Eugenio Beltrami

Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant curvature, the pseudosphere, and in the interior of an n-dimensional unit sphere, the so-called Beltrami–Klein model. He also developed singular value decomposition for matrices, which has been subsequently rediscovered several times. Beltrami's use of differential calculus for problems of mathematical physics indirectly influenced development of tensor calculus by Gregorio Ricci-Curbastro and Tullio Levi-Civita.

Life

Beltrami was born in 1835 in

In the second memoir published during the same year (1868), "Fundamental theory of spaces of constant curvature", Beltrami continued this logic and gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension. He accomplished this by introducing several models of non-Euclidean geometry that are now known as the Beltrami–Klein model, the Poincaré disk model, and the Poincaré half-plane model, together with transformations that relate them. For the half-plane model, Beltrami cited a note by Joseph Liouville in the treatise of Gaspard Monge on differential geometry. Beltrami also showed that n-dimensional Euclidean geometry is realized on a horosphere of the (n + 1)-dimensional hyperbolic space, so the logical relation between consistency of the Euclidean and the non-Euclidean geometries is symmetric. Beltrami acknowledged the influence of Bernhard Riemann's groundbreaking Habilitation lecture "On the hypotheses on which geometry is based" (1854; published posthumously in 1868).

Although today Beltrami's "Essay" is recognized as very important for the development of non-Euclidean geometry, the reception at the time was less enthusiastic. Luigi Cremona objected to perceived circular reasoning, which even forced Beltrami to delay the publication of the "Essay" by one year. Subsequently, Felix Klein failed to acknowledge Beltrami's priority in construction of the projective disk model of the non-Euclidean geometry. This reaction can be attributed in part to the novelty of Beltrami's reasoning, which was similar to the ideas of Riemann concerning abstract manifolds. J. Hoüel published Beltrami's proof in his French translation of works of Lobachevsky and Bolyai.

Works

• Beltrami, Eugenio (1868). "Saggio di interpretazione della geometria non-euclidea". Giornale di Matematiche. 6: 284–312.
• Beltrami, Eugenio (1868). "Teoria fondamentale degli spazii di curvatura costante". Annali di Matematica Pura ed Applicata. Series II. 2: 232–255.
  S2CID 120773141
  .
• Sulla teoria dell'induzione magnetica secondo Poisson (in Italian). Bologna. 1884.{{cite book}}: CS1 maint: location missing publisher (link)
• Opere matematiche di Eugenio Beltrami pubblicate per cura della Facoltà di scienze della r. Università di Roma (volumes 1–2) (U. Hoepli, Milano, 1902–1920)^[1]
• Same edition, vols. 1–4

Notes