## Abstract

KK: You will probably know this famous quote by Walther von Dyck from 1929: “To show the essence, content, and aims of mathematical research in its entirety cannot be the task of a museum.”

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**KK**: You will probably know this famous quote by Walther von Dyck from 1929: “To show the essence, content, and aims of mathematical research in its entirety cannot be the task of a museum.”^{Footnote 1} Nevertheless, we would like to speak with you today about mathematics exhibitions in the long nineteenth and the early twentieth century. If you think specifically about the small exhibition of mathematical models at a mathematics convention in Göttingen in 1873, or about the exhibition of scientific instruments and apparatuses in the South Kensington Museum in London in 1876, or the important 1893 exhibition in Munich commissioned by the German Mathematical Society, or finally the opening of the mathematics section at the Deutsches Museum in 1925,^{Footnote 2} how would you describe the relation between mathematics and exhibition from a historical perspective? And how do mathematical models fit into this history of mathematics exhibitions?

**UH**: The first thing that needs to be said is that the nineteenth and the early twentieth century was an era of exhibitions, in which regional and national trade and industry exhibitions as well as world exhibitions played a major role in economic, cultural, and public life. Yet with the emergence of engineering colleges, the nineteenth century was also characterized by collections of ‘disciplinary artifacts,’ that is, by scholarly collections that were specific to a particular discipline. These two contexts had a profound influence on the development of mathematics exhibitions, and thus also on the development of the interrelationship between mathematics, exhibitions, and mathematical artifacts.

The 1873 exhibition on the occasion of the mathematics convention in Göttingen you mentioned was certainly an initial event for German mathematics. It was probably the first time that mathematical models—some of which came from Paris—as well as mathematical instruments and calculating machines were exhibited as part of a mathematics convention.^{Footnote 3} From today’s point of view, this group of mathematicians showed an astonishing eagerness to engage with the ‘mathematical artifacts,’ that is, with objects that, in a tangible form, represented mathematics as a discipline. In the late nineteenth century these mathematical artifacts were primarily mathematical models, such as string or plaster models, that were produced in the context of contemporary geometrical research and constructed by mathematicians. These models and apparatuses come from a tradition that can be traced back above all to the string models used by Gaspard Monge and Théodore Olivier in their geometry classes and to the subsequent interest in mathematical models in France (see Fig. 1).^{Footnote 4}

The next large scientific exhibition in which mathematics played a role took place at the South Kensington Museum in 1876 (see Fig. 2). This was in fact a very large, successful exhibition on apparatuses and instruments in science and technology.^{Footnote 5} Here, mathematics only formed a small subsection of the exhibition, and only two branches were exhibited: arithmetic and geometry. The section on arithmetic chiefly displayed slide rules and calculating machines, including contemporary models and historical machines such as the Pascaline from the seventeenth century. The section on geometry, on the other hand, exhibited instruments that were used for geometric drawing or to produce specific mathematical curves—but also three-dimensional mathematical models. This seems to have made a great impression on a number of German mathematicians, since in Germany the interest in geometric models and their production was only just emerging, and models from Germany could also be seen in the London exhibition. With this exhibition one can see how mathematical artifacts (i.e., mathematical models and mathematical instruments) were used within a public science exhibition to ‘construct’ mathematics as a discipline. One can also see the context—also the national context—in which mathematical artifacts acquired their respective meaning. Many of the German mathematicians were extremely enthusiastic to discover such an extensive exhibition of mathematical models from France, Britain, and Germany—nothing like it had been seen before.

This enthusiasm also provided the incentive for the next large exhibition on mathematical models, apparatuses, and instruments, planned for the annual conference of the Gesellschaft Deutscher Naturforscher und Ärzte (Society of German Natural Scientists and Physicians) in 1892 in Nuremberg. Due to the 1892 cholera epidemic, the exhibition had to be cancelled and was eventually shown independently of the convention of natural scientists as part of the annual conference of the German Mathematical Society in Munich in 1893 (see Fig. 3). The Munich exhibition showed roughly the same group of mathematical artifacts—namely, the calculating machines and slide rules for arithmetic, and the drawing instruments and geometric models for geometry—but now in a different context than in 1876 in South Kensington. It is worth mentioning here that this led to a different interpretation not only of the artifacts but also of mathematics as a whole.^{Footnote 6} The exhibition consisted of four large sections—analysis, geometry, mechanics, and mathematical physics—each of which was presented in a space of its own and named after an eminent scientist. The exhibition was organized and curated by Walther Dyck,^{Footnote 7} who in 1884 was made a professor of higher mathematics and analytical mechanics at the Technische Hochschule München (Munich Engineering College). With this exhibition, Dyck not only set out to present mathematics as an academic discipline but also to show mathematics’ potential for application and its utility for technology. Thus, the exhibition was addressed not only to mathematicians but also to natural scientists and engineers. In addition, Dyck wanted to show how mathematical devices and artifacts could be used to practice mathematics and to visualize objects in teaching and research. It is reasonable to assume that in the concept and implementation of this exhibition, the mathematical instruments and models played an entirely different role than in England in the mid-1870s.

**KK**: This orientation to applied mathematics and, conversely, to the mathematization of engineering teaching—was that not already part of the French tradition earlier on, that is, following the founding of the École polytechnique? And was this mathematization of engineering practice one of the reasons for the strong interest in mathematical models and the exhibitions containing these models?

**UH**: First of all, it has to be said that mathematical education at the German engineering colleges founded in the course of the nineteenth century ultimately tended to be oriented to the model of the École polytechnique in Paris. However, before 1850, the mathematical level of engineering courses at the polytechnic schools in Germany (the predecessor institutions of the engineering colleges) was comparatively low and the academic level of the Paris model remained unrivaled. It was only in the 1860s, with attempts made by German engineers to turn the polytechnic schools into engineering colleges, and the higher expectations connected with this, that there was a real improvement in the standard of mathematics teaching.^{Footnote 8} Here, however, mathematics had a dual purpose: first, to secure the scholarly standard of the engineering colleges and the formal education of the students; and, second—in analogy with its relation to the exact sciences—to serve as the scientific basis for the technical sciences.^{Footnote 9} However, in what measure and in what manner the tradition developed by Gaspard Monge of using geometric models in the classroom was transferred to or taken up by the German polytechnic schools and later by the engineering colleges has still not been systematically investigated by historians of mathematics. The first larger model collection at a polytechnic school or engineering college in Germany was almost certainly the one assembled in Karlsruhe by the mathematician Christian Wiener, who taught descriptive geometry there. It is probably safe to say that in the 1870s one still looked at mathematical models from France, but that in the 1890s that was no longer the case.

Your second question on the relation between the mathematization of engineering practice and the strong interest in mathematical models and exhibitions of models merits closer examination. I will attempt to give an answer with the example of the Technische Hochschule München and a mathematician who has already been mentioned, Walther von Dyck. Von Dyck received his scientific education at the Technische Hochschule München (the Königlich Bayerische Technische Hochschule München, today’s TUM or Technical University of Munich), where he studied mathematics from 1875 to 1879. His teachers at the Technische Hochschule München were the mathematicians Alexander Brill and Felix Klein, the main founders of the mathematical model tradition in Germany. Brill and Klein encouraged mathematics students as well as engineering students, including Rudolf Diesel,^{Footnote 10} to build these models in their classes. It is worth mentioning here that they both saw a great need to convey an increasingly abstract ‘pure’ mathematics to the engineering students, who showed little appetite for this. Therefore, Brill and Klein developed the didactic principle of generating an interest in and of conveying complex and increasingly abstract mathematical concepts by means of geometric intuition [*Anschauung*]—not only in geometry but also in analysis.

However, the teaching conditions in mathematics at the Technische Hochschule München in the 1870s under Brill and Klein differed fundamentally from the situation in the 1890s when Dyck was teaching there and planning his first mathematics exhibition. Already during the early 1890s, Dyck felt a growing pressure to show the application potential and the real utility of the higher mathematics and analytical mechanics he was teaching. Whereas the model making continued by Dyck following the example of Brill and Klein was motivated at least in part by the teaching of mathematics students at the Technische Hochschule München, one has to see the 1893 exhibition conceived and curated by Dyck in the context of these general developments.

In the years that followed, a number of lucky coincidences came together that led Dyck to continue to occupy himself intensively with mathematics exhibitions and eventually to become the most important organizer of mathematics exhibitions of the nineteenth and early twentieth century. Thus, in 1903, Dyck’s former schoolmate Oskar von Miller founded the Deutsches Museum in Munich, and Dyck as the rector of the Technische Hochschule München played a crucial role in its founding. Together with the engineer Carl von Linde, Oskar von Miller and the recently ennobled Walther von Dyck made up the first board of directors of the Deutsches Museum.^{Footnote 11} In the newly founded museum, mathematics was then presented in another way, in the form of a permanent exhibition. The mathematics section of the Deutsches Museum was shown for the first time in 1905 in a provisional building, and then permanently following the completion of its current home in 1925.

If one compares the photos of the 1893 exhibition with the photos of the 1925 exhibition (Figs. 3, 4 and 5), one can see, on the one hand, that Dyck learned a lot about how exhibitions can be used as a medium to convey mathematical concepts and, on the other, the extent to which the didactic ambition had grown. Of course, one of the tasks of the Deutsches Museum was to make mathematics accessible to the general public. For this reason, the museum attempted to present mathematics in different ways—for example, by using examples taken from art (see Fig. 5). This meant, however, that the mathematical models started to play a less prominent role. In photos of the period, one can see how these models were exhibited at a height of about two meters and moved into the corner. This can be compared—certainly a little exaggeratedly—to the way products are displayed in supermarkets: the important items are placed at the front. In our case, what happens is that, alongside the mathematical instruments and calculating machines, it is the didactic aspect (i.e., whatever should be conveyed with the images and with the help of art) that comes to the fore. The mathematical models, on the other hand, tend to be seen further back.

Of course, that is also related to the fact that, toward the end of the nineteenth century, the production of mathematical models became less and less important. Models became less important in research, and in mathematical institutes they increasingly acquired the status of a merely decorative accessory. In Germany around 1870, however, models still had a significant role to play: at German universities and engineering colleges, these mathematical artifacts were used for the institutionalization of mathematics, that is, to found mathematical institutes. In a physics institute with it measuring instruments and diverse apparatuses this was relatively straightforward. But to find a way to convey mathematics in a tangible manner using concrete objects and instruments was more problematic. For a certain period in the second half of the nineteenth century, this constitutive role was played by mathematical models. This can be seen, for example, in the founding of the mathematics department by Klein at the Universität Leipzig in 1881. Here, Klein first requested the acquisition of a collection of geometric models and only then the founding of his own mathematics institute. Later, at the Universität Göttingen, Klein continued the model tradition in a similar way. In exhibitions and mathematics congresses after 1893, however, including the International Congresses of Mathematicians, which occurred regularly after 1897, mathematical models began to play a less prominent role. In Zürich in 1897, in Paris in 1900, as well as in later International Congresses of Mathematicians, such as in Cambridge in 1912, one sees fewer mathematical models, which were increasingly replaced by mathematics books.^{Footnote 12} Hence, mathematics started to be presented differently, and that reflects a shift within the discipline. In a broader context, the mathematical artifacts and the mathematics exhibitions in Munich in 1893 and in the Deutsches Museum in 1905 can probably be interpreted as part of a general fin de siècle scientific culture.^{Footnote 13}

**MF**: I would like to come back to von Dyck’s remark from 1929 that we quoted at the beginning. Von Dyck claims that mathematics—or, more precisely, “the essence” of mathematics—cannot be exhibited. On the other hand, more than forty years earlier, in 1886, Brill stated: “Often […] the model prompted subsequent investigations.”^{Footnote 14} Hence, in Brill’s time, mathematical models were still considered as epistemological objects. As you have already pointed out, at the beginning of the twentieth century, if not earlier, models are either exhibited as museal objects or disappear from view. How is the model’s change of status (from scientific artifact to museal object) to be reconciled with the transformation of mathematics, or of the image of mathematics?

**UH**: To answer this question, let us perhaps first stay in the Munich context. Munich was a center of mathematics—here, the discipline was supported by two academic institutions, by the Ludwig-Maximilians-Universität and the Technische Hochschule München, although the latter only had the right to award doctorates after 1901. However, already in these two institutions, mathematical models were used in different ways. The model tradition developed principally at the Technische Hochschule, where it was introduced by Klein and Brill, who designed and constructed many models with their students and doctoral candidates. These models were then sold by Alexander Brill’s brother Ludwig Brill through his publishing house—and later also by the Verlag Martin Schilling.^{Footnote 15} The model tradition had a heyday in Munich from 1875 to 1880 because Klein and Brill were working there. When Brill was offered a position at the Universität Tübingen in 1880, he took this tradition with him, and again established a model collection. In Munich the model tradition was principally continued by Dyck, who was made a professor at the Technische Hochschule München in 1884. Dyck’s efforts were supported and continued by Sebastian Finsterwalder when the latter became a professor of mathematics at the Technische Hochschule at the beginning of the 1890s. In this way, the model tradition in Munich survived almost into the 1930s. Models continued to be actively built—on the one hand, by Finsterwalder and, on the other, by his student Robert Sauer, who was a privatdozent at the Technische Hochschule München.

Hence, Dyck and Finsterwalder, who remained professors in Munich into the 1930s, practiced a mathematics that did not submit to the modernization of mathematics represented by Bartel Leendert van der Waerden, Emmy Noether, David Hilbert, or Nicolas Bourbaki;^{Footnote 16} rather, they practiced what might be described as a traditional mathematics, a mathematics of the nineteenth century. Their geometric objects were less abstract and formal than, above all, concrete and intuitive. Intuition (*Anschauung*) played an important role both in the research process and in the publications. Whereas Dyck made his mark principally with mathematics (museum) exhibitions and carried out research on mathematical instruments (integraphs), for Finsterwalder, who was concerned with concrete and application-based mathematical problems—and who besides Carl Runge was probably the most important applied mathematician of his time—the models remained above all a research instrument with which he attempted to visualize and interpret mathematical problems and concepts (see Fig. 6).^{Footnote 17}

At the Ludwig-Maximilians-Universität, however, the situation was different. Here, there was a much smaller model collection compared to the Technische Hochschule München, and far from all the mathematicians showed an interest in models: whereas the mathematicians Gustav Bauer and Ferdinand Lindemann created a collection of models, and Karl Doehlemann had his students make numerous card and string models, which were shown in the cabinets of the seminar rooms, Alfred Pringsheim (an epigone of Weierstrassian analysis) firmly rejected the use of intuition [*Anschauung*] in the teaching of mathematics in colleges.^{Footnote 18} Moreover, these models initiated by the—in scholarly terms—probably least important mathematician at the university, largely lacked the research context of the models by Brill, Klein, and Finsterwalder at the Technische Hochschule München. They were also not as elaborately produced as the models at the Technische Hochschule München and nor were they sold as copies in large quantities by the publishers Ludwig Brill and Martin Schilling, but remained for the most part unique pieces produced by prospective schoolteachers.^{Footnote 19}

In other places and at other colleges, the way in which model collections were developed and deployed in the classroom depended likewise both on the institutional conditions and on the mathematicians teaching there—we still know relatively little about this. If one looks at the surviving model collections, then, besides the one found at todays Technical University of Munich, one finds larger collections above all in Dresden,^{Footnote 20} Karlsruhe, Tübingen, and Göttingen. At the engineering college in Karlsruhe it was the mathematician Christian Wiener who built the models himself and whose example was followed by his son Hermann Wiener, who later taught at the engineering college in Darmstadt. In Tübingen it was Alexander Brill who established and developed the collection, having switched to the university there from Munich.^{Footnote 21} The most remarkable collection, however, was probably the one at the Georg-August-Universität Göttingen.^{Footnote 22} This is principally due to the fact that Klein brought his concept of collections of mathematical models to Göttingen from Munich via Leipzig in 1886, and in Göttingen (as already in Munich and Leipzig) he again built up a large model collection. This was presented in glass cabinets in the mathematics institute (see Fig. 7) and included mathematical instruments and calculating machines. In Göttingen, however, it is questionable whether the models were still used for research and teaching purposes as had been the case for Klein at the beginning of his career in the 1870s.^{Footnote 23} It is reasonable to assume that from a certain point, perhaps at the turn of the century, that was only the case to a very limited extent.

**MF**: In this connection, one should perhaps also mention that Hilbert and Stefan Cohn-Vossen’s book *Anschauliche Geometrie*, published in 1932, contains numerous photos of mathematical models.^{Footnote 24} The book was based on a lecture given by Hilbert. For Hilbert, however, the models reproduced in the book were not research objects but objects whose purpose was above all to trigger enthusiasm, and in this way they lose their heuristic, research-oriented status.

**UH**: I can basically only agree with this assessment of Hilbert’s book *Anschauliche Geometrie*. While Hilbert grants an important role to geometric intuition [*Anschauung*] in the research process, the vast number of drawings and photographed mathematical models from the Göttingen collection were principally intended to promote mathematics as an academic discipline.

If I may add an autobiographical observation: Beginning in the mid-1980s I studied mathematics in Göttingen myself. The cabinets with the models were found in the mathematics institute on the second floor, in the central hall. However, I never perceived these models as mathematical objects, at least not before I became a historian of mathematics. They were decoration in the best sense of the word. The most interesting for me was one in which Klein had mapped and traced the curves on a bust of the Apollo Belvedere. He wanted to prove that the classical Greek beauty ideal corresponded to the shape of certain mathematical curves (see Fig. 8). That has stayed with me, but I have almost no recollection of the other objects or models, since at the time I hardly paid attention to them.

Of course, what you said in connection with Hilbert’s book about the function of the models as a trigger for enthusiasm could already be applied to the nineteenth century. The mathematicians teaching at German universities and engineering colleges at that time had to find a way to promote their discipline, and indeed not only to the students; they also had to ‘sell’ mathematics with respect to the other disciplines. Moreover, in the German Empire, the physics and chemistry professors as well as the professors of engineering increasingly received large new institute buildings with a relatively extensive assistant personnel. The size of the institute as well as the collection of teaching materials and the number of assistants that were employed to look after them were an expression of the importance the college attributed to the respective discipline.

In connection with your question, it seems remarkable that at the International Congresses of Mathematicians it was no longer primarily models that were exhibited, but books, even though the models were more striking and were able to ‘enthuse.’ That may be due in part to the increasing orientation of mathematics to another branch of scholarship, namely the humanities, as well as the increasing orientation to pure mathematics.

**MF**: The issue of the books leads me to a question that moves away a little from mathematics as an academic discipline. How did the publishers benefit from having their books shown? And what was their role in relation to the changes in the model tradition? Here, I am not only referring to the publishing houses of Brill and Schilling, which sold mathematical models, but in particular to the publishers whose books were shown at the world congresses.

**UH**: There is of course an important exhibition that we have not yet talked about, namely the mathematics exhibition that was shown in 1893 as part of the World’s Columbian Exposition in Chicago (see Fig. 9).^{Footnote 25} This presentation came about because the Prussian government wanted to present German science at the world exhibition in Chicago. The Prussian Ministry of Education found a partner in Dyck, since at the time he was busy preparing the exhibition that was intended for Nuremberg in 1892, but which was eventually shown in Munich in 1893. The mathematics exhibition in Chicago showed mathematical models just as they could also be seen in the 1893 exhibition in Munich. In addition, however, the Chicago exhibition also presented the production of German mathematics publishers in order to highlight the research achievements of German mathematics.^{Footnote 26} This was done, on the one hand, by displaying the collected works of famous German mathematicians and, on the other, by presenting the famous German mathematics journals, such as August Leopold Crelle’s *Journal für die reine und angewandte Mathematik*, and *Mathematische Annalen*, while also showing portraits of eminent German mathematicians—and a bust of Carl Friedrich Gauß. So, in the case of this mathematics exhibition, one worked not only with models but also with important publications and publication series as well as with major names. The self-display at this large public exhibition in Chicago was therefore different from the self-display of mathematics in the 1876 exhibition at the South Kensington Museum, which was distinct in turn from the mathematics exhibitions accompanying the congresses of mathematicians, for which there was also a different public.

Remarkably, in 1925 in the Deutsches Museum, one then finds a presentation that in part has a strong connection to the conception of world exhibitions and also displays the great mathematicians as statues, but does not incorporate the books. As I mentioned earlier, the concept of the great mathematician was also found in Dyck’s 1893 exhibition in Munich, where each exhibition space bore the name of a famous scientist, and each of the names stood for a different branch of mathematics. The personification of science through the names and busts of great scientists corresponds in a way to the focus on the great artists in art exhibitions. The books and scientific objects, on the other hand, were shown in order to present the respective discipline.

To come to your actual question: did the exhibitions also bring benefits to the publishers? They certainly brought major benefits to the publishing houses. The congresses attracted a large specialist public that saw what was being published or the publisher with which as mathematicians they could be published themselves. At least until World War I, publishing mathematics books was quite a profitable business. That was true, for example, for the Teubner Verlag, which in the late nineteenth and early twentieth century not only published numerous mathematics textbooks but also the *Encyklopädie der mathematischen Wissenschaften mit Einschluß ihrer Anwendungen*. In addition, in 1912, the Teubner Verlag endowed a prize for the promotion of mathematics, which certainly also served to attract attention to the house and helped it to extend its network. After the war and with the onset of inflation, the business of the Verlag was thrown into a crisis. However, the Springer Verlag, which had already been founded in the nineteenth century, then followed in the footsteps of Teubner and developed into the next major mathematics publisher in Germany with an international standing.^{Footnote 27}

**KK**: I would like to come back to the question of the overall concept and guiding image of mathematics that was created in these exhibitions and who was actually responsible for this conception. In the case of the 1873 exhibition in Göttingen and the 1893 exhibition in Munich, it was the mathematicians who assembled and presented the objects and who therefore assumed the role of curators. But when the mathematical models end up in the museum context, they leave the academic context of teaching and research. In France it was even the case that a few models were made at the suggestion of the Musée des Arts et Métiers. How would you describe the influences and responsibilities during the conception of the first mathematics section in the Deutsches Museum?

**UH**: For the Deutsches Museum, which would become one of the world’s largest museums of science and technology, one can say that its conception in the founding phase was shaped above all by the world exhibitions, and that it was also influenced by the engineering college in Munich. The influence of the Technische Hochschule München can be seen in the—in a few major points—similar basic conception of these two Munich institutions. The museum should serve to show the interaction between science and technology, and that corresponds to the function of mediating between research and application that in Dyck’s view the engineering colleges should have at that time.

In this connection, it is perhaps interesting to note that, for Dyck’s 1893 exhibition, this relation is presented somewhat differently again. Think of the scientists who met at the annual meetings of the Gesellschaft Deutscher Naturforscher und Ärzte (Society of German Natural Scientists and Physicians). What did they find there? Exhibitions. In that case, these were exhibitions of medical and scientific instruments, but one can immediately see why Dyck and others might have thought, in 1892 or 1893, of planning a mathematics exhibition.

In the founding phase of the Deutsches Museum at the end of the century, it was certainly also the case that scientists at that time did not only want to present the results of their research to a general bourgeois or aristocratic public; they also wanted to use the history and the heroic figures of their respective science for the sake of self-display. For the most part, the first exhibition curators and consultants at the Deutsches Museum were professors, who were brought into the museum to conceive exhibitions relating to particular disciplines and to select the artifacts. The museum continued this tradition for decades, as can be seen with the exhibitions on computer science and mathematics curated by the Munich computer science professor Friedrich L. Bauer in the 1980s and 1990s.

**MF**: This leads, for me, to a number of fundamental questions: How, at that time, did one understand the act of exhibiting? What was the task of exhibitions, in particular in museums? Were exhibitions intended to instruct and educate or to trigger enthusiasm? What role did ‘visualization’ play in these exhibitions?

**UH**: The short answer to your question is that to exhibit meant to ‘construct’ mathematics. If we look again at Dyck’s conception from 1893, with this exhibition, he wanted to show the relation between mathematics and its application, and thus not only pure science but also applied mathematics. However, this occurred in a period of the arithmetization of mathematics and thus parallel to a process in which mathematical concepts that were introduced in a purely formal way began to appear whose practical application was initially rather unclear. When, beginning in the mid-1890s in the so-called anti-mathematics movement at the German engineering colleges, the engineers protested vehemently against this arithmetization and a mathematics that had become incomprehensible to them (such as Weierstrassian analysis), the Technische Hochschule München played a special role in these debates. Together with other professor colleagues, Dyck developed the concept of a strongly ‘science-oriented’ engineering college that was committed to a strong interaction between mathematics, physics, and the theoretical disciplines of engineering on one side and technical applications on the other.^{Footnote 28} Dyck was one of the first to recognize this crisis and he endeavored to counteract it with his exhibition on an applied and material mathematics. In this sense, he was, among other things, exhibiting his own conception of mathematics.

**MF**: Dyck’s catalogue to the 1893 exhibition is called *Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente* (Catalogue of Mathematical and Mathematical-Physical Models, Apparatuses, and Instruments). How do models, instruments, and apparatuses relate to one another in the different exhibition contexts (congress, world exhibition, museum)?

**UH**: First of all it is quite amazing that one sees these very different artifacts next to one another. Since what, for instance, does a calculating machine have to do with a geometric plaster model? They were made in completely different contexts, and they also operate in completely different contexts.

Included among the mathematical instruments, for example, are simple, familiar artifacts such as a compass, and less familiar devices such as the planimeter, a device used to measure a surface area by tracing a curve around the boundary of the shape to be measured. Examples of these are illustrated and thus visualized in Dyck’s catalogue. One illustration shows a planimeter that geodesists use to calculate surface areas, another shows the harmonic analyzer that Arnold Sommerfeld constructed with Emil Wiechert (Fig. 10). In the catalogue one also finds entries on and illustrations of stepped drum calculating machines, like those used by the engineer Carl von Linde.^{Footnote 29} These instruments and apparatuses thus refer to mathematical, natural-scientific, and technical practitioners.

The models on the other hand stand for a kind of *Versinnlichung* [sensible rendering], or rather for the materialization of a geometric, or physical, intuition [*Anschauung*], as was the case with the mechanistic models for the visualization of electrical processes exhibited in 1893 by Ludwig Boltzmann.^{Footnote 30} The models were part of a teaching practice, whereas the instruments and apparatuses referred to the mathematician or physicist who used them in his or her scientific practice. All of that was encompassed by mathematics—that is how I would understand this exhibition context around 1890.

If one looks at the later literature on instruments in (applied) mathematics, then one sees that the models eventually disappear. Thus, Friedrich Willer’s 1926 book is called *Mathematische Instrumente* (Mathematical Instruments).^{Footnote 31} This now deals only with the application context and with building devices that can then be used by natural scientists, engineers, and, possibly, mathematicians. In the *Zeitschrift für Instrumentenkunde* (Journal for the Study of Instruments), the major journal for scientific instruments in Germany published from 1881 onward, these mathematical instruments and calculating machines are placed in another larger context, where they are found next to astronomical, geodetic, meteorological, nautical, and other instruments and apparatuses. Although one also finds didactic devices such as demonstration apparatuses, mathematical models are missing here to my knowledge.

**KK**: Mathematics drew inspiration for model making from physics among other places. But can one say, conversely, that physics also profited from geometric graphic representations and that this in turn elevated the scientific status of representations as a whole? Here, I am thinking, for instance, of James Clerk Maxwell, who utilized two-dimensional representations in magnetism as the basis for further analyses and calculations. In this connection, Maxwell speaks around 1850 of ‘mechanical analogies’ in physics and attributes a special value to these analogies.^{Footnote 32} Later, Ernst Mach also draws on Maxwell in an article on analogy and similarity as a ‘leitmotif of research’ and thus as a heuristic method. Can Maxwell’s valorization of analogical representation and description be transferred to mathematical models?^{Footnote 33} Or, to put it another way, how did physics and mathematics behave toward each other in the late nineteenth century, when one thinks of the scientific deployment of three-dimensional representations, and thus also of models?

**UH**: If we look again at the 1893 Munich exhibition, at that time the physicists were completely aware that they were building models for the description of physical processes, and thus for the visualization of wave propagation and for the mechanical *Versinnlichung* [sensible rendering] of electrical processes. In this respect, Boltzmann’s model for the mechanical representation of electricity is a very interesting exponent of this history. It seems to me, however, that what was happening at the same time in mathematics was much closer to mathematics itself. The mathematicians visualized or materialized the objects of their geometric or function-theoretical research. Dyck, for example, had models of the Weierstrass ℘-function built (see Fig. 11), which is to say he visualized this function in three-dimensions. Boltzmann, on the other hand, attempted to visualize electrical processes via something mechanistic, and thus via something with which the people of the time were already familiar and therefore better able to imagine. In the case of Boltzmann’s model, a kind of detour is implied, whereas the mathematicians used their mathematical models to directly explain geometric problems or to interrogate these. While similarity also plays a role in the mathematical models, there is an epistemic difference between the way in which Boltzmann deployed his model and the way mathematicians deployed their models. This becomes clearer with the later works of Dyck’s colleague Finsterwalder, since Finsterwalder questioned, for example, how certain geometric properties of surfaces could be realized through deformable, mechanical models, and in this way he established that these models could lead to new discoveries in geometric research.^{Footnote 34} Although both Boltzmann and Finsterwalder took the path of a mechanical *Versinnlichung* (of geometric structures or of electrical processes), the value of Boltzmann’s model for the process of acquiring scientific knowledge is probably much more limited and only plays a role, if at all, at a certain stage.

**KK**: What you have just said reflects what Boltzmann writes in 1892 in his contribution to Dyck’s catalogue: “At first, of course, all these mechanical models [by Maxwell] existed only in thought; they were dynamic illustrations in the imagination, and they could not be executed with this generality in practice.”^{Footnote 35} And in 1902, in his article “Models” in the *Encyclopædia Britannica*, Boltzmann characterizes mathematical models in relation to their tangibility, and thus their concrete three-dimensional execution.^{Footnote 36}

**UH**: When we talk about the relation between mathematics and physics, one has to bear in mind that Dyck’s 1893 exhibition was the only exhibition in which he cast the net so widely and integrated physics, and thus also brought Boltzmann into the exhibition. That was unusual. The physicists were also included in the 1876 exhibition in the South Kensington Museum, but they were represented by their own section, because all sections could be seen there. In this respect Dyck’s exhibition was special, which is partly explained by the fact that Dyck, at that time, was sitting in on Boltzmann’s lectures, where he was grappling with mathematical physics, and also of course saw the engineers and their teaching models. To some extent, Dyck created his own universe, which can be seen in his work at the Technische Hochschule München and in the emerging engineering movement of the 1890s. But, in general, physics and mathematics do not seem to me to have had much to do with one another with regard to their models, and nor does this change later on. The mathematical instruments are then mostly used and described by so-called applied mathematicians such as Runge or Friedrich Adolf Willers or later Alwin Walther. One builds these instruments or machines—that is, planimeters, integraphs, or calculating machines—but one no longer sees physical models.

**MF**: You just mentioned applied mathematics and its special importance for the 1892/1893 exhibition. What was the relation between theoretical mathematics and applied mathematics in the other exhibitions we have spoken about?

**UH**: If we look at the 1876 exhibition at the South Kensington Museum, this included two branches of mathematics: arithmetic and geometry. Arithmetic was elementary mathematics in the best sense, and the machines exhibited were actually elementary mathematical machines. In addition, geometry was also to be seen there, but the two together did not make up the whole world of mathematics in 1876. Analysis, for example, was completely missing. Nor did one speak explicitly of applied mathematics there, but one simply assigned to the objects—that is, to the geometric models and the arithmetical machines—certain mathematical subdisciplines. In Dyck’s 1893 exhibition, on the other hand, it was already a matter very clearly of applied mathematics, and this is also found as an explicit category in the accompanying catalogue.^{Footnote 37} That was due to the fact that, at that date, Dyck was already aware of the tension between the mathematics teaching at the Technische Hochschule and what the engineers could actually utilize from mathematics, but also from analytical mechanics. Thus, Dyck’s exhibition anticipated certain aspects of a development that manifested itself later on—for instance, with Runge’s appointment to the chair of applied mathematics at the Georg-August-Universität Göttingen in 1904. Through Runge’s appointment, applied mathematics was institutionalized at the Universität Göttingen. At the same time, however, Runge should not only be seen as a precursor of modern ‘numerical mathematics,’ as this has predominantly been the case in the historiography of mathematics; he should also be interpreted as a precursor of ‘practical mathematics’ (*praktische Mathematik*), since, in Göttingen, besides numerical and graphic methods, he also integrated instrumental mathematical methods into his teaching and research. Beginning around 1910, the expression ‘practical mathematics,’ which is now no longer commonly used, stood for the triad of graphic, numerical, and instrumental methods for solving mathematical problems and has become known especially through the Institut für Praktische Mathematik at the Technische Hochschule Darmstadt, founded in the 1930s by Alwin Walther.^{Footnote 38}

Between 1893 and 1914, and thus in the period leading up to World War I, the development of applied mathematics was strongly promoted by the fact that the prevailing pure mathematics increasingly split off from the application contexts. In the ‘agenda’ of mathematics as a discipline, analysis was much more important than the application of analysis. Here, I am deliberately using the concept of agenda developed by the science historian Michael S. Mahoney^{Footnote 39} to emphasize what a discipline focuses on and thus what it pits itself against or the central tasks it has to solve. And these tasks were increasingly defined within mathematics as inner-mathematical problems, which ultimately led to mathematicians no longer expressing such a keen interest in application. This split and the counter movement was also apparent around 1900 in Klein’s reorientation of the *Zeitschrift für Mathematik und Physik* (Journal of Mathematics and Physics), which was now called *Zeitschrift für angewandte Mathematik* (Journal of Applied Mathematics), and it led in effect also to the establishment of the first professorship of applied mathematics at the Georg-August-Universität Göttingen as well as to the establishment of further professorships for applied mathematics—for example, in Jena. Starting in 1895 mathematics faced the growing problem that its position at the engineering colleges was becoming endangered by the anti-mathematics movement I mentioned earlier. It had to prove that it was still important to engineers. In Munich one was aware of this problem, and the artifacts in Dyck’s 1893 exhibition can all be seen in this context.

**KK**: To finish we would like to speak about visualization techniques, precisely because to exhibit mathematics also means to visualize it. If one looks at the photo of the 1925 exhibition in the Deutsches Museum (Fig. 4), then one sees that mathematical panels and graphic displays are very present. For the catalogue of the earlier exhibition, one sees the same thing: graphic displays and panels occupy a large space. What is the significance of the two-dimensional visualization methods in relation to the three-dimensional models?

**UH**: One of course has to consider what and how much one can visualize in a book or in an exhibition in 1850, 1890, 1910, or 1930. That is connected to the technical constraints of printing—for example, the possibility of reproducing photos. If you look at Dyck’s catalogue from 1892/1893, you will see that it is relatively rich in illustrations, but it does not include any photos. That changes completely with Hilbert’s book *Anschauliche Geometrie* almost forty years later. In Dyck’s catalogue one finds many drawings, such as the hypocycloids (Fig. 12),^{Footnote 40} and in this sense the catalogue makes full use of the potential to visualize objects, processes, or instruments. Furthermore, Dyck carries the model tradition forward in his own way by, as already mentioned, having function-theoretical models produced. But when he has the Weierstrass ℘-function modeled, this certainly does not take place within a research context. It is a purely didactic model, whose purpose is to show what this function looks like.

On the other hand, drawing on Henri Poincaré’s research of the 1880s, he also worked on the qualitative theory of differential equations.^{Footnote 41} There too he attempted to visualize differential equation curves and thus, for example, to show the shape of the solution curves. In a letter to Dyck, Klein wrote that the latter should make use of his “artistic imagination” [“künstlerische Phantasie”] in his mathematical works.^{Footnote 42} Here, one sees a strong tradition of understanding mathematics visually and of concretely visualizing it, whether through artifacts (i.e., via tangible objects) or through drawings. And this occurs at a time in which modern mathematics was beginning to move away from *Anschauung* (intuition). There was therefore also the opposite movement that attempted to extract anything geometric from the theory of functions and to proceed in a purely arithmetical way, as with Karl Weierstraß.

Dyck and his colleague Finsterwalder were in the truest sense of the word counter-modern, if with ‘modern’ one thinks of Herbert Mehrtens’s *Moderne—Sprache—Mathematik*.^{Footnote 43} But one can also consider this from another angle, since one of the greatest innovations of the German science system was that it managed between 1900 and 1925 to create a link between mathematics and engineering. That not only led in 1921 to the founding of the *Zeitschrift für angewandte Mathematik und Mechanik*, but also established a link between engineering, mathematics, and physics. What is classified as counter-modern from the perspective of the modernization of mathematics now becomes modern.

Translated by Benjamin Carter.

**Ulf Hashagen** is the head of the Research Institute for the History of Science and Technology at the Deutsches Museum and teaches at the Ludwig-Maximilians-Universität München (LMU). Following the study of mathematics in Marburg and Göttingen he obtained his doctorate in the history of science at the Ludwig-Maximilians-Universität München. From 1993 to 2000 he helped as a curator to set up the Heinz Nixdorf MuseumsForum in Paderborn and was jointly responsible for the scientific conception and realization of the permanent exhibition on the history of information technology. Since 2001 he works and carries out research at the Research Institute for the History of Science and Technology at the Deutsches Museum principally on the history of computing and the history of mathematical sciences in the nineteenth and twentieth centuries. A further area of research is the history of scientific and mathematical artifacts.

## Notes

- 1.
“Wesen, Inhalt und Ziele der mathematischen Forschung in ihrer Gesamtheit vor Augen zu führen, kann nicht Aufgabe eines Museums sein.” Walther von Dyck, “Mathematik,” in

*Das Deutsche Museum: Geschichte/Aufgaben/Ziele*, 2nd ed., ed. Verein deutscher Ingenieure/Conrad Matschoss (Berlin and Munich: VDI Verlag and R. Oldenbourg, 1929), 169–78, here 169. - 2.
On the London exhibition, see: South Kensington Museum, ed.,

*Handbook of the Special Loan Collection of Scientific Apparatus 1876*(London: Chapman and Hall, 1876) (on the geometric models, esp. 34–53). On the 1893 Munich exhibition, see: Walther Dyck, ed.,*Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente*(Munich: Wolf und Sohn, 1892); Walther Dyck, ed.,*Nachtrag zum Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente*(Munich: C. Wolf und Sohn, 1893); Walther Dyck, “Einleitender Bericht über die mathematische Ausstellung in München,”*Jahresbericht der Deutschen Mathematiker-Vereinigung*3 (1894): 39–56. On the genesis of the 1893 exhibition, see: Ulf Hashagen,*Walther von Dyck (1856–1934): Mathematik, Technik und Wissenschaftsorganisation an der TH München*(Stuttgart: Franz Steiner Verlag, 2003), 419–35. - 3.
See: August Gutzmer,

*Geschichte der Deutschen Mathematiker-Vereinigung von ihrer Begründung bis zur Gegenwart*(Leipzig: Teubner, 1904) 19–24. - 4.
See: Felix Klein,

*Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert*, vol. 1 (Berlin: Springer, 1926), 78; Dyck, ed.,*Nachtrag*, 51–52. - 5.
See: Robert Bud, “Responding to Stories: The 1876 Loan Exhibition of Scientific Apparatus,”

*Science Museum Group Journal*1 (Spring 2014), 10.15180; 140104, online journal, http://journal.sciencemuseum.ac.uk/browse/2014/responding-to-stories/ (accessed November 23, 2021). - 6.
The four rooms were named after Gottfried Wilhelm Leibniz, René Descartes, Galileo Galilei, and Isaac Newton, and also showed their busts. See: Dyck, ed.,

*Katalog*; see, in particular: the introduction of Dyck (“Einleitung”), in ibid., iii–vi. - 7.
On this exhibition, see: Hashagen,

*Walther von Dyck*. - 8.
For the general development of engineering education in the nineteenth century, see also: Jonathan Harwood, “Engineering Education between Science and Practice: Rethinking the Historiography,”

*History and Technology*22, no. 1 (2006): 53–79. - 9.
See: Susann Hensel, “Die Auseinandersetzungen um die mathematische Ausbildung der Ingenieure an den Technischen Hochschulen Deutschlands Ende des 19. Jahrhunderts,” in

*Mathematik und Technik im 19. Jahrhundert in Deutschland*, ed. Susann Hensel, Karl-Norbert Ihmig, and Michael Otte (Göttingen: Vandenhoeck & Ruprecht, 1989), 1–111. - 10.
See: Hashagen,

*Walther von Dyck*, 58–61; Stephan Finsterbusch, “Die ersten Diesel-Modelle,”*Frankfurter Allgemeine Magazin*52 (August 12, 2017), 42–43. See Fig. 11 for a series of models (of the real part of the Weierstrass ℘-function) constructed by Dyck’s assistant Heinrich Burkhardt and the teacher trainee Adolf Wildbrett. - 11.
See: Ulf Hashagen, “Ein unbekannter Mitbegründer des Deutschen Museums: Zum 150. Geburtstag des Mathematikers Walther von Dyck,”

*Kultur und Technik*30 (2006), 43–45. - 12.
For an extensive overview on the International Congress of Mathematicians from the end of the nineteenth century to the present, see: Guillermo P. Curbera,

*Mathematicians of the World, Unite!*(Wellesley: A K Peters, Ltd., 2009). Curbera notes that, at the International Congress of Mathematicians in Heidelberg in 1904, “the congress organized an ‘Exhibition of Literature and Models,’ with mathematical books, models, and apparatus that concentrated on recent materials, less than ten years old. The bibliographic exhibition consisted of scientific literature. More than 900 publications were exhibited […] [and] [m]ore then 300 mathematical models were shown […].” (Ibid., 32). At the 1912 congress in Cambridge, an “exhibition of books, models and machines (chiefly calculating machines) [was organized], which was arranged in two rooms […].” (Ibid., 51). - 13.
See: Ulf Hashagen, “Mathematics on Display: Mathematical Models in Fin de Siècle Scientific Culture,”

*Oberwolfach Report*12 (2015), 2838–41. - 14.
“Dem Verfertiger eines Modells stand es frei, eine Abhandlung zu demselben zu schreiben, deren Veröffentlichung […] nicht wenig dazu anreizte, die oft mühsamen Rechnungen und Zeichnungen, welche der praktischen Ausführung zu Grunde lagen, durchzuführen. Öfter veranlaßte umgekehrt das Modell nachträgliche Untersuchungen über Besonderheiten des dargestellten Gebildes.” [“The maker of a model was free to write a paper on this, whose publication […] played no small part in encouraging one to carry out the often-arduous calculations and drawings at the basis of the practical execution. Often, conversely, the model prompted subsequent investigations.”] Alexander Brill, “Über die Modellsammlung des mathematischen Seminars der Universität Tübingen,”

*Mathematisch-naturwissenschaftliche Mitteilungen*, vol. 2 (1887), 69–80, here 77. - 15.
The first three editions of the Brill Verlag’s catalogue of models were published in 1881, 1882, and 1885: Ludwig Brill, ed.,

*Catalog mathematischer Modelle für den höheren mathematischen Unterricht*(Darmstadt: L. Brill, 1881); Ludwig Brill, ed.,*Catalog mathematischer Modelle für den höheren mathematischen Unterricht*, 2nd ed. (Darmstadt: L. Brill, 1882); Ludwig Brill, ed.,*Catalog mathematischer Modelle für den höheren mathematischen Unterricht*, 3rd ed. (Darmstadt: L. Brill, 1885). Between the second and the third edition (between 1882 and 1885) hundreds of models were added to the catalogue, as well as elaborate theoretical explanations. - 16.
On the rise of modern algebra during this period, see: Leo Corry,

*Modern Algebra and the Rise of Mathematical Structures*(Basel: Birkhäuser, 2004). - 17.
See, for example: Sebastian Finsterwalder, “Mechanische Beziehungen bei der Flächendeformation,”

*Jahresbericht der Deutschen Mathematiker-Vereinigung*6 (1899): 45–90. - 18.
Alfred Pringsheim, “Über den Zahl- und Grenzwertbegriff im Unterricht,” Jahresbericht der Deutschen Mathematiker-Vereinigung 6 (1899), 73–83; Micheal Toepell,

*Mathematiker und Mathematik an der Universität München: 500 Jahre Lehre und Forschung*(Munich: Institut für Geschichte der Naturwissenschaften, 1996), 251–55. - 19.
Martin Schilling, ed.,

*Catalog mathematischer Modelle für den höheren mathematischen Unterricht*(Leipzig: Martin Schilling, 1911). - 20.
See: https://www.math.tu-dresden.de/modellsammlung/ (accessed August 18, 2021).

- 21.
See: https://www.unimuseum.uni-tuebingen.de/de/sammlungen/mathematische-modellsammlung.html (accessed August 18, 2021); Ernst Seidl, Frank Loose, and Edgar Bierende, eds.,

*Mathematik mit Modellen: Alexander von Brill und die Tübinger Modellsammlung*(Tübingen: Museum der Universität Tübingen—MUT, 2018). - 22.
See also: Anja Sattelmacher, “Zwischen Ästhetisierung und Historisierung: Die Sammlung geometrischer Modelle des Göttinger mathematischen Instituts,”

*Mathematische Semesterberichte*61 (2014): 131–43. - 23.
David E. Rowe, “Mathematical Models as Artefacts for Research: Felix Klein and the Case of Kummer Surfaces,”

*Mathematische Semesterberichte*60 (2013), 1–24. - 24.
See, for example: David Hilbert and Stefan Cohn-Vossen,

*Anschauliche Geometrie*(Berlin: Springer-Verlag, 1932), 14, 17, 175, 191, 193, 194, and passim. - 25.
Karen H. Parshall and David E. Rowe, “Embedded in the Culture: Mathematics at the World’s Columbian Exposition of 1893,”

*Mathematical Intelligencer*15 (1993): 40–45; Hashagen,*Walther von Dyck*, 425–36. - 26.
See: Walther Dyck, ed.,

*Special-Katalog der Mathematischen Ausstellung*(Munich: Wolf, 1893). See also the series of lectures delivered by Klein at the 1893 exposition: Felix Klein,*Lectures on Mathematics*(New York: Macmillan, 1894); and especially his appendix to the lectures, Felix Klein, “The Development of Mathematics at the German Universities,” 99–109, esp.: 108–9: “In conclusion a few words should here be said concerning the modern development of university instruction. The principal effort has been to reduce the difficulty of mathematical study by improving the seminary arrangements and equipments. […] Collections of mathematical models and courses in drawing are calculated to disarm, in part at least, the hostility directed against the excessive abstractness of the university instruction.” See also: Parshall and Rowe, “Embedded in the Culture,” 40–45. - 27.
Volker R. Remmert and Ute Schneider,

*Eine Disziplin und ihre Verleger: Disziplinenkultur und Publikationswesen der Mathematik in Deutschland, 1871–1949*(Bielefeld: transcript, 2010). - 28.
Ulf Hashagen, “Der Mathematiker Walther von Dyck und die ‘wissenschaftliche’ Technische Hochschule,” in

*Oszillationen: Naturwissenschaftler und Ingenieure zwischen Forschung und Markt*, ed. Ivo Schneider, Helmuth Trischler, and Ulrich Wengenroth (Munich: Oldenbourg, 2000), 267–96. - 29.
On the planimeter, harmonic analyzer, and arithmometer, see: Dyck, ed.,

*Katalog*, 100ff., 214ff., and 150; Joachim Fischer, “Instrumente zur Mechanischen Integration: Ein Zwischenbericht,” in*Brückenschläge: 25 Jahre Lehrstuhl für Geschichte der exakten Wissenschaften und der Technik an der Technischen Universität Berlin 1969–1994*, ed. Hans-Werner Schütt and Burghard Weiss (Berlin: Engel, 1995), 111–56. - 30.
Dyck, ed.,

*Katalog*, 405–8. Here, Boltzmann describes an “Apparat zur mechanischen Versinnlichung des Verhaltens zweier elektrischer Ströme (Bicycle)” [“Apparatus for the mechanical*Versinnlichung*of the behavior of two electrical currents (bicyclic)”] (ibid., 405). - 31.
Friedrich Adolf Willers,

*Mathematische Instrumente*(Berlin: de Gruyter, 1926). - 32.
James Clerk Maxwell, “On Faraday’s Lines of Force,” [1855/6] in James Clerk Maxwell,

*The Scientific Papers of James Clerk Maxwell*, vol. I, ed. William Davidson Niven (Cambridge: Cambridge University Press, 1890), 155–59. - 33.
Ernst Mach, “Die Ähnlichkeit und die Analogie als Leitmotiv der Forschung,” in

*Annalen der Naturphilosophie*, vol. 1, ed. Wilhelm Ostwald (Leipzig: Veit & Comp., 1902), 5–14, here 5: “Die Analogie ist jedoch ein besonderer Fall der Aehnlichkeit. Nicht ein einziges unmittelbar wahrnehmbares Merkmal des einen Objectes braucht mit einem Merkmal des anderen Objectes übereinzustimmen, und doch können zwischen den Merkmalen des einen Objectes Beziehungen bestehen, welche zwischen den Merkmalen des anderen Objectes in übereinstimmender, identischer Weise wiedergefunden werden. […] [M]an könnte dieselbe auch eine abstrakte Ähnlichkeit nennen.” [“Analogy is, however, a special case of similarity. Not a single immediately perceptible feature of one object need coincide with a feature of another object, and yet relations can exist between the features of one object in exactly the same way as those between features of the other object. […] [O]ne might also call this an abstract similarity”]. - 34.
Finsterwalder, “Mechanische Beziehungen bei der Flächendeformation,” 46: “Man kann sich versuchsweise auf den Standpunkt stellen, die geometrischen Eigenschaften der Flächen nach der Möglichkeit, sie auf mechanischem Wege herzustellen, zu beurteilen, und es ist interessant zu sehen, wie eine Reihe geometrisch wichtiger Eigenschaften auch einer einfachen Realisirung auf mechanischem Wege fähig sind. Andererseits aber wird man durch die Möglichkeit der mechanischen Realisirung auf die Betrachtung von geometrischen Eigenschaften aufmerksam, welche sich sonst der Untersuchung leicht entziehen würden, so dass die hier eingeschlagene Methode auch einigen heuristischen Wert beanspruchen darf.” [“By way of experiment, one can attempt to judge the geometric properties of the surfaces according to the possibility of producing them by mechanical means, and it is interesting to see how a number of geometrically important properties are also capable of a simple realization by mechanical means. On the other hand, the possibility of mechanical realization makes one attentive to geometric properties that would otherwise easily escape from investigation, so that the method adopted here may also claim some heuristic value”].

- 35.
“Alle diese mechanischen Modelle [von Maxwell] bestanden vorerst freilich nur in Gedanken, es waren dynamische Illustrationen in der Phantasie, und sie konnten in dieser Allgemeinheit nicht praktisch ausgeführt werden.” Ludwig Boltzmann, “Über die Methoden der theoretischen Physik,” in

*Katalog mathematischer und mathematisch-physikalischer Modelle, Apparate und Instrumente*, ed. Walther Dyck (Munich: C. Wolf und Sohn, 1892), 89–98, here 97. - 36.
Ludwig Boltzmann, “Models,” in:

*Encyclopædia Britannica*, 10th ed., vol. 30 (Edinburgh, 1902), 788–91, here 788–89: “The term model denotes a tangible representation, whether the size be equal, or greater, or smaller, of an object which is either in actual existence, or has to be constructed in fact or in thought. […] In pure mathematics, especially geometry, models constructed of*papier-mâché*and plaster are chiefly employed to present to the senses the precise form of geometrical figures, surfaces, and curves.” - 37.
Indeed, the closing passage of the introduction of the 1892 catalogue, written by Dyck, emphasizes both pure and applied mathematics: “Möge Interesse und Mitwirkung Aller, die in diesem Jahre unser Vorhaben gefördert haben, uns dabei nicht fehlen, mögen insbesondere die Fachgenossen mit Rat und That den Zweck des ganzen Unternehmens durchführen helfen, ein vollständiges Bild zu geben, all’ der mannigfachen Hilfsmittel, wie sie heute in Gestalt von Modellen, Apparaten und Instrumenten dem Unterricht und der Forschung in der reinen und angewandten Mathematik dienen!” [“May the interest and participation of all those who in this year have supported our undertaking not fail, may in particular our colleagues help to accomplish in word and deed the purpose of the whole undertaking, to provide a complete picture of all the manifold aids as they serve today in the form of models, apparatuses, and instruments for teaching and research in pure and applied mathematics!”] (Dyck, ed.,

*Katalog*, vi). The third section of the catalogue is titled “Angewandte Mathematik” (ibid., 307–420). See also: Dyck, “Einleitender Bericht über die Mathematische Ausstellung in München,” (1894); Hashagen,*Walther von Dyck*, 254–64. - 38.
For a range of interpretations, cf.: Gottfried Richenhagen,

*Carl Runge (1856–1927): Von der reinen Mathematik zur Numerik*(Göttingen: Vandenhoeck & Ruprecht, 1985); Ulf Hashagen, “Rechner für die Wissenschaft: ‘Scientific Computing’ und Informatik im deutschen Wissenschaftssystem 1870*–*1970,” in*Rechnende Maschinen im Wandel*, ed. Ulf Hashagen and Hans Dieter Hellige (Munich: Deutsches Museum, 2011), 111–52, here 116; Ulf Hashagen, “Computers for Science-Scientific Computing and Computer Science in the German Scientific System 1870*–*1970,” in*The German Research Foundation 1920–1970*, ed.: Mark Walker, Karin Orth, Ulrich Herbert, and Rüdiger vom Bruch (Stuttgart: Steiner, 2013), 135–50. - 39.
Michael S. Mahoney, “Computer Science: The Search for a Mathematical Theory,” in

*Science in the 20th Century*, ed. John Krige and Dominique Pestre (Amsterdam: Harwood Academic Publishers, 1997), 617–34; Michael S. Mahoney, “Software as Science – Science as Software,” in*History of Computing: Software Issues*, ed. Ulf Hashagen, Reinhard Keil-Slawik, and Arthur Norberg (Berlin: Springer Verlag, 2002), 25–48. - 40.
A hypocycloid is generated by the passage of a point

*P*on a circle*k*with radius*r*when rolling inside a circle*K*with radius*R*. - 41.
Hashagen,

*Walther von Dyck*, 265–97. - 42.
Ibid., 280.

- 43.
Herbert Mehrtens,

*Moderne—Sprache—Mathematik: Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme*(Frankfurt am Main: Suhrkamp, 1990).

## Acknowledgements

Michael Friedman and Karin Krauthausen acknowledge the support of the Cluster of Excellence “Matters of Activity. Image Space Material” funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2025—390648296.

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Friedman, M., Hashagen, U., Krauthausen, K. (2022). Interview with Ulf Hashagen: Exhibitions and Mathematical Models in the Nineteenth and Twentieth Centuries. 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_14

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