Vergleichende Betrachtungen über neuere geometrische Forschungen

Erlangen: Andreas Deichert, 1872. First edition. THE FAMOUS 'ERLANGEN PROGRAM': THE UNIFICATION OF GEOMETRY.

First edition, rare in the original printed wrappers, of Klein's 'Erlangen Program,' his most famous and influential work. 'Klein's most important achievements in geometry, however, were the projective foundation of the non-Euclidean geometries and the creation of the 'Erlangen Progamm'" (DSB). After lecturing at Göttingen for a year, Klein joined the faculty at the University of Erlangen in 1872. "As was the custom, Klein had to present an Inaugural Address ... It is commonly confused with the Erlangen Program, but that was not the Address. Rather, the Erlangen Program was a pamphlet printed by Deichert in Erlangen and distributed to those who came to the Inauguration. A few copies were doubtless distributed to friends and colleagues abroad, and to some libraries, because that was customary at the time; but the informal nature of the publication partially accounts for the negligible response to the Erlangen Program in 1872" (Gray in Landmark Writings, p. 546). It "comprised a proposal for the unification of Euclidean geometry with the geometries that had been devised during the nineteenth century by mathematicians such as Karl Gauss, Nicolai Lobachevsky, Janos Bolyai and Bernhard Riemann. He showed that the different geometries are each associated with a separate 'collection' or 'group' of transformations. Seen in this way, the geometries could all be treated as individual members of one overall family, and from this very connection conclusions and inferences could be drawn. Klein ... demonstrated that every individual geometry could be constructed purely projectively; he produced projective models for Euclidean, elliptic, and hyperbolic geometries. Much later in his life, Klein returned to the Erlangen progamme to apply it to problems in theoretical physics, with special relevance to the theory of relativity" (Hutchinson DSB, p. 393). "For Klein, his Erlangen Program was an attempt to create an underlying unity for what had become the fragmented discipline of geometry. He did this through his innovative use of the group concept, which was not then widely known ... Mathematicians were accustomed to using transformations of figures, say to replace a figure with an equivalent but simpler one, or to choose more convenient coordinate axes. Klein shifted attention from the figures to the transformations, and argued that henceforth geometry should be about groups as well as the properties of shapes. So a geometric property was one that was invariant under all the operations of the group associated to that geometry. He specifically employed the idea of one group being a subgroup of another. This enabled him to fix a space but vary the group, either to introduce a new geometry or to recognise a known one in an unexpected setting ... Klein recognised that by selecting a figure in a space and considering the subgroup that maps that figure to itself was a fundamental way to inter-relate geometries and so to find a unifying principle that would encompass all of geometry ... Klein ended his Erlangen Program with a series of seven notes of varying length and significance. One is worth picking out. Note 5 referred to what Klein cautiously continued to call the 'so-called non-Euclidean geometry' in order to avoid debates with non-mathematicians. Non-Euclidean geometry was the subject of two important memoirs by Klein written on either side of the Erlangen Program which probably did more to convey the message of the Program than did his obscurely published pamphlet" (Gray, Worlds Out of Nothing (2011), pp. 235-6). RBH lists two copies, neither in the original printed wrappers.

"Klein chose the title to his essay carefully: he intended first to review, and then to compare, a number of recent researches in different areas of geometry. He claimed no novelty for the way he treated specific topics; what was original was the unified viewpoint he offered and its suggestions for the direction of future work. This viewpoint centred on the group-theoretic classification of the different geometrical methods then in use, and this emphasis owes a lot to Lie's influence. The Program was written while Klein was in daily contact with Lie, and reviewing its origins as he did when it was reprinted in his Collected Works he wrote that Lie was very much persuaded of the merits of the idea.

"It is easiest to understand this idea in its paradigm example: the way non-Euclidean geometry appears as a sub-geometry of projective geometry, done in an Appendix to the Erlangen Program. Klein had learned of non-Euclidean geometry from Otto Stolz when in Berlin, but initially had had a hard time understanding it, and then in persuading Weierstrass of the value of his new point of view.

"Inspired by a serendipitous reading of a paper by the English mathematician Arthur Cayley, who had had a glimmer of the same idea, Klein argued as follows. The model of non-Euclidean geometry developed by Eugenio Beltrami draws the entire non-Euclidean plane inside a circle, and it draws straight lines in non-Euclidean geometry as straight lines inside the circle. This suggested to Klein that the allowable transformations of figures in non-Euclidean geometry ought to be those which are projective transformations mapping the circle to itself, because they will automatically map each straight line to a straight line. Now, a non-Euclidean transformation is one that maps a line segment in non-Euclidean geometry to another line segment of equal non-Euclidean length. Projective geometry, on the other hand, can map any two points to any two points; the most important property of projective transformations is that they map four points on a line to four points on a line if and only if the four points have the same cross-ratio. For Klein's idea to work, he had to find a way of expressing non-Euclidean distance, which only involves two points, in terms of the four-point projective invariant, cross-ratio. He did this by observing that two points inside the circle define a non-Euclidean straight line that meets the circle in two more points, thus giving him four points. It remained for Klein to define non-Euclidean distance between the original two points in terms of the cross-ratio of the four points, and this he did by a straightforward technical argument.

"The upshot of all of this was that Klein had two geometries, projective geometry and non-Euclidean geometry, and each geometry had a family of allowable transformations. In fact, each of these families is what is technically called a group of transformations. Moreover, the space of non-Euclidean geometry is a subspace of projective geometry, consisting of the points inside a fixed but otherwise arbitrary conic (for convenience, a circle) and the transformation group of non-Euclidean geometry is a subgroup (in the technical sense of the term, to be defined below) of the transformation group of projective geometry. So, simply but accurately, non-Euclidean geometry could be called a sub-geometry of projective geometry.

"Klein began the Erlangen Program by alluding to the recently discovered relation between the metrical and projective properties of figures, but in the more natural case of Euclidean geometry. Then came the geometry, as he called it, of reciprocal radii vectores, today called inversive geometry, and then birational geometry, which will be defined below.

"Having introduced the main geometries, Klein then introduced the group concept. This was not widely known in 1872. His presentation, as he himself noted when the work was reprinted in the 1890s, was less than perfect: the only property of a group that he insisted upon was closure. Perhaps the simplest example of a group is the integers with the operation of addition. More generally, and in modern terms, a set of objects forms a group if:

closure holds: any two objects can be combined to form a third which also belongs to the group (in symbols A + B = C);
there is an identity object, often denoted e, with the property that the combination of any object with the identity returns that object (A + e = A);
every object, say A, in the group has an inverse, usually denoted −A, which is an object such that the combination of an object and its inverse is the identity (A + −A = e); and
the so-called associative law holds: A + (B + C) = (A + B) + C.

"A subgroup is a subset of a group that is also a group. The integers form a group because the sum of two integers is a group, the integer 0 is the identity element, the inverse of the integer n is the integer −n, and the associative law holds: k + (m + n) = (k + m) + n. In Euclidean geometry, the group of all distance-preserving transformations consists of the familiar rotations, translations, reflections and their composites.

"Klein may have skimped on the definition of a group because he did not do something later generations were to do, namely he did not distinguish between an abstract group and a transformation group. For groups whose elements are transformations it is, for example, automatic that the associative law holds, and it is usually easy to see if a set of transformations contains an identity element and an inverse for every element. But these imperfections should not obscure the magnitude of the point Klein was insisting upon. No-one had disputed for centuries that geometry, any geometry, involved the use of transformations, for example to replace a figure with an equivalent but simpler one, or to choose more convenient coordinate axes. Klein shifted mathematicians' attention from the figures to the transformations, and argued that geometry was about groups as well as the properties of shapes.

"In Sections 1-3 of the Erlangen Program, Klein discussed the use of the group concept, in particular the idea of one group being a subgroup of another. This enabled him to fix a space and vary the group, either to introduce a new geometry or to recognise a known one in an unexpected setting. This enabled him to distinguish usefully between the invariants of one group and another, and therefore between their geometrical properties, as the paradigm example of non-Euclidean geometry and projective geometry demonstrates. But Klein also pointed out that spaces could not only be of any dimension, but the points of one could be objects in another: one might study the space of all lines in a projective plane, or the space of all conics. This transition took one into what his contemporaries called higher geometry, which Klein had learned from Plücker and then greatly extended. The example of introducing imaginary elements (those described by complex coordinates) was mentioned explicitly, and motivated by the desire to bring geometry into line with algebra.

"In Section 4, Klein discussed a way in which two seemingly distinct geometries can be shown to be the same. This is the method of transfer, as he called it, which applies when there is an invertible transformation, t say, between two spaces A and A', with transformation groups B and B', respectively. The map between B and B' that sends an element b of the group B to the element tbt-1 (where t-1 is the inverse of t) in the group B' transfers the group acting on the space A to the space A'. If the action of the transferred group is identical with that of the group B' - for which the technical term is isomorphic - then the geometries could reasonably be said to be equivalent. Similarly, such a transfer can introduce a geometry onto a space by allowing one to transfer the action of a group on a space to the action on a new space ...

"Klein then discussed the transfer of the projective geometry of the plane onto a quadric by stereographic projection. As a good 19th-century geometer, he did not care if the spaces were real or complex: his example is easier for us to understand if the space is real and the quadric is a hyperboloid of one sheet. The transfer map picks as a centre of the projection a point on the hyperboloid, and maps the two straight lines of the hyperboloid through that point onto points at infinity in the image projective plane. Any other point of the hyperboloid is mapped onto the projective plane by the line joining it to the centre of projection. Thus, elementary plane projective geometry and the geometry of a quadric with a marked point are the same. In Section 5, the same process of transfer was illustrated in more elaborate spaces, and Klein carefully pointed out now that it is not enough that the spaces have the same dimensions; the group actions must also agree.

"In Section 6 Klein looked at inversive geometry. In the plane (or, respectively, space) this is the geometry of circles (respectively, spheres) in the plane, and the allowed transformations are those mapping circles to circles (respectively, spheres to spheres), which can also be characterised algebraically. The inversive geometry of the plane, he pointed out, is identical to projective geometry on a quadric, while the inversive geometry of space coincides with the projective geometry on a quadric in projective five-dimensional space.

"Section 7 of the Erlangen Program was devoted to mention of Lie's sphere geometry. This was an ingenious way of studying all spheres in three-dimensional space, by showing that the geometry was equivalent to the geometry of all lines in three-dimensional space. Lie's line-sphere transformation turned out to have implications for differential geometry that enabled both Klein and Lie to have interesting new results in the early 1870s. Section 8 was yet more ambitious; Klein indicated that one might hope for a birational geometry of space, where the transformations are quotients of polynomials; and for topology (which he called 'analysis situs'), where the transformations are invertible continuous maps. He even held out the prospect of a geometry of all invertible differentiable maps, which would map (in modern language) the tangent space of a surface to itself. This idea led him in Section 9 to the study of contact transformations, which was to become a major theme of Lie's work in future years.

"The 10th and final Section was devoted to some a few short remarks about other possibilities. One was the study of manifolds of constant curvature; another was derived from a comparison with the Galois theory of equations, leading to the suggestion that it would be fruitful to pass down a chain of subgroups.

"The Erlangen Program ended with a series of seven notes of varying length and significance. Note 5 was on the 'so-called non-Euclidean geometry', as Klein cautiously continued to call it in order to avoid debates with non-mathematicians. Note 6 discussed Klein's work on line geometry, and clarified the tricky point that the projective geometry of space with respect to a fixed quadric does not impose a geometry on the quadric itself. The final Note hinted at a theme of growing importance to Klein: the geometrical interpretation of the invariants associated to a binary form.

"The central message of the Erlangen Program is that every geometry is to be thought of as a space and a group of transformations acting on that space that preserve the essential features of that geometry. These might be metrical, as they are in Euclidean geometry or non-Euclidean geometry, the property of being a circle (inversive geometry in the plane) or of being a straight line (projective geometry). Two implications follow from this insight. One is that all the different geometries known at that date can be thought of as special cases in a hierarchy of geometries, thus re-unifying geometry, a subject that Klein thought had diversified too much. The other is that, by the principle of transfer, seemingly different geometries can be seen to be essentially the same, and a space can perhaps be given a geometry where previously it had none" (Gray, in Landmark Writings, pp. 546-9).

"In his later years Klein returned to the 'Erlanger Programm' and, in a series of works, showed how theoretical physics, and especially the theory of relativity, which had emerged in the meantime, can be understood on the basis of the ideas presented there. The 'Programm' was translated into six languages and guided much work undertaken in the following years" (DSB).

Parkinson, Breakthroughs, p. 381. Gray, 'Felix Klein's Erlangen Program, 'Comparative considerations of recent geometrical researches' (1872)', Chapter 42 in Landmark Writings in Western Mathematics 1640-1940 (Grattan-Guinness, ed.), 2005.

8vo (224 x 144 mm), pp. 48. Original printed wrappers (neatly rebacked, small repairs to upper inner and lower outer corner of front wrapper).