Felix Klein

I. Line-segment, Area, Volume as Relative Quantities

You will notice by the heading of this section that I am following the intention announced above, of examining simultaneously the corresponding magnitudes on the straight line, in the plane, and in space. At the same time, however, we shall take into account the principle of fusion by making use at once of the rectangular system of coordinates for the purpose of analytic formulation.

III. The Grassmannian Principle for Space

We shall now carry out the corresponding investigations for space in complete analogy with the foregoing considerations for the plane. We start therefore from the matrices, which can be formed with the coordinates of 1, 2, 3, or 4 points.

V. Higher Configurations

This completes what I wished to say here about elementary configurations of geometry, and I shall now turn to the higher configurations, which arise by combination of these. I shall do this in a historical form, so that you can get an idea of the development of geometry in the different centuries.

I. Affine Transformations

The name, which goes back to Mobius and Leonhard Euler, implies that, in such a transformation, infinitely distant points correspond again to infinitely distant points, so that, in a sense, the “ends” of space are preserved. In fact, the formulas show at once that x´, y´, z´ become infinite with x, y, z. This is in contrast to the general projective transformations, which we shall study later, in which x´, y´, z´ are fractional linear functions, and by which, therefore, certain finite points will be moved to infinity. These affine transformations play an important role in physics under the name of homogeneous deformations. The word “homogeneous” implies (in contrast to heterogeneous) that the coefficients are independent of the position in space under consideration; the word “deformation” reminds us that, in general, the form of any body will be changed by the transformation.

IV. Tansformations with Change of Space Element

The most obvious cases are those correspondences, which interchange point and line in a two-dimensional region, or point and plane in a three-dimensional region. I shall restrict myself to the first case, and I shall follow the line of thought, which Plucker first used in 1831 in the second part of his Analytisch-geometrische Entwickelungen, which we mentioned earlier (p. 61). For it, the analytic formulation constituted the point of beginning.

II. The Teaching in France

The conditions here are the more interesting for us since they have influenced in various ways the developments in Germany. Here, a situation fundamentally different from England is revealed. While the strictly conservative Englishman adheres to the old institutions, the Frenchman loves the new and achieves it even if often – rather than by continuous transformation of the Old – by sudden reformation, which somewhat constitutes a revolution. The organisation of teaching is entirely different: in France, there is not only centralisation of the exam – due to entrance examinations to higher education institutions, especially those in Paris – but also generally a strictly centralised organisation of teaching. The supreme authority, the Conseil d’Instruction Superieure (amongst its members being always mathematicians of the first rank) is the absolute ruler and is entitled to decree, at its discretion, far-going reforms and changes as often as it wishes. Such reforms have to be realised throughout the country immediately, and the teachers must see how to cope. The individual freedom to a high degree of each teacher, to which we in Germany are accustomed is less in practice here. One might even speak of a “system of revolution from above”.

II. The Grassmannian Determinant Principle for the Plane

Let us recall the fundaments of the considerations of the first chapter. There, using the coordinates of three points, we set up the determinant and interpreted it as twice the area of a triangle, i.e., as the area of a parallelogram.

I. The Systematic Discussion

In this chapter, we shall at first use geometric transformations to achieve a systematic division of the entire field of geometry, one which will enable us, from one standpoint, to overlook the separate parts and their interrelations.

IV. The Teaching in Germany

In principle, I want to consider all the German-speaking countries, such as Germanspeaking Switzerland and Austria. In Germany, the ways in which the teaching of geometry have evolved show completely different patterns to those in the other countries; especially, due to the lack of uniformity, as it was achieved in other countries – be it by strict governmental organisation or by the intervention of a strong personality. Here in Germany public education became established in each individual state according to proper conceptions; moreover, also at the individual institution, for individual teachers always retained a relatively large degree of freedom for independent practice. Thus, a great number of various suggestions from different sources achieved realisation concurrently; usually, their efficiency could be established, even before they had been sanctioned in official curricula. I shall be able, of course, to select out just a few aspects which became particularly important for the development in the last decades – say from about 1870 on. For additional information, I recommend to you the extensive presentation of the general lines of development in the book Klein-Schimmack.

III. The Teaching in Italy

In Italy, we note another highly characteristic development that reveals quite different patterns than those in England and France; in their typical forms it can at the extreme be placed in parallel with Meray. I want to concern myself only with modern Italy from about 1860 onwards. The greatest influence on the uniform restructuring of mathematics teaching in the then newly unified state was Luigi Cremona, the same person whom you all know for his scientific importance in the development of modern geometry; actually, he is the founder of the independent algebraic-geometrical research in Italy, which has provided such excellent results. Due to his scientific activity, Cremona has exerted a lasting impact on higher education, by connecting projective geometry with descriptive geometry and graphical statics. Engineers everywhere in the world speak today of Cremona’s force diagram, and if this name may be historically unjustified, it shows clearly Cremona’s great influence.