This is in response to comments and the answer by user studiosus to this question:

As for Beltrami's work: Consistency of a geometry from (post) Hilbert viewpoint has nothing to do with existence of an (isometric) embedding in a particular space. For instance, for a Riemannian manifold to exist, it suffices to define it via an atlas of charts together with a Riemannian metric tensor. What Beltrami did was to embed isometrically a proper open subset of the hyperbolic plane in $R^3$. Since a proper open subset of $H^2$ violates axioms of hyperbolic geometry (it is not homogeneous!), existence of such an embedding does not prove consistency of hyperbolic geometry.

I have Stillwell's translation of Beltrami's papers with me and as I understand them, Beltrami in his first paper uses differential geometry to show that the surface of the pseudosphere admits hyperbolic two-dimensional geometry as described synthetically by Lobachevsky, in his second paper he generalizes the result of the first paper to other dimensions. Beltrami does not make claims about consistency (and especially not that hyperbolic geometry was "as consistent" as Euclidean geometry, since the latter was unquestioned at the time) but many authors claim that by showing that a surface in Euclidean space admits a part of the hyperbolic plane, Beltrami shows the consistency of hyperbolic geometry. For example:

In the first of the two papers published that year Beltrami pointed out that the trigonometry of the geodesics of the pseudosphere [...] was identical with the trigonometry of the hyperbolic plane. Consequently any self-contradiction that might arise in hyperbolic geometry would of necessity also constitute a self-contradiction of Euclidean geometry. In other words, Beltrami proved that that hyperbolic geometry was just as consistent as Euclidean geometry.

$-$ Saul Stahl, The Poincaré Half-plane: A Gateway to Modern Geometry

I have seen many other authors make similar claims. Is this an incorrect interpretation? Can someone clarify?

  • $\begingroup$ Sid: As I said before, I think that the argument given by Stahl (and others) is completely mistaken (hyperbolic geometry does not reduce to trigonometry!), it is as if he (Stahl) missed all the post-Hilbert mathematics. To reiterate: embedding a piece of a hyperbolic plane in $R^3$ is neither necessary nor sufficient in order to establish that consistency of hyperbolic geometry reduces to the one for Euclidean geometry. $\endgroup$ Commented Feb 6, 2014 at 14:04
  • $\begingroup$ @studiosus I am writing a term paper on hyperbolic geometry and I would like to discuss this apparant misunderstanding by modern authors. Do you know of any sources that discuss how exactly Beltrami managed to show the consistency of hyperbolic geometry, if it was not by embedding a piece of the hyperbolic plane in $R^3$? $\endgroup$
    – Sid
    Commented Feb 6, 2014 at 14:11

3 Answers 3


Beltrami's Models

Beltrami's Models of Non-Euclidean Geometry by Nicola Arcozzi might be of interest. It does not start with the pseudosphere in the sense of the tractricoid, which is a finite surface of constant negative curvature embedded into three-dimensional Euclidean space. Instead, it describes planar models, one of which Arcozzi calls the projective model but which is known to me as the Beltrami-Klein model.

Quoting Arcozzi:

Beltrami calls psedudospheres the surfaces bijectively parametrized by the coordinates $(u,v)$ and endowed with the metric (1) (below, we will use idifferently [sic] psedosphere, hyperbolic plane, non-Euclidean plane)

So even though Beltrami started describing the tractricoid (and other surfaces of revolution with constant negative curvature, iirc), here he apparently is using the term in a different meaning, and keeping these two meanings apart is important.

So what Beltrami did was come up with a model: a way to translate terms of the axioms into geometric representations. Namely a hyperbolic point shall be modeled by a point inside a given disc (or other conic, at least in Klein's version), and a hyperbolic line shall be modeled by a segment of that disc. He also redefines metric, in particular he defines lengths (this is the equation (1) the above quotation refers to). He then shows that this model has all the properties of hyperbolic geometry.

So if his Euclidean geometry is consistent, then his model works, therefore it has the properties it demonstrate it should have, therefore hyperbolic geometry is consistent. Or formulated the other way round, if there was a problem with hyperbolic geometry, then there would be some problematic configuration in this model, and since he deduced that proper Euclidean geometry can not cause any such problems, this would imply that there can be no proper i.e. consistent Euclidean geometry either.


Greenberg's Euclidean and Non-Euclidean Geometries states that

Beltrami proved the relative consistency of hyperbolic geometry in 1868 using differential geometry (see The Pseudosphere, Chapter 10).

At first I read this as supporting your claim that Beltrami did use the tractricoid directly to prove that consistency. But reading that chapter 10, I'm not so sure any more. It starts by mentioning that the hyperbolic plane cannot be isometrically embedded, but a portion of it can.

So I guess that Beltrami might have recognized that he can carry the metric of the tractricoid over to a portion of the plane, and then extend it to the whole disk in the consistent way expressed by that equation (1) in Arcozzi's text. So the tractricoid would serve as a tool to demonstrate that the metric he chose is sane and relates to constant negative curvature, but the hyperbolic plane used for the consistency proof goes beyond the tractricoid.

Arcozzi again

Reading more of Arcozzi suggests a different interpretation, though:

On the other hand, he seems to worry that the surface with the metric (1) might not be considered wholly “real”, since it is not clear in which relation it stands with respect to Euclidean three space (the strictest measure of “reality”). Then, he will show that, after cutting pieces of it, the pseudosphere can be isometrically folded onto a “real” constant curvature surface in Euclidean space.

However, reading even further, one finds section 3.3 where Arcozzi speculates on how Beltrami might have thought of his geometry. The image presented there (at least the way I understand it) is more that of a 3D curved surface, quite like the tractricoid, rather than of a flat surface equipped with some strange artificial metric computation. However, due to differences in how things were done at the time, self-intersection apparently was little concern, and similar for the fact that only a limited portion was representable this way. Particularly after Beltrami had demonstrated the possibility of isometric motions.

Beltrami himself

Skimming Beltrami's Saggio di interpretazione della geometria Non-Euclidea myself, I recognize that equation for the distance element of the Beltrami-Klein model. It is indeed numbered (1) in his work as well. At first glance I see no reference to the pseudosphere at all, only references to constant negative curvature. I don't speak Italian, but here is what it has to say about pseudospheres (this is from a different version which lacks illustrations but was typeset in $\TeX$):

Per evitare circonlocuzioni ci permettiamo di denominare pseudosferiche le superficie di curvatura costante negativa, e di conservare il nome raggio alla costante $R$ da cui dipende il valore della loro curvatura.

By Stillwell's translation:

To avoid circumlocution, we call the surface of constant negative curvature pseudospherical, and we retain the term radius for the constant $R$ on which its curvature depends.

This in my opinion supports Arcozzi's view on how this term was used.

  • $\begingroup$ I guess that settles it: (some) modern authors are mistaken to think that realizing hyperbolic geometry on the tractricoid proves anything by itself, and the model that Arcozzi is referring is what turned out to be useful to prove consistency. Regarding the psuedosphere, there is an English translation of Beltrami's papers by Stillwell, in his introduction he says: "The surface we call the psuedosphere is not given a name by Beltrami. It is only a special case of what he calls a 'psuedospherical surface,' an arbitrary surface of constant negative curvature [...]". $\endgroup$
    – Sid
    Commented Feb 6, 2014 at 14:57
  • $\begingroup$ @Sid: that sounds familiar. I have a statement to that effect in some unpublished text of mine, with a reference to Stillwell, but I don't have that book at hand just now. If you do, can you edit my answer to include Stillwell's translation of that sentence from the Saggio, instead of the automatic one by Google? It should be somewhere after Eq. (8). $\endgroup$
    – MvG
    Commented Feb 6, 2014 at 15:08
  • $\begingroup$ I found the passage and did so. Google's translation was surprisingly accurate. $\endgroup$
    – Sid
    Commented Feb 6, 2014 at 15:19
  • $\begingroup$ MvG, I posted another question about hyperbolic geometry: math.stackexchange.com/questions/669553/… I just wanted to show you this in case you know of any references and since you have given such extensive and good answers to my questions in the past. $\endgroup$
    – Sid
    Commented Feb 9, 2014 at 13:41

There are at least four relevant articles of Beltrami. In the first one (Ann. Mat. Pura App.I (1865), no. 7, 185-204) he computes all two dimensional Riemannian metrics for which a coordinate system exists where all geodesics are represented as straight lines: he finds out that only surfaces of constant curvature have this property. I have little doubts that at this point he had the expression for the non-Euclidean metric, as it is in the disc "Klein model". In 1867 he had written "Delle variabili complesse sopra una superficie qualunque", in which he shows that all surfaces have a complex structure (they all are locally conformal to Euclidean plane). Many calculations and coordinate charts we find in the articles on non-Euclidean geometry, probably owe much to this paper. Then came "Saggio di interpretazione della geometria non-euclidea", where the Klein model is introduced at the very beginning of that article, as equation (1). In the Nota II of that paper, Beltrami shows that the disc is homogeneous and isotropic w.r.t. the metric; he has already shown that there is exactly one geodesic connecting two distinct points; it is obvious that a geodesic splits the disc into two parts. These properties are enough, for us, to identify Beltrami's model with non-Euclidean geometry. They were enough for him, too, although he lacked the language of modern mathematical logics, in which this can be precisely stated. There remains the residual possibility that some geometric property which is true in the model, might not be decidable in non-Euclidean geometry (a problem of completeness). Also, Beltrami proves in his model a number of Theorems of non-Euclidean geometry, as to convince the reader that, the axioms being satisfied, their consequences are, as well. Of course, from our viewpoint, the problem of completeness would affect Euclidean geometry in equal measure, and there is would be no need to mention that given the axioms, theorems follow no matter what the model is. But in the 1860's it was difficult to find words and concepts to express this.

In my opinion, Beltrami was not at this stage completely satisfied by having just an analytic object with a not wholly certain geometric interpretation. This is why he locally identifies his disc model with a surface in three-dimensional Euclidean space. He does so (and we know that it can't be done otherwise) in three different ways, obtaining three different (locally isometric) surfaces in three-space. One of them is the tractroid. But he was wholly aware that the surfaces only encoded local information, and were not enough to provide a model for the global axioms of non-Euclidean geometry (for which he had, in fact, a very convincing analytic model).

Finally, in 1868 "Teoria fondamentale degli spazii di curvatura costante", Beltrami, after having read Riemann's habilitation essay, in which the viewpoint on geometry is turned upside-down (the distinction between "purely analytic" and "geometric" is totally erased), Beltrami introduces several different models, in all dimensions. It is easy to guess that some of these models he already had on his desk, and that he published them now, because he felt philosophically backed by Riemann's essay.


My understanding was that Beltrami also originated the upper-half-plane model of the hyperbolic plane, probably in the same paper with an upper-half-space model for $\mathbb H^3$ and so on. The upper half plane is very convincing for me, you get easy parametrizations for geodesics, either $(A, e^t)$ or $(A + B \tanh t, B \operatorname{sech} t)$ for $B > 0.$ This gives an acceptably easy way to find the distance between two points. Circles are geodesic circles, although the geodesic center is not quite where you expect. Anyway, http://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model I haven't done this, but an arc-length parametrization of a circle could be found by mapping to the Poincare disc (take the original center as $i,$ map that to $0,$ use sine and cosine, then map back).

Here we go, a constant speed, but not unit speed, parametrization of a geodesic circle in the upper half plane of $\mathbb C,$ with geodesic center $i,$ is $$ \frac{2 B \cos t + i (1 - B^2)}{1 - 2 B \sin t + B^2}, $$ with $0 < B < 1. $ There is a diameter along the imaginary axis, the center is $i,$ while the two ends of that diameter are $$ i \, \left( \frac{1+B}{1-B} \right) $$ and $$ i \, \left( \frac{1-B}{1+B} \right), $$ with the imaginary parts evidently being reciprocals.

Knowing the actual circumference would give us a constant with which to adjust the variable $t$ I use above, so as to get unit speed. With geodesic radius $ r =\log (1+B) - \log (1-B)$ and circumference $2 \pi \sinh r,$ I get circumference $$ \left( \frac{4 \pi B}{1-B^2} \right) $$


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