*Bollettino dell'Unione Matematica Italiana*(3)

**12**(4) (1957), 727-730. We present below an English translation by Fiona Spencer (a University of St Andrews student) of this obituary:

**Ugo Amaldi**

On Monday 11 November, in the early hours of the morning, Ugo Amaldi Emeritus professor of the science faculty of the University of Rome, died in Rome, in his home.

Born in Verona on 18 April 1875, he completed his university studies in Bologna, where he was a pupil of Federigo Enriques, of Cesare Arzelà and, above all, of Salvatore Pincherle, who guided him in his first research in Mathematical Analysis. Graduating in 1898 with a thesis on the Laplace transformation, in the same year he attained the ability to teach mathematics in secondary schools. Between his first publications emerged the treatise, that took time, entitled: *Le operazioni distributive e le loro applicazioni all'Analisi *(Bologna, 1901) (Distributive operations and their applications in Analysis), written in collaboration with Pincherle. In 1902 he became an assistant lecturer in Complementary Algebra and Analytical Geometry, and in 1903 he won the national Algebra & Analytical Geometry competition for a professorship at the University of Cagliari where he taught for two years. He transferred to Modena in 1906, where he was a professor in Analytical and Projective Geometry (until 1919), then in Padua he was a professor of Descriptive Geometry with Applications (1919-1922) and of Analytical Geometry (1922-1924). Finally in Rome, he was a professor of Mathematical Analysis and Analytic Geometry in the Faculty of Architecture (1924-1942) and of Algebraic and Infinitesimal Mathematical Analysis in the Faculty of Science (1942-1950).

Among the Italian analysts, Ugo Amaldi was without doubt the most profound expert and cultivator of the theory of continuous groups of transformations, to which he dedicated, without interruption, almost twenty years of study. His research in this field was primarily published in the *Mathematical Journal of Battaglini* (1901), in the *Essays of the Turin Academy of Science* (1905-1906) and in the *Documents and Essays of the Academy of Modena* (1906-1913). They were found outstanding and culminated in an awarding ceremony in 1917 at the National Academy of Sciences of Italy (known as the XL), which awarded him the gold medal for Mathematics in the following year. Such research connects directly to the fundamentals of Lie theory and makes a remarkable contribution to the resolution (today still unsolved) of the problem of determining and classifying all the continuous groups of transformations, that may be finite or infinite. This problem has been resolved by Sophus Lie only in the cases of the straight line and the plane, with the characterisation of particular classes of *types *of groups. Fundamentally, in Lie's theory, was the result that the finite irreducible groups of contact transformations of the plane fall into only three types, represented by the group ∞^{10} of the contact transformations that transform circles into circles, and from two subgroups of this group ∞^{10}, respectively ∞^{7} and ∞^{6}.

For much of three-dimensional space, Lie had only begun a classification, limited to vaguely claiming to have "in principle" a solution to the "exceedingly difficult problem" of determining all the finite groups of transformations, and that under this illusion, completing this, would only leave "some calculations of detail" to be done.

Subsequent work by the aforementioned Lie, of Scheffers, of Oseen and of Kowalewski, then demonstrated that the difficulties were still far greater than those that had first appeared to Lie, especially those concerning infinite continuous groups. However Amaldi succeeded in determining all the types of actually existing continuous groups of transformations, whether they be finite or infinite, not only in S_{3}, but also in S_{4}.

Of acute and versatile talent, Amaldi was able to bring to the study of mathematics all the richness of an exceptionally high culture, far wider than the boundaries of the pure sciences. His gifts emerge in every page of the works in which he collaborated but especially in the numerous lithographed courses of his lectures, in which, in a fluid and brilliant style, his unsurpassed teaching art is demonstrated. In his lectures, in fact, Amaldi knew how to balance, with fine psychological intuition, the technique of analytical tools, the deepening of the philosophical significance of the concepts and the interpretation of their historical origin. And I don't believe that a university professor has ever earned more admiration from their students. All unanimously always recognised in him the happiest fusion of all the true gifts of a great teacher, the clarity and precision of speech, the enthusiasm and fervency of his conviction, the breadth and depth of the art, the sense of duty, and the sense of justice. But, perhaps, more than the latter, they admired in him his sense of equality, because justice was never seen by him in a formal or cold or abstract way, instead it was always enlivened with a warm human understanding.

Among those courses meriting special mention are those of *Lezioni di Analisi matematica, algebrica e infinitesimale* (Lectures in Mathematical, Algebraic and Infinitesimal Analysis), a real jewel of simplicity and elegance, and *Introduzione alla Teoria dei gruppi infiniti di transformazioni* (Introduction to the theory of infinite groups of transformations), in which the seminars he held at the National Institute of Advanced Mathematics in Rome between 1942-1944 are summarised. But where Amaldi showed himself a distinguished Master of Mathematical Pedagogy, are in the numerous texts for secondary schools, above all those of geometry, written in collaboration with Enriques. Those texts are destined to remain for a long time as a model of their kind, both in Italy and abroad. Every page of these texts was at length, deeply mediated with respect to the psychology of young people, every argument was thoroughly covered and the most interesting aspects highlighted, in particular for students of licei classici [secondary schools that concentrate their teaching in classical arts subjects]. No one has since done this better than Amaldi, a passionate connoisseur of classical culture, who knew how to relate the ideas, in an educationally inspiring way, to their historical origin that of Greek geometrical thinking. And that the struggle was the fusion of mentalities, that's distant, that's diverse, of the two distinguished collaborators! From Enriques' volcanic mind flowed, as just outlined, the scientific ideas and the most original and daring speeches, but he did not write anything, everything had to come to be realised, put into consideration of the teaching experience, elaborated with the greatest love, tweaked down to the most minute details by his colleague and friend.

With Enriques he collaborated and published, in subsequent editions (beginning from 1900), the *Questioni riguardanti le matematiche elementari (*Questions concerning elementary mathematics), the articles *Sui concetti di retta e di piano *(On the concepts of the straight-line and the plane) and *Sulla teoria dell'equivalenza* (On the equivalence theory). But his greatest work, as an essayist, is the famous *Lezioni di meccanica razionale *(Rational Mechanics Lectures) in two volumes, in which he collaborated with Tullio Levi-Civita.

In these one feels the ties of friendship and deep lifelong adoration, beginning in the years in which they were colleagues at the University of Padua, and that veneration by Amaldi always remained unconditional and enthusiastic, as attested by the splendid commemoration of him delivered in the Lincei Academy in 1946 and the noble words that he wrote in the preface of the second edition, revised and corrected, of the aforementioned *Lezioni di meccanica razionale *personally edited in 1951-52. The second edition of this book was not the last of his labours: to another much larger he embarked on fervently until his last months, lavishing his outstanding competence in the treatment of the national edition of the works of Levi-Civita and of Volterra (under the wishes of the Lincei Academy), a monumental work unfortunately not yet completed.

He was a member of the Pontifical Academy of Sciences, a member of the Accademia dei Lincea, a member of the XL, a member of the academies of Modena, Padua, Turin and Catania, and a corresponding member of the Veneto Institute of Sciences, Literature and Arts.

Amaldi was extraordinarily kind and sensitive, he was of a fine spirit and highly refined, a warm and vibrant soul that attracted and captivated whoever had the good fortune to be near to him. I met him more than twenty years ago, one Saturday evening in Guido Castelnuovo's house: as soon as he knew who I was, he came to look for me and spoke to me immediately, with great friendliness, about my mathematical work, which he knew very well, in particular of one of my notes on projective geometry. In the spring of 1937 he accepted me to be his assistant to the professorship of Mathematical Analysis at the University of Rome, and since then I remained at his side continuously until 1950, that is until the end of his teaching. In many years of affectionate intimacy, the teacher and unparalleled friend always knew how to clearly read my mind, finding in every important moment of my life dedicated to study and scientific research, the pertinent and decisive words, to praise good results, to offer encouragement about repeated failures, and to gently reproach when idle periods occurred. But always (and this seems to me to be the salient trait of his personality) he avoided bringing himself up as an example and, if he spoke of his own experiences, of which there were also many, it was preferably to warn young people about falling into the errors that he had committed, errors "despite which and by sheer luck" - he said - he considered himself to have happily overcome. He was in fact profoundly modest, on more than one occasion I heard him say that his own work seemed to him to be no more than "long exercises".

The pessimistic upshot, to which one cannot but raise a negative point, is that of the danger of despair from which in fact he seemed several times to be challenged by. But maybe rather he felt, at a certain moment in his life, a sense of dismay and almost of inadequacy in the face of complexities and tremendous difficulties of the problems of the theory that he had greatly studied and loved, of the problems that he glimpsed beyond those that were able to be patiently and brilliantly solved. And moreover his most noble mind, always knew how to react to every pessimism and every dejection, indeed he seemed to draw from this reaction new incentive to study and new interest and admiration for the work of high mathematical priests: he studied no one's work with as much interest and admiration as he did that of Élie Cartan, which he felt to be close to his own work, and that he had the privilege of being able to assess and appreciate like few others.

An exemplary husband and father, he had the combined luck and ineffable joy of seeing a splendid and large family growing up around him. In his final years, when going to find him in his large, beautiful, welcoming home in Rome or in his villa at Carpaneto Piacentino, surrounded by children, grandchildren and great-grandchildren in a tender atmosphere of affection and gentleness, he seemed to be making the visit of a patriarch and tapping into the source of biblical wisdom. Because in his home and in the bosom of his family, that is in every moment of his life, he was at the centre of the heart of the family on which he lavished all the treasures of his large spirit, and one could really feel his great personality and understanding expanding. One could also feel it when he entertained the children, especially the youngest ones, towards whom he showed great patience, indulgence, and admirable finesse. He was happy when I brought him my children, and I brought them to him very often, making them happy as well as me. And I remember many times that I marvelled that they frequently would prefer to abandon the games that they would be going to play with his grandchildren, in order to come up to us and listen to our great, incomprehensible and boring discussions. I also remember, when my wife or I would let off steam confiding our concerns about teachers and educators, he almost always took the side of our children and he managed to convince us of the errors that we had committed: he managed to do that easily especially since we well knew which teacher had taught his children, a teacher who had never felt the need to resort to force, a very strict teacher, but never apparently so.

None of this was, after all, an expression of his large, intense, warm devoutness and piety. He was in fact not only a believer, but strictly observant, and his Catholic faith had not even the slightest shade of fanaticism or bigotry, since in others he only saw and appreciated righteousness, nobility, strictness of manners, and respect for other's ideas or opinions. That is why I found, and I still find, perfectly natural his friendship towards people of other religions, in particular towards some Jews whom I repeatedly heard him admire as "the most evangelical people he had ever known".

Humility, the true, authentic Evangelical humility, he knew and practised all of his life. I will never forget the last words he said to me, a few days before his death that took him suddenly, saving him from the greatest sufferings of an illness that had long affected him and from which he could not be cured. It was at the end of a brief telephone conversation, in which he was immensely pleased with my most recent and major professional success. He had a clear mind, but a tired voice. When I told him that I had an diseased leg, that I was walking with difficulty, so I had preferred to postpone to a better time my visit, he replied that he would have forced himself to come to me, that he wanted to see me at all costs and congratulate me on my success in person!

His highly clear mind had completely divorced questions of faith from science. He was in disagreement, on this, with his colleague Fantappiè, who he held in high regard, feeling that faith and science were like two activities of a completely different spirit, between the two of them one should not, one could not, have any interference. He was an aristocrat in the English sense of the word, that is to say liberal though also democratic, but Guelph - he said - not Ghibelline [Guelphs supported the Pope, Ghibellines supported the Holy Roman Emperor]. He saw the distinction between the social classes on the grounds of education and culture, and many times I heard him praise a book that had impressed him in his youth and to whose ideas he had enthusiastically adhered to, *L'étape* by Paul Bourget. Because culture and above all education can, in his opinion, only be achieved slowly, through continued effort from one generation to another, from father to son, only being permissible to very few individuals, extraordinarily gifted by nature, skipping the stages of ascent.

Such a man could not naturally be fascist and that in fact he never was. When the collusion between fascism and Nazism began to take shape, all of his anger exploded and his spirit was plagued by deep sorrow at the time of racial persecution and war, although his conservative nature kept him distanced from any rebellion against the established authorities.

His long, hard-working, and we could say, happy existence, ended in full harmony and consistency with the ideas that he always strongly expressed, leaving a final goodbye to his descendants who had the rare privilege of surrounding him and making a loving circle around him in the painful moment of his passing.