# George Bryan's obituary of Eugenio Beltrami

Eugenio Beltrami died on 18 February 1900 and George Bryan published his obituary in the 12 April 1900 issue of

*Nature*. The full reference is G H Bryan, Eugenio Beltrami,*Nature***61**(1589), 568-569. We note that Bryan was a frequent visitor to Italy, a country he greatly loved. We give a version of the obituary below.**Eugenio Beltrami**

By the death, on 18 February, of Prof Eugenio Beltrami, after a long illness followed by an unsuccessful surgical operation, Italy has lost a mathematician who did much to bring his country to the forefront in the mathematical world almost simultaneously with the ascendancy of Italy in the world of politics.

Eugenio Beltrami was born at Cremona on 16 November 1835, of a well-known and highly-cultured Italian family. After completing his school curriculum in his native town, he went to Pavia, and then studied mathematics for three years under Brioschi. For some years Beltrami had to earn his own living, and an appointment in the Administration of the Italian Railways, which he held first at Verona, and then at Milan, if it afforded him no scope for his mathematical abilities, at any rate furnished him with the means of subsistence. At Milan, in 1860, Beltrami became acquainted with Cremona, whose influence, combined with a study of the works of Gauss, Lagrange and Riemann, opened the way for his development of higher geometry, in which branch of mathematics Beltrami published his first papers, in 1862, in the

*Annali di Matematica*.

In the same year he was appointed professor extraordinarius in algebra and analytical geometry at Bologna, and in the following year he became professor ordinarius of geodesy at Pisa, where he enjoyed the friendship of Riemann and Betti. In 1866, Beltrami returned to Bologna. where he occupied the chair in rational mechanics. Two years later appeared what has been aptly regarded as Beltrami's masterpiece, the "Saggio d'interpretazione della geometria non euclidea," published in the

*Giornale matematico di Napoli*. We learn that Beltrami's attention was first attracted to this subject by an observation of Lagrange on maps, in which geodesics are represented on a plane by straight lines, and was thus led to consider the properties of surfaces on which the geodesics are represented by linear equations in curvilinear coordinates. Beltrami found that such surfaces were the same as surfaces of constant curvature.

He was thus led to examine the properties of the surface of constant negative curvature, to which he gave the name of pseudosphere, and the geometry of such a surface was found to be identical with the geometry of Gauss and Lobachevsky. As his old pupil and successor at Pavia, Prof Carlo Somigliana, remarks, "It can thus be said that although the germs of his results can be traced back to some of his predecessors, and, in particular, can be found in the profound considerations of Riemann, and other advances have come subsequently, yet his work represents and synthesises the most decisive step that has been made in modern times by the geometric conception of real space."

Nor was the "Saggio d'interpretazione" by any means Beltrami's only contribution to mathematical literature at the period under consideration. We find him extending the properties of surfaces of constant curvature to n dimensional space; and his papers on differential parameters, on the flexure of ruled surfaces, and on the general theory of surfaces, published a few years previously to the "Saggio," are well known to mathematicians.

In 1873, Beltrami migrated to Rome as professor of rational dynamics and higher analysis, and was elected a Fellow of the Italian equivalent of our Royal Society, the Reale Accademia dei Lincei. His sojourn in Rome was of brief duration; for, much to the regret of his friends there, he went to Pavia in 1876, where he lectured on mathematical physics and higher mechanics, and it was not until 1891 that an opportunity offered itself for him to return to Rome. It was only two years ago that Beltrami was prevailed on to accept the office of President of the "Lincei," and last year he was unanimously elected to the senatorial rank. As a general rule, however, he avoided all public appointments, and the only other post he held was on the Italian Council of Education. He preferred to devote his entire energies to the studies in which he was interested, and sought no scientific distinctions; still, the laurels which he had well earned were freely showered on him by the academies of Bologna, Lombardy, Turin, Naples, Paris, GĂ¶ttingen, Brussels, Munich and Berlin; and the London Mathematical Society was also proud to place his name on its list of foreign mathematicians.

We have hitherto spoken chiefly of Beltrami's work as a pure mathematician, but his later investigations tended more especially in the direction of applied mathematics. Hydrodynamics, theory of potential, elasticity, physical optics, electricity and magnetism, conduction of heat and thermodynamics were all made the subject of papers, each of which "shed a bright light on some difficult or controversial point." In the theory of the potential considerable simplifications of method were made, and the papers on potentials of symmetric distributions and on the attractions of ellipsoids are described by Somigliana as "true models of classical elegance." In the theory of elasticity, LamĂ©'s equations were shown to be intimately related to the euclideity of space, and the generalisations for spaces of constant curvature opened up a new field for research, of which Beltrami endeavoured to make use in accounting for the uncertainties in Maxwell's theory, which substitutes action in a continuous medium for action at a distance.

The last period of his researches was devoted to developing Maxwell's theories of electro-magnetic phenomena, a difficult task, for which Beltrami's mathematical knowledge well fitted him. All who have read Maxwell's treatise realise that it contains many obscure points and demonstrations of hardly a rigorous nature, and most of those who have failed to follow his arguments have preferred to regard the results as statements of Maxwell's views, rather than inquire into the validity of the reasoning on which they were based. Beltrami, on the other hand, being well versed in the art of exact expression and the elegances of neatness of analytical form, was not contented with Maxwell's rough-and-ready methods, but devoted long hours of deep thought to coordinating and perfecting the ideas which he regarded as incomplete. Among his latest contributions to the

*Atti dei Lincei*we notice a paper on thermodynamic potentials published in 1895.

As a professor, Beltrami's lectures are said to have been characterised by the same perfection of style and exactness of form which are so conspicuous in his writings. His genial manner and high culture made him a centre no less in general society than in the scientific world. Shakespeare's epithet, "Cunning in music and in mathematics" well applies to Beltrami, and we learn from Signor Pietro Cassani's obituary address to the Venetian Academy, that having been taught music in his early clays by his mother, and afterwards under Ponchielli, he would often delight his friends by his renderings on the piano of the masterpieces of Bach, Mendelssohn and Schumann.

The life that has been brought to such a sad close must have been in many respects an ideal life. Beltrami had every opportunity for devoting himself to the studies which he chose as his life's work; he knew nothing of rivalries and petty jealousies, as he made no enemies; but, on the other hand, we cannot but suppose that his experience of the necessities of making the best of somewhat uncongenial surroundings during his years of railway work had a beneficial influence on his after life, in preventing Beltrami from attempting to live up to a false ideal. His loss adds another to the many gaps in the mathematical world, but his published works form a fitting memorial of their author, and several of them bid fair to be handed down to posterity among the mathematical classics.

We are indebted to Prof Blaserna, of Rome, for much valuable information on which this account is based.

Last Updated January 2021