**Onorato Nicoletti**was born in Rieti, a hilltop town in central Italy. At the time when Onorato was born the town was part of Umbria, only being annexed by Lazio in 1923. He attended the R. Scuola Normale Superiore in Pisa before entering the University of Pisa where he graduated with his laurea in 1894. Leaving Pisa, he moved to Rome where he taught at the Technical Institute.

In 1897 he entered a competition for the chair of infinitesimal calculus at the University of Modena. A committee, with Ulisse Dini as president, considered Nicoletti's application along with that of seven other candidates, namely Italo Zignago, Giulio Vivanti, Mineo Chini, Rodolfo Bettazzi, Domenico Amanzio, Orazio Tedone and Giuseppe Lauricella. A report of the findings of the committee appeared in [3] and we give a version of their evaluation of Nicoletti:-

The committee ranked Vivanti first followed by Nicoletti and Lauricella. However, Vivanti did not take up the position and, in January 1898, Nicoletti was appointed to the chair of infinitesimal calculus at the University of Modena. After two years at Modena, Nicoletti was called to the Chair of Algebra at the University of Pisa where he became a colleague of Ulisse Dini, who held the chair of infinitesimal analysis, of Luigi Bianchi and of Eugenio Bertini. In 1913 Giacomo Albanese became Dini's assistant but, in 1918, Dini died. At this time Nicoletti took over Dini's courses on infinitesimal calculus moving to the chair of infinitesimal analysis. Albanese then became Nicoletti's assistant. Nicoletti remained in Pisa for the rest of his career, but died at the age of 57, well before he reached the age of retiring.Dr Onorato Nicoletti graduated with a Iaurea in Pisa in1894. He presents twelve interesting works. The first relates to the surface of minimum arc. The second one, which is pure analysis, completes the results obtained by Appell for the expression of the functions in lacunary spaces in the parallelogram of periods. The other works relate to ordinary differential equations, or to those with partial derivatives also of a higher order than the2nd, and they all have particular importance for the completion of the results obtained with the method of successive approximations and with that of Riemann as well as for the studies that are taking place on the2nd order equations of hyperbolic type, in which the Laplace series is finite. In the work "On Ordinary Differential Equations" he examines under what conditions the integrals of these equations are continuous functions and derivable from the initial values. If the author had also knowledge of other works, which refer to the subject, he could, in a simpler way, reach even more complete results. The work on the transformation of the2nd order linear differential equations, and the other analogues, demonstrate the talent of the competitor. Nor is it necessary to take into account some very slight and rare blemishes or impropriety of language(work No.6), things that do not detract from the interest of the studies made by him and its scientific value. Indeed, the Commission is convinced that these works show sure promise of a splendid scientific career, if he perseveres, as is not to be doubted, with equal ardour in his studies. Nicoletti's good attitude is also known, which is also proven by a certificate from the director of the Pisa Scuola Normale Superiore, relating to the courses given in it.

Nicoletti published works in various fields of mathematics, including algebraic analysis, infinitesimal analysis, equations related to Hermitian matrices and differential equations. For example, on Hermitian matrices he wrote *Sulla caratteristica del determinante di una forma di Hermite* Ⓣ (1909), on ordinary differential equations *Sugli integrali delle equazioni differenziali ordinarie, considerati come funzioni dei loro valori iniziati* Ⓣ (1895) and on partial differential equations *Sull'estensione dei metodi di Picard e di Riemann ad una classe di equazioni a derivate parziali * Ⓣ (1896). He gave original contributions to Max Dehn's theory of the equivalence of polyhedral sets, extending it and generalizing it with an entire class of new relationships in three papers *Sulla equivalenza dei poliedri* Ⓣ in 1913, 1914 and 1915.

For a list of papers and books by Nicoletti, see THIS LINK.

The *Enciclopedia Hoepli delle Matematiche Elementari* Ⓣ was edited by Luigi Berzolari, Duilio Gigli and Giulio Vivanti and published between 1923 and 1950. Nicoletti contributed two articles to this encyclopaedia, namely *Forme razionali di una o più variabili* Ⓣ and *Proprietà generali delle funzioni algebriche* Ⓣ. He also wrote popular mathematics school books such as *Lezioni di analisi matematica* Ⓣ (1926) written in collaboration with Guido Ascoli and Francesco Cecioni. Perhaps equally important was his school textbook *Aritmetica e algebra* Ⓣ written with Giovanni Sansone. Editions of these two books appeared long after Nicoletti's death. Another school textbook, this time written in collaboration with Arturo Maroni, was *Aritmetica razionale: per l'istituto magistrale* Ⓣ (1925). He also jointly edited a series of mathematics textbooks for secondary schools along with Roberto Marcolongo which was published by Perrella in Naples. His two books with Maroni and with Sansone appear in this series.

We end by quoting from an English translation of Francesco Cecioni's moving article [2]:-

On31December1929our university family suffered another mourning, adding to those, numerous and painful, of these last years; a short illness cut short the life, still flourishing with passionate energy, of Onorato Nicoletti.For some months the first signs of illness had shown themselves, but all of us, our colleagues and friends, deluded ourselves that it was a transitory phenomena, and we hoped that a short rest would give him back to his family and to us, in the fullness and exuberance of the life force that we used to see in him. But the total affection that we brought to him had veiled our eyes, and the sad reality was quite different.

A man of a frank and loyal character, of great moral rectitude, of a vivacious and exuberant spirit, he inspired in those who knew him closely a feeling of deep affection; he was at times ready to scorn, which then his great goodness and his deep sense of justice immediately dismissed.

In his exact and precise lessons, but also clear, compelling, animating, and vivacious, he transfused his whole intellect as a scientist, all his refined temperament enthusiastic about the innumerable and deep harmonies of his science, all his love for the art of teaching. Enthusiastic, he knew, both as professor and in the relations with his numerous students, how to train and to fascinate.

His colleague professor Puccianti gave well-said words at his final salute at the funeral: "He felt his teaching, more than as a duty, as a mission, as a necessary exercise, an irrepressible expansion, I would dare to say an unbreakable outburst of his lively talent, of his fervent, restless activity." Those who have had the good fortune to approach and cooperate with him can fully know what he meant by his duty as head of the school, with what meticulous care he prepared his courses and his lessons in detail, never seeming to have achieved what didactically and scientifically he wanted. So much was his love for the school that for several years he considered it necessary to hold, beyond his teaching of Algebra, the free assignment of the teaching of Calculus. He also held for several years, after the death of Dini, the task of managing the School of Engineering, in the difficult moments requiring his support.

As he accomplished his duty with respect to the school, so he performed it in his family. He carried on, still young, when he lost his beloved wife, but he, sustained by his Christian faith, replaced her by reuniting in himself, for his son and his three daughters, even at an early age, the affections and cares of both mother and father, and always he devoted himself to their education and to their well-being, receiving in return the purest family joys of which he had too little time for him to enjoy. And when the hour of misfortune came, we could see with what strength and with what true Christian spirit, the fruit of the education they received, the sons, making strength out of the ineffable pain, assisted to the last their dying father, together with the priest called in time to his bed.

The merits of Onorato Nicoletti as a scientist will be hinted at here only briefly; in this brief mention, it was more dear to me to have spent a little time reviewing his figure as a man, still as alive among us as when he was with us.

Honored Nicoletti was born in Rieti on21June1872. A student of the R. Scuola Normale Superiore and the University of Pisa, he graduated in1894. Rapidly the young scientist was established after two years of improvement, and after just over a year, teaching in the Technical Institute of Rome, he won with a group of twelve important works, almost all on differential equations, the competition for the chair of Calculus in Modena, where he was nominated in January of1898. After two years he was called by his former teachers to the Chair of Algebra in Pisa, and in our University of which he felt the glorious tradition, and to which he linked the profound reverent affection for Dini, Bianchi, and Bertini, he later moved to the chair of Calculus, not wishing to move away.

He was of vast and solid culture, of ingeniousness in a non-ordinary, prompt, acute, profound way; admirable the ease with which he penetrated questions that were also proposed for the first time. He had great power of analysis; examining questions dealt with by others, even the most extreme, he often succeeded in overcoming large-scale hypotheses and unnecessary conditions, thus reducing his treatment to a very simple logical scheme, which allowed him, sometimes overcoming serious difficulties with uncommon abilities and deductive force, to obtain far more general results than those already known.

His contribution to science is truly remarkable; but even greater this would have been if his family and school occupations, and the activity he intelligently lent to various public administrations, had given him more time to devote to scientific research; his major production is in fact grouped in certain periods of his life.

A remarkable group of works, forming an organic complex, refers to the theory of partial differential equations, and precisely to extensions of the method of successive approximations and to the theory of transformations; another group refer to that of ordinary differential equations. Other publications concern differential geometry, Taylor's multiple series, the theory of limits, that of analytic functions; two of these works were also translated abroad.

In the field of algebra, there are many subjects with which he dealt. Perhaps his most noteworthy works in this field, due to the novelty and interest of the results, are those on the convergence of multiple-variable iteration algorithms(and some of this research he left unfinished). Also important are the works concerning certain classes of equations with real roots, the theory of determinants and matrices, the Weierstrass theorem on the equivalence of two bundles of bilinear forms; this latter work was to be followed by two other memoirs that he could not complete.

Finally, his works on the equivalence of polyhedral sets are very interesting, in which he adds a whole class of relationships to the well-known relationship found by Dehn; and this is achieved through the introduction, made with ingenious abstraction and generalization, of a symbolic calculation on the pairs of quantities, which may perhaps also be useful in other matters.

Then there are vast monographic articles to remember.

A profound connoisseur of all the issues concerning secondary schools, he had been editing for several years, together with professor Roberto Marcolongo, a fine collection of mathematics texts for middle schools; and to this, he had directly collaborated on two popular books, one in collaboration with Professor Sansone, the other with Professor Maroni.

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