Abstract
In two articles published in 1868 and 1869, Eugenio Beltrami provided three models in Euclidean plane (or space) for non-Euclidean geometry. Our main aim here is giving an extensive account of the two articles’ content. We will also try to understand how the way Beltrami, especially in the first article, develops his theory depends on a changing attitude with regards to the definition of surface. In the end, an example from contemporary mathematics shows how the boundary at infinity of the non-Euclidean plane, which Beltrami made intuitively and mathematically accessible in his models, made non-Euclidean geometry a natural tool in the study of functions defined on the real line (or on the circle).
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Notes
- 1.
In a letter to Genocchi in 1868 ([15] p. 578–579), Beltrami says that he had the manuscript of the Saggioready in 1867, but that, faced with criticism from Cremona, he had postponed its publication. After reading Riemann’s Habilitationschrift, he felt confident in submitting the article to the Neapolitan Giornale di matematiche, emended of a statement about three dimensional non-Euclidean geometry and with some integration “which I can hazard now, because substantially agreeing with some of Riemann’s ideas.”
- 2.
In another letter to Genocchi ([15] p. 588), Beltrami writes that this fact clearly follows from his Saggio, and that “in the note of Hoüel I do not find further elements to prove it”. Beltrami being generally rather unassuming about his own work, it is likely that he had already thought of this consequence of his model, but that he thought it prudent to leave it to state explicitely to the reader.
- 3.
Interestingly, the work of Saccheri, which had an indirect role in the development of non-Euclidean geometry and was then forgotten, was re-discovered by Beltrami [10].
- 4.
The obtuse hypothesis holds on a sphere, using geodesics (great circles) as straight lines; but on a sphere we do not have uniqueness of the geodesic through two points. This was considered to be a major problem by Beltrami, who was looking for a geometry in which all principles of Euclidean geometry hold true, but the uniqueness of parallels, but not for Riemann. In the Habilitationschrift, Riemann offered the sphere as a model for a geometry in which no parallel existed. Some of the principles used by Saccheri had to be abandoned: uniqueness of the line through two points, it seemed; but also the infinite extension of lines (Riemann seems to be the first to point out that the right geometric requirement is not that the straight lines have infinite length – what he calls “infinite extent of the line”, translated in “unboundedness” in modern netric space theory –, but that one finds no obstructions while following a straight line – a property he calls “unboundedness”, translated nowadays in “metrically complete and without boundary”). Clifford used the two-to-one covering of the real projective plane by the sphere to exhibit a geometry with positive constant curvature in which (1) there was just one line through two points; (2) space was homogeneous and isotropic; and (3) there are no distinct, parallel straight lines. Kelin realized the role of this model in the discussion of non-Euclidean geometries. Of course, in the projective-space model, often called Riemann’s non-Euclidean geometry, one not only has to give up the infinite extension of straight lines, but also the fact that a line divides the plane into two parts; or, which is about the same, orientability of the plane.
- 5.
This is true also for geodesics not in the form u = constand v = const. See the formulae at p. 381 in the Saggio.
- 6.
This amounts to say that, for each given value of Rthere is, but for isometries, exactly one non-Euclidean plane having “radius of curvature” equal to R(where Ris connected to the universal geometric constants envisioned by Lobachevsky, Bolyai, and Gauss).
- 7.
As a matter of fact, later Hilbert showed that no regular surface in Euclidean three space is isometric to the whole pseudosphere. Much later, Nash showed that any surface can be isometrically imbedded in a Euclidean space having dimension large enough.
- 8.
At this point only I disagree with Gray’s account of Beltrami’s work. He writes, p. 208 in [19]: “...but all Beltrami did was hint that the pseudosphere [the tractroid, in this paper’s terminology] must be cut open before there is any chance of a map between it and the disc [Beltrami’s projective model]”. In fact, Beltrami’s hint at p. 390 follows his formula (12), which can be used to find (after a cut) a surface of revolution in Euclidean space, what we call here S 3. Formula (14) at p. 393 leads to our S 2and, finally, formula (17) at p. 394 leads to the tractroid, our S 1below. The tractrix, called by Beltrami linea dalle tangenti costanti(constant tangent curve) is explicitly mentioned at p. 395. I find it indicative of Beltrami virtuosism, that he managed to exhibit three different surfaces of revolution by using the projective model, which is not especially amenable to this sort of calculations.
- 9.
The main reason for this is that, infinitesimal balls centered at the point (r, 0) in \({\mathbb{B}}_{{\mathbb{C}}^{2}}\)(0 < r < 1), have two (real) large directions and two (real) small directions. Let \(z = x + iy\): end consider (z, 0) in the small ball the x(normal) direction is small exactly as in the Beltrami–Klein model. Due to the holomorphic coupling of the variables in the metric (21), the (tangential) direction yis small as well. We have then two large and one small tangential directions, causing an interesting degeneracy of the metric in the limit.
- 10.
A similar remark is in Teoria, p. 427.
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Arcozzi, N. (2012). Beltrami’s Models of Non-Euclidean Geometry. In: Coen, S. (eds) Mathematicians in Bologna 1861–1960. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0227-7_1
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