Geiser’s Style and Method

In a nutshell, Synthetische Geometrie is well written and in fact quite enjoyable to read. Never having studied synthetic geometry myself, I found the material that Geiser covers very accessible; his explanations are easy to follow and to understand. Furthermore, a number of figures illustrate some of the concepts and constructions in the book. It is thus conceivable that budding Polytechnic students would have found Synthetische Geometrie useful in giving them the necessary basics of synthetic geometry.

However, it is very theoretical and requires the reader to engage in independent study. Whilst the geometric constructions are explained well and the steps are easy to follow, one does have to sit down with pen and paper and work the problems out for oneself. This can be seen as an example of Geiser’s teaching talents: he does not hand his readers everything on a silver plate, but encourages them to discover geometry themselves. Curiously, however, Geiser includes no exercises at all, as is common practice in the majority of introductory textbooks; he does not even refer the reader to a collection of problems, for example. There are a few worked examples, but they are rather abstract and take the form of definitions and proofs rather than exercises [15; e.g. p. 12-14; p. 19-20; p. 49-51; p. 62-63; p. 72-73; p. 139-146; § 25; § 28]. A student of pure mathematics should not have any problems with this, but for school leavers and engineering students the material may have seemed rather academic and not particularly tangible – similarly to his lectures, which were not very popular with the engineers (see section 2.1). Moreover, Geiser hardly includes any references to practical applications of the material covered (an exception is §27, where he explains how stereographic projection is used to create maps).

Unfortunately, I have not been able to find any books that I could compare Synthetische Geometrie to. A large number of geometry textbooks were published during the second half of the 19th

century, but most of them are either more advanced than Geiser’s book, or treat different branches of geometry, or both. Holzmüller places Synthetische Geometrie into a historical context in [23], but covers several areas of geometry (see section 5.1.4).

Books that serve a similar purpose in their respective fields would include T Reye’s Synthetische Geometrie der Kugeln … [31] and Geometrie der Lage [30] as well as W Fiedler’s Darstellende Geometrie … [14]. A much more recent example would be C Durell, Projective Geometry [13]. However, note that all of the above books are aimed at university students and require a more profound mathematical knowledge than Synthetische Geometrie.

An example that predates Geiser’s book, but caters to a similar audience, would be M Ohm’s Die reine Elementar-Mathematik, in particular volume III

[29]. It is much more application-oriented than Synthetische Geometrie and contains a wealth of exercises and examples.

From a modern perspective, Geiser’s style is rather wordy, and he keeps algebraic expressions to a minimum. However, this is due in part to the subject matter as well as to the era. Contemporaries of Geiser should not have found his style unusual.

To summarise, Synthetische Geometrie is a coherent, comprehensive introduction to synthetic geometry. It contains the basic tools needed to engage in further study of the subject, as well as some more curious and complicated problems, which provide the necessary inspiration for this. A greater emphasis on practical applications and a few historical notes would have been desirable, though. In general, the book is well structured, the explanations are concise, and the constructions are easy to reproduce. However, a few additional figures, a clearer division into paragraphs and a collection of practice problems would have made the book even better.

5.1.3.1 §18: Pole and Polar with Respect to a Circle

As an example of Geiser’s approach, let us look at Chapter 6, Section 18, on poles and polars [15, p. 106-111]. Here Geiser combines concepts that he derived previously, such as harmonic properties of points and rays. At six pages it is of middle length, compared to the other sections. It contains two figures, which Geiser uses in his proofs, but it is rather theoretical, like most of the book.

In the first paragraph Geiser defines two harmonic conjugate points p and p’ on a line with respect to (w.r.t.) the points of intersection of the line with a given circle M. He explains why there are infinitely many harmonic “poles” assigned to a point p [cf. 15, p. 106]. He then wants to prove that ‘the harmonic conjugate poles of a point p are always distributed on a certain straight line’ [15, p. 106], and adds that if this is true, then the line is perpendicular to the

line through p and the centre of the circle M ‘because of symmetry’ [ibid.]. The proof of this is purely geometrical; Geiser uses harmonic ranges, harmonic rays, and angles between them. Adding the observation that it does not matter whether the point p is inside or outside of the circle M, and using a theorem (without proof) to explain why the harmonic conjugate poles of p lie on a tangent of M if p lies on the circle, Geiser arrives at the main theorem of the section: ‘The harmonic poles that are conjugate to a point p w.r.t. a circle M lie on a straight line P, which is called the polar of the point p’ [15, p. 107]. He then summarises how to find the polar geometrically, and the three cases depending on the location of p. He does similar summaries for his next theorem, which he gives without proof, that every polar has one and only one pole [cf. 15, p. 108]. Furthermore, he repeats an earlier observation [e.g. cf. 15, p. 52-53; p. 70-71] by stating that the pole of a straight line at infinity is the centre of the circle.

Next, Geiser explains how to construct the polar of a point w.r.t. a given circle using a ruler only, and, using a complete quadrilateral, proves that the line one obtains is indeed a polar. From this he deduces another theorem: ‘If a point moves along a straight line P, then its polar [w.r.t.] a circle M rotates about a fixed point p, which is the pole of P’ [15, p. 110]. Remarking that the proof has already been given, he uses a lemma, without proof, to show that the proof is easier if P does not cross the circle [cf. 15, p. 110]. He then gives a stereometric proof of the theorem, which does not rely on harmonic properties.

Geiser states the converse of the theorem that the polars of all the points on a straight line go through its pole, noting that it ‘does not require any specific proof’ [15, p. 111]. However, he gives special cases of the two theorems, again without proof. Finally, Geiser briefly explains how to construct the pole of a given straight line w.r.t. a circle using a ruler only.

As mentioned above, the section is rather abstract, but this is in accordance with the rest of the book and could well be due to the nature of synthetic geometry. The proofs are purely geometrical, which is typical of Geiser here; there are only a few proofs that require equations. Whilst Geiser states and

uses a number of results without proof, he gives alternative proofs for a couple of results. Other sections display a similar approach. This is quite useful for students as it offers a different angle to the problem, thus broadening their mathematical horizon, but it also provides a perhaps more intuitive proof. Some readers might feel that he omits information in using theorems without proof, but then the book was intended as an introduction. As it stands, there are quite a lot of proofs for students to learn already.

Bearing in mind that synthetic geometers sought to purge all metric aspects and use an axiomatic method, Geiser’s book gives a good introduction into the synthetic approach, which is illustrated by this chapter. Whether it was because of the nature of the subject or because of his intended audience, or both, Geiser explains how to construct certain results using pen and paper. This makes the material a bit more accessible and helps the reader to understand the subject material. It would also have been useful for future engineers who had to learn technical drawing. However, for engineering students in particular, some explanations as to the real-life applications of poles and polars, in this example, would have been desirable. Geiser leaves his readers with a lot of information, but they would have to turn to other, possibly more advanced, books in order to understand the significance of the material treated in Synthetische Geometrie.