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?
Best Answer
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:
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
Greenberg's Euclidean and Non-Euclidean Geometries states that
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:
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 $\TeX$):
By Stillwell's translation:
This in my opinion supports Arcozzi's view on how this term was used.