# Models of hyperbolic geometry

Wikipedia states the following:

[The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Riemann)

But it doesn't give any sources. I would like to know about the real history of the models of hyperbolic geometry, i.e. Beltrami-Klein, Poincaré disk model, Poincaré half-plane model. When and by whom where they discovered?

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The book by Saul Stahl (referenced to on the Wikipedia page) contains a chapter of about 7 or 8 pages on the history of non-Euclidean geometry at the end (I only skimmed it, and it looks quite nice). There's also a second edition. Also, you can find some of the original papers linked to on the Wikipedia page. –  t.b. Jul 12 '11 at 18:01
The book by Marvin Jay Greenberg has reached the status of an authorative source on the history of (non-)Euclidean geometry over the last twenty years. –  t.b. Jul 12 '11 at 18:07
I am slightly uneasy about that quote from Wiki, since I had the impression (tho' do not recall the source) that Beltrami used the "straight-line model" in the disk, while Poincare used the "arc-of-circle" model. Both versions do persist into higher-dimensional balls, the straight-line model corresponding to modeling balls in terms of orthogonal groups $O(n,1)$, and the arc-model on even-dimensional balls only, modeled by $U(n,1)$. I'd be interested to know the true history. –  paul garrett Jul 12 '11 at 18:16

## 2 Answers

The relevant papers by Beltrami, Klein and Poincare are all available in English translation with excellent introductions in Stillwell, Sources of Hyperbolic Geometry (AMS, 1996). Basically, both models can be found in germinal form in Beltrami's 1868 papers (where they are treated from the point of view of differential geometry) although Klein and Poincare added important insights (Klein from the point of view of projective geometry and Poincare from the point of view of complex analysis). The reference to Riemann in the Wikipedia quote is nonsense.

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Thanks for the illumination! –  paul garrett Jul 12 '11 at 20:28

Actually, one of the very few formulas in Riemann's Habilitation provides a metric $\sqrt{\sum dx^2}/(1+\alpha/4\sum x^2)$ which, for various values of $\alpha$, gives hyperbolic, spherical, and Euclidean metric.

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