**Faa di Bruno** is best known for his formula for the _n_th derivative of a composition of functions. He was beatified in 1988

- The village of Bruno is about 18 km south west of Alessandria, a major city roughly equidistant from Turin, Milan and Genoa.
- The revolutions which broke out across Europe in 1848 led to the First Italian War of Independence in which Faà di Bruno participated in the Piedmontese Brigata Guardie commanded by Vittorio Emanuele.
- For six weeks the Brigata Guardie attacked the Austrians at Peschiera del Garda and were victorious on 30 May 1848.
- Faà di Bruno's main task during these months was to draw up maps and he produced his major map of the Mincio area.
- He undertook his duties with great enthusiasm, realising that the generals commanding the Piedmontese armies did not have recent accurate maps of the Lombardo-Veneto region.
- Faà di Bruno's horse was killed by Austrian rifle fire, so he mounted another horse only to have that killed too.
- He had two young sons and, realising Faà di Bruno was both an extraordinary person and an exceptional scholar, he asked him to tutor his two young sons.
- However, Faà di Bruno was known as a devout Catholic and the King immediately came under pressure to appoint a secular tutor for his sons.
- Then Faà di Bruno asked permission to leave the army and take up the study of mathematics.
- At the Sorbonne Faà di Bruno was in the same classes as Charles Hermite and the two became close friends.
- However, his time in Paris was not all spent on academic studies for he also assisted in the parish of Saint Sulpice and visited the homes of the poor.
- While in Paris he began publishing mathematical papers: Note sur un nouveau procédé pour reconnaître immédiatment, dans certains cas, l'existence de racines imaginaires dans une équation numérique Ⓣ(Note on a new method for recognizing immediately, in some cases, the existence of imaginary roots in a numerical equation.) (1850); Démonstration d'un théorème de M Sylvester, relatif à la décomposition d'un produit de deux déterminants Ⓣ(Demonstration of Sylvester's theorem on the decomposition of a product of two determinants) (1851); and Démonstration d'un théorème relatif à la réduction des fonctions homogènes à deux lettres à leur forme canonique Ⓣ(Demonstration of a theorem on the reduction of homogeneous functions in two letters to their canonical form) (1852).
- The Church of San Massimo, in Borgo Nuovo, was completed in June 1853 and, in the following year, Faà di Bruno set up a choral school for women in the church where they were trained every Sunday by Faà di Bruno who played the organ.
- In May 1855 Faà di Bruno returned to Paris to undergo further training at the astronomical observatory at Brera.
- The authorities in Turin promised him that, after he returned from Paris, he would be employed at the Observatory in Turin - a promise they did not keep.
- However, in Paris Faà di Bruno studied astronomy under Urbain Le Verrier and also undertook research with Cauchy on mathematics.
- He continued to publish papers in both French and in Italian: Sullo sviluppo delle Funzioni Ⓣ(On the development of functions) (1855), Sulle Funzioni Isobariche Ⓣ(On isobaric functions) (1856) and Note sur une nouvelle formule de calcul différentiel Ⓣ(Note on a new calculus formula) (1857) were all written while he was undertaking research in Paris.
- Both the first and the third of these papers contain the result for which he is best known, namely Faà di Bruno's formula.
- Riordan and Comtet saw it as part of combinatorial analysis, a term that seems to be going out of fashion; the subject subsumed in algebraic combinatorics, the books of Riordan and Comtet largely superseded by Stanley's monumental 'Enumerative combinatorics' (1997, 1999) where Faà di Bruno's formula is mentioned, but not stated.
- Several authors have pointed out that Faà di Bruno was not the first to either state or prove this result.
- However, Faà di Bruno did give a form of the formula using determinants which nobody had found earlier.
- From 1859 Angelo Genocchi held the Chair of Algebra and Complementary Geometry at Turin, then in the following year he moved to the Chair of Higher Analysis and Faà di Bruno was appointed as his deputy.
- After his time studying in the Sorbonne, Faà di Bruno did much charity work on his return to Turin.
- He had seen food being prepared and distributed to the poor while in Paris and, back in Turin, he began to organise a similar scheme during the winter months.
- At this time Faà di Bruno came in contact with Giovanni Bosco.
- Faà di Bruno, following Bosco's example, founded the Pia opera di Santa Zita in the San Donato area of Turin on 2 February 1859.
- He used his own money together with funds he had collected standing at the doors of churches.
- Also in 1859 Faà di Bruno founded the Opera per la santificazione delle feste, a Society to promote Sunday observance and to protect workers who were being forced to work on Sundays.
- Faà di Bruno was incredibly energetic in his work with the poor.
- In 1860 he founded the Infermeria di San Giuseppe, an infirmary for poor women and the sick where people had an opportunity to convalesce after an illness.
- In 1862 he founded a boarding-house for the elderly and disabled women.
- Frequent epidemics, especially of typhus and cholera, were the result of poor hygiene and Faà di Bruno was able to set up wash rooms in the San Donato area.
- But his charity work was not restricted to caring for the sick, for he was also passionate about providing educational opportunities for the young people from poor families.
- In 1864 he set up classes providing training in home economics and, in 1866, he organised courses to train people to become elementary school teachers.
- Building the church took around three years and much of the architectural design was due to Faà di Bruno himself.
- Despite the many long hours he spent undertaking charity work, Faà di Bruno did not ignore his mathematical research.
- This fame, however, did not lead to rapid promotion within the University of Turin.
- The reason for this was the secular nature of the Italian Independence movement with discrimination against those active in the Church.
- In 1871 he was put in charge of teaching calculus and analytic geometry and he was appointed as an extraordinary professor of higher analysis in 1876.
- Although he is usually remembered today because of "Faà di Bruno's formula", his most influential mathematical work was his book Théorie des formes binaires Ⓣ(Theory of binary forms) on binary forms which he published in 1876.
- The book was based on lectures that Faà di Bruno had given at the University of Turin.
- The book became better known in 1881 when Max Noether published a German edition.
- Not only did Faà di Bruno have difficulties in obtaining promotion within the university, but he also had difficulty in becoming ordained.
- Faà di Bruno had to make a special plea to pope Pius IX to overrule the archbishop before he was ordained a Roman Catholic priest in Rome on 22 October 1876.
- The religious order he had founded, the Suore Minime di Nostra Signora del Suffragio, supported girls in a house called the Conservatorio del Suffragio.
- In order to provide work for the girls, Faà di Bruno had the idea that they could train as typesetters.
- There a number of mathematics books were published including one by Faà di Bruno himself on elliptic functions.
- In 1898, ten years after Faà di Bruno's death, the printing press was purchased by Giuseppe Peano for 407 lire and he printed the Rivista di Matematica Ⓣ(Journal of mathematics) on it for several years.
- He was of a solitary disposition and spoke seldom (and not always successfully in the classroom).
- In fact, to give a little more information on his musical talents, he composed scared melodies which were highly thought of by Franz Liszt.
- We have mentioned above his interest in scientific instruments, but let us add that he invented, among other things, a differential barometer, described in a publication of 1870, and an electric alarm clock.
- He died suddenly, two days short of his 63rd birthday, from an intestinal infection.
- Bosco, who had been an inspiration to Faà di Bruno and had died less than two months before him, was made a Saint on 1 April 1934.
- Already by this time there was a movement to canonise Faà di Bruno and in 1955 the Sacred Congregation of Rites officially accepted the claim for Faà di Bruno to be canonised.

Born 29 March 1825, Alessandria, Piemonte (now Italy). Died 27 March 1888, Turin, Italy.

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Astronomy, Origin Italy

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive