Solid mechanical problems in the “Risorgimento” generation

In the classical work of Isaac Todhunter and Karl Pearson A History of Elasticity and of the strength of materials [7] more than twenty-three pages are dedicated to Gabrio Piola with a critical and detailed descriptions of six papers published in Opuscoli (1833) and in Memorie della Società Italiana di Scienze moderne di Modena.

It is assumed that Carlo Cattaneo, who was exiled in Switzerland, can be considered as the promoter of the enterprise.

The first Italian railway line was the Napoli-Portici in 1839; in 1857 the connection between Milan and Venezia was completed. In 1890 the Italian railway line was 1369 km and the European mean velocity had reached 100 km/h.

The congress “Francesco Brioschi (1824–1897) Convegno di studi matematici” has been dedicated to him in 1997 to the centennial celebration of his death. The proceedings have been published in 1999. In the following year appeared two volumes “Francesco Brioschi e il suo tempo, 1824–1897”. (Vol. I Saggi edited by C.G. Lacaita and A. Silvestri, Vol. II Inventari edited by C. Brunati, D. Franchetti, P. Papagna, P. Pozzi). They constitute an important evidence too. It’s interesting to look over the Inventari, as they offer a rich exchange of personal correspondence on the most different subjects, not only the scientific, but also the political and organizing ones with many important personalities.

I think it’s worth to mention [11] Brigaglia that in his Brioschi, Cremona e l’insegnamento della geometria nel Politecnico (Saggi, vol. cit., pp. 408–409) states “there is no doubt … that in the first year after the Unity of Italy Brioschi thought not only to the formation of a technically clever engineer, but a new ruling class too.

The Lamé’s formulations go back to 1852.

De Saint Venant himself, in one of the long notes to the French translation of the Clebsch’s Treatise, though accepting the Lamé’s and Green formulation, believes that for a isotropic compact material, the elastic constants can be reduced to one only.

Enrico Betti (Pistoia 1823–Soiana 1892), mathematician and physicist, was appointed Member of the Class of Physical, Mathematical and Natural Sciences of the Istituto Lombardo on 25.08.1864. He has been Professor of Higher Algebra, later on of Analysis and Higher Geometry and, at last, of Physics-Mathematics at the University of Pisa, where he taught Astronomy and Celestial Mechanics. He was headmaster of the Scuola Normale from 1865 to his death. He was one of the XL of the Società Italiana delle Scienze and Member of the Accademia dei Lincei; he was Member of the Society of Mathematics of London, Göttingen and Berlin too. Brioschi and Cremona worked together for a new foundation to teach geometry in middle school. Cremona fought in the University Battalion during the 1848 war. From 1874 to 1876 he was General Secretary of the Ministry of Education, he was Member of the Council of Education and since 1884 Senator of the Kingdom.

This issue is quoted by A.E.H. Love in the famous book published in New York in 1926. In this work it’s underlined that the Betti’s Theorem is noteworthy from a methodological and demonstrative point of view.

Luigi Cremona (1830–1903) (brother of the painter Tranquillo Cremona) had appointed Member of the Class of Physical, Mathematical and Natural Sciences of the Istituto Lombardo, Correspondent 1864 and Ordinary Member in 1868. He was a Francesco Brioschi’s student. He began his career as Professor in 1860 at the University of Bologna, where he taught Higher Geometry and continued it at the Politecnico di Milano with Higher Geometry and Graphical Statics courses, and at the end he was in Rome since 1873 as Higher Geometry and Mathematics Professor. In 1875 he was appointed Senator of the Kingdom. From 1861 to 1879 he presented 13 memoirs to the Istituto Lombardo. L. Berzolari gave the commemoration speech to the Istituto Lombardo.

These works were mentioned with emphasis by Culmann (1821–1881) of the ETH Zürich, and translated in English by T.H. Beare in 1890.

The engineers mastered quickly the graphical methods and these methods became effective and often irreplaceable instruments, handled with certainty but with disinterest to the advanced higher Geometry that had produced them. And this point for Cremona, who conceived geometry as a training for the spirit, should have been surely disappointing.

Eugenio Beltrami (1835–1900), was student of Mathematics of the Collegio Ghisleri di Pavia, when, in 1855, he was expelled because he had showed sentiments of Italian character. After 1859 he introduced himself to Brioschi, who groomed him for the scientific research, though he hadn’t a degree (he will never get it). Mathematician, Brioschi’s student, he began his Professor career at the University of Bologna in 1862 and continued it in Pisa, Pavia and, at the end in Rome. He was Senator of the Kingdom and President of the Accademia dei Lincei. He has been Correspondent Member of Class of Physical, Mathematical and Natural Sciences of the Istituto Lombardo since 20.02.1868 and has been Ordinary Member since 1887. He presented 24 memoirs to the Istituto Lombardo from 1868 to 1887. In 1892 he delivered the commemoration speech of Betti (Rend. Circolo Matem. di Palermo). He gave the commemoration speech of Brioschi and a Speech honoring Kelvin at the Accademia dei Lincei. He succeeded to Clausius at the Academy of France. He participated in the life of the Istituto Lombardo with 32 memoirs, 4 of which concerning the Solid Mechanics, with reference to the Theory of Elasticity. Somigliana pronounced his necrology. Maurice Levy published Beltrami’s commemoration in the Comptes Rendus de l’Académie des Sciences.

The Beltrami’s cap, that is the pseudo sphere at Gauss curvature negative constant, is hold by the Mathematics Department of Pavia University.

In 1885 Menabrea published the theorem based on the principle of “least work”, but he does not offer an acceptable proof of this principle. Castigliano’s thesis for the engineering’s degree at the Politecnico di Torino contains a statement of his famous theorem with some applications to the theory of structures. In a more extended form, the work was published by the Accademia delle Scienze di Torino in 1875 (from S. Timoshenko op. cit).

We must remember the Beltrami’s memoirs: Sulle equazioni generali dell’elasticità. Annali di Matematica pura ed applicata, Milano 1881; Sulla teoria del potenziale, R. Ist. Lomb., 1883; Sulla rappresentazione di forze newtoniane per mezzo di forze elastiche. R. Ist. Lomb. 1884; Sull’uso delle coordinate curvilinee nelle teorie del potenziale e dell’elasticità. Regia Accademia delle Scienze di Bologna, serie IV, tomo VI, 1885; Intorno al mezzo elastico di Green nota II. R. Ist. Lomb, 1891.

… Taluni … pongono … [nello studio delle scienze fisiche e matematiche] … ogni cura nella parte pratica di esse, e, formandosi un’idea troppo ristretta della loro utilità, rifuggono da ogni ricerca che non si presti ad una immediata applicazione. … Sappiamo di alcune dottrine, credute sul principio infeconde di conseguenze, che nel progresso riuscirono ad ottimo fine …”. Gabrio Piola, Prefazione, in “Opuscoli matematici e fisici di diversi autori, 1832.

“Condizione … [per] … concepire un … rinnovamento della partecipazione nazionale all’evoluzione scientifica e tecnologica Europea … [è] … la piena partecipazione ai settori più avanzati nella ricerca”.