By Dr George Bruce Halsted.
The great vindicator and interpreter of non-Euclidean geometry, Eugenio Beltrami like its two creators János Bolyai and Nikolai Lobachevsky, was noted as a student for his insubordination.
Beltrami was born at Cremona on November 16th, 1835, and attended the elementary schools, gymnasium and lyceum of that city, except for the scholastic year 1848-49 in which he was at that gymnasium of Venice which now bears the name of Marco Polo.
Having finished the lyceal studies in the summer of 1853, in the following November he inscribed himself as student in the Mathematical Faculty of the University of Pavia, after having obtained there a place on the Castiglioni Foundation in the Collegio Ghislieri (founded 1567 to support students at the University of Pavia).
But in the succeeding year, accused of having promoted disorders against the Abbot Leonardi, rector of this college, he was expelled from it, together with five others of his college mates. Thus, like Lobachevsky, not even being a charity beneficiary could restrain his irrepressible independence.
This expulsion worked a terrible hardship and disappointment on the ambitious youth. As Gino Loria says in his notice of Beltrami, on which we draw here, "This measure - perhaps not absolutely without grounds, but certainly too rigorous - had disastrous consequences for its victim."
For, though the University was still open to him, and though he had the great good fortune to attend certain lectures of Francesco Brioschi, yet the poverty of his family constrained him to return home before having passed the examination which precedes the doctorate. In November, 1856, he went to Verona where he had obtained the humble position of secretary to the engineer Diday in the government service of Lombardy-Venice. Here he remained until the 10th of January, 1857, when for political reasons he was bruskly dismissed by the director-general, Busche.
Fortunately the annexation of Lombardy to Piedmont, happening soon after, allowed Diday to transfer his office to Milan, taking his secretary with him. At Milan Beltrami undertook all over again his mathematical education. Here he was so fortunate as to have access again to his former professor, Brioschi, and also to Luigi Cremona.
The friendship of these great men was of decisive influence for his life, opening to him at a stroke the very career for which he had longed. Thus he who for lack of a degree had seen himself disbarred from the secondary schools and from the corps of military engineers, was, on the basis of his publications in the Annali di Matematica, named (18 October 1862) "Professore straordinario" in the University of Bologna.
It is to the honour of Cremona to have suggested this appointment, and of Brioschi, then secretary-general to the Minister of Public Instruction, to have adopted the suggestion. By it Beltrami was at one stroke liberated from the shackles of a humble administrative occupation and put in position to consecrate himself wholly to the genial occupation for which he was by nature so pre-eminently fitted.
The very next year, on the proposal of Enrico Betti, he was offered the professorship of geodesy in the University of Pisa. From Pisa he returned to Bologna in September, 1866, as professor of rational mechanics.
But it was ideas of a geodetic character which gave that turn to his creative genius destined to be crowned with such brilliant fame. In the exordium of a memoir dated Pisa, 31 May 1866, Beltrami remarks that in treating of a map destined to serve for measurements of distance it would be most convenient to determine, that to the geodetics of the surface should correspond the straights of the plane, because, such a representation accomplished, the questions concerning geodetic triangles would be reduced to simple questions of plane trigonometry. He concludes that "the only surfaces capable of being represented on a plane so that to every point corresponds a point and to every geodetic a straight are those whose curvature is everywhere constant."
Now at the very time that Beltrami was working on surfaces of constant curvature, Heinrich Richard Baltzer (1818-1887) in Germany, Jules Hoüel in France and Giuseppe Battaglini in Italy had set to work to diffuse the revolutionary ideas of Bolyai and Lobachevsky, while Dedekind published the "Habilitationsvorlesung" of Riemann: "On the hypotheses which lie at the basis of geometry."
Of this Beltrami made an annotated translation of which he speaks in two letters to Angelo Genocchi:
... The manuscript of the translation of a posthumous memoir of Riemann, most interesting for the extreme importance and vastness of its subject, to which, it is necessary to say, the brevity of the development is little adequate. As to my translation I ought to mention in the first place that I have tried to be most faithful to the text, for fear that doing otherwise I might alter the true sense of a composition which in many points gives place for doubts of interpretation. At the end of the translation I have inserted some annotations, made currenti calamo and for internal use (employing the phrase of the apothecaries), that is to say destined properly for myself alone. Hence such notes either should not be read, or not without the greatest indulgence, because most of them are rather indications destined to serve me as guides to call attention to those points that occasioned me the most difficulty, than comments properly so called and definitive. There are then very many other points which merit elucidation. You will find in two of my annotations certain allusions to a mode of mine of interpreting the results of the non-Euclidean geometry. If you wish any hint about this you might turn to Professor Cremona, who has had in his hands a manuscript of mine containing the complete real construction of the planimetry of Lobachevsky. The ground of the researches of Lobachevsky lies beyond doubt in the doctrine touched by Riemann. I intend to publish some of my studies on this subject, encouraged by the support which they find in the work of Riemann, only, lately come to my cognizance."
"Bologna, 23 July, 1868.
The past year, when no one knew of this fundamental work of Riemann, I had communicated to Cremona a paper of mine in which I gave an interpretation of the non-Euclidean planimetry, which seemed to me satisfactory. Cremona did not judge differently. At the end I had risked a judgment on the non-Euclidean stereometry, which now I do not think correct. But the article, freed from this slip, will appear in the Giornale of Naples, in its original form, save certain additions that I can hazard now, because substantially concordant with some of the ideas of Riemann."
Its genial idea, for which Beltrami had been prepared by his researches on surfaces of constant curvature, an irresistible idea to silence opposers of the non-Euclidean geometry, was to present the example of a surface, regular as the plane and the sphere, in which the lines corresponding to the straights of the plane and to the great circles of the sphere, that is the geodetics, comported themselves as the straights of the non-Euclidean plane. This was a surface of constant negative curvature, a pseudo-sphere.
But very soon after the publication of this memoir, Hermann Helmholtz and Felix Klein expressed grave doubts on certain points of Beltrami's reasoning and mathematics, saying that he had not made certain the existence of surfaces of the type to which he has recourse to represent, without changing its nature, the new system of geometry. It seems that Beltrami tried to dissipate these doubts. At least it seems that to such attempts we owe a memoir where is studied with scrupulous care the surface generated by the rotation of the tractrix about its asymptote with the aim of deducing the elements by a construction simple and exact of the surface itself.
The end aimed at not having been reached, Genocchi repeated the objections in a way still more particularized and energetic, maintaining that it had not been demonstrated that the partial differential equation, characteristic of the surface of constant negative curvature, admits at least one integral satisfying all the conditions imposed on the pseudo-sphere to serve the representation of Beltrami.
There is no trace of response from Beltrami, not even in his correspondence with Genocchi. Perhaps he perceived that unfortunately those objections were well founded.
But the final establishment of that fact, the complete demonstration of the non-existence of regular surfaces of constant negative curvature in all their extent, was not accomplished until after the death of Beltrami, by David Hilbert in 1901.
The representation imagined by Beltrami is therefore sufficiently limited, and the beautiful edifice, if it does not fall to the ground, shows itself in solidity and extension less than what was believed. Fortunately the non-Euclidean geometry is now completely vindicated in many simpler ways. Moreover Beltrami's pseudo-sphere had always done harm to the many people who took up the false idea that geodesic geometry on the pseudo-sphere was Bolyai's non-Euclidean geometry, instead of only being an interesting representation of it in Euclidean space; just as Bolyai's geometry of geodesies on a limit surface in Lobachevsky's space is a representation of Euclidean geometry in Bolyai space.
From this epoch (1868) the interest of Beltrami in non-Euclidean geometry never flagged.
He made an application of the expression given by Lobachevsky for the
angle of parallelism. In his correspondence with Genocchi he points out the flaw in the argumentation proposed by Jules Carton [Vrais principes de la géométrie euclidienne et preuves de l'impossibilité de la géométrie non euclidienne] for demonstrating the postulate of Euclid, presented by Bertrand to the Institute of France December 20, 1869, he discusses the enquiries undertaken to determine who was Ferdinand Karl Schweikart (1780-1859), now so well known as an independent creator of the non-Euclidean geometry, and he gives an appreciation of Hoüel's article in "Sur l'impossibilité de démontrer par une construction plane le postulatum d'Euclide." A public proof of this interest is also the charming communication made to the Accademia dei Lincei March 17, 1889, to present in the proper light the work, whose value and significance were then unknown, of Saceheri, "Euclides ab omni naevo vindicatus."
Beltrami never ceased to meditate on the non-Euclidean geometry even when concentrating all his powers to the study of natural phenomena. A proof of this is his discovery that the general equation of elasticity is bound to the Euclidean postulate. Moreover one of his gifted disciples observes "show he shows, in a certain passage, that he had turned his attention to the way in which physics would be able to profit from hypotheses of a diverse geometric nature of space, a difficult conception, more explicitly advanced by William Clifford; nor was he ever able to lose from view those curved spaces, with which he had commenced so triumphantly."
Settled finally at Rome, member of the most celebrated scientific societies of the world, successor to Brioschi as President of the Accademia dei Lincei, Senator of the Realm of Italy, many times chosen by public vote to sit in the Council Superior of Public Instruction, acclaimed master by the entire body of scientists, happy with a devoted wife, yet from 1896 he was undermined by a mysterious malady, and died February 18th, 1900.