TITLE: How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?
QUESTION [17 upvotes]: 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?

REPLY [6 votes]: 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.