This chapter illustrates Geiser’s interest in teaching. First, his schoolbook, one of his major works, is analysed. An account of the life of Julius Gysel, a schoolteacher, headmaster, and Geiser’s friend, follows. Gysel can be seen as an example of a late 19th/early 20th century schoolmaster with whom Geiser
worked in order to improve school education (see section 2.3).
5.1 Einleitung in die synthetische Geometrie
5.1.1 Background and Motivation
In 1863, Geiser habilitated at the Polytechnic as a Privatdozent. As part of his teaching duties he offered an introductory course on synthetic geometry for a number of years (see chapter 2). As a result of his lectures and in the hope of improving mathematics education in general, Geiser wrote his textbook Einleitung in die synthetische Geometrie. Ein Leitfaden beim Unterrichte an höheren Realschulen und Gymnasien1, published in 1869 [15]. He explains his motives for
writing the book in the very interesting preface, given in full here:
When a new publication emerges from the mighty stream of geometry textbooks, which did not emanate from the circle of well-versed educationalists, but traces its origin back to a junior lecturer, then a justification of the same may only be found in itself. But may the author at least be permitted to explain and to account for its purposes and objectives in a preface.
For several years now, the author of this book has been entrusted with the obligatory instruction in synthetic geometry, which is supposed to initiate the students at the Department for Mathematics Teachers at the Swiss Polytechnic in the afore-mentioned science. In his lectures he has continually experienced a series of gaps in the preparatory training of his audience. These gaps had to
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“Introduction to Synthetic Geometry. A Guide for Instruction at Higher Realschulen and Gymnasien”. Henceforth referred to as Synthetische Geometrie.
be filled in the first instance, before any attention could be given to the actual subject matter.
Out of this necessity a course of lectures emerged: “Introduction to Synthetic Geometry”. Several repetitions and diverse revisions of this course supplied the content for this little book at hand.
To begin with, the need to train the visual-spatial ability of the audience with as little prerequisites as possible prevailed. Therefore, upon completion of the section on plane geometry, the derived theorems in the first main section on the “Theory of Transversals” were already transferred into space as far as possible. The theorems on the planar triangle are followed by the corresponding theorems on the solid triangle and the tetrahedron. In addition to harmonic points and rays, harmonic planes are examined as well. Furthermore, a separate chapter is dedicated to linear transformations in the plane and in space. The relationship between planar and three-dimensional shapes turns out to be even more intimate in the second main section on “Circle and Sphere”. In this section, every chapter contains theorems from both plane geometry and solid geometry, which illustrate one another. When deriving the fundamental theorems on radical axes, points of similarity and the harmonic properties of circles, it is demonstrated that in some cases the three-dimensional observations can even lead to the desired result more easily than the calculations required by plane geometry for proving these theorems. The author admits that the chosen approach, though in fact only presupposing the basic elements of plane geometry and solid geometry (apart from a few simple trigonometric formulae), will not be an easy one for pupils to follow, as it demands full attention and a sufficient knowledge of the preceding course material at every moment. In his lectures, he has experienced again and again that just the first steps in synthetic geometry are the hardest ones. However, he believes that through a thorough treatment of the material presented a sufficient understanding can be achieved, also on the level for which this “introduction” has been written.
Sure enough, this will necessitate geometry becoming more important in school than is currently the case. Realschulen, which prepare their pupils for polytechnic schools, will be more inclined to this expansion of instruction in geometry, since technical education is primarily constructive and therefore requires a developed visual-spatial ability. This will increasingly have to be the case, as descriptive geometry in its natural development draws more and
more on the purely synthetic direction. Therefore, this book may also be regarded as a resource for descriptive geometry; in particular if the theories, theorems and constructions contained in it are always accompanied by a practical implementation in the form of figures, which require meticulous drawing.
Admittedly, at Gymnasien this last consideration will be omitted. However, one can state reasons no less substantial to support the view that instruction in geometry should be expanded at these institutions as well; even be treated as the centre of mathematical instruction in the higher years. Only when this happens will it be possible to achieve and retain the proper position of mathematics as a discipline that stimulates and trains the mind amongst the classical-philosophical sciences.
May the author be allowed yet a personal remark: In strictly scientific circles, achievements such as the one at hand are often regarded with great contempt and disdain. This has not kept him from daring to publish the same. He is aware of the fact that he has not taken on this task as a result of the, nowadays admittedly fairly widespread, addiction to prolific writing. In fact, he believes to be serving science by attempting to smooth and to alleviate the paths leading to science to the best of his humble abilities. Incidentally, surely he will be allowed to point out that even the greatest mathematicians of his home country, Switzerland, did not disdain to see to spreading science in the wider population. But surely even the most rigorously minded will not want to reproach men such as Leonhard Euler and Jakob Steiner for this endeavour of theirs.
May this attempt to make synthetic geometry accessible to school be recommended to teachers and pupils of this science, for consideration free of prejudice; and if the Swiss educational establishments in particular receive it favourably, then a gladly held wish of the author will come true.
C. F. G.
[15, p. iii-vi]
Interestingly, Geiser does not go into much detail about the actual book, but offers his personal opinions on the place of mathematics in school education and on academic arrogance. References to the latter frame the preface, in a manner of speaking: First, Geiser suggests that educationalists might not approve of his work due to his young age and lack of relevant (at
least in the eyes of the experts) qualifications and experience. In the penultimate paragraph, he hints at the contemptuousness of his colleagues, who might have regarded such a work as beneath them. One could argue that neither of these remarks has lost their relevance more than 150 years later. A number of universities in the UK now run outreach projects, but a look at the University of St Andrews’s access webpage [40] reveals that many of these projects focus on encouraging pupils to apply to university, as well as offering guidance for the application process. Other outreach projects involve explaining (science) research to the public, e.g. at science fairs. The ETH, Geiser’s alma mater, organises events, such as public lectures and departmental visits, which introduce the wider public to scientific concepts and research methods [38]. None of these aim at improving a pupil’s scientific training before embarking on their studies (at least not explicitly), as opposed to Geiser’s book. With regard to potential attacks from educationalists, which Geiser hints at, education experts still argue about how best to teach and what to teach, developing new frameworks in the process, such as the Curriculum for Excellence in Scotland.
It is not surprising that Geiser felt that he had to defend his book against possible accusations, or indeed that he feared such attacks. He was only 20 years old when he started teaching, and 26 years old when the book was published. It is not unreasonable to assume that some veteran teachers would have questioned his expertise, or indeed that his university colleagues would have wondered why he did not invest his time into doing research and publishing papers. However, Geiser did do research in the 1860s; in addition to his thesis he published nine papers in the period 1866-1869, which together account for 40% of his research publications [cf. 24, p. 526-528]. Furthermore, he also edited a few of Steiner’s papers, including volume I of Jacob Steiners Vorlesungen über synthetische Geometrie, in the years 1866-1868 (see section 2.2). It is possible that Steiner’s lecture notes inspired him to write his own book, but if that was the case, he certainly does not say so in Synthetische Geometrie. Unsurprisingly, Steiners Vorlesungen covers more advanced topics than Synthetische Geometrie, but one could regard Geiser’s book as a prequel, which
introduces the student to the fundamental concepts of synthetic geometry that are essential for understanding Steiner’s lectures.
Whatever influence the editing process might have had on Geiser’s own writings, he gives an indication of his heavy workload in the preface to Steiners Vorlesungen: apologising for a lack of coherence, he remarks that ‘in the last two years in particular, the author [Geiser] was deprived of his best working hours due to his own research and an often burdensome teaching load, so that he could only attend to editing his first draft every now and then’ [16, p. vi]. Writing an entire textbook on top of this indicates his interest in education and his commitment to raising its standards, which remained apparent throughout the rest of his life.
Looking at Geiser’s remarks about the place of geometry in mathematics education in schools, he touches upon a heated debate that occupied mathematics teachers and mathematics professors as well as engineers and educationalists in Switzerland, but predominantly in Germany, during the late 19th and early 20th centuries. Geiser gives an account of the so-called
“Engineers Movement” (“Ingenieursbewegung”) at German universities and polytechnics, which started in the late 1890s, in his biography of Theodor Reye [17, p. 166-171] (see appendix E.3.2). As he wrote the biography about half a century after Synthetic Geometry, it is interesting to note that his opinion did not change significantly during the years. Admittedly, he does not explicitly state his views on the Engineers Movement in the Reye biography, but it seems that he thought that mathematics deserved a prominent position in education. See also appendix B for a short account of the corresponding debate at the Polytechnic2
.
Whilst mathematics and engineering professors at higher education institutions argued about the place of mathematics in relation to the applied sciences, the debate took a different shape at secondary schools. There, the question was where mathematics stood in relation to the humanities. Looking specifically at geometry, Geiser acknowledges that whilst it has real-life applications, it is also an art and trains the mind (similarly to classics, for
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example). Even today mathematics is generally considered a science, although (pure) mathematics can in fact be seen as an art, much closer to philosophy than to, say, chemistry.
5.1.2 Structure and Content
Returning to Synthetic Geometry, let us now look at the content of the book. Geiser splits it into two parts: I) “Theory of Transversals”, and II) “Circle and Sphere”. Each part contains four chapters, numbered consecutively:
- Chapter 1: “Transversals in a Triangle”
- Chapter 2: “Triangle and Tetrahedron. Complete Planar Figures” - Chapter 3: “Harmonic and Involutory Structures”
- Chapter 4: “Linear Dependencies in the Plane and in Space” - Chapter 5: “Powers. Similarity Points”
- Chapter 6: “Harmonic Properties of Circles and Spheres” - Chapter 7: “Applications”
- Chapter 8: “The Principle of Conjugate Radii”
Each chapter will be summarised in the subsequent paragraphs. Then, some examples from the book will be given, with comments, before Geiser’s style and approach will be discussed.