Abstract
In comparison with France, the development of descriptive geometry in Austria started with a delay of approximately 40 years and reached a first culmination in education and research in the era of Emil Müller , during the first decades of the twentieth century. With respect to education, emphasis was mostly placed on the practicability of descriptive geometry methods, and ‘learning by doing’ was seen as an important methodological principle. At some schools and in variable degrees, the syllabus of descriptive geometry was extended by closely related geometric subjects like kinematics, photogrammetry, nomography, or elementary differential geometry.
In view of research, during the nineteenth century, the synthetic method of reasoning dominated; descriptive geometry was seen as a counterpart to analytic geometry. Later this puristic point of view became obsolete. Descriptive geometry found its justification as a method to study three-dimensional geometry through two-dimensional views, thus providing insight into structure and metrical properties of spatial objects, processes, and principles. This is independent of the tools and still valid when computers take over computational and drawing labour.
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Keywords
- Austro-Hungarian empire
- Cyclography
- Descriptive geometry
- Differential geometry
- Kinematics
- Projective geometry
- Relief perspective
- Johann Hönig
- Josef Krames
- Erwin Kruppa
- Emil Müller
- Gustav Peschka
- Rudolf Staudigl
- Walter Wunderlich
1 Introduction
It was Gaspard Monge’s revolutionary merit
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to extract the geometric methods from their various applications in fields like architecture, the art of painting, stone cutting, civil and mechanical engineering,
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to put them onto a common scientific basis, and
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to combine all in a separate discipline under inclusion of related mathematical topics.
Immediately after its foundation, the new science “descriptive geometry” was implemented in the curricula of French polytechnic schools. In AustriaFootnote 1 it took a couple of years until this new discipline found acceptance in the curricula of the new Austrian polytechnic schools, which were founded around 1810 in Prague, Vienna, Graz, Brünn (today Brno/Czech Republic), and Lemberg (today Lviv/Ukraine).
The delayed acceptance by these new Austrian schools was mainly caused by their founders’ belief that not science but experience in practice should be the ultimate goal for their students. Consequently, during the first decades of the nineteenth century, the geometric methods necessary for engineering and architecture were still taught within traditional courses like engineering drawing or architectural drawing (Fig. 11.1). However, soon it became obvious that there is no progress in technical practice without scientific achievements. Shortcomings were also observed in education: teaching mere practice without any scientific background was not satisfying. In this sense, the original mission statement of the polytechnic schools was recognized as being too narrow.
In 1834, Johann Hönig (1810–1886) became the first to teach an optional course on descriptive geometry in Vienna, and in 1843 he became the first professor of descriptive geometry at a new chair (German: Lehrkanzel) that had been founded 1842 at the Polytechnicum in Vienna. In Prague first optional lectures had already been held 1830; a new chair was established in 1853. This was the beginning of a flourishing era for descriptive geometry in Austria, in education as well as in research.
Below we provide an overview on how the contents of descriptive geometry education and research have changed in Austria over the course of time. Further details can be found in Benstein (Chap. 9, this volume) and in the references therein, or in Loria (1908). For more details concerning Vienna, the reader is referred to Binder (Chap. 12, this volume).
2 Descriptive Geometry Education at University Level
In the nineteenth century, all professors in Austria were totally free in the selection of topics to teach in their lectures. Therefore, the development of descriptive geometry can be studied only on the basis of related textbooks. Below we concentrate mainly on books of authors who worked at the descriptive geometry chair in Vienna, which was more or less the leading institution in Austria. It is worth to notice that about 50% of the authors originated from Bohemia, which today is part of the Czech Republic.
There is a visible difference between the drawings displayed in the literature before and after Monge (compare Figs. 11.1 and 11.2). Figure 11.1, selected from Table 3 in the book by Rittinger (1839) , shows the strategies to produce simple axonometries or perspectives. It is a purely planar process without any attempt to reveal the included geometric relations, though perspective affine and projective transformations clearly play a role in these routines. The labeling of points corresponds to the planar construction and has no relation to the spatial situation. Before Monge, the main purpose of drawings was to produce pictures which came close to a real impression of the depicted object. Therefore, even in mechanical engineering, there was a priority of axonometric views and perspectives.
Monge defined representing and analysing three-dimensional objects as the two main objectives of the science of descriptive geometry.Footnote 2 Drawings reduce spatial geometry problems to planar problems. Hence, in the period after Monge, drawings served as a tool to determine metric properties, but also, to create objects that satisfy given conditions. The latter can be traced to the usage of graphical methods in the design of military fortifications, on account of which it is reported that for some time descriptive geometry methods were even handled as a military secret. By virtue of Monge’s theories, priority was given to the principal views: top view, front view, and side view.
2.1 Hönig and Staudigl
The first Austrian textbook on descriptive geometry was published by Johann Hönig (Hönig 1845). It shows a consequent use of spatial coordinates (x, y, z) and a consistent labeling of points and lines (cf. Fig. 11.2): one prime for the top view, two primes for the front view, and three primes for the side view. This tradition is still valid in Austria and Germany. The axis between the image planes, the so-called “hinge line” (German: Rissachse, denoted by x in Fig. 11.2), played an important role. Various constructions were based on the traces of planes and the trace points of lines (see Barbin, Chap. 2, this volume). It is surprising that Emil Müller already recommended (Müller 1911, p. 62) to omit the hinge line for several reasons: This axis is not visible in technical drawings, and constructions based on trace points of lines or traces of planes often fail since the trace elements lie far beyond the limits of the drawing board. Moreover, a translation of a single image plane does not change the corresponding view. This meets Felix Klein’s general definition, according to which geometry has to study invariants. It turned out that, from Müller onward, the hinge line was more or less banned at university level but not in high schools. This may be due to a didactical reason: This axis supports the pupils’ imaginations as something that is fixed in space and pupils can rely upon.
Let us return to Hönig’s book on descriptive geometry (Hönig 1845): the figures in the 26 tables, which are added as a supplement, also show various curves, such as trochoids, evolutes, and involutes, as well as surfaces of revolution and helical surfaces, defined as trajectories of points or lines under particular movements. Differential-geometric aspects are not addressed: there are no tangents drawn to the depicted curves. Moreover, the contours of displayed surfaces are often missing, and no contour points have been constructed. Ellipses are depicted without indicating their axes of symmetry. Of course, it was not until 1845 that the famous construction of vertices from given conjugate diameters was found by the Swiss mathematician David Rytz von Brugg .
This changed soon. Twenty years later, Rudolf Staudigl recommended in his textbook (Staudigl 1875) to construct not only points but also tangents to the displayed curves, in order to obtain a higher precision. In Fig. 64 of Staudigl’s book already the Rytz construction is presented, but no mention is made of the name Rytz. Staudigl’s textbook also contains a detailed description of how to find the contours of surfaces of revolution and even possible cusps. Since, for these surfaces, the shade lines, i.e., the boundaries of shades, are also constructed, we already recognize something which is characteristic of descriptive geometry: a clear distinction between the ‘true’ contour on the surface in space and its image, the ‘visual’ contour in the image plane. The same distinction exists between shade lines and shadow lines, terms which already date back to Monge. In his textbook, Staudigl also shows the construction of tangents to the visual contour, while tangents to the true contour are missing, though in an earlier paper (Staudigl 1843) the author already presented a pertinent result. However, he did not mention that, due to Charles Dupin (Dupin 1813, p. 48), these tangents are conjugate to the lines of sight.
2.2 Descriptive Geometry and Projective Geometry
Staudigl was the first Viennese descriptive geometer who focussed also on projective geometry. In his textbook (Staudigl 1870), he called it ‘Neuere Geometrie’, and according to the book’s preface, he considered it as “geometry of position”. However, his book is actually a comprehensive introduction to projective geometry, treated in a “synthetic” way, i.e., without any computation. The book starts with perspectivities in the plane and ends with projective properties of spatial cubics.
The incorporation of projective geometry into descriptive geometry was perfectly done by Wilhelm Fiedler (1832–1907), who was professor at the Polytechnicum in Prague and later in Zürich. The presentation in his textbook (Fiedler 1871) deviated in several respects from the other ones: He approached the topic in a deductive way. Beginning with central projections and projective transformations, the usual mappings, transformations, and constructive methods of descriptive geometry are developed step by step, in a top-down approach, by successive specialization. Fiedler also broke with another tradition. He was the first to include analytic representations. More details about Fiedler can be found in Volkert (Chap. 10, this volume).
In Austria, exactly in Brünn and later in Vienna, descriptive geometry and projective geometry have been bound together by Gustav A.V. Peschka (1830–1903). His comprehensive textbook (Peschka 1883–1885) consists of four volumes, which amount to a total of 2553 pages and additional tables with 1140 figures. Each single volume is dedicated to emperor Franz Joseph’s son, Kronprinz Rudolph, and, as proudly stated in the book’s subtitle, “in keeping with the latest scientific developments”. Volume 1 presents the traditional topics of descriptive geometry and the basics of projective geometry, often structured as a series of more or less ‘academic’ exercises, for example: Find graphically the position of a balloon which is seen from three given points on earth under given slope angles. In Austria, it was for the first time that descriptive geometry was presented without direct engineering applications.
Volume 2 continues with a synthetic treatment of algebraic curves and surfaces of arbitrary degree. This is pure “geometry of position”. Volume 3 focusses on surfaces of 2nd degree and presents their projective, metric, and differential-geometric properties. Finally, volume 4 treats algebraic ruled surfaces, surfaces of revolution and helical surfaces, and on approximately 200 pages, shades, and shadows. The last exercise deals with the curves of constant illumination on a Dupin cyclide (however, without identifying this class of surfaces).
2.3 Further Development
With Emil Müller (1861–1927), the successor of Peschka on the chair in Vienna, the focus of descriptive geometry education in Vienna returned again to engineering applications. Müller’s most successful era will be presented in the fourth section.
After one century of descriptive geometry education, another characteristic of descriptive geometry became visible: Its algorithms never remain restricted to generic elements, but they always focus on particular cases like contour points or singularities of curves or surfaces, too. Till today, such a point of view is advantageous for the development of computational algorithms, because it forces to look carefully for cases, where general algorithms fail.
In the nineteenth century, additional topics were implemented into descriptive geometry courses. Contour maps of geometric elements, in particular of surfaces, combined with marked altitudes, were the basis of a topic, which in German is called Kotierte Projektion (topographic mapping). Here the students of civil engineering learned graphical methods to solve geometrical and topographical problems, e.g., for the design of roads. It needs to be noted that Monge was already familiar with this method, not least because of its military importance. The latter is confirmed by the fact that the first textbook on this topic (Noizet 1823) was written by a Captaine du Génie (captain of the corps of engineers).
On the other hand, at some universities, in particular in Brünn, the students of mechanical engineering learned some basics of kinematics, i.e., about point trajectories and instantaneous poles or axes of planar or spatial motions. In Josef Krames’ textbook on descriptive geometry (Krames 1947), the word ‘kinematics’ even appears in the title.
3 Scientific Progress in and Around Descriptive Geometry
It is quite natural that some topics of scientific research during the early days of descriptive geometry were later included into descriptive geometry courses for engineers. This holds, for example, for the theory of “shadows and shading”, and more general, for geometric lightning models, the illumination of surfaces including isophotic lines, i.e., curves of constant illumination (Fig. 11.3), or for the detection of brightest points.
This happened hand in hand with research on “differential geometry”. Based on Dupin’s results on the curvature of surfaces (Dupin 1813), the asymptotic lines and the curvature lines of particular surfaces inclusive torsal and non-torsal ruled surfaces were studied. However, in differential-geometric research, synthetic methods soon reached their limits, and with Carl Friedrich Gauß and Bernhard Riemann , the intrinsic differential geometry of surfaces came into the focus of interest. A late example of a textbook with constructive applications of differential geometry was published by Kruppa (1957). He used Dupin indicatrices, for instance, to determine tangents at singularities of curves of intersection.
Another topic intimately connected with descriptive geometry was the “relief perspective” (in Italian: prospettiva solida), used, e.g., on stages. Here, a perspective collineation maps a half space, which includes the depicted scene, onto a layer bounded by two parallel vertical planes (Fig. 11.3). The construction of a relief perspective of any given object is based on two theorems, which are attributed to Gournerie (1859) and Staudigl (1868) , respectively. The first addresses the front view of the wanted relief, the other the top view. Both coincide with particular perspective views of the given object.
A new field of research in descriptive geometry started with Pohlke’s theorem, which states that each axonometry is the composition of a parallel projection and a scaling. For more details about Karl Pohlke see Benstein (Chap. 9, this volume). Subsequently, new proofs of this theorem were given (cf. Müller and Kruppa 1923), and the underlying problem, i.e., the decomposition of any transformation into a product of simpler ones, could, of course, lead to various generalizations.
A topic that has its origin in descriptive and projective geometry became famous under the name “geometry of position” (German: Geometrie der Lage). It was developed as a counterpart to analytic geometry and focussed on geometric theorems which are independent of any metric. However, it exceeded the borders of projective geometry toward algebraic geometry, since the question of constructability with ruler and compass had no importance. The proofs of algebraic statements were mainly based on results of the French mathematician Michel Chasles concerning algebraic (m, n)-correspondences. Prominent German-speaking representatives were K. G. Christian von Staudt , Fiedler , and Theodor Reye . Among Austrian’s descriptive geometers, only Peschka was involved. Von Staudt’s occupation with questions of algebraic geometry was probably also the origin for his work on imaginary elements. He demonstrated that even pairs of complex conjugate elements are accessible for graphic constructions, which even became standard in Müller’s teacher training.
Another topic which evolved from descriptive geometry and separated soon was “photogrammetry” (today also known under the name “remote sensing”). It started with the question of how to recover metrical data from perspectives and reached high actuality with the invention of photography. Two fundamental theorems are attributed to Sebastian Finsterwalder , a mathematician and surveyor in Munich. Soon the economical importance of this field was recognized, also in view of military applications, and much effort was made in the design of mechanical devices for transforming aerial photographs into maps. In Vienna, Eduard Doležal was a pioneer in photogrammetry.
In the field of kinematics, we owe remarkable progress to descriptive geometers. For example, even in the present day, the contributions of Ludwig Burmester in Dresden (the same, who designed the relief perspective shown in Fig. 11.3) and Martin Disteli in Karlsruhe are well known in the scientific community. Prominent kinematicians originating from the Austrian school of descriptive geometry include Wilhelm Blaschke , Josef Krames , Hans Robert Müller , and Walter Wunderlich .
In “cyclography”, descriptive geometers studied a new type of mapping, where points were no longer sent to points but to oriented circles (called “cycles”) in the plane. By virtue of the fundamental theorem, points belonging to the same line in space with an inclination of 45∘ correspond to circles with oriented contact. This mapping, which is attributed to Fiedler (Fiedler 1882), was the beginning of the geometry of circles and spheres, and further on of conformal differential geometry. It also opened a door to pseudo-Euclidean geometry (or classical Minkowski geometry), which gives spacetime, i.e., the geometric standard model of Albert Einstein’s special relativity, in four dimensions.
Finally, it must be mentioned that, at the end of the nineteenth century, German companies like M. Schilling and B.G. Teubner started to produce mathematical models of curves and surfaces, for instance, from gypsum or brass and strings (cf. von Dyck 1892).Footnote 3 The intention behind these collections was to support the student’s spatial ability and intuition as well as to demonstrate the beauty of mathematics. Many models visualize results of descriptive geometry; Fig. 11.3 shows one example out of these collections.
4 The High Standard of Descriptive Geometry in Emil Müller’s Era
Emil Müller was an engaged and inspiring teacher, famous also for his ingenious drawings on the blackboard. His academic career started rather late, after a 10-year career as a teacher at the Baugewerkschule in Königsberg i. Pr./Germany. In 1902, he was appointed professor for descriptive geometry at the Technische Hochschule Vienna.
In a couple of papers, he presented his ideas on education in descriptive geometry (e.g., Müller 1910b or Müller 1911). He emphasized its importance in civil and mechanical engineering, and he was convinced that spatial ability could only be trained with objects of our physical world. Therefore, in his eyes, projective geometry was of less importance. And he avoided too much abstraction or even a flavour of an axiomatic treatment.
Müller gradually published a collection of applied descriptive geometry exercises based on his ideas (Müller 1910–1926), which consists of six volumes with 60 exercises.Footnote 4 Some of the examples included in the collection were incredibly rich in detail (note Fig. 11.4 or Fig. 10.2). Certain solutions, produced by professionals, have even been printed in large format and made available as pieces of fine art (Wildt 1895, 1902, note Fig. 11.5). Parallel to this collection of examples, mainly for civil engineering and architecture, Müller’s colleague in Vienna, Theodor Schmid , edited a collection of 25 examples for mechanical engineering (Schmid 1911, note Fig. 10.4).
Müller’s textbook on descriptive geometry (Müller 1908, 1916) became a standard reference work in Austria, where it continued to be used as such until the sixties of the twentieth century. From the 4th edition onward, it was edited by Kruppa , who would only publish one volume; the last edition appeared in 1961. Later it was replaced with Wunderlich’s pocket-books (Wunderlich 1966, 1967) and Fritz Hohenberg’s textbook (Hohenberg 1956). The first one was outstanding because of its precise formulations and elegant reasoning, the latter because of the high-quality figures and its focus on various applications of geometry recovered in almost all branches of engineering.
Müller was the first one to create a particular program for high-school teachers in descriptive geometry. While in former time this training consisted only of standard lectures and exercises for civil and mechanical engineers and occasional courses on projective geometry, Müller gave lectures on the ‘geometry of mappings’, on ‘cyclography’, ‘ruled surfaces’, and ‘constructive treatment of helical and translational surfaces’. Three of Müller’s special courses were later published as lecture notes, which confirmed the successful evolution of descriptive geometry from a mere technique to a science.
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I.
The first volume was co-edited by Müller and Kruppa (1923) and presents the theory of linear mappings in a visual and constructive way, quite contrary to today’s treatment in linear algebra. Besides, the mapping of lines onto their trace points in given planes is considered from a general synthetic point of view. A comprehensive synthetic treatment of the “kinematic mapping” is also included. It was discovered in 1911 by Josef Grünwald and Blaschke , independently of each other. Due to this intuitively introduced mapping, points in space are in one-to-one correspondence to planar displacements. This was the forerunner of a method which continues to be of great significance in robotics: curves in a 7-dimensional space correspond to one-parameter movements of the end effector. Volume 1 of Müller’s lecture notes concludes with a survey on Sophus Lie’s “line-sphere-transformation”.
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II.
The courses of type two were later elaborated and edited by Krames (Müller and Krames 1929) under the title Die Zyklographie (Cyclography). This is an intuitive introduction into the Möbius-, Laguerre-, and Lie-geometry of oriented circles and spheres, but also into pseudo-Euclidean geometry with its indefinite metric, here under the name “C-geometry”. This volume also provided extensive information on the cyclographic images of curves and surfaces and hence, e.g., caustics.
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III.
The third volume, again elaborated and edited by Krames , treats ruled surfaces (Müller and Krames 1931). It provides a synthetic differential geometry of ruled surfaces, including striction curves, as well as bendings of ruled surfaces. Besides, this volume presents a unique graphics-oriented analysis of algebraic ruled surfaces of degrees 3 and 4. It demonstrates Müller’s and Krames’ mastery in synthetic reasoning; for Krames, descriptive geometry was “die Hohe Schule des räumlichen Denkens und der bildhaften Wiedergabe” (the high art of spatial reasoning and its graphic representation) (Krames 1947, p. 1). This book is still a storehouse for experts on computer graphics who like to produce colourful realistic pictures of spectacular ruled surfaces.
While in education Müller avoided excessive abstraction and preferred the synthetic method, his scientific publications demonstrate a mastery of analytic reasoning. This is why Kruppa in his obituary (Kruppa 1931, p. 50) characterized Müller as a geometer in the middle between the purely synthetic treatment and the analytic method.
Müller’s scientific œuvre reveals him to be an expert in Grassmann’s theories on multi-dimensional geometry. He authored an article (Müller 1910a) on coordinate systems in the prestigious Enzyklopädie der Mathematischen Wissenschaften, which outlined the state of the art of mathematics at the beginning of the twentieth century. Another subject, where Müller’s scientific achievements have not lost their significance, is “relative differential geometry”, where, instead of the unit sphere, an appropriate surface is used to define the normalization.
5 Conclusion
The scientific foundation of descriptive geometry by G. Monge had a tremendous impact on the education in Austrian schools and polytechnical institutes. From the mid to the nineteenth century until the end of the twentieth century descriptive geometry was, beside mathematics, mechanics, and physics, one of the basic sciences, which were taught in the first semesters of almost all technical studies. Furthermore, till today descriptive geometry is a topic in vocational high schools and selected gymnasia, and about 50% of Austrian pupils in the age of 13 or 14 years become acquainted with a light version of descriptive geometry in a subject called Geometrisches Zeichnen (geometric drawing).
However, a worldwide scan at the begin of the twenty-first century reveals that outside of France the name of Gaspard Monge is almost forgotten and the topic descriptive geometry is more or less unknown. There are only a few exceptions: it was in Ukraine that the national Association of Applied Geometry devoted its 1995 annual meeting to the 200th anniversary of Monge’s Géométrie descriptive. Moreover, the Serbian Society for Geometry and Graphics continues to use the sophisticated name moNGeometrija for their biannual international scientific conferences.Footnote 5
With the rise of computers, manual constructions have been replaced by CAD software. Instead of sheets with drawings, we use 3D-databases, and with 3D printers, we can produce 3D models of virtual shapes of any complexity. Nevertheless, only people with a profound knowledge of descriptive geometry are able to make extended use of CAD programs since the interface is usually based on 2D images only. The more powerful a modeling software, the higher the required geometric knowledge. Although the name “descriptive geometry” is gradually vanishing, the science is still in use (cf. Cocchiarella 2015), and parts of it are included in different fields like engineering drawing, architectural drawing, computer graphics, computer vision, virtual reality, or computer-aided design. Moreover, in a graphics-oriented world, a specific training of spatial ability is inevitable for many professions.
Notes
- 1.
In this context, ‘Austria’ stands until 1918 for the Austro-Hungarian Empire, mainly for the German-speaking part, and afterwards for the country with its today’s extension.
- 2.
G. Monge (Monge 1811, p. 1): “La Géométrie descriptive a deux objets:
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Le premier, de donner les méthodes pour représenter sur une feuille de dessin qui n’a que deux dimensions, savoir, longueur et largeur, tous les corps de la nature qui en ont trois, longueur, largeur et profondeur, pourvu néanmoins que ces corps puissent étre définis rigoureusement.
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Le second objet est de donner la maniére de reconnaître, d’aprés une description exacte, les formes des corps, et d’en déduire toutes les vérités qui résultent et de leur forme et de leurs positions respectives.”
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- 3.
A collection of mathematical models is, e.g., provided at http://www.geometrie.tuwien.ac.at/modelle/.
- 4.
Later editions, co-edited by Erwin Kruppa, were still available in the 1950s of the last century.
- 5.
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Stachel, H. (2019). The Evolution of Descriptive Geometry in Austria. In: Barbin, É., Menghini, M., Volkert, K. (eds) Descriptive Geometry, The Spread of a Polytechnic Art. International Studies in the History of Mathematics and its Teaching. Springer, Cham. https://doi.org/10.1007/978-3-030-14808-9_11
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