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Strength of Materials and Theory of Elasticity in 19th Century Italy pp 123–177Cite as

The Mathematicians of the Risorgimento

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Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 52))

Abstract

The constituting phase of the Kingdom of Italy was a time of recovery of mathematical studies. The political unity facilitated the inclusion of Italian mathematicians in the context of European research, in particular the German one. The internationalization of Italian mathematics is customarily associated with a trip taken in 1858 by some young mathematicians including Francesco Brioschi, Enrico Betti and Felice Casorati in Europe. In a few years we assist in the development of some schools that will maintain their role even in the 20th century. Among them, those promoted by Enrico Betti and Eugenio Beltrami were undoubtedly the most important. In this chapter we present briefly the contribution of two of the leading pioneers and their students.

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Notes

  1. 1.

    p.   230. In fact there was already a first founding of Scuola normale by Napoleon in 1813, along the lines of the French Écoles.

  2. 2.

    Witnessed by a letter of August 25th, 1847 of Mossotti, who from Viareggio, dissuaded Betti from pursuing his initial interest in descriptive geometry [24], p. 231.

  3. 3.

    Curtatone and Montanara are two places near Mantua where in 1848 important battles against the Austrian army were fought and lost by Italian volunteers.

  4. 4.

    p. 233.

  5. 5.

    Georg Friedrich Bernhard Riemann (Breselenz 1826-Selasca 1866) was a German mathematician and physicist. He grew up in poverty, which interfered with his education. He moved to Lüneburg to study and found a friend in his instructor Schmalfuss who gave him free access to his private library. Thus he was able to read the books of Gauss and Legendre. Riemann left Lüneburg and, after a year spent at the university of Göttingen, in 1847, moved to Berlin. Here he was in contact with some of the most prominent German mathematicians of the time, and he studied inter alia Jacobi’s and Dirichlet’s papers. He returned to Göttingen to finish his graduate work; his first argument went back to 1851 and concerned a new theory of functions of a complex variable, a nascent branch of mathematics at that time, that thanks to his contribution received a major boost. In 1854 he read for his qualification for teaching, his second thesis, entitled Über die Hypothesen, welche der Geometrie zu Grunde liegen, published posthumously in 1867 which introduced the concepts of variety and curvature of a manifold, in non-Euclidean spaces.

  6. 6.

    The proposal of Betti and Riemann’s rejection are documented in some letters referred to in [24].

  7. 7.

    pp. 283–290.

  8. 8.

    Betti’s biography is taken from [52]. More details about the scientific aspects of the work of Betti can be found in [28].

  9. 9.

    p. 161.

  10. 10.

    p. 163. Our translation.

  11. 11.

    p. 45. Our translation.

  12. 12.

    p. 179. Our translation.

  13. 13.

    Betti knew that, if desired, he could rewrite all the ‘less rigorous’ steps developed with the use of infinitesimals with a strict mathematics.

  14. 14.

    p. 3. From now on the quotations from the Teoria della elasticità by Betti refer to the offprint by Soldaini of 1874 [20].

  15. 15.

    p. 391.

  16. 16.

    p. 18. Let \( E\) be the tensor of deformation, \(\Theta \) and \(\Lambda \) are respectively the trace of \(E\) and \( E^2\).

  17. 17.

    p. 253. The tilde distinguishes the two Lamé constants from Betti’s \(\mathrm{{\lambda }}\) and \({\upmu }\).

  18. 18.

    p. 20. Our translation.

  19. 19.

    p. 22.

  20. 20.

    p. 40. Our translation.

  21. 21.

    p. 381.

  22. 22.

    p. 379 Our translation.

  23. 23.

    In [53] Green enounced a more general theorem than (3.14), today known as the second Green’s identity:

    $$\begin{aligned} \int _{{{\upsigma }}} v\frac{\partial u}{\partial n}d{\upsigma }+\int _{{S}} u\triangle v\;dS=\int _{{{\upsigma }}} u\frac{\partial v}{\partial n} d{\upsigma }+\int _{{S}} v\triangle u\; dS. \end{aligned}$$

    Functions \(u\) and \(v\) are (whatever their form) endowed with the necessary conditions of regularity ([53], p. 23, par. 3, not numbered equation); by imposing that \(u\) and \(v\) be harmonic functions the Eq. (3.14) is obtained, not made explicit by Green.

  24. 24.

    Equations 32–33, pp. 33–34.

  25. 25.

    Green’s integral formula provides the function \(v\), solution of (3.14), at a point \(P\) internal to \(S\) starting from the knowledge of \(v\) on \({\upsigma }\). On the basis of the (3.15) it is given by:

    $$\begin{aligned} v=\frac{1}{4{\uppi }}\int _{{{\upsigma }}}\overline{v}\frac{\partial }{\partial n}\left( v'+\frac{1}{r}\right) \; d{\upsigma }. \end{aligned}$$

    Here \(r\) is the distance of \(P\) from the points \(Q\) of \({\upsigma }\). The function \(v'(P,Q)\), sometimes called a Green function, satisfies Laplace’s equation and is such that \(u =(v'+1/r)=0\) on \({\upsigma }\) [53], p. 29. More frequently, one calls the whole expression \(u\) a Green function. Among the authors which individuate in \(v'\) Green’s function to signal Betti, Rudolf Otto Sigmund Lipschitz, and Carl Neumann. Green seems to prefer the use of the function \(u\) [53], p. 31, § 5, Eq. 5.

  26. 26.

    p. 297.

  27. 27.

    Equation 43, p. 55.

  28. 28.

    Equation 45, p. 56–57

  29. 29.

    p. 62.

  30. 30.

    p. 63.

  31. 31.

    Equation 48. p. 63.

  32. 32.

    Betti’s passages contain many typos, partially emended in the Opere, edited by da Orazio Tedone.

  33. 33.

    pp. 79–80.

  34. 34.

    Equation 56, p. 81.

  35. 35.

    p. 81.

  36. 36.

    p. 83.

  37. 37.

    Equation 59, p. 84.

  38. 38.

    This is done in all groups of the not numbered equations enclosed among the (59) and (60) of the Chap. 10. The first group represents the linear elastic homogeneous and isotropic relationship between the components of stress and the partial derivatives of the components of the displacement, the second group expresses the local equations of static equilibrium, the third group characterizes the components of the normal (oriented toward the interior according to the convention in the 19th century) to the outer surface of the cylinder, and the fourth group expresses the boundary conditions on the specialized components of the normal just characterized.

  39. 39.

    Equations 60–62, p. 85.

  40. 40.

    Betti did not use a semi-inverse method based on the so-called hypothesis of Clebsch-Saint-Venant, Eq. (62). The vanishing of the stresses on the plane of the section (today indicated with the symbols \({\upsigma }_x,\,{\upsigma }_y,\,{\uptau }_{xy}\)) is a condition for the auxiliary field of displacement useful to be introduced in the reciprocal theorem.

  41. 41.

    Second not numbered equation after the equation 62, p. 86.

  42. 42.

    A discussion in absolute form in the subject is for example found in [76].

  43. 43.

    Equations 67–69, p. 91.

  44. 44.

    p. 91.

  45. 45.

    p. 91.

  46. 46.

    Equation (70) in Chap. 10.

  47. 47.

    Equations (71) and (72 ) of Chap. 10.

  48. 48.

    Last equation of Chap. 11, p. 91.

  49. 49.

    A college with the aim of promoting studies at the university of Pavia, supporting pupils, chosen on the basis of merit, with logistic and cultural opportunities.

  50. 50.

    It is apparent that Beltrami talked about his stay in Verona.

  51. 51.

    Beltrami said, in Latin, tamquam tabula rasa.

  52. 52.

    Tome 1, p. XI. Our translation.

  53. 53.

    Before getting the permanent position as full professor (Professore ordinario), candidates should undergo a period of apprenticeship, during which they were referred to as professors in charge (Professore straordinario).

  54. 54.

    Tome 1, p. XIII. Our translation.

  55. 55.

    p. 319.

  56. 56.

    The first four postulates of Euclidean geometry are:

    1. 1.

      Let it be postulated to draw a straight line from any point to any point.

    2. 2.

      and to produce a limited straight line in a straight line,

    3. 3.

      and to describe a circle with any center and distance,

    4. 4.

      and right angles are equal to one another [66] (p. 318.)

    Besides the postulates, there are also five common notions:

    1. 1.

      Things equal to the same thing are also equal to one another.

    2. 2.

      And if equals are added to equals, the wholes are equal.

    3. 3.

      And if equals are subtracted from equals the remainders are equal.

    4. 4.

      And things which coincide with one another are equal to one another.

    5. 5.

      And the whole is greater than the part [66] (p. 319.)

  57. 57.

    The angle of parallelism \({\upalpha }\) is the angle that a straight line \(s\) forms with the perpendicular \(p\) to a given straight line \(r\) such that all straight lines forming with \(p\) an angle greater than \({\upalpha }\) do not meet \(r\); in Euclidean geometry, \({\upalpha }={\uppi }/2\).

  58. 58.

    p. 290.

  59. 59.

    Remark that \(Q_1,Q_2,Q_3\) in general depend on \(q_1, q_2, q_3\), even if Beltrami did not state it explicitly.

  60. 60.

    pp. 384–385. Our translation.

  61. 61.

    p. 386.

  62. 62.

    Note by Beltrami: Leçons sur les coordonnées curvilignes, Paris, 1859, p. 272.

  63. 63.

    p. 389. Our translation.

  64. 64.

    p. 398.

  65. 65.

    p. 403. Our translation.

  66. 66.

    p. 170.

  67. 67.

    p. 194.

  68. 68.

    Note by Beltrami: For the proof of the sufficiency of these equations, please look at the note at the end of the present Memoir.

  69. 69.

    p. 195.

  70. 70.

    p. 192.

  71. 71.

    Appendix III.

  72. 72.

    p. 75.

  73. 73.

    They are known as the implicit compatibility equations.

  74. 74.

    Note by Beltrami This most known relation already results from the definition formulas (a): however, for the present scope it was necessary to remark that it is included in the nine integrability condition of which word is here.

  75. 75.

    pp. 221–223. Our translation.

  76. 76.

    p. 327. Our translation.

  77. 77.

    p. 329. Our translation.

  78. 78.

    p. 511. Our translation.

  79. 79.

    p. 512. Our translation.

  80. 80.

    pp. 704–714. Our translation.

  81. 81.

    Journal de l’École Polytecnique, cahier XLVIII (1880), p. 1 (note by Beltrami). The paper is entitled: Sur l’équilibre des surfaces flexibles et inextensibles.

  82. 82.

    pp. 420–421.

  83. 83.

    p. 427.

  84. 84.

    p. 425.

  85. 85.

    p. 427. Our translation.

  86. 86.

    p. 429; Beltrami referred to [3].

  87. 87.

    The average curvature is \(\frac{1}{2}\left( \frac{1}{R_1}+\frac{1}{R_2}\right) \), with \(R_1,R_2\) the radii of curvature in the directions \(u,v\).

  88. 88.

    p. 450. Our translation.

  89. 89.

    p. 453. Our translation.

  90. 90.

    Between parentheses we report the year of their beginning of studies.

  91. 91.

    p. 58. Our translation.

  92. 92.

    p. 43. Our translation.

  93. 93.

    Our translation.

  94. 94.

    “The theory of the so-called distortions developed by prof. Volterra contemplates the stresses developing in a not simply connected body, when, once made a cut that does not interrupt the connection, the edges of the cut itself are subjected to rigid relative displacements, after which the continuity of the material is restored by a suitable addition, or subtraction, of material”. [83], p. 350.

  95. 95.

    A wide bibliography and a comment on Volterra’s papers may be found in [64].

  96. 96.

    p. 154. Our translation.

  97. 97.

    p. 159. Our translation.

  98. 98.

    The proof had already been given by Weingarten in [101].

  99. 99.

    Somigliana proved afterwards that this depends on Volterra’s very strong assumption on deformations, supposed regular up to the second derivative.

  100. 100.

    They are Eq. (3.57).

  101. 101.

    They are the six parameters \(l, m, n, p, q, r\) defining the relation (3.57).

  102. 102.

    We often refer to the characteristics, or parameters, of the distortions as being “distortions”, tout court. On the contrary, to Volterra the distortion is a state of the body, defined by the distortion parameters.

  103. 103.

    pp. 165–167. Our translation.

  104. 104.

    p. 875.

  105. 105.

    p. 1005. Our translation.

  106. 106.

    p. 1015. Our translation.

  107. 107.

    p. 37. Our translation.

  108. 108.

    Somigliana’s most important memoirs on the subject are [83, 84].

  109. 109.

    pp. 350–351. Our translation.

  110. 110.

    Cesaro’s biography is extracted from [52].

  111. 111.

    See, for instance, the textbook by Guidi [56].

  112. 112.

    p. 213. Our translation.

  113. 113.

    In [14], v. 2, pp. 329–334.

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  1. Dipt di Ingegneria Strut. e Geotecnica, Università di Roma “La Sapienza”, Rome, Italy

    Danilo Capecchi & Giuseppe Ruta

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Capecchi, D., Ruta, G. (2015). The Mathematicians of the Risorgimento. In: Strength of Materials and Theory of Elasticity in 19th Century Italy. Advanced Structured Materials, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-05524-4_3

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