Wilhelm Fiedler and His Models—The Polytechnic Side

Volkert, Klaus

doi:10.1007/978-3-030-97833-4_3

Klaus Volkert³

Part of the book series: Trends in the History of Science ((TRENDSHISTORYSCIENCE))

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Abstract

In this article I will present some information on the history of mathematical models.

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Modeling for Introducing Students to New Tools

In this article I will present some information on the history of mathematical models (in the sense of concrete objects referring to something mathematical and abstract) in the world of polytechnic schools, in particular at what is known today as the Eidgenössische polytechnische Schule (ETH) in Zurich. My story is focused on the person and the work of Wilhelm Fiedler (1832–1912), professor of descriptive geometry and geometry of position at that polytechnic from 1867 until 1907. I will provide some unique information on the role of models in everyday life of a polytechnic school by looking at Fiedler’s large correspondence and at his publications as well as at some other documents. What I want to emphasize is the importance of the tradition of polytechnic schools concerning the production and the use of models. This tradition differed profoundly from that of universities; in particular, it paved the way for the establishment of models as scientific and pedagogical objects. A threefold continuity characterized Fiedler’s use of models: It was a continuation of the teaching he probably had as a pupil, it was in accordance with the traditions of the polytechnic teaching in general, and it was in continuity with the conception and practice of the Polytechnikum in Zurich.

Wilhelm Fiedler

Otto Wilhelm Fiedler was born April 3, 1832 in Chemnitz. In those days, Chemnitz was an important industrial center called the Manchester of the east. Because his parents lacked the financial means to pay the fees of a Gymnasium, Wilhelm went first to the Bürgerschule, then to the Gewerbeschule in Chemnitz. Afterwards, he studied at the Bergakademie at Freiberg—not at a university because he had no Abitur and no money—where the founder of scientific mine surveying J. Weisbach influenced him. After finishing his studies, he worked as a teacher for a short time in Freiberg, then at his former school in Chemnitz. In 1859, he presented his thesis on descriptive geometry (“Die Zentralprojektion als geometrische Wissenschaft,” published 1860 in Chemnitz in the Wissenschaftliche Abhandlung of his school) at the university of Leipzig; its referee was August Ferdinand Möbius. In 1864, Fiedler was called to the Polytechnic in Prague where he was deeply involved in the struggles between ‘German’ and ‘Czech’ professors. He left Prague in 1867 in order to go to Zurich. Here he was appointed professor of descriptive geometry and geometry of position (darstellende Geometrie und Geometrie der Lage) or what is known today as projective geometry. Fiedler was a member and for many years also head of the VI. Abtheilung (VI. department), also called Fachlehrerabtheilung. This department was responsible for the training of future teachers and for providing courses on basic subjects like descriptive geometry, mathematics and physics for the students of the other departments—typically future engineers. Besides Fiedler there were two chairs for German speaking professors of (pure) mathematics in the department. Here we find illustrious names like Bruno Elwin Christoffel, Georg Frobenius, Adolf Hurwitz, Hermann Minkowski, Friedrich Prym, Friedrich Schottky, Hermann Amandus Schwarz, Heinrich Weber—to cite only a few. Fiedler taught for forty years in Zurich. He was responsible for descriptive geometry, but he also delivered courses on projective geometry and other geometrical subjects for future teachers. He had two assistants who helped him with his teaching. In particular, they were occupied with the practical work done by the students in the classrooms for drawing.

These assistants were not supposed to do research; they were hired for teaching tasks only. It is important to note that the Polytechnikum in Zurich—like many other polytechnics—offered an atmosphere that was very favorable to the use, the production and the presentation of material models of all kinds. Models were an integral and highly estimated part of the polytechnic tradition.^{Footnote 1} In Gottfried Semper’s building at Zurich, they were ubiquitous and displayed in prominent places.^{Footnote 2} When entering the building through its main entrance, one ran into the collection of sculptures. The role of the collections is explained in a report on the polytechnic written on the occasion of the world-exposition at Paris (1889):

These [the collections of the Polytechnikum] are primarily intended for teaching and study in the relevant fields, for which purpose they are made available to students at all times for private study. In addition, they serve to further the research of the teaching staff, advanced students, and scholars outside the institute. Finally, they are used for the instruction of the general public, and it is to this end that at least the more important collections are made available at certain hours to all visitors, and their exhibits displayed and labelled in such a way that the layperson without expert guidance is still able to benefit from them.^{Footnote 3}

Nowadays, Fiedler is remembered—if he is remembered at all—as the German editor of several books written by George Salmon: the combination Salmon–Fiedler was well known during the second half of the nineteenth century. It should be noted that Fiedler did much more than simply translate Salmon’s book; he inserted a lot of new texts, made corrections, added problems, notes, and so on.

But he published also a lot of papers and three books of his own:

Die Elemente der neueren Geometrie und der Algebra der binären Formen. Ein Beitrag zur Einführung in die Algebra der linearen Transformationen (Leipzig: Teubner, 1862). [The elements of the newer geometry and algebra of binary forms. A contribution to the introduction to the algebra of linear transformations.]

Die darstellende Geometrie in organischer Verbindung mit der Geometrie der Lage. Für Vorlesungen an technischen Hochschulen und zum Selbststudium (Leipzig: Teubner, 1871); 2nd ed. (Leipzig: Teubner, 1875); 3rd ed. in 3 vols.: vol. 1 (Leipzig, 1883), vol. 2 (Leipzig, 1885), vol. 3 (Leipzig, 1888); 4th ed. of the first vol. (Leipzig, 1904).^{Footnote 4} [Descriptive geometry in organic relation with geometry of position. For lectures at technical universities and for self-study]

Cyclographie oder Construction der Aufgaben über Kreise und Kugeln und elementare Geometrie der Kreis- und Kugelsysteme (Leipzig: Teubner, 1882). [Cyclography or construction of tasks on circles and spheres and elementary geometry of circle and sphere systems]

The teaching of geometry to future architects and civil engineers at the Polytechnikum in Zurich was minimized around 1885—in particular, projective geometry^{Footnote 5} was eliminated in those courses. But Fiedler always provided a substantial teaching of geometry to future teachers. To the students of the VI. department, geometry was taught by a series of four courses:

Descriptive geometry. Part one: parallel projection (3 hours a week, winter term 1879/80);

Descriptive geometry. Part two: central projection (2 hours a week, summer term 1880);

Geometry of position (4 hours a week, winter term 1880/81);

Projective coordinates (2 hours a week, winter term 1880/81 and summer term 1881).^{Footnote 6}

By means of this teaching, Fiedler could survive—so to speak—as a researching geometer combining his research with instruction.^{Footnote 7}

Some Remarks on Teaching and Early Models

In order to understand Fiedler’s occupation with models better, it is helpful to look closer at the teaching delivered at Gewerbeschulen^{Footnote 8} and polytechnics, in particular of descriptive geometry. The latter was considered a hybrid theme: on one hand mathematics, on the other hand practice. Often, it was not understood as a genuine part of mathematics itself. This is illustrated by the fact that the Polytechnikum in Zurich—and most other polytechnics in the German speaking area—had both professors for mathematics and descriptive geometry. Descriptive geometry did not have a lot of theorems with proofs but many, many problems to solve by drawing. Often, these solutions are only graphic approximations, which can be checked by drawings.

It should be remembered that the teaching of drawing had a long-standing tradition going back to the eighteenth century with drawing schools for craftsmen. In that teaching, material models were used to present the objects to be depicted (a cube, for example) to the pupils—or to control the drawings made by the students.^{Footnote 9} Of course, these models were rather simple and crude, but the tradition existed. Traces of this tradition are visible in some Schulprogramme; these were reports on the academic year, often supplemented by a paper written by a teacher.^{Footnote 10} Concerning models, I want to cite three examples where they were discussed: explicitly three books by J. Gierer (1847), Franz Harter (1847) and Paul Wiecke (1873).^{Footnote 11} This is to give but a few examples. Obviously, this strand of the history of models deserves further investigation. The Polytechnikum in Zurich owned several models of elementary character like cubes made out of glass or metal,^{Footnote 12} even as late as 1930, a dodecahedron made out of marble.

At the Gewerbeschule, which gradually replaced the drawing schools during the nineteenth century, descriptive geometry was taught at a basic level. Fiedler grew up in this tradition. As a professor of the Gewerbeschule, he himself taught descriptive geometry. This was not his first occupation when he came to Chemnitz, but in 1857 he was obliged to do so because his colleague, who was responsible for teaching descriptive geometry, fell ill.^{Footnote 13} In order to understand the teaching of descriptive geometry, it is important to remember that there were other more concrete courses in drawing—for example, in technical drawing or on perspective. This means that descriptive geometry was not understood as cultivating drawing itself but as a theory basic for different types of drawing used by engineers, architects, etc. Nevertheless, the students had to do a lot of drawing for their courses on descriptive geometry, but normally of a more abstract nature (no machines or buildings, etc.) Descriptive geometry is more than drawing as is indicated by its name.^{Footnote 14} This statement was very important to Fiedler.

As noted above, at polytechnic schools there were a lot of models in use and in production; hence, to use models in mathematical training was in complete conformity with the practice of these schools. Let me note that elementary mathematical models of the type normally used at Gewerbeschulen were produced and sold by a company named Polytechnisches Arbeits-Institut J. Schröder in Darmstadt (see Fig. 1); they were explicitly called Unterrichtsmodelle (‘models for teaching’). Around 1840, Jakob Schröder,^{Footnote 15} a former teacher, invented a model to illustrate plane and elevation (see Fig. 2). Composed of three planes meeting orthogonally, the planes were mobile, such that they could be rotated into one single plane.^{Footnote 16}

Shortly after his arrival in Zurich, Fiedler bought a collection of forty models in four series produced by J. Schröder at a price of 40 sfr.^{Footnote 17} These models were made out of wood (two series), by rods (one series) or by strings (one series). Some of them still exist in Zurich (in the Sammlungen of the ETH and at the Kantonsschule Rämibühl).

The focus of Schröder’s company was on producing models of machines and instruments, but the company also sold models of solids (see Fig. 3) and ‘models of descriptive geometry’ like the plane and elevation of a straight line, two cylinders penetrating each other, and shadows. There were also wooden cones that could be decomposed into two parts in order to illustrate the resulting conic section and wooden ellipsoids that could be decomposed into two parts showing the range of circles on the surface. Schröder’s models are rather robust^{Footnote 18}—really usable in everyday teaching. Despite the fact that they were widespread in the nineteenth century,^{Footnote 19} seemingly most of these models have disappeared since then, perhaps because they were considered to be neither of high monetary (compared to other models later sold by L. Brill or Martin Schilling they were rather cheap) nor scientific value. In the recent discussions on mathematical models, this aspect, which one may call ‘sub-scientific’ or ‘elementary’, was and is often neglected.^{Footnote 20} One of the rare traces of Schröder’s firm can be found in a talk given by Alexander Brill on the occasion of the opening of the collection of mathematical models at the University of Tübingen (November 7, 1886):

For a long time in Germany too there have been models whose purpose has been to serve the needs of mathematical education. Probably in most universities and technical schools there is a cupboard or dusty corner somewhere containing cardboard models of polyhedra, cones with planar sections, curves of intersection of cones and cylinders, etc., mostly of indeterminate provenance, which in the current state of higher education are no longer required, and thus no longer replaced by the newer more beautiful forms that numerous suppliers of teaching aids (the oldest being the estimable Schröder model factory in Darmstadt) habitually give these elementary teaching materials.^{Footnote 21}

Wooden models produced by J. Schröder are presented in Figs. 3, 4, and 8.

There was another usage of the term ‘model’ which was important (and current) in the second half of the nineteenth century—at least for German speaking scholars working in descriptive geometry. This usage is related to the problem of representing a part of space^{Footnote 22} by another part of space such that the impression is that of seeing the scene by one eye. In the language of descriptive geometry, this is the problem of projecting a part of space from a point to another part of space—that is, a central projection. If the spaces were imbedded in a four-dimensional space, everything could be done in strong analogy to the case of the plane.^{Footnote 23} If not, things become complicated. A practical situation in which this problem arises is the construction of a stage design in a theater. Jean-Victor Poncelet, the first who studied the problem from an abstract point of view, coined the term ‘relief-perspective.’^{Footnote 24} In German, it was common to speak of Reliefperspektive or Modellierung (see Fig. 5), its product being a Modell. Note that this type of model is an abstract entity. It is an image of a mapping (in modern terms), not a material object.

Here is a passage in which Fiedler explains the significance of this construction^{Footnote 25} and the motivation behind it:

The significance of this construction is that it includes the geometric basis of artistic modelling, that is, one based on pleasing deception, and that it [the construction] encompasses all other technically feasible modeling methods, and finally the methods of presentation through drawings on planes as special cases. [...] When, in addition, the models constructed in this way are lit by light radiating from a suitably chosen point of the opposing plane Q, the eye C believes, instead of its collinear transformation, to see the object itself endowed with the correct sun shadows. (Stage scenery—the space between the planes S and Q is the space of the stage, S the plane of the curtain.)^{Footnote 26}

This text is taken from an autographed^{Footnote 27} version of Fiedler’s course on descriptive geometry. Here, he is more explicit than in his book on descriptive geometry and he provides more background information than in his textbook. One of the first models produced in Fiedler’s surroundings was that of a relief-perspective constructed by his assistant Rafael Morstadt in Prague.^{Footnote 28} Seemingly, this model has disappeared, but models of this type, constructed by Burmester, still exist (see Fig. 6).^{Footnote 29}

The interest in the construction of models by using the Reliefperspektive is underlined by the fact that R. Staudigl published a book on that theme in 1868.^{Footnote 30} In its preface, he emphasizes the fact that the Reliefperspektive is of a great practical value and that it was neglected by the geometers. In a letter dated Riga, 27 March / 8 April,1877,^{Footnote 31} Fiedler’s former assistant, Alexander Beck, reported:

Some time ago I had a plaster model made as an example of a relief-perspective, for which I provided the sculptor with a drawing. [...] The model has turned out very well and has become a centerpiece of my collection. I also intend to acquire the new Munich models.^{Footnote 32}

The Reliefperspektive is a recurrent theme in Fiedler’s writing—not the least because it is so near to the practice of painters and architects. In a paper published in 1882, Fiedler presented the history of this subject, in particular the forgotten book Versuch einer Erläuterung der Reliefperspective zugleich für Maler eingerichtet von J. A. Breysig, Professor der schönen Künste und erstem Lehrer an der k. Kunstschule zu Magdeburg, 1798, written by Johann Adam Breysig.^{Footnote 33} Fiedler remarks that careful models of a Reliefperspektive were built by his former assistant Morstadt in Prague, by Burmester in Dresden and by his assistant Johannes Keller.^{Footnote 34} The Reliefperspektive was also discussed in a Schulprogramm.^{Footnote 35}

In his paper “Über die niedere Sphärik” [On Elementary Spherics] (1832), Christoph Gudermann described another early model (or apparatus) for teaching purposes.^{Footnote 36} In this paper, Gudermann announces his treatise on spherical geometry, published in 1835, and discusses the theory of area for spherical triangles based on—in modern terms—equidecomposability. At the end of the paper, he deplores the state of the art of spherical geometry, that is, the fact that it had fallen into neglect for some time. One reason for this—following Gudermann— were the difficulties of representing the situation on the sphere by plane drawings. So Gudermann constructed a model of a sphere including tools to draw great circles and to transport distances such that constructions can be performed. He called it a “sphärographischer Apparat.”^{Footnote 37} This model was produced and sold by J. V. Albert, a firm in Frankfurt. It contained a sphere, a ruler (an instrument to draw great circles on the sphere), and a transporter (an instrument to move segments or lengths from one place to another, a compass, so to speak). Gudermann underlined the fact that his model is also beneficial for the teaching of geography and astronomy. In sum, we can state that Gudermann’s model was a paradigmatic case for the construction of models motivated by pedagogical needs.

In concluding this section, I want to add that Fiedler also used wall charts in his teaching in order to illustrate themes (like the common perpendicular of two skew straight lines in space).^{Footnote 38} Seemingly, the use of wall charts was a widespread custom in (elementary) teaching in that period. They were produced and sold by different companies.

Models in Fiedler’s Correspondence

In this paragraph, I will present some traces in Fiedler’s work left by his preoccupation with models, in particular, in his large correspondence. When Fiedler came to Zurich in 1867, he replaced Joseph Wolfgang von Deschwanden, the first professor of descriptive geometry at the recently founded Zurich Polytechnikum (1855). Deschwanden was a man with highly developed abilities as a painter, but he was also an expert in the construction of machines.^{Footnote 39} It is not known whether Deschwanden used models or not. But, at least in other places in the school, Fiedler encountered many models.

There are two main types of sources concerning Fiedler’s relation to models: on one hand there are documents by the administration and the inventories cited above (see Fig. 7),^{Footnote 40} on the other hand there are letters to Fielder. Let us start with the first. The highest level of administration of the Polytechnikum was the Schulrath (board of the school) with five members—among them the president of the board, the Schulrathspräsident.^{Footnote 41} In the records of the Schulrath, we find remarks that Fiedler received money in order to buy models: January 19, 1871 and May 8, 1877 he got 100 sfr. In 1871, the sum was dedicated to buy models in gypsum, 1877 to purchase models to be used in teaching. Thus, it is clear that the officials approved the acquisition of models.

Luckily, we can rely on a more detailed document. The Polytechnikum, that is, the Schulrath, reported once a year to the Bundesrath in Bern on the state of the affairs of the school.^{Footnote 42} In the annual report for the year 1878, we find a long passage on the collection of mathematical models^{Footnote 43}: it is stated first that the collection of mathematical models is presented in the report for the first time,^{Footnote 44} and that it is not intended to present all objects in the collection in the report.

Furthermore, however, for a number of years wire or rod models of algebraic surfaces etc. have been made by the various assistants of descriptive geometry under the direction of the professor [that is, Fiedler]—which should be mentioned here as a speciality. It is due to the great and regular workload of the assistants that work on these models has been slow to advance, and that the number of models has grown only slowly.^{Footnote 45}

Then, some special models (or types of models) are mentioned in more detail^{Footnote 46}:

1.
“A wire model of an orthogonal system in the bundle.”

2.
“A model of the main points and main planes of the central collineation of spaces in connection with the central projection of planar systems.”

3.
“A string model of the developable surfaces of a cylindrical helix.”

4.
“Wire models of cubic surfaces with 27 real straight lines with systems of parallel cross sections”—three types of this model were present in the collection.

“Models of quartic surfaces”—two types of this model were present in the collection.

“A model of a curve of intersection of two second-degree surfaces and their developable surfaces with representation of their double curves is in progress.”^{Footnote 47}

This list is interesting for several reasons. First, it is clear that the cited models in the collection were not elementary (like a cube, etc.) In particular, the models of the fourth type, the famous cubic surface with 27 real straight lines (among them Alfred Clebsch’s Diagonalfläche), were rather sophisticated. They were the result of contemporaneous advanced research.^{Footnote 48} Their existence presupposed a certain level reached in the theory of surfaces. After the discovery by Arthur Cayley and Salmon (1849) that a cubic surface has 27 straight lines, several mathematicians worked on that surface—among them Clebsch. In 1868, Christian Wiener (Karlsruhe) constructed a material model of such a surface.^{Footnote 49} Adolf Weiler’s example (1872) was a substantial improvement (see below).^{Footnote 50} It is clear that the progress of geometric theory played a decisive role here: only after the progresses made by algebraic (in those times often called analytic) geometry^{Footnote 51} did it become possible to think of models of sophisticated surfaces. A discovery of a certain importance was Schläfli’s double-six; it enabled Wiener to construct his model.^{Footnote 52} Therefore, it is surprising that the story of these models began only in the 1840s.

It is commonly accepted that Wiener’s model was the first model of this type of surface. But Fiedler reported the story in a different way:

The editor [Fiedler] has owned a rod model of the generic cubic surface with 27 real straight lines since 1865 [sic!]; he has provided information about its simple construction in vol. 14 of the Zeitschrift für Math. (Literaturzeitung). A plaster model resulting from a suggestion by Clebsch and constructed by Wiener has been in wide circulation since 1869. Calculations based on this have been communicated by Cayley in Transactions of Cambridge, vol. 12 (1873), pp. 366–383. Recently, the cubic surfaces with four nodes and the diagonal surface (also as a rod model) have also been modeled. The principle of the rod model can be applied without difficulty in numerous other cases.^{Footnote 53}

In his paper mentioned in the citation above, Fiedler gave a detailed description of stereoscopic photographs of Wiener’s model published by Wiener himself. Then, Fiedler presents the story of his own model, beginning in 1861. After failing with his attempt to construct the model using calculations, which turned out to be too difficult, Fiedler decided to construct it with the help of the Reliefperspektive. Of course, the result is only an approximation of the geometric surface. The model was brazed together by Fiedler’s assistant Rafael Morstadt in Prague.^{Footnote 54} Fiedler called the result a Stabmodell (model made by metal wiring). Note that in the report cited above it was called a “speciality” of Fiedler’s approach.^{Footnote 55} Fiedler concluded:

In this way the model in the form of a rod model provides a complete view of the surface with its very curious openings. It has a width, length, and height of approximately 0.8 m. I have always found it extremely serviceable as a means to clarify general conceptions regarding the theory of algebraic surfaces; its accuracy is entirely adequate. I have occasionally indicated a pair of conjugate Steiner’s tritangent planes [Trieder]; I have determined the double points of the involutions on the lines of a tritangent plane and thereby shown [anschaulich gemacht] how the six parabolic points of the surface lie on the same plane four times in threes on a line, etc.

The relative ease of construction makes a repetition for the purpose of the formation of varieties and special cases possible; perhaps this communication will encourage this. May it at least awaken a broad interest in the models and photographs of Herr Professor Wiener.

Fluntern, near Zürich, W. Fiedler.^{Footnote 56}

Second, some of the examples cited in the report are directly linked to Fiedler’s specific interests: the first one to his rather original way of constructing duality in a (projective) plane,^{Footnote 57} the second is related to the Reliefperspektive discussed above.

Who were Fiedler’s assistants in the period from 1867 (Fiedler’s arrival at Zurich) to 1876 (publication of the report just cited)? With the help of the course catalogue,^{Footnote 58} we find the following names: Albert Fliegner (1842–1928),^{Footnote 59} J. J. Hemmig, Alexander Beck (he became professor of descriptive geometry in Riga), Adolf Weiler^{Footnote 60} and Johann Keller. Marcel Grossmann, besides Martin Disteli and Giuseppe Veronese perhaps the best-known pupil of Fiedler, was his assistant after 1900 and later (1907) his successor to the chair for descriptive geometry at the Polytechnikum.

As mentioned above, Adolf Weiler (1851–1916) is well known for having constructed a model of Clebsch’s surface during his stay in Göttingen.^{Footnote 61} After studying at the Polytechnikum in Zurich (1867–1871), where he got a diploma of the Fachlehrerabteilung,^{Footnote 62} Weiler left Zurich for Göttingen. It seems plausible that Fiedler proposed this to Weiler.^{Footnote 63} Weiler remained in contact with his “Sehr geehrter Herr Professor” via letters. In his second letter from Göttingen, dated June 15, 1872, Weiler reported:

Last week I spoke at the colloquium on ruled surfaces & made models for this. These were rather small & and made of small cigar boxes with sheets of paper stuck in holes & strings threaded between.

That led me to execute a similar model on a slightly larger scale & to send it to you & it is the first of its kind. Through trial and error I have obtained rather favorable results. On the surface the double curves with two cuspidal points as well as two fixed tangent planes are clearly visible.^{Footnote 64}

Weiler spoke of common work with Felix Klein some months later (January 14, 1873):

I’m currently working on an assignment in line geometry & if there is enough time I will produce a few models of cubic surfaces with Klein. Klein is also very busy still, so the plan will perhaps be put into action later.^{Footnote 65}

On the basis of the correspondence with Fiedler, we may state that Weiler was an important pioneer in the construction of models, inspired in Zurich by Fiedler. After his return to Switzerland, Weiler served for a short time as a math teacher at Stäfa, 1874 he returned to Zurich becoming one of Fiedler’s assistants. He received his habilitation at the Polytechnikum in 1875 and at the university in 1891. In 1899, he was promoted to professor there. Note that there is a difference here: the well-known model by Weiler is one in gypsum (or plaster),^{Footnote 66} whereas Fiedler’s models of type 4 were made by metal wiring.^{Footnote 67}

For every semester, the Polytechnikum published the catalogue of courses offered to the students. In those catalogues, some institutions and collections are described under the heading “Sammlungen und Institute” [collections and departments]. Here, we find hints at the collection of models for mathematics and descriptive geometry in the catalogues for the winter terms 1892/93, 1894/95, and 1896/97. It is more or less obvious that this collection was not considered to be very important because it is mentioned only on few occasions. The most important collection, getting a lot of money from the Schulrath, was the Maschinensammlung (collection of models of machines). The collection of mathematical models was probably stored at the second floor of Semper’s building in the rooms reserved for descriptive geometry.^{Footnote 68}

It should be mentioned that most polytechnical schools run Modellirwerkstätten (workshops for modeling), where students could construct models in wood, clay, or metal—mainly of machines, buildings, and so on. Qualified persons with a great degree of practical competence headed these workshops. Thus, the Modellirwerkstätten were also a good place to construct mathematical models. But we have no traces for such an activity. It seems plausible that products of the Werkstätten were occasionally presented to the public—e.g. in the context of exhibitions.^{Footnote 69} Fiedler hired and paid sculptors to also produce models for him in plaster (Lobde [Berlin], Spiess).^{Footnote 70}

Let us look at Fiedler’s letters. Among his correspondents there are many geometers—often working at polytechnic schools—like Christian and Hermann Wiener, Alexander Beck, Ludwig Burmester, Oskar Schlömilch. A very interesting correspondence is that with Klein. It started in 1872 when Klein sent his now famous ‘Erlanger Programm’ to Fiedler. In his letter, dated Erlangen, February 2, 1873, Klein speaks of models in the context of an intended meeting of mathematicians in Göttingen.^{Footnote 71} In particular, he invites Fiedler to participate in the planned exhibition of mathematical models:

We want to combine this with a model exhibition. The committee will pay for the cost of postage and packing and guarantee safe transport. Could you send us models from Zürich, perhaps also from the models of third-degree surfaces you’ve had made? These should be sent to Dr. Riecke in Göttingen, assistant at the Physics Institute. We would be greatly in your debt.^{Footnote 72}

Alexander Brill wrote about this meeting:

These endeavors [the construction of models] were further stimulated by a gathering of mathematicians in Göttingen (1873), where an exhibition of mathematical models with broad participation aroused interest. Alongside the collections of Plücker, Klein, and Muret, there were several models by Schwarz and Wiener, as well as a cubic surface (‘diagonal surface’) constructed by Weiler (then at Göttingen) at the instigation of Clebsch. In addition, there were several models representing singular points of a cubic surface (one with 4 nodes) constructed in collaboration with Klein, and an elliptic paraboloid represented by assembled pieces of card in the form of semicircles from its circular sections—and much else besides.^{Footnote 73}

Hence, no trace of Fiedler.

In a second letter, dated Erlangen, November 11, 1873, Klein mentions “nice models of minimal surfaces,” which were constructed following instructions by Schwarz^{Footnote 74} and which were presented at the Göttingen meeting. He asks Fiedler for the address of the craftsman who produced his models. In a third letter, dated Erlangen, November 21, 1873, Klein thanks Fiedler for his explications of Schwarz’s models. He is pleased by the fact that Adolf Weiler—who had just got his doctorate with Klein in Erlangen—returned to Zurich in order to become Fiedler’s assistant. Some years later, after Klein was appointed at the polytechnic in Munich (1875), he wrote to Fiedler (dated Munich, October 22, 1876) about the importance of improving the teaching of mathematics for future engineers: “When I was in Zurich a year ago, I saw the thread models of F₃ and of F_n, and its associated. I would like to have these for our institute as well.” [“Ich sah, als ich vor einem Jahr in Zürich war, die Fadenmodelle der F₃ und der F_n und dazugeh. \(\tilde{C}_2\). Ich möchte für unser Institut sehr gerne diese ebenfalls haben.”]^{Footnote 75} It is well known that Klein, in collaboration with Alexander Brill, created an important collection of models at the polytechnic in Munich.^{Footnote 76}

L. Brill,^{Footnote 77} a firm that specialized in producing and selling mathematical models later sold Weiler’s sophisticated models. In 1899, L. Brill was sold to the firm Martin Schilling and moved from Darmstadt to Halle and later to Leipzig. Many of the models still existing are models sold by those firms. Fiedler himself bought models from Schilling, as is proved by letters from Schilling to Fiedler conserved in Zurich.^{Footnote 78} In a letter to Schilling, dated Zurich, November 25, 1901, Fiedler remarked that he himself had constructed models of the type of nr. 234^{Footnote 79} in Schilling’s catalogue long ago. He also remarked that he had bought some other models from L. Brill’s firm.^{Footnote 80} In his letter, Fiedler also remarks that the collection of mathematical models in Zurich was reopened recently.

A letter written by Walther Dyck to Fiedler (dated Munich, December 28, 1891) expressed the interest of wanting to organize an exposition of models in Nuremberg in 1892 in the context of the planned meeting of the Deutsche Mathematiker-Vereinigung.^{Footnote 81} He asked Fiedler to participate: “Now I know that your institute in particular contains a large number of specifically interesting models and apparatuses that are not available elsewhere.”^{Footnote 82}

Dyck asked Fiedler to write a paper on his collection in order to describe it for the public. The meeting at Nuremberg was canceled because of an epidemic there; the exposition was displaced and presented in Munich (September 1 to September 20, 1893).^{Footnote 83} In a second letter, Dyck asked Fiedler to write a paper on his collection for the planned exposition in Chicago—once again in vain. In his letter, Dyck praised Fiedler’s importance for the development of the collection in Zurich by his “excellent teaching activities.”

In sum, we get the impression that Fiedler kept a certain distance to the mainstream movement in Germany promoting models during the 1890s. This might have been due to his age (he was in his sixties) or maybe it was the result of a certain distrust that he felt against the activities of university professors taking up ideas from the world of polytechnics. In a letter by Klein, dated Munich, July 30, 1880, speaking of his ideas to introduce courses on descriptive geometry at the university (of Leipzig), Fiedler noted (August 13, 1880):^{Footnote 84}

Method of execution—not exclusively graphical-technical—not modern geometry adrift from the Mother of Geometry—only body, only spirit! He won’t fall short, but let the effort be rewarded. I didn’t expect the decision [to take up geometry again] after his recent papers.^{Footnote 85}

Fiedler was deep-rooted in the tradition of polytechnic education. He was not—like the great majority of the other mathematicians at the polytechnics, including his own colleagues in Zurich, Klein being a typical example—a mathematician trained at a university who took up a job at a polytechnic for a certain time exporting ideas from there to here.^{Footnote 86}

In conclusion, I want to emphasize that the correspondence was an important instrument for Fiedler to get and to provide information (of all kinds). Partly, this was due to his geographic position, but partly it was also a consequence of his style. In 1905, looking back at his career, Fiedler wrote^{Footnote 87}: “Durch die briefliche Verbindung mit vielen der Besten unter den Mathematikern der Zeit, wie Möbius, Plücker, Hesse, Aronhold, Clebsch, Kronecker, Cayley, Salmon, Brioschi, Beltrami, Cremona, um nur bereits Abgerufene zu nennen, hat mich aber meine einsame Arbeit immer beglückt.” Fiedler spoke on the topic of isolation on several occasions, e.g. in his correspondence.^{Footnote 88}

Models in Fiedler’s Teaching and Publishing

Of course, there are other places in Fiedler’s large correspondence that hints at models, and, of course, we do not know all this correspondence in detail. Nevertheless, I stop here in order to look at Fiedler’s published books and papers, including some materials found in the ETH archive.

There are three types of publications in Fiedler’s large body of work: first, there are the books by Salmon and Fiedler, Fiedler’s greatest success. Whereas Fiedler reworked the books by Salmon with additions, changes, and so on, they remained the books of their author. We find only few personal notes by Fiedler in them. Second, there are Fiedler’s own books. Among them, only his treatise on descriptive geometry had a certain success with different editions. In it, we find a lot of personal remarks underlining the fact that for Fiedler research and teaching were strongly linked. Thirdly, there are his papers. During his life in Zurich, Fiedler published many papers in the Vierteljahrsschrift der Naturforschenden Gesellschaft Zürich. He was an active member of the society, serving even as its president for a certain time. These papers often have the character of reports on work in progress. In one of these papers, models play a notable role: “Herr Prof. Fiedler hält einen Vortrag über Geometrisches mit Vorweisungen” (Prof. Fiedler gives a talk on geometric themes with demonstrations).^{Footnote 89} Fiedler brought two models with him and commented on them. The two models illustrated how two cones of second degree penetrate each other—in particular, the curves of intersection.^{Footnote 90} Each has a different size—one large, one small—and they were made of different materials—strings and rods, or rods only. The discussion of the nature of the curves of penetration is very detailed; in particular, it becomes clear how the research on the abstract nature of the curve and the construction of the concrete object interacted. Fiedler does not mention the name of the person who constructed the models; perhaps he did it himself.

There was a second report on a talk delivered by Fiedler (February 19, 1883): “Herr Prof. Fiedler spricht unter Vorweisung von Modellen über eine Singularität algebraischer Oberflächen” (Prof Fiedler speaks about a singularity of algebraic surfaces and shows some models).^{Footnote 91} Also, in this talk, Fiedler presented some models. First, he showed a model of Clebsch’s surface, then he referred to a Diplomarbeit (master’s thesis) written 1875 in the VI. department^{Footnote 92} and to some investigations of a former pupil of his at the school in Chemnitz, Eckart, now professor at Chemnitz. Here again, we see that the construction of models was closely linked to Fiedler’s research and that of his students. Fiedler presented also a second model of a surface—the first of a surface of third degree, the second of fourth, both made by strings—referring once again to a Diplomarbeit. It was a Hungarian student, Béla Tötössy, who presented this thesis in 1880. Fiedler was very content with it^{Footnote 93} and, due to his contacts with Klein, he arranged its publication in the Mathematische Annalen. Tötössy had constructed a model of a ruled surface of fourth degree made out of strings; it was given to the collection of the Polytechnikum. In his talk, Fiedler explained this model in some detail. At the end of his talk, he announced some results presented by Tötössy at the “mathematische Seminar des Polytechnikums” (June 14, 1882).^{Footnote 94} By all these remarks, one gets the impression that Fiedler worked in a busy and productive situation at the Polytechnikum. It seems that the years around 1882 were a sort of peak in this respect.

In 1891, Fiedler presented a summary of his studies on curves of penetration to the Naturforschende Gesellschaft^{Footnote 95}—without any hint at models. He provides many, many details and, in sum, one gets the impression that Fiedler’s mathematics was rather old fashioned in the 1890s. Contemporary research in mathematics with its tendency for abstraction, axiomatics, etc.^{Footnote 96} no longer had such an interest in details; it was increasingly oriented towards general aspects. In other words, Fiedler had lost the connection to the new trends.

Fiedler was not only a researcher and an engaged teacher, he also contributed to the didactical discussion (to use a modern term, Fiedler himself preferred the term “pedagogical”). In 1877, he published a paper “Zur Reform des geometrischen Unterrichts.”^{Footnote 97} This was a plea to integrate descriptive and projective geometry into the teaching of geometry at school in order to provide a clear structure (“system”). Fiedler discusses the contents of the teaching of geometry, not its methods. But there is one place where we get a glimpse of methods too—that is, the use of models. Here, he speaks of drawings made on the base of models made of rods (“Stabmodell”)^{Footnote 98}; he concludes:

Actual descriptive geometry is not needed to deal with such problems; they form a simple connection between the practice of drawing after rod models and the fundamental definitions of geometry, and automatically lead to the correction of failings in perception, for instance, and to the realization of the indispensability of such corrections in all cases in which it is a matter of mathematically defined forms.^{Footnote 99}

It should be mentioned that there are some objects in Fiedler’s estate, preserved at the ETH archive, which are directly linked to his teaching: there are photographs of star polyhedra (see Fig. 8) and of Clebsch’s surface and drawings of little rods which perhaps served to build up polyhedra.^{Footnote 100} There are also many notes, written by Fiedler himself or taken by his students, reproducing his lectures or parts of them, but there are no hints as to the use of models (Fig. 9).

Conclusions

At the end of this article, I want to conclude by making some remarks. First, it seems important to note that the use of mathematical models was not new in the second half of the nineteenth century. At least in the context of technical education, in the drawing schools, the Gewerbeschulen, and so on, models were known and used for a long time; models were an integral and important aspect of polytechnic tradition. For Fiedler, coming from that tradition, it was quite natural to use models in his teaching. And, of course, the Polytechnikum in Zurich with all its collections and Werkstätten was favorable to the use of models of all kinds; skilled experts were present in the Werkstätten. In particular, students were familiar with the production and handling of models, they knew how to understand things through them. As we have seen, Fiedler produced and used models rather early in his career. Afterwards, the construction of models was an important part of his teaching and of his research. In his eyes, the construction of models offered a possibility to control and correct the results that were deduced in a theoretical way; the integration of students and assistants into the process of research became possible by models. And, of course, they were a means to train spatial intuition.

During his first years, Fiedler took part in a network of researchers interested in the construction of models. It must be added, however, that later on, he did not participate in the public activities of this network—such as the expositions in Göttingen and Munich; nor did he engage himself in the ‘mass production’ (to follow David E. Rowe) of models. Fiedler felt more or less isolated, also at his Polytechnikum. The only colleague who worked on mathematical models there was Schwarz, who left Zurich in 1875.^{Footnote 101} Fiedler was interested mainly in ‘home-made’ models;^{Footnote 102} apparently, he did not want to see them produced by a firm.^{Footnote 103}

He installed an important collection of mathematical models at the Polytechnikum in Zurich, also by buying models from firms like L. Brill, Martin Schilling, and J. Schröder. The models were presented to a restricted public^{Footnote 104}; thus, one may state that they served also as publicity for his chair and his field, that is, descriptive geometry. The way that the models were presented also allowed for their use (at least in a passive way—by looking at them) and to understand things through them in this way.^{Footnote 105} The presentation of models was a common practice in other collections of the Polytechnikum; the collections were places of learning. We may suppose that the students of the Polytechnikum were accustomed to learn in this way (“lehrerlos”^{Footnote 106}). Fiedler used his models also to address a larger public including non-mathematicians by presenting them to the Naturforschende Gesellschaft.

In sum, we may state that the triad research–teaching–representation was realized by Fiedler’s way of working with models—for him, the most important aspect was teaching.^{Footnote 107}

Notes

1.
See: Uta Hassler and Torsten Meyer, “Die Sammlung als Archiv paradigmatischer Fälle,” in Kategorien des Wissens: Die Sammlung als epistemisches Objekt, ed. Uta Hassler and Torsten Meyer (Zurich: vdf Hochschulverlag AG, 2014), 7–74, as well as Uta Hassler and Christine Wilkening-Aumann, “‘Den Unterricht durch Anschauung fördern’: Das Polytechnikum als Sammlungshaus,” in Kategorien des Wissens: Die Sammlung als epistemisches Objekt, ed. Ute Hassler and Torsten Meyer (Zurich: vdf Hochschulverlag AG, 2014), 75–98, and Klaus Volkert, “Mathematische Modelle und die polytechnische Tradition,” Siegener Beiträge zur Geschichte und Philosophie der Mathematik 10 (2018): 161–202.
2.
See: Hassler and Meyer, “Die Sammlung als Archiv,” 13, for a list of collections presented at the Polytechnikum in Zurich.
3.
“Dieselben [die Sammlungen des Polytechnikums] sind in erster Linie für den Unterricht und das Studium in den betreffenden Fächern bestimmt, wofür sie den Studirenden jederzeit zum Privatstudium offen stehen; daneben dienen sie auch den wissenschaftlichen Fortschritten der Lehrkräfte, vorgerückter Studirender und von Gelehrten ausserhalb der Anstalt; endlich sind sie noch der Belehrung des Publikums gewidmet, indem wenigstens die wichtigeren Sammlungen zu bestimmten Stunden jedermann geöffnet und deren Gegenstände so aufgestellt und etiquettirt sind, dass auch der Laie, ohne fachmännische Führung, daraus Belehrung schöpfen kann.” Wilhelm Fiedler, Die Eidgenössische Polytechnische Schule in Zürich: Herausgegeben im Auftrage des Schweizerischen Bundesrathes bei Anlass der Weltausstellung in Paris 1889 (Zürich: Zürcher & Furrer, 1889), 59.
4.
Translation into Italian: Wilhelm Fiedler, Trattato di geometria descrittiva, tradotto da Antonio Sayno e Ernesto Padova. Versione migliorata coi consigli e le osservazioni dell’Autore e liberamente eseguita per meglio adattarla all’insegnamento negli istituti tecnici del Regno d’Italia (Firenze: Successori Le Monnier, 1874).
5.
In the 1860s, Karl Culmann, a famous civil engineer and creator of graphical statics, established courses on projective geometry (“Geometrie der Lage”) for his students. His idea was that these courses should provide the basis for understanding the methods of graphic statics developed by himself. After a short period when Theodor Reye provided this teaching, it was taken over by Fiedler after his arrival at Zurich.
6.
This cycle was held 1879 to 1881; Fiedler’s eldest son, Ernst, was among the students. He took notes and prepared excellent booklets with his notes written in shorthand (ETH-Bibliothek, Hochschularchiv, Hs 110 Mappe 1).
7.
Fiedler himself explained this in a letter to Friedrich Kick (ETH-Bibliothek, Hochschularchiv, Hs 87: 498a).
8.
Gewerbeschulen (or Gewerbsschulen) were secondary schools preparing pupils for technical and business professions or for entering a polytechnic. They provided no Abitur, so their graduates could not enter a university. During the nineteenth century these two orientations were separated step by step. For the sake of simplicity, I will speak simply of technical schools.
9.
See: Antonius Lipsmeier, Technik und Schule: Die Ausformung des Berufsschulcurriculums unter dem Einfluß der Technik als Geschichte des Unterrichts im technischen Zeichnen (Wiesbaden: Steiner, 1971), Wolfgang König, Künstler und Strichezieher: Konstruktions- und Technikkulturen im deutschen, britischen, amerikanischen und französischen Maschinenbau zwischen 1850 und 1930 (Frankfurt: am Main Suhrkamp, 1999), and Kerrin Klinger, Zwischen Gelehrtenwissen und handwerklicher Praxis. Zum mathematischen Unterricht in Weimar um 1800 (München: Fink, 2014). It should be remarked that Pestalozzi was a great advocate of intuition and its role in learning mathematics. Later, in the second half of the nineteenth century, Kant’s ideas on pure intuition were broadly discussed in connection with geometry.
10.
There are thousands of such papers devoted to mathematics and its teaching, see: Gert Schubring, Bibliographie der Schulprogramme in Mathematik und Naturwissenschaften (wissenschaftliche Abhandlungen) 1800–1875 (Bad Salzdetfurth: Franzbecker, 1986). A more recent and more complete source on Schulprogramme (on all themes) is provided by Franz Kössler (see: Franz Kössler, “Personenlexikon von Lehrern des 19. Jahrhunderts,” preprint, published 26th Sept, 2008, http://geb.uni-giessen.de/geb/volltexte/2008/6106/ (accessed November 23, 2021)).
11.
See: Paul Wiecke, “Ueber die abwickelbaren Normalflächen an Hyperboloiden,” (attachment to Programm der Königlichen Höheren Gewerbeschule zu Kassel, Kassel: Hof- und Waisenhausdruckerei, 1873), J. Gierer, “Der Zeichenunterricht nach Körpern,” (Schulprogramm Landwirthschafts- und Gewerbschule Fürth [Bayern], 1847), and Franz Harter, “Ueber die Ausführung der Körperschnitte in Modellen als Beitrag zum Unterrichte in der descriptiven Geometrie,” (Programm der Landwirthschafts- und Gewerbschule Amberg, 1840). Wiecke is referring in his paper to models presented by his school at the world-exhibition taking place in Vienna in 1873. I thank Hellmuth Stachel (Vienna) for providing me this information.
12.
See: ETH-Bibliothek, Hochschularchiv, Hs 1196: 30 and Hs 1196: 50.
13.
See: Wilhelm Fiedler, “Meine Mitarbeit an der Reform der darstellenden Geometrie in neuerer Zeit. Schreiben gerichtet an den Herausgeber dieses Jahresberichts,” Jahresbericht der Deutschen Mathematiker Vereinigung 14 (1905): 493–503, here 493.
14.
It is neither called ‘constructive geometry’ nor ‘linear drawing.’
15.
On Jakob Schröder’s work and life see: Lipsmeier, Technik, 216–17.
16.
See: Lipsmeier, Technik, 290. Some years before Schröder, Théodore Olivier at Paris constructed a similar model called “Omnibus.” Concerning this model see: João P. Xavier and Eliana M. Pinho, “Olivier string models and the teaching of descriptive geometry,” in “Dig where you stand” 4, Proceedings of the Fourth International Conference on the History of Mathematics Education, ed. Kristín Bjarnadóttir, Fulvia Furinghetti, Marta Menghini, Johan Prytz, and Gert Schubring, (Roma: Edizioni Nuova Cultura, 2017), 399–413.
17.
See: ETH-Bibliothek, Hochschularchiv, Hs 1196: 30, 1 and Hs 1196: 50, 1. At that time, Fiedler earned 4,300 Swiss francs a year.
18.
See the more detailed discussion of the nature of Schröder’s models and their dispersion by Lipsmeier, Technik, 217 n. 600. The great importance of Schröder’s company in the field of technical education is also underlined by Hermann Ludewig, Das technische Unterrichtswesen auf der Weltausstellung in Wien 1873, mit besonderer Berücksichtigung des maschinentechnischen Unterrichts (München: Ackermann, 1875), 16, 25–26, 32–33, 38, 41.
19.
See: Lipsmeier, Technik, 217, for more information on the dissemination of Schröder’s models.
20.
By the way, there was also a long-standing tradition to use materials which one may call models in the elementary teaching of mathematics—like rods presenting numbers, and so on.
21.
“Auch in Deutschland giebt es seit Langem Modelle, welche den Bedürfnissen des mathematischen Unterrichts dienen sollen. Wohl an den meisten Universitäten und technischen Schulen finden sich in irgend einem Schrank oder einem staubigen Winkel Pappmodelle von Polyedern, Kegeln mit ebenen Schnitten, Durchdringungskurven von Kegeln mit Cylindern u.s.w. von meist unbestimmter Herkunft, die bei dem heutigen Stande des Hochschul-Unterrichts nicht mehr gebraucht werden und deshalb durch die neueren schöneren Formen, welche zahlreiche Lehrmittel-Anstalten (darunter als älteste die verdiente Schröder’sche Modellfabrik in Darmstadt) diesen elementaren Unterrichtsmitteln zu geben pflegen, nicht mehr ersetzt werden.” Alexander Brill, “Über die Modellsammlung des mathematischen Seminars der Universität Tübingen,” Mathematisch-naturwissenschaftliche Mitteilungen 2, no. 1 (1889): 69–80. See also: Walther Dyck, “Einleitender Bericht über die mathematische Ausstellung in München,” Jahresbericht der Deutschen Mathematiker-Vereinigung 3 (1892–93): 39–56, here 51, who cites the “altbekannte Werkstätte von Schröder in Darmstadt” [well-known workshop of Schröder in Darmstadt].
22.
Often, this was the half-space behind a plane—the front of the stage, so to speak. In that case, the other plane is the vanishing plane—the back plane of the stage.
23.
Thus, here the motivation can be found to introduce four-dimensional space or to create models for that situation. See: Klaus Volkert, In höheren Räumen: Der Weg der Geometrie in die vierte Dimension (Berlin: Springer, 2018). Henrik de Vries, a former student of Fiedler, discussed that possibility at length in his book Die Lehre von der Zentralprojektion im vierdimensionalen Raum (Leipzig: Göschen, 1905).
24.
Perspective en relief. But note that the name was used before Poncelet; see below on Breysig’s book: Johann Adam Breysig, Versuch einer Erläuterung der Reliefsperspective, zugleich für Mahler eingerichtet (Magdeburg: Keil, 1798).
25.
The construction itself is indicated in Fig. 5. The principal idea behind it is to reduce the spatial problem to the plane by using plane cuts passing through the center of projection.
26.
“Die Bedeutung dieser Construction liegt darin, dass sie die geometrische Grundlage künstlerischer d. h. auf angenehme Täuschung ausgehende Modellirung mit enthält, und dass sie alle anderen technisch brauchbaren Modellirungsmethoden u. schliesslich die Darstellungsmethoden durch Zeichnungen auf Ebenen als spezielle Fälle umfasst. […] Wenn dem entsprechend hergestellte Modelle noch beleuchtet werden durch Licht, das von einem geeignet gewählten Punkte der Gegenebene Q ausgeht, so glaubt das Auge C statt seiner collinearen Umformung das mit dem richtigen Sonnenschatten ausgestattete Object selbst zu sehen. (Decoration der Schaubühne—der Raum zwischen den Ebenen S und Q ist Raum der Bühne, S die Vorhangebene.)” Wilhelm Fiedler, Darstellende Geometrie: Autographierte Ausarbeitung einer Vorlesung (Zürich: E. Zimmer, 1894), 48. An exemplar of Fiedler’s lecture notes can be found at the library of the ETH (Rar 9099). I thank Urs Stammbach (Zurich) for providing me with an exemplar of these lecture notes.
27.
Autography or lithography was a technique of printing invented by Alois Senefelder around 1800, which was used to produce lecture notes. Well-known examples are due to Felix Klein, but he was not the only person to see the utility of that technique for teaching. As early as 1874, Fiedler ordered 400 copies of an autographed page on the method of least squares in order to distribute them to the students (see: ETH-Bibliothek, Hochschularchiv, Hs 1796: 50, 2). Later, he bought also autographies by Klein.
28.
See: Rafael Morstadt, “Ueber die räumliche Projection (Reliefperspective), insbesondere diejenige der Kugel,” Zeitschrift für Mathematik und Physik 12 (1867): 326–39. We will see below that Morstadt constructed also another model for Fiedler. Note that Morstadt (and Burmester) produced a material model of the abstract model constructed by a Reliefperspektive.
29.
I thank Ulf Hashagen (Munich) for providing me this information.
30.
Staudigl, Grundzüge.
31.
The two dates are noted because the Julian calendar was still in use at Riga.
32.
“Ich habe mir vor einiger Zeit ein Gypsmodell als Beispiel einer Reliefperspective anfertigen lassen, zu welchem ich dem Bildhauer die Zeichnung lieferte. […] Das Modell ist sehr gut gelungen und bildet ein Prunkstück meiner Sammlung. Ich gedenke auch, die neuen Münchner Modelle anzuschaffen.” ETH-Bibliothek, Hochschularchiv, Hs 87: 40.
33.
It is specific to the early history of descriptive geometry in German speaking countries that it was closely linked to the arts of painting and drawing. See: Nadine Benstein, “A German Interpreting of Descriptive Geometry and Polytechnic,” in Descriptive geometry, The Spread of a Polytechnic Art: The Legacy of Gaspard Monge, ed. Évelyne Barbin, Marta Menghini, and Klaus Volkert (Cham: Springer 2019), 139–66.
34.
See: Wilhelm Fiedler, “Zur Geschichte und Theorie der elementaren Abbildungs-Methoden,” Vierteljahrsschrift der Naturforschenden Gesellschaft Zürich 27 (1882): 125–75, here 135–36.
35.
Wilhelm Burghardt, “II. Beitrag für den Unterricht in der Reliefperspective,” (Programm Nordhausen Realschule, 1861) [Published together with: “I. Anleitung zur Analyse vermittelst des Lötrohres”].
36.
That is synthetic geometry of the sphere (in contrast to spherical trigonometry). At the time this paper was written, Gudermann was a teacher at the Gymnasium in Cleve (now: Kleve [Niederrhein]).
37.
Christoph Gudermann, “Ueber die niedere Sphärik,” Journal für die reine und angewandte Mathematik 8 (1832): 363–69, here 369.
38.
See: ETH-Bibliothek, Hochschularchiv, Hs 1196: 30 and Hs 1196: 50 for more details.
39.
He served also as director of the Polytechnikum and as professor at the university.
40.
One of them (ETH-Bibliothek, Hochschularchiv, Hs 1196: 30) was an official document approved by the authorities, the other (Hs 1196:50) was a private document in the possession first of Fiedler, then of Grossmann. The first was established in 1907 when Fiedler was replaced by his successor Grossmann; it was controlled annually until the 1930s by the authorities. It seems plausible that the inventories cite only models which were bought or which resulted in other costs (e.g. for the material). Home-made models produced for free are not mentioned at all. I thank Wibke Kolbmann for providing me information on the inventory of: ETH-Bibliothek, Hochschularchiv, Hs 1196: 50.
41.
For a long time this was Karl Kappeler.
42.
It should be remembered that the Polytechnikum in Zurich was a foundation by the Swiss Federation whereas all other Swiss schools and universities were (and are) run by the cantons—except, of course, private institutions and the French-speaking counterpart of the ETH in Lausanne.
43.
Bericht des Eidgenössischen Polytechnikums an den Bundesrath 1876 (Bibliothek ETH-Archiv), 10–11.
44.
It was also the last time.
45.
“Ueberdies aber sind seit einer Reihe von Jahren von den jeweiligen Assistenten der darstellenden Geometrie unter Leitung des Professors Stab- oder Drahtmodelle algebraischer Flächen usw. gefertigt worden, welche als Specialität hier einmal Erwähnung finden mögen. Es ist in der großen regelmäßigen Arbeitsbelastung der Assistenten begründet, daß diese Modellarbeiten nur langsam gefördert werden können und daß die Zahl der Modelle nur langsam wächst.” Bericht des Eidgenössischen Polytechnikums, 10.
46.
Bericht des Eidgenössischen Polytechnikums, 10–11.
47.
1. “Ein Drahtmodell des Orthogonalsystems im Bündel.” / 2. “Ein Modell der Hauptpunkte und Hauptebenen der centrischen Collineation der Räume im Zusammenhang mit der Centralprojection ebener Systeme.” / 3. “Ein Fadenmodell der entwickelbaren Flächen einer cylindrischen Schraubenlinie.” / “Drahtmodelle von Flächen dritter Ordnung mit 27 reellen Graden mit Systemen paralleler Querschnitte” / “Modelle von Flächen vierter Ordnung” / 4. “Ein Modell der Durchdringungscurve von zwei Flächen zweiten Grades und ihrer entwickelbaren Fläche mit Darstellung ihrer Doppelcurven ist in Arbeit.” Bericht des Eidgenössischen Polytechnikums, 11.
48.
See: David E. Rowe, “Mathematical Models as Artefacts for Research: Felix Klein and the Case of Kummer Surfaces,” Mathematische Semesterberichte 60 (2013): 1–24, for more details on that research.
49.
An exemplar of it was bought by Fiedler in 1870 at a price of 115 Swiss francs (see: ETH-Bibliothek, Hochschularchiv, Hs 1796: 30, 9).
50.
Concerning the history of those models which we may call scientific (“artefacts for research” [David Rowe], that are not primarily constructed for teaching purposes), see: Oliver Labs, “Straight Lines on Models of Curved Surfaces,” The Mathematical Intelligencer 39, no. 2 (2017): 15–26; Rowe, “Mathematical Models” and David E. Rowe, “On Building and Interpreting Models: Four Historical Case Studies,” The Mathematical Intelligencer 39, no. 2 (2017): 6–14. The story of Clebsch’s models and other surfaces is reported by Klein himself several times, see: Felix Klein, Gesammelte mathematische Abhandlungen, vol. 2, ed. Robert Fricke and Hermann Vermeil (Berlin: Springer, 1922), 3–5. Concerning the history of the surface itself, see: François Lê, “Entre géométrie et théorie des substitutions: une étude de cas autour des vingt-sept droites d’une surface cubique,” Confluentes Mathematici 5, no. 1 (2013): 23–71, and Labs, “Straight Lines.”
51.
We should also think of differential geometry with its central notion of curvature as an important source for new and interesting examples of modeling—a fact often underlined by Brill in his autobiography.
52.
See: Labs, “Straight Lines,” 19–21.
53.
“Von der allgemeinen Fläche dritter Ordnung mit 27 reellen Geraden besitzt der Herausgeber [Fiedler] ein Stabmodell seit 1865 [sic!], über dessen bequeme Herstellung er Auskunft gegeben hat in Bd. 14 der ‘Zeitschrift für Math.’ (Literaturzeitung). Seit 1869 ist ein von Wiener construirtes Gypsmodell mehrfach verbreitet, das auf Anregung von Clebsch entstanden ist. Berechnungen aus Anlass derselben hat Cayley mitgetheilt in ‘Transactions of Cambridge’ Bd. 12 (1873), p. 366–383. Neuerdings sind auch die Flächen dritter Ordnung mit vier Knotenpunkten und die Diagonalfläche (auch als Stabmodell) modelliert worden. Das Prinzip der Stabmodelle ist auf sehr viele andere Fälle bequem anwendbar.” George Salmon and Wilhelm Fiedler, Analytische Geometrie des Raumes, vol. 2: Analytische Geometrie der Curven im Raume und der algebraischen Flächen, 2nd ed. (Leipzig: Teubner, 1874), 662–63 [Fiedler is referring to the Zeitschrift für Mathematik und Physik which included a “Literaturzeitung” with a separate pagination]. This book contains also a very detailed description of another model made out of paper; see: Salmon and Fiedler, Analytische Geometrie, 624–25. Both texts are in the notes, written by Fiedler. They are not in Salmon’s original version of his book.
54.
Felix Klein knew of the existence of this model and other models in Zurich; this is shown by his letter, dated Erlangen, February 2, 1873, cited below.
55.
Fiedler’s model does not exist anymore. There are no pictures of it, only Fiedler’s report and some traces in his correspondence are left.
56.
“So giebt das Modell als Stabmodell doch die vollständige Anschauung der Fläche mit ihren höchst merkwürdigen Oeffnungen. Es ist circa 0,8 m breit, lang und hoch. Ich habe es immer ganz trefflich brauchbar gefunden, um die allgemeinen Anschauungen der Theorie algebraischer Flächen daran zu verdeutlichen; seine Genauigkeit reicht dafür völlig aus. Ich habe gelegentlich ein Paar der Steiner’schen conjugirten Trieder markirt, ich habe die Doppelpunkte der Involutionen auf den Geraden einer dreifach berührenden Ebene bestimmt und dadurch anschaulich gemacht, wie solche sechs parabolische Punkte der Fläche in derselben Ebene viermal an dreien in einer Geraden liegen etc. / Die relative Leichtigkeit der Herstellung macht eine Wiederholung derselben behufs Bildung von Varietäten und Specialfällen möglich, vielleicht regt diese Mittheilung dazu an. Möge sie wenigstens nicht verfehlen, für die Modelle und Photographien des Herrn Professor Wiener vielseitig Interesse zu erwecken. / Fluntern, bei Zürich, W. Fiedler.” Wilhelm Fiedler, “Besprechung von Stereographische Photographien des Modelles einer Fläche dritter Ordnung mit 27 reellen Geraden. Mit erläuternden Texten von Dr. Chr. Wiener, Professor am Polytechnikum zu Carlsruhe,” Zeitschrift für Mathematik und Physik 14 (1869), Literaturzeitung: 34.
57.
It is described in Fiedler, Die darstellende Geometrie, 2nd ed., 113–14.
58.
Officially, these catalogues were “Anhänge zu den Protokollen der Sitzungen des Schulrathes,” usually they are called “Polyprogramme.”
59.
Fliegner became professor of mechanics at the Polytechnikum in 1872.
60.
Not to be confused with August Weiler, a math teacher at Mannheim and author of textbooks. On Adolf Weiler, see: Universität Zürich. Rektoratsreden und Jahresberichte. April 1916 bis Ende März 1917 (Zürich: Orell Füssli, 1917), 54–55.
61.
This model was presented to the Göttingen Academy by Clebsch in 1872. See: Alfred Clebsch, “Ueber Modelle von Flächen dritter Ordnung,” in Nachrichten von der K. Gesellschaft der Wissenschaft und der Georg-August-Universität aus dem Jahre 1872 (Göttingen: Dieterich, 1872), 402–4. Weiler is cited by Klein in his letter to Fiedler dated Erlangen, February 2, 1873, mentioned above. In his Gesammelte mathematische Abhandlungen (vol. 2), Klein explains that Weiler brought with him the “tradition of Fiedler” (Klein, Gesammelte mathematische Abhandlungen, 3).
62.
The department concerned with future teacher training.
63.
Fiedler maintained an important correspondence with Clebsch, whom he considered as an authority; see: ETH-Bibliothek, Hochschularchiv, Hs 87: 160–70.
64.
“Letzte Woche sprach ich im Colloquium über Regelflächen & verfertigte dazu Modelle. Diese waren ziemlich klein & bestanden aus Zigarrenkistchen, in denen Blätter in Löchern steckten & dazwischen Fäden durchgezogen.
Das führte mich darauf, ein solches Modell in etwas größeren Dimensionen auszuführen & Ihnen zu übersenden & es stellt dasselbe die erste Gattung dar, durch Ueberlegen und Probieren habe ich ziemlich günstige Verhältnisse erhalten. Auf der Fläche sind die Doppelcurven mit zwei Cuspidalstellen sowie zwei stationären Tangentialebenen deutlich sichtbar.” Weiler to Fiedler, June 15, 1872, ETH-Bibliothek, Hochschularchiv, Hs 87: 1491, https://doi.org/10.7891/e-manuscripta-85659 (accessed November 23, 2021). This model is noted in Fiedler’s inventory as no. 7, p. 2, together with the commentary: “(von Herrn Weiler ausgeführt in Begleit. seiner Preisbewerbungsschrift 1872).”
65.
“Gegenwärtig arbeite ich an einer Aufgabe aus der Liniengeometrie & wenn Zeit dafür bleibt, werde ich mit Klein einige Modelle über Flächen dritter Ordnung ausführen. Noch ist auch Klein sehr beschäftigt, so daß der Plan vielleicht später angegangen wird.” Weiler to Klein, January 14, 1873, ETH-Bibliothek, Hochschularchiv, Hs 87: 1494, https://doi.org/10.7891/e-manuscripta-85682 (accessed November 23, 2021). In the meantime, Weiler followed Klein to Erlangen, where he received his doctorate in 1873. Its theme was a problem from line geometry.
66.
See: Göttinger Sammlung mathematischer Modelle, Modell 135. The firm of L. Brill later sold a plaster model of Clebsch’s surface designed by Rodenberg based on Weiler’s model. It was bought later also by Fiedler for his collection.
67.
Not to be confused with rods—a material often used by Fiedler.
68.
See the plan of the Polytechnikum reproduced in: Hassler and Meyer, “Die Sammlung als Archiv,” 18. The area reserved for descriptive geometry in Semper’s plan was situated at the south-east corner of the building on the second floor. The way in which mathematical models were presented is discussed in Anja Sattelmacher, “Zwischen Ästhetisierung und Historisierung: Die Sammlung geometrischer Modelle des Göttinger mathematischen Instituts,” Mathematische Semesterberichte 61 (2014): 131–43. Here one finds also some photographs. The order in which the models were presented is an interesting problem that is wide open to my knowledge.
At the Polytechnikum in Zurich, there was also a room with a library for the assistants of descriptive geometry.
69.
See: Hassler and Meyer, “Die Sammlung als Archiv” and Hassler and Wilkening-Aumann, “Das Polytechnikum” for more information on the Maschinensammlung and the Modellierwerkstatt.
70.
ETH-Bibliothek, Hochschularchiv, Hs 1796: 50.
71.
This meeting took place in Göttingen from April 16 to April 18, 1873. Partly, it was held to commemorate Clebsch who died just before. It is also considered as a step towards the foundation of the Deutsche Mathematiker-Vereinigung (1890).
72.
“Wir möchten mit derselben eine Modell-Ausstellung verbinden. Das Comité übernimmt die Versendungs- als auch die Verpackungskosten, so wie die Garantie für ungefährdeten Transport. Könnten Sie von Zuerich aus uns Modelle zukommen lassen, vielleicht auch von den Modellen betr. Flächen dritten Grades, die sie haben anfertigen lassen? Diesselben wären an Dr. Riecke in Goettingen, Assistent am physikalischen Institute, zu adressieren. Sie würden uns dadurch sehr verpflichten.” Klein to Fiedler, February 2, 1873, ETH-Bibliothek, Hochschularchiv, Hs 87: 576, https://doi.org/10.7891/e-manuscripta-84491 (accessed November 23, 2021). Thanks to Peter-Maximilian Schmidt (Zurich) for transcribing Klein’s letters to Fiedler.
73.
“Eine weitere Anregung wurden diesen Bestrebungen [zur Konstruktion von Modellen] zu Teil auf einer Versammlung von Mathematikern in Göttingen (1873), wo eine vielseitig beschickte Ausstellung von mathematischen Modellen Interesse erregte. Es waren dort ausser den Collectionen von Plücker, Klein, Muret mehrere Modelle von Schwarz, Wiener u. A. auch eine gewisse Fläche 3. Ordnung (‘Diagonalfläche’), welche auf Clebschs Veranlassung von Weiler (damals Göttingen) hergestellt worden war, ferner einige Modelle zur Repräsentation singulärer Punkte einer Fläche 3. Ordnung (eine solche namentlich mit 4 Knoten), unter Mitwirkung von Klein konstruiert, dann ein elliptisches Paraboloid, das durch zusammen gesteckte Kartonscheiben in der Form von Halbkreisen aus seinen Kreisschnitten dargestellt war, u. A. m.” Brill, “Über die Modellsammlung,” 75–76. In a letter (ETH-Bibliothek, Hochschularchiv, Hs 87: 1495, https://doi.org/10.7891/e-manuscripta-85665 (accessed November 23, 2021)), dated Erlangen, April 20, 1873, Weiler informed Fiedler on this meeting, in particular on the models presented there. He told Fiedler that three of his models were presented at the meeting, one model by Neesen (Bonn), some models by Johann Eigel Sohn (Cologne [Weiler says Bonn, which seems to be false]) and by Schwarz. He mentioned also “French” models made of strings. Eigel produced models designed by Plücker and Klein; see: Rowe, “Mathematical models,” 6.
74.
Schwarz was professor of mathematics at the Polytechnikum in Zurich from 1869 to 1875. So, Fiedler and Schwarz were colleagues—seemingly, on good terms: Schwarz allowed Fiedler to use a manuscript on minimal surfaces written by him as an appendix to the second edition of Salmon and Fiedler, Analytische Geometrie des Raumes, vol. 2, 683–90. Fiedler was also one of the godfathers of the first daughter of the Schwarz couple, Kronecker was the other (see: letter to Cremona, October 30, 1869).
75.
Klein to Fiedler, October 22, 1876, ETH-Bibliothek, Hochschularchiv, Hs 87: 583, https://doi.org/10.7891/e-manuscripta-84497 (accessed November 23, 2021). F₃ is a cubic surface, F_n a surface of degree n. C means here a conic surface, 2 indicates that the class of the surface is reduced by 2 (because of the presence of knot-points). The creation of an adequate nomenclature including symbolic representations is a typical feature linked to exhibiting models; see: Hassler and Meyer, “Die Sammlung als Archiv,” 10.
76.
Some information on Brill, Klein and their models can be found in Brill’s unpublished autobiography “Aus meinem Leben” (conserved at the library of the Technische Universität Munich). I thank this library for providing me a copy of Brill’s text.
77.
Ludwig Brill was a brother of the mathematician Alexander Brill just cited. Alexander. Brill proposed to his brother the idea to sell mathematical models in order to save his firm which was in financial difficulties. The firm was the heritage of their father; in its beginnings it was a printing house.
78.
Fiedler to Schilling and Schilling to Fiedler, December 28, 1900 to January 15, 1904, ETH-Bibliothek, Hochschularchiv Hs 87: 1082a–1084. The acquisition took place in 1900 and 1901. See also the inventories Hs 1196: 30 and Hs 1736: 50.
79.
These models are string models of curves in space (Series XXI in: Martin Schilling, ed., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, 6th ed. (Halle an der Saale: Martin Schilling, 1903), 145–151).
80.
This is confirmed by the inventories ETH-Bibliothek, Hochschularchiv, Hs 1796: 30 and Hs 1796: 50.
81.
The Deutsche Mathematiker-Vereinigung (DMV) was founded in 1890. The planned exhibition was perhaps intended to promote the newly founded association in public.
82.
Dyck to Fiedler, December 28, 1891, ETH-Bibliothek, Hochschularchiv Hs 87: 249, https://doi.org/10.7891/e-manuscripta-84098: “Nun weiß ich, daß besonders Ihr Institut eine große Reihe spezifisch interessanter und auswärts nicht vorhandener Modelle und Apparate enthält.”
83.
Dyck published a long paper describing the exposition (Walther Dyck, “Einleitender Bericht über die mathematische Ausstellung in München,” Jahresbericht der Deutschen Mathematiker-Vereinigung 3 (1894): 39–56) and a catalogue of the exposition (Walther Dyck, ed., Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente (Munich: Wolf & Sohn, 1892)). Fiedler is not mentioned in the catalogue (but Schwarz). In describing the history of mathematical models, Dyck cites long passages from Brill, “Über die Modellsammlung.”
84.
Klein to Fiedler, July 30, 1880, ETH-Bibliothek, Hochschularchiv Hs 87: 584, https://doi.org/10.7891/e-manuscripta-84531 (accessed November 23, 2021).
85.
“Art u. Weise der Ausf.—nicht ausschl. graph.-techn.—nicht mod. Geom. losgelöst von der Mutter d. G.—bloß Körper, bloß Geist! Er wird’s nicht verfehlen, aber die Mühe möge ihren Lohn finden. Erwartet habe ich s. Entschl. nicht nach seinen neueren Abh.”
86.
This aspect is discussed in detail in the dissertation by Nadine Benstein (Wuppertal): Zwischen Zeichenkunst und Mathematik: Die darstellende Geometrie und ihre Lehrer an den Technischen Hochschulen und deren Vorgängern in ausgewählten deutschen Ländern im 19. Jahrhundert (2019; available at UB Wuppertal: http://elpub.bib.uni-wuppertal.de/edocs/dokumente/fbc/mathematik/diss2019/benstein).
87.
Fiedler, “Meine Mitarbeit,” 503.
88.
See the letters to Cremona, April 25, 1864, August 5, 1866, June 22, 1867, in Correspondence of Luigi Cremona (1830–1903), ed. Giorgio Israel, 2 vols. (Turnhout: Brepolis, 2017), vol. I, 642–43, 647–49, 653–55).
89.
Wilhelm Fiedler, “Vortrag über Geometrisches mit Vorweisungen,” Vierteljahresschrift der Naturforschenden Gesellschaft Zürich 28, no. 2–3 (1883): 289–90. It was the secretary of the society, Robert Billwiller, who wrote the published report—not Fiedler himself. In most cases, the Vierteljahrsschrift published papers written by Fiedler himself. The talk was delivered July 23, 1883.
90.
A typical problem of descriptive geometry, going back at least to the days of Monge, was to construct such curves of intersection. A model of that type was announced in the report for the Bundesrath in 1876.
91.
Wilhelm Fiedler, “Vortrag über eine Singularität algebraischer Oberflächen unter Vorweisung von Modellen,” Vierteljahrsschrift der Naturforschenden Gesellschaft Zürich 28, no. 2–3 (1883): 290–93.
92.
Because the information provided by Fiedler is meager, it is not absolutely clear to which thesis he is referring here. There is a list of candidates in Fiedler’s papers (ETH-Bibliothek, Hochschularchiv, Hs 87a: 71); in it we find two entries which can fit here: Johannes Keller “Über reciproke Fundamentalgebilde der zweiten und dritten Stufe” and E. Mosiman “Über die Eigenschaften von Flächen dritter Ordnung.” Keller was Fiedler’s assistant from 1875 to 1887. He received his PhD in 1879 from the University of Zurich (in collaboration with Fiedler): “Die einander doppelt conjugirten Elemente in allgemeinen reciproken Systemen”; later he passed his habilitation at the Polytechnikum. As mentioned by Fiedler on another occasion, Keller produced a model of a Reliefperspektive.
93.
This is the only model cited by Fiedler in his correspondence with Cremona; see: letter to Cremona dated February 12, 1882.
94.
Wilhelm Fiedler, “Vortrag,” 292. Tötössy later became a professor at the polytechnic at Budapest.
95.
Wilhelm Fiedler, “Ueber die Durchdringungen projectivischer Kegel,” Vierteljahrsschrift der Naturforschenden Gesellschaft Zürich 36 (1891): 87–113.
96.
This trend was often related to the Berlin School (Kronecker, Weierstrass), Hilbert’s Grundlagen der Geometrie (1899) later reinforced it.
97.
Wilhelm Fiedler, “Zur Reform des geometrischen Unterrichts,” Vierteljahrsschrift der Naturforschenden Gesellschaft Zürich 22 (1877): 82–97. This paper was also translated into Italian. Three letters written by Fiedler to the editor, Torelli, were included in the Italian edition. In the first of these letters, Fiedler speaks of teaching drawing with the help of models.
98.
Fiedler, “Zur Reform,” 88.
99.
“Eigentliche descripitive Geometrie ist zur Ausführung solcher Probleme nicht erforderlich; sie bilden eine einfache Verbindung der Uebung im Zeichnen nach Stabmodellen mit den fundamentalen Definitionen der Geometrie und führen sofort zur Correctur des etwa der Wahrnehmung nicht treu genug Abgesehenen und zur Einsicht von der Unentbehrlichkeit einer solchen Correctur in allen Fällen, wo es sich um mathematisch bestimmte Formen handelt.” Fiedler, “Zur Reform,” 89.
100.
ETH-Bibliothek, Hochschularchiv, Hs 87a: 29.
101.
After arriving at Göttingen, Schwarz engaged himself in building up a collection of models there; see: Sattelmacher, “Zwischen Ästhetisierung und Historisierung,” 131. Models in the possession of Schwarz are also mentioned in the inventory ETH-Bibliothek, Hochschularchiv, Hs 1796: 50.
102.
See Figure 10 for an example.
103.
In Dyck’s catalogue (1892), we find a firm in Zurich named Coradi, a Mathematisch-mechanische Werkstatt, which produced mathematical instruments like integraphs (see: Dyck, ed., Catalog, 197–201). Hence, it would not have been difficult for Fiedler to find a partner for the production of models in his own town—if he would have wanted to. Coradi presented its instruments at the International Congress in Heidelberg (1904). The exhibition of models and instruments at that congress was organized and presented by Martin Disteli (then professor in Strasbourg i.e., before Karlsruhe, later he became professor in Dresden and, once again, in Karlsruhe and finally at the University of Zurich), a former student and assistant of Fiedler.
104.
That means students, professors and external experts. Other collections at the Polytechnikum, such as the famous entomological collection, were open to a general public. Therefore, they had a warden.
105.
Fiedler considered himself as an autodidact. See his letter to Klein, Hottingen–Zürich XII 1885, Staats- und Landesbibliothek Göttingen, Cod. Ms. F. Klein 19. Hence, learning on his own was familiar to him.
106.
Hassler and Meyer, “Die Sammlung als Archiv,” 12. Of course, some indications concerning the models could be made by the teachers during their courses.
107.
Note added in proofs: some letters to and by Fiedler are now accessible in Sara Confalonieri, Peter-Maximilian Schmidt, and Klaus Volkert, eds., Der Briefwechsel von Wilhelm Fiedler mit Alfred Clebsch, Felix Klein und italienischen Mathematikern, series Siegener Beiträge zur Geschichte und Philosophie der Mathematik, vol. 12 Siegen: universi, 2019.

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Klaus Volkert

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Karin Krauthausen

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Volkert, K. (2022). Wilhelm Fiedler and His Models—The Polytechnic Side. In: Friedman, M., Krauthausen, K. (eds) Model and Mathematics: From the 19th to the 21st Century. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-97833-4_3

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Wilhelm Fiedler and His Models—The Polytechnic Side

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Hermann Laurent

Modeling for Introducing Students to New Tools

Stoker, Bram

Wilhelm Fiedler

Some Remarks on Teaching and Early Models

Models in Fiedler’s Correspondence

Models in Fiedler’s Teaching and Publishing

Conclusions

Notes

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