roman Dodecahedra", Mathematical Intelligencer, Vol. 15, pp. 52-53, 1993.

Articles in Encyclopedia Brittanica -- I have found the 9th already (MIT), the 11th (BU) has no article on models, the 14th (BPL) also has an article on models. There are more at the BPL.

National Council of Teachers of Mathematics. Multi-Sensory Aids in the Teaching of Mathematics the 18th yearbook of the NCTM.

MIT has yearbooks 1926-1984. Humanities Stacks. QA1.N227.

Axonometrical Projections of the Most Important Geometrical Surfaces: Drawings in Descriptive Geometry by Ferdinand Engel, H. Goenebler, NY, 1855. Includes a catalogue of wood and plaster models.

Harvad Cabot Science Oversize Math 5808.54F

The Teacher's Guide to Illustration: A Manual to Accompany Holbrooks's School Apparatus by F.C. Brownell, pub by Holbrook School Apparatus Co, 1857.

Harvard Gutman Education EducT 5038.57.237 also has 4th 3d 1861 Educ 2325.58.8

William Ladd. Catalogue of optical, mathematical, and philosophical instruments. London, 1868.

Harvard. Countway Meidicine. Rare Books QC645.N66 1868.

G. Salmon. A Treatise on Analytic Geometry of Three Dimensions. 4th Edition. 1882. (BU Science Library has this)

B. Segre. The Non-Singular Cubic Surfaces: A New Method of Investigation with Special Reference to Questions of Reality.

BU libraries apparently ordered it in 1995, but never recieved. MIT Science Stacks, QA 573.S4 1942.

P. Fermat Ad Locos Planos et Solidos Isagoge (Introduction to Plane and Solid Loci). Written in 1629, published in 1679.

Description of model of Christian Wiener of Cubic surface with 27 real lines: Trans Camb Philos. Soc, Vol XII 1871 Part I pp 366-383. Description of surface by Cayley.

MIT RSC.

Blythe, On Models of Cubic Surfaces -- campbridge U Press, 1905

RSC HDC QA573.B66x. Also at Harvard Cabot Science, Math 8509.05.5. Also at the BPL.

Martin Schilling, Catalog mathematischer Modelle fur den hoheren mathematischen Unterricht, 1911.

Harvard Widener Math 9909.03

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

Uber die gleicheckig... is in MIT, RSC HDC, QA491.B7x 1906.

It would be great to take a look at Rodenberg's thesis or the pamphlet describing the series of models that came with it. See Barth and Knorrer's refs.

Look into the mathematical association's visual aids exhibition in 1947 (from Cundy p 193)

Geometrical Tools, A Mathematical Sketch and Model Book by R.C. Yates. Educational Publishers 1949. Earlier edition from 1941 has same title, minus the word "Geometrical" printed by the author.

I have a review of this book from Am Math Monthly 1942, vol49 issue 7.

Casting Geometric Models in Plaster-of-Paris by Wallace L. Hainlin. 1955.

Stalking Solid Geometry With Knife and Clay by Jack Price.(1961)

Mathematsche Abhandlungen aus dem Verlange Mathematicscher Modelle von Martin Schilling, Halle a.S. 1904.

Mensuration Taught Objectively with Lessons on Form by W. W. Ross (self-published?) c 1891.

Apparatus Designed for Families, Schools, Lyceums and Academies by J. Holbrook Allen and Ticknor, Boston, 1833.

Illustrated Catalogue of Mathematical, Optical and Philosophical Instruments and School Apparatus, J.W. Queen and S.L.Fox 1859, Philadelphia. Has glass models of polyhedra inscribed in cubes pp 52-53.

Pike's Illustrated Descriptive Catalogue of Optical, Mathematical and Philosophical Instruments B. Pike, Jr, 1848, NY.

Catalogue des Collections, Conservatoire National des Arts et Metiers. Paris, 1882.

The Olivier Models by W.C. Stone. Friends of the Union College Library, Schenectady, NY, 1969.

Priced and Illustrated Catalogue and Descriptive Manual of Mathematical Instruments and Materials for Drawing, Surveying and Civil Engineering, J.W. Queen & Co, Philadelphia, 1881.

"List of Models of Mathematical Surfaces Belonging to the Johns Hopkins University" in Johns Hopkins University Circular, January 1885 # 36.

Johns Hopkins University Circular January 1885. Article by William E. Story.

Richard Baker, Mathematical Models, Iowa City, 1931.

Benno Artmann, "Symmetry Through the Ages: Highlights from the History of Regular Polyhedra", in In Eves' Circles, ed by Joby Milo

Translation of the Erlanger Program into English: Bulletin of NY math society 2 (1892-1893) p. 215-249. Apparently has some mention of models.

A. Brill, Ueber die Modellsammlung des mathematischen Seminars der Universitat Tubingen, 1886.

Mathematisch-naturwissenshaftliche Mittheilungen von O. Boklen, 1887.

W. Dyck. Spezial-Katalog der Mathematischen Ausstellung, Berlin, 1893. Is this the catalog made for the chicago worlds fair??

C. Schilling. Die Minimalflachen funfter Klasse mit dem Stereoscop-Bild eines Modells derselben. Dissertation Gottingen, 1880.

R. Kummer. Die Flachen mit undendlichvielen Erzeugungen durch Translationen von Kurven. Dissertation Leipzig. 1894

Bradley's Practical Geometry. According to the Catalogue of the South Kensington exhibition, this has drawings of many of the surfaces represented by the models in the exhibition.

From same source, Leroy, Adhemar, Lefebure de Fourcy, De Laa Gournerie (treateses on practical and descriptive geometry, and "in their treatises on Sterotomy and Stone cutting).

From same source, Prof. Schellbach's "Darstellende Optik" has drawings of models made by Ferdinand Engel. According to catalog entries, Ferdinand Engel is the author of this book. It is at Harvard, I believe.

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