**Ugo Amaldi**'s parents were Edoardo Amaldi and Emilia Vettori. Edoardo was an Austrian magistrate who was near Padua when the Second Italian War of Independence broke out in 1859. This war involved Piedmont, who were supported by the French, fighting against the Austrians who controlled the region. After the defeat of Austrian forces the war ended with Piedmont supported by France in control of much of northern Italy. Edoardo was appointed by Piedmont as a magistrate in Como. He served as a magistrate in several different cities over the following years, mostly in Pavia before ending his career as Attorney General in Bologna. He married Emilia Vettori who came from a family who strongly supported the partisans. One of her brothers had supported Giuseppe Garibaldi during the failed insurrection of 1834 and followed Garibaldi into exile in South America. Ugo was the youngest of his parents' three children, having two older brothers, Italo (1864-1940) who became an assistant to Felice Casorati and Ernesto Pascal at Pavia before becoming a professor at various technical institutes, and Paolo (1865-1956) who became a medical doctor and the first director of the Neuropsychiatric hospital in Mendrisio in 1898.

It was in Pavia that Amaldi completed his secondary school studies at the high school "Ugo Foscolo". Felice Casorati had been a pupil at this school forty years before Amaldi. At this school Amaldi was taught mathematics by Luigi Berzotari who was both an excellent mathematician and teacher. Berzotari had published *Sulla superficie del quarte ordine avente una conica doppia* Ⓣ in the *Annali di Matematica Pura ed Applicata* in 1885. After graduating from the high school in 1894, Amaldi entered the University of Bologna in 1894 where he was taught by Cesare Arzelà, who held the chair of Infinitesimal Calculus, and Salvatore Pincherle who held the chair of Algebraic Analysis and Analytic Geometry. He was also taught by Federigo Enriques who was only four years his senior. Amaldi always considered himself a student of Pincherle, but regarded Enriques as his second master and the two later collaborated.

When he had began his undergraduate studies Amaldi was undecided whether to specialise in engineering or in mathematics. However, although he enjoyed both practical and theoretical courses, it was Enriques' course on projective geometry that really inspired him. In his second year (1895-96) he took courses by Pincherle on group theory where he met both discrete groups and continuous groups. In his third year he attended Enriques' seminar on higher geometry, where he became interested in non-euclidean geometry. On 28 November 1898 Amaldi graduated after submitting a thesis on the Laplace transform entitled *La trasformazione di Laplace e le equazioni differenziali lineari, a coefficienti razionali, di rango 1* Ⓣ. After graduating he passed the examinations to qualify as a secondary school teacher.

While collaborating with Pincherle on writing a treatise, Amaldi published *Sulla trasformazione di Laplace* Ⓣ (1898) and *Sulle sostituzioni lineari commutabili* Ⓣ. His 490-page treatise, co-authored with Pincherle, was entitled *Le operazioni distributive e le loro applicazioni all'analisi* Ⓣ and it was published in 1901. After this publication Amaldi published a remarkable number of papers: *Contributo alla determinazione dei gruppi continui finiti dello spazio ordinario* Ⓣ and *Le superficie con infinite trasformazioni conformi in sè stesse* Ⓣ appeared in 1901, then five more papers in 1902 as well as a 655-page textbook *Elementi di geometria ad uso delle scuole secondarie superiori* Ⓣ co-authored with Enriques.

In 1902 he became an assistant lecturer in Complementary Algebra and Analytical Geometry at the University of Bologna. On 3 September of that year a national competition was announced for a chair of Algebra and Analytical Geometry at the University of Cagliari with a deadline for applications on 6 October 1902. He won the competition and taught at the University of Cagliari for two years beginning in 1903. In February 1903 he married Luisa Basini who he had known for many years. They had three children, Adalgisa (known as Gisina), Mercedes and Edoardo. Gisina married the magistrate Vittorio Olivieri Sangiacomo and had two sons: Giorgio and Corrado. Mercedes married the engineer Lodovico Marchesi and had five children: Maria Luisa, Camillo, Ugo, Giovanna, and Ludovica. Edoardo (1908-1989) studied under Enrico Fermi and became a leading physicist. He married Ginestra Giovene and had four children: Ugo, Paola, Francesco and Daniela.

Amaldi transferred to Modena in 1906, where he was appointed a professor in Analytical and Projective Geometry, then in 1919 he moved to Padua where he was a professor of Descriptive Geometry with Applications until 1922. Remaining in Padua, he was appointed to the chair of Analytical Geometry in 1922, a position he held for two years. Finally in 1924 he moved to Rome where he was a professor of Mathematical Analysis and Analytic Geometry in the Faculty of Architecture until 1942 when he became professor of Algebraic and Infinitesimal Mathematical Analysis in the Faculty of Science for the rest of his career. He retired at the age of 75 in 1950.

The greatest of Amaldi's research contributions is his classification of infinite dimensional Lie groups acting on three-dimensional space. Amaldi had attended a course by Pincherle on Lie groups which he found fascinating and he was also fascinated by Pincherle's theory of functionals. Advised by Pincherle, Amaldi decided to apply the theory of functionals to deal with infinite dimensional Lie groups. Over the next years he worked on the problem, read thoroughly the work of Élie Cartan on the subject, and corresponded with Cartan. He produced important results on infinite dimensional Lie groups acting on 3-dimensional space as Enrico Rogora explains in [7]:-

[These are impressive results, but somehow Amaldi's achievement seems to have gained surprisingly little attention. Enrico Rogora tries to explain why this happened [7]:-Amaldi]completed the classification of punctual and contact transformation groups acting in three dimensional space, both finite and infinite dimensional.

A very personal account of Ugo Amaldi's personality is given by Tullio Viola, who was his colleague for many years, in [8]. We present an English translation by Fiona Spencer (a University of St Andrews student) of Viola's obituary of Amaldi at THIS LINK.Part of the responsibility for the oblivion of Amaldi's work is due to Amaldi himself. At a certain point of his career he decided to accept a position at the "School of Architecture" in Rome, which, contrary to his expectations, never became a university institution. Hence he had been out of the mainstream of Italian mathematical research for years. Amaldi also realized the loss of centrality of the classification problem for the mathematics of his period, due to the necessity to put Lie's theory of transformation groups on firmer basis and he always considered his work with excessive modesty, as is shown, for example, in the excerpt of the following letter[written in June1918]to Levi-Civita, where he thanks his friend for his considerations on his most important work. "I heartfully thank you for your kind words about my huge memoir. By experience I perfectly know that l need to soften your judgments about my work because of your great benevolence toward myself. Even so, I have greatly appreciated your good words, since now that I am in front of this huge volume, I feel concerned, especially with respect to the Society XL, by the responsibility to publish such a work for which I have honestly to admit the imbalance between size and interest. After the half commitment I look in a previous work, it was a kind of point of honour for me to complete this classification: in any case, this work is the end of my researches on the classification of continuous groups, on which I have already insisted too much."

Perhaps Amaldi's name is remembered more today for his textbooks written with Federigo Enriques. We mentioned above their textbook *Elementi di geometria ad uso delle scuole secondarie superiori* Ⓣ, published in 1903, which was aimed at the teaching of geometry in secondary schools. This was the first of a series of textbooks written by Amaldi and Enriques for use in secondary schools. We list some of these books: *Nozioni di geometria ad uso dei ginnasi inferiori* Ⓣ (1910); *Nozioni di matematica ad uso dei lici moderni* Ⓣ (2 volumes, 1914-15); and *Geometria elementare per le scuole secondarie superiori* Ⓣ (2 volumes, 1925-26). Problems with these books occurred in 1938 when the Italian Fascist government passed a law which:-

It is hard to believe that many textbooks would be available for use in Italian schools after this remarkably all-embracing law. Amaldi was not Jewish, but his co-author Enriques was Jewish and was forced to resign from his teaching position in 1938. The law, drafted by the Minister of National Education Giuseppe Bottai, meant that the jointly authored Amaldi-Enriques books could no longer be used in schools. In an attempt to allow these books to be able to continue being used in schools, editions were published with only Amaldi's name as author. However, the Ministry of National Education quickly spotted that an attempt was being made to have a textbook, written in part by a Jew, still available in schools and warnings were issued to the publisher. After the Italian Fascist government fell in 1944, editions of the books were published with the two authors.... prohibited the adoption of textbooks by authors of the Jewish race. The ban also extends to books that are the result of the collaboration of several authors, one of whom is of the Jewish race; and works that have been commented on or reviewed by people of Jewish race.

Another very important work which we should mention was *Lezioni di meccanica razionale* Ⓣ which Amaldi wrote in collaboration with Tullio Levi-Civita. A hand-written version of part of their text was published in 1920 and the book was printed in two volumes, the first being published in 1923. The authors wrote in the Preface to this volume:-

The second volume appeared in two parts, the first in 1926 and the second in the following year.The work of which we present the first volume, has come about through the teaching of rational mechanics, which one of us has professed for over twenty years in Padua and Rome, the other for six years in Modena and Padua. Originally derived from teaching, the work has a specific elementary character, so it cannot, nor does it pretend to be, a real treatise. However, we dare to hope that it will also be welcomed as a reference book, because we, while carrying out our institutional discussion, have consistently and thoughtfully considered the various and diverse needs of those who study the rational mechanics, according to the needs of mathematicians or physicists or astronomers or surveyors, or, on the other hand, technicians, or mechanical manufacturers, or civil engineers, or naval engineers, or hydraulic engineers or industrial engineers or electrical engineers.

Amaldi died after much suffering with an incurable illness. For an English translation of the obituary [8] which gives a very personal account of Amaldi's life, see THIS LINK.

**Article by:** *J J O'Connor* and *E F Robertson*

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