# Giacinto Morera

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Giacinto Morera (18 July 1856 – 8 February 1909), was an Italian engineer and mathematician. He is known for Morera's theorem in the theory of functions of a complex variables and for his work in the theory of linear elasticity.

## Biography

### Life

He was born in Novara on 18 July 1856, the son of Giacomo Morera and Vittoria Unico.[1] According to Template:Harvtxt, his family was a wealthy one, his father being a rich merchant. This occurrence eased him in his studies after the laurea:[2] however, he was an extraordinary hard worker and he widely used this ability in his researches.[3] After studying in Turin he went to Pavia, Pisa and Leipzig: then he went back to Pavia for a brief period in 1885, and finally he went to Genova in 1886, living here for the next 15 years. While being in Genova he married his fellow-citizen Cesira Faà.[4] From 1901 on to his death he worked in Turin:[5] he died of pneumonia on 8 February 1909.[6]

Francesco Siacci, who tutored Giacinto Morera in the early stage of his career.

He earned in 1878 the laurea in engineering and then, in 1879, the laurea in mathematics, both awarded him from the Politecnico di Torino:[7] According to Template:Harvtxt, the title of his thesis in the mathematical sciences was: "Sul moto di un punto attratto da due centri fissi colla legge di Newton".[8] In Turin he attended the courses of Enrico d'Ovidio, Angelo Genocchi and especially the ones of Francesco Siacci, who was acknowledged by him as his mentor in science and life.[9] After graduating, he followed several advanced courses: he studied in Pavia from 1881 to 1882[10] under Eugenio Beltrami, Eugenio Bertini[11] and Felice Casorati. In 1883 he was in Pisa under Enrico Betti, Riccardo de Paolis and Ulisse Dini: a year later, he was in Leipzig under Felix Klein, Adolph Mayer and Carl Neumann.[12] In 1885 he went in Berlin in order to follow the lessons of Hermann von Helmholtz, Gustav Kirchhoff, Leopold Kronecker[13] and Karl Weierstrass at the local university: later in the same year, he went back to Italy, briefly working at the University of Pavia as a professor in the then newly established "Scuola di Magistero".[14] In 1886, after passing the required competitive examination by a judging commission,[15] he became professor of rational mechanics at the University of Genova: he lived there for 15 years, serving also as dean and as rector.[16] In 1901 he was called by the University of Turin to hold the chair of rational mechanics, left vacant by Vito Volterra.[5] In 1908 he passed to the chair of "Meccanica Superiore"[17] and was elected dean of the Faculty of Sciences.[18]

### Honours

He was member of the Accademia Nazionale dei Lincei (first elected corresponding member on 18 July 1896, then elected national member on 26 August 1907)[19] and of the Accademia delle Scienze di Torino (elected on 9 February 1902).[20] Template:Harvtxt refers that also the Kharkov Mathematical Society elected him corresponding member during the meeting of the society held on 31 October 1909 (Old Calendar), being apparently not aware of his death.

### Tracts of his personality and attitudes

Carlo Somigliana, close friend and biographer of Giacinto Morera.

In his commemorative papers, Carlo Somigliana describes extensively Morera's personality:[21] according to him, he was a devoted friend and precious colleague,[22] capable of serenely judging men and facts.[23] On the very personal level, he remembers him s a cheerful person and a witty talker.[24]

His intelligence is described as sharp and penetrating,[25] his mind as uncommonly lucid,[26] himself as possessing analytic and critical abilities and being versatile, capable to grasp and appreciate every kind of manifestation of the human intellect.[27] Nevertheless, Somigliana also states that he was not interested in any scientific or other kind of field outside of his own realm of expertise.[28] Template:Harvtxt himself, in the inaugural address as the rector of the University of Genova, after quoting a statement attributed to Peter Guthrie Tait,[29] revealed the reason behind his views:[27] "In science, the one who has a sound and solid knowledge, even in a narrow field, holds a true strength and he can use it whenever he needs: the one who has only a superficial knowledge, however wide and striking, holds nothing, and indeed he often holds a weakness pushing him towards vanity".[30]

Acknowledged as honest, loyal and conscientious,[31] gifted with exquisite qualities of temperament and intellect,[32] his simple manners earned him affection even when performing the duties of dean and rector at the University of Genoa.[33] Also Template:Harvtxt describes him as a man of high moral value, and ascribes to such qualities the reason of his success in social relations and in performing his duties as a civil servant.

However, despite of being successful in social relations, he did not cured nor appreciated much appearances and was not interested in activities other than teaching and doing research: consequently, he was not well known outside the circle of his family and relatives and the circle of his colleagues.[32] He did not make a display of himself, careless of being not acknowledged by everyone for his true value: he also had a serious conception of life and strongly disliked vanity and superficiality.[22]

According to Somigliana,[27] his entire life was devoted to the higher unselfish ideal of scientific research: and Template:Harvtxt also remarks that only his beloved family shared the same attentions and cares he reserved to his life–long ideal.

## Work

### Research activity

Una quantità di quistioni egli chiarì, semplificò o perfezionò, portando quasi sempre il contributo di vedute ingegnose ed originali. Talchè la sua produzione scientifica può dirsi critica nel senso più largo e fecondo, cioè non dedicata allo studio di minuziosi particolari, ma alla penetrazione e soluzione delle quistioni più difficili e complicate. Questa tendenza del suo ingegno si rivelò anche in un carattere esteriore di molte sue pubblicazioni, che egli presentò in forma di lavori brevi e concettosi; dei quali poi particolarmente si compiaceva, ed in conformità del suo carattere sincero, la sua compiacenza non si tratteneva dal manifestare apertamente.[34]

According to Somigliana,[32] he was not gifted by a strong inventiveness: he did not create any new theory since this was not his main ability.[35] Instead, he perfected already developed theories:[36] nearly all of his researches appear as the natural result of a deep analysis work on theories that have already reached a high degree of perfection,[35] clearly and precisely exposed.[37] He had an exquisite sense for the applicability of his work, derived from his engineering studies,[38] and mastered perfectly all known branches of mathematical analysis and their mechanical and physical applications.[39]

He authored more than 60 research works: nearly complete lists of his publications are included in the commemorative papers Template:Harv, Template:Harv and Template:Harv. In particular Template:Harvtxt classifies Morera's work by assigning each publication to particular research field: this classification is basically adopted in the following subsections.[40]

#### Complex analysis

File:Morera's Theorem.png
A curve C in a domain D, as required by the statement of Morera's theorem.

Morera wrote eight research works on complex analysis:[41] the style he used for their writing probably inspired Somigliana the quotation introducing the "Research activity" section.[42] Morera's theorem, probably the best known part of his scientific research, was first proved in the paper Template:Harv.[43] The theorem states that if, in the complex plane ℂ, the line integral of a given complex–valued function f satisfies the equation

${\displaystyle \oint _{C}f(z)\,dz=0}$

for every closed curve C in a given domain D, then f is holomorphic there.

#### Differential equations

This section includes all his works on the theory of differential equations, ordinary or partial ones: Template:Harvtxt classifies this contributions as works in the theory of the equations of dynamics, in the theory of first-order partial differential equations and in the theory of exact differential equations.[44] He wrote twelve papers on this topic: the results he obtained in these works are well described by Template:Harvtxt. In the paper Template:Harv he gives a very brief proof of a transformation formula for the Poisson brackets first proved by Émile Léonard Mathieu, while in the paper Template:Harv he simplifies the proof of a theorem of Francesco Siacci which is substantially equivalent to Lie's third theorem: the paper Template:Harv is concerned with the Pfaff problem, proving a theorem on the minimum number of integrations to be performed in order to solve the problem.

#### Equilibrium of continuous bodies in elasticity theory

Template:Harvtxt classifies four of his works within the realm of elasticity theory: his contribution are well described by Template:Harvtxt and by Template:Harvtxt in their known monographs. The works within this section are perhaps the second best known part of his research, after his contributions to complex analysis.

#### Mathematical analysis

Template:Harvtxt classifies four of his works under the locution "Questioni varie di Analisi".[45]

#### Potential theory of harmonic functions

His contribution of this topics are classified by Template:Harvtxt under two sections, named respectively "Fondamenti della teoria della funzione potenziale"[46] and "Attrazione dell'elissoide e funzioni armoniche ellissoidali".[47] The work Template:Harvtxt deals with the definition and properties of ellipsoidal harmonics and the related Lamé functions.

#### Rational mechanics and mathematical physics

Template:Harvtxt includes in this class twelve works:[48] his first published work Template:Harv is included among them.

#### Varia: algebraic analysis and differential geometry

This section includes the only two papers of Morera on the subject of algebraic analysis[49] and his unique paper on differential geometry:[50] they are, respectively, the papers Template:Harv, Template:Harv and Template:Harv.

### Teaching activity

References Template:Harv, Template:Harv and Template:Harv do not say much about the teaching activity of Giacinto Morera: Somigliana[51] describes once his teaching ability as incisive. However, his teaching is also testified by the litographed lecture notes Template:Harv: according to the OPAC, this book had two editions, the first one being in 1901–1902.[52]

## Publications

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. "On the motion of a point attracted by two fixed centers according to Newton's law", is Morera's first published paper, probably including material from his laurea thesis in mathematics.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. English translation of the title: "The "Fundamental theorem in the theory of canonical equations of motion" of prof. Siacci".

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. This paper was read by Enrico D'Ovidio at the meeting of the Class of Mathematical, Natural and Physical Sciences of the Accademia delle scienze di Torino, while Template:Harvtxt finished it on the February 1883 in Pisa, according to the date reported on the last page of the paper.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. This paper was read by Francesco Siacci at the meeting of the Class of Mathematical, Natural and Physical Sciences of the Accademia delle scienze di Torino, while Template:Harvtxt finished it on 15 April 1883 in Pisa, according to the date reported on the last page of the paper.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. Morera's unique paper in differential geometry.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. "A fundamental theorem in the theory of functions of a complex variable": the paper containing the first proof of Morera's theorem.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. (English translation of the title: "A little contribution to the theory of quadratic forms").

• {{#invoke:citation/CS1|citation

|CitationClass=citation }} (requires DjVu plu-gin). "On the Cauchy integral": a paper containing the studies of Morera on the boundary values of the Cauchy integral.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. The inaugural address pronounced in occasion of the beginning of the academic year 1888–1889 at the University of Genoa, published in the form of a pamphlet: a translation of the title reads as:-"The teaching of mathematical sciences in Italian universities".

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. "Proof of an integral calculus formula", contains a short proof of Stoke's formula in the plane.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. Part II was read at the meeting of the Class of Mathematical, Natural and Physical Sciences of the Accademia delle scienze di Torino held a few weeks later, on 11 March 1906: see here for a brief description of the meeting, and here to access part II of the paper directly.

## Notes

1. According to Somigliana ([[#Template:Harvid|1910]], p. 573; [[#Template:Harvid|1910a]], p. 605): these commemorations include also a list of Morera's published works.
2. According to Template:Harvtxt and to Somigliana ([[#Template:Harvid|1910]], p. 573; [[#Template:Harvid|1910a]], p. 605).
3. According to Template:Harvtxt and Template:Harvtxt, while not gifted by a strong inventiveness, he nevertheless approached many difficult questions, introducing original views that simplified considerably the theories he worked to.
4. There is a discrepancy between the statement of source Template:Harv and the ones of sources Template:Harv, Template:Harv, Template:Harv: the former one refers that he lived in Genova for 14 years, while the others quantify the duration of the same period as 15 years. The version of the second group of references has been adopted, also on considering that Vito Volterra went to Rome in 1901.
5. Template:Harvtxt and Template:Harvtxt refer that he died in few days, notwithstanding his strong constitution.
6. According to Template:Harvtxt and Template:Harvtxt.
7. "On the motion of a point attracted by two fixed centers according to Newton's law". Somigliana ([[#Template:Harvid|1910]], p. 573 and [[#Template:Harvid|1910a]], p. 605) does not say if it was published as his first paper Template:Harv: however, the title is the same and the dates nearly coincide.
8. According to Template:Harvtxt, who uses precisely the Italian respectful title "maestro". Somigliana ([[#Template:Harvid|1910]], p.574 and [[#Template:Harvid|1910a]], p. 605) and Template:Harvtxt refer also that it was Francesco Siacci who guided Morera towards the study of rational mechanics.
9. According to Template:Harv and Template:Harv.
10. Template:Harvtxt reports "Eugenio Berbini" (see also Template:Harvnb) which is obviously a typo.
11. According to reference Template:Harv. Since Adolph Mayer and Felix Klein were teaching in universities outside Leipzig, it is not clear from the reference if the courses Morera attended to in Germany were privately held or were advanced university courses. Nevertheless, Template:Harvtxt states precisely these dates, names and places, as does Template:Harvtxt.
12. Only Template:Harvtxt cites Kronecker as one of his teachers.
13. According to Template:Harvtxt. The "Scuola di Magistero", literally "Teaching School", was a particular University school aimed to the training of teachers.
14. Template:Harvtxt states that the examination was "onorevolmente vinto" which literally means "won in honorable way", perhaps alluding to a honorable mention awarded to him by the examining commission.
15. Precisely, according to Template:Harv he served the University of Genova as dean for the periods 1891–1892 and 1896–1897, and as rector in the two years following his last dean mandate.
16. "Higher Mechanics": the locution identifies an advanced course on rational mechanics.
17. According to the [[#Template:Harvid|yearbook of the academy]], p. 494.
18. Cossa ([[#Template:Harvid|1902]], p. 252) also describes briefly his election ceremony to resident member, i.e. "socio residente".
19. Template:Harvtxt states that they were friends for more than twenty years and also colleagues from 1901 onward, talking about their scientific researches almost every day. In Template:Harv and Template:Harv he complains about the pain of commemorating him, nevertheless aiming to do this in order to widespread the knowledge of his personality and work.
20. See Template:Harv and Template:Harv.
21. See Template:Harv, Template:Harv and Template:Harv. Somigliana exactly states that he possessed "Serenità nel giudicare uomini e cose".
22. According to Somigliana ([[#Template:Harvid|1910]], p. 580; [[#Template:Harvid|1910a]], p. 610) and Template:Harvtxt.
23. Template:Harvtxt goes further stating also that "(nella sua mente) non trovavano mai posto idee vaghe o incomplete" (English translation: "(in is his mind) confused and incomplete ideas did not find any place").
24. See Template:Harv and Template:Harv.
25. This was a consequence of his particular opinions, again according to Somigliana ([[#Template:Harvid|1910]], p.580; [[#Template:Harvid|1910a]], p. 610): he excluded, and almost feared, everything not being classifiable as complete strictly scientific knowledge.
26. "Schivate la scienza popolare, essa è tanto più perniciosa, quanto più pretenziosi sono quelli che la diffondono" (English translation: "Beware of popular science, it is as much as pernicious, as pretentious are the ones who spread it"), as also reported by Somigliana ([[#Template:Harvid|1910]], p. 580; [[#Template:Harvid|1910a]], p. 610).
27. The exact words of Template:Harvtxt are:-"Nella scienza chi ha cognizioni salde e profonde, in un campo anche ristretto, possiede una vera forza e all'uopo sa giovarsene; chi invece ha solo cognizioni superficiali, anche molto estese ed appariscenti, possiede nulla, anzi spesso ha in sè un elemento di debolezza, che lo sospinge alla vanità".
28. See Template:Harv.
29. Again according to Template:Harvtxt.
30. (English translation) "He cleared, simplified or perfected, a number of questions, bringing almost always the (personal) contribution of ingenious and original views. Therefore his scientific production can be defined a critical review in the wider, prolific sense, not aimed to the study of minutious particulars, but to the deep understanding and solution of the most difficult and complex questions. This tendency of his skill revealed itself in the formal character of many of his publications, that he presented in the form of brief, pregnant works; he was particularly satisfied of them, and according to his sincere nature, he did not refrain to manifest his satisfaction frankly".
31. See Template:Harv.
32. See Template:Harv.
33. See Template:Harv.
34. According to Template:Harv: his first university studies were in the engineering field, as briefly described in the "Education and academic career" subsection of this entry.
35. See Template:Harv and Template:Harv.
36. However, Maggi's terminology is not strictly followed: a modern terminology is used when needed in order to ease the comprehension.
37. According to Template:Harvtxt's classification, these works belong to "analytic function theory" i.e. "Teoria delle funzioni analitiche".
38. According to Template:Harvtxt himself, "Tipiche fra quelle sue numerose note, brevi e concettose, sono alcune che riguardano la definizione di variabile complessa", i.e. (English translation) "Typical examples of his numerous brief and pregnant notes, are some dealing with the definition of a complex variable".
39. Template:Harvtxt gives a short account of the history of the theorem, and refers also to the later paper Template:Harv. There Morera defines holomorphic functions using his theorem, and then derives some interesting consequences.
40. He precisely names this section "Equazioni della Dinamica, equazioni alle derivate parziali del primo ordine ed equazioni ai differenziali totali".
41. An English translation reads as:-"Various topics in mathematical analysis".
42. Literally, "fundamentals of the theory of the potential function" Template:Harv.
43. "Attraction by an ellipsoid and ellipsoidal harmonics" Template:Harvtxt.
44. He classifies those works exactly as "Questioni varie di Meccanica e di Fisica matematica (Various topics in Mechanics and Mathematical Physics)" Template:Harv.
45. According to Template:Harvtxt.
46. According to Template:Harvtxt.
47. See Template:Harv.
48. This first edition is the one which Template:Harv, Template:Harv and Template:Harv refer to.

## Biographical references

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. The "Yearbook" of the renowned Italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of informations about its academic and scientific activities.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. "The Italian contribution to the mathematical theory of elasticity" is a survey paper describing the Italian contributions to the field of elasticity, including brief sketches of the biographies of the main scientists involved.

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. This paper is an Italian translation by the author of an original commemorative paper in Russian, published in the {{#invoke:citation/CS1|citation |CitationClass=citation }}

• {{#invoke:citation/CS1|citation

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• {{#invoke:citation/CS1|citation

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• {{#invoke:citation/CS1|citation

|CitationClass=citation }}. Available from the website of the Società Italiana di Storia delle Matematiche.

## References

• {{#invoke:citation/CS1|citation

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• {{#invoke:citation/CS1|citation

|CitationClass=citation }} ISBN 0-387-13161-2.

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• {{#invoke:citation/CS1|citation

|CitationClass=citation }} (reviews of the first 1953 English edition).

• {{#invoke:citation/CS1|citation

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