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Computations by Means of Drawings

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Strength of Materials and Theory of Elasticity in 19th Century Italy

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

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Abstract

The second half of the 19th century saw a very quick diffusion of graphical statics. Lectures on graphical statics were given in Switzerland (Zurich); in Germany (Berlin, Munich, Darmstadt, Dresden); in the Baltic regions (Riga); in the Austrian-Hungarian empire (Vienna, Prague, Gratz, Brunn); in the United States; in Denmark. The author that mainly developed its techniques was the German scholar Carl Culmann, who placed graphical statics besides the newborn projective geometry. Culmann’s approach was enthusiastically followed in Italy, where, first in Milan at the Higher technical institute, then, after 1870, in many Schools of application for engineers, among which those of Padua, Naples, Turin, Bologna, Palermo, Rome, and, eventually, also in the universities of Pisa and Pavia, courses of graphical statics were activated. The Italian scholar who collected Culmann’s inheritance, and extended it, was Luigi Cremona.

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Notes

  1. 1.

    The issue was anticipated by the print of handouts of his lectures on graphical statics at the Polytechnic of Zurich, in 1864 and 1865. In 1875 the first volume of the second edition, designed in two volumes, was issued, but Culmann died in 1881, before completing the second volume. A French translation of the first volume of the second edition was issued in 1880 [29].

  2. 2.

    The expression graphical calculus was introduced for the first time by Louis Ézéchiel Pouchet  [70].

  3. 3.

    Projective geometry after Poncelet would be developed in France by Chasles, in Germany by Karl Georg Christian von Staudt, Plücker, Möbius, Steiner, and Clebsch.

  4. 4.

    We report, as examples, the references quoted by Cremona:

    K. Von Ott, Die Grundzüge des graphischen Rechnens und der graphischen Statik, Prag, 1871; J. Bauschinger, Elemente der Graphischen Statik, München, 1871; F. Reauleaux, Der Constructeur (3rd edition) (2 Abschnitt), Braunschweig, 1869; L. Klasen, Graphische Ermittelung der Spannungen in den Hochbau-und Brückenbau-Construction, Leipzig, 1878; G. Hermann, Zur graphischen Statik der Maschinengetriebe, Braunschweig, 1879; S. Sidenam Clarke, The principles of graphic statics, London, 1880; J. B. Chalmers, Graphical determination of forces in engineering structures, London, 1881; K. Stelzel, Grundzüge der graphischen Static und deren Anwendung auf den continuirlichen Träger, Graz, 1882; M. Maurer, Statique graphique appliquée aux constructions, Paris, 1882 [22, pp. 341–342].

  5. 5.

    p. XI. Our translation.

  6. 6.

    p. XXIV. Our translation.

  7. 7.

    p. X. Our translation.

  8. 8.

    Introduction. Our translation.

  9. 9.

    Hamilton was talking about himself in third person.

  10. 10.

    p. 3.

  11. 11.

    p. 3.

  12. 12.

    p. 12. Our translation.

  13. 13.

    Chapter 3.

  14. 14.

    This volume was privately issued in New Haven in two parts, the first in 1881, the second in 1884.

  15. 15.

    Besides the volumes [50], Heaviside published several papers on the subject in the journal “The Electrician” during the 1880s.

  16. 16.

    v. 1, p. 16. Our translation.

  17. 17.

    p. 251.

  18. 18.

    p. 250.

  19. 19.

    Italics is ours.

  20. 20.

    p. 251.

  21. 21.

    Remark that the first one corresponds to Euler’s theorem for polyhedra in space; the second, mechanically interpreted, provides the necessary condition for a plane truss to be kinematically and statically uniquely determined.

  22. 22.

    p. 258.

  23. 23.

    p. 258.

  24. 24.

    p. 258.

  25. 25.

    p. 7.

  26. 26.

    We have examined some features of this school in [9].

  27. 27.

    In this book the uncertainty on the correct writing of Culmann’s Christian name (with a capital K or C) is also told to derive from the non-uniqueness of German language of the first half of the 19th century.

  28. 28.

    The second edition of the work is dated 1875.

  29. 29.

    (*) Varignon mentioned it in its Nouvelle mécanique published in 1687 (Note by Culmann).

  30. 30.

    (**) It is by chance that in 1845 an autographed course without the name of the author, having title: Instruction on the stability of constructions, has fallen into our hands. He who gave it to us attributed it to Mr. Michon. That course contains six lessons on the stability of vaults and four on that of coating walls (Note by Culmann).

  31. 31.

    pp. IX–XII. Our translation.

  32. 32.

    Culmann’s intentions, as we read in the preface in French to the second edition, reported above, were: “The second volume will contain a series of applications to beams, to frameworks, to arches, and to retaining walls”.

  33. 33.

    The copy of the German edition of Graphische Statik that we have consulted was property of Saviotti, as written in pen in one of the first pages, and contains some annotations, quite likely by Saviotti himself.

  34. 34.

    Still until twenty or thirty years ago, almost identical techniques of graphic integration and derivation were taught in some engineering schools in Italy.

  35. 35.

    p. XIII. Our translation.

  36. 36.

    p. 291. Our translation.

  37. 37.

    Remember that a one-to-one correspondence exists among the matrices that, with respect to a basis, are the image of symmetric tensors of order two, and plane conics. As a consequence, this representation is a natural graphic representation of Cauchy’s theorem on the state of stress at a point.

  38. 38.

    The standard font is ours. It is a central ellipsis similar to that of inertia, the coefficients of proportionality being different; \(\epsilon \) is the elastic modulus, \({\mathfrak I}\) is a moment of inertia, \(\Delta s\) is an element of arc.

  39. 39.

    pp. 530–531. Our translation.

  40. 40.

    Gaudenzio Cremona had already had three children from his preceding marriage with Caterina Carnevali.

  41. 41.

    Barnaba Tortolini (1808–1874), priest and mathematician, professor of mathematics at the University of Rome, founded the first Italian scientific journal with international diffusion.

  42. 42.

    In the following we will consider the two notions of polarity and reciprocity as equivalent, even though in projective geometry they are distinct in general. Let us give some definitions:

    A reciprocity in a plane, where any two homologous elements correspond to each other in a double way (by an involution), that is a reciprocity equivalent to its inverse, is called a polar system, or a polarity; a point and a straight line corresponding to each other in a plane polarity are called pole and polar one of the other.

    Polarity in a plane may also be defined as a one-to-one correspondence among points and straight lines, such that: if the straight line corresponding to a point A (its polar) passes through a point B, the corresponding (polar) of B passes through A.

    Remark. Analogously (in space) we may define polarity in a star [31], p. 186. Our translation. (A.5.11)

    To each polarity we may associate a conic, ellipsis, hyperbole, or parabola. “A set of the points and of the straight lines conjugated with themselves is said fundamental conic of the polarity” [31, p. 204.

  43. 43.

    If the resultant vanishes, the central axis coincides with the line at infinity of the plane.

  44. 44.

    See also [5, 6, 29, 60].

  45. 45.

    p. 505. Our translation.

  46. 46.

    Explicit figures, practically coinciding with those drawn also nowadays, are on pp. 190–191. Varignon had already expressed these ideas in [80].

  47. 47.

    p. 202.

  48. 48.

    vol. 1, pp. 190–191. Our translation.

  49. 49.

    Conversely, a force applied at a point may be decomposed in the sum of two forces applied at the same point, along two assigned directions.

  50. 50.

    A couple is a system composed by two parallel forces of equal magnitude and opposite sign, the distance of their lines of actions being called arm of the couple. Just like a force causes the variation of a translational motion (that is it is the prototype of interactions spending power on translations), a couple causes the variation of a rotary motion (that is it is the prototype of interactions spending power on rotations). A couple with null arm is thus a system composed by two collinear opposite forces, trivially equivalent to a null system.

  51. 51.

    This operation is, however, implied by the operation 2.

  52. 52.

    The idea is that the sum of the two forces provides in any case the resultant of the system, while the arbitrary choice of the line of action of one of the two let it be posed at a distance such as to warrant the equivalence of the resultant moment \(M\).

  53. 53.

    vol. 2, p. 72. Our translation.

  54. 54.

    p. 345. Our translation.

  55. 55.

    p. 348. Our translation.

  56. 56.

    p. 535.

  57. 57.

    Cremona’s reciprocal figures were investigated until relatively recent years, See, for instance, [5, 58, 74, 83].

  58. 58.

    p. 352. Our translation.

  59. 59.

    p. 356. Our translation.

  60. 60.

    v. 1, pp. 540–541.

  61. 61.

    vol. I, p. XI. Our translation.

  62. 62.

    vol. II, footnote on p. 21.

  63. 63.

    vol. I, p. X.

  64. 64.

    vol. II, pp. V–IX.

  65. 65.

    vol. I, p. XI. Our translation.

  66. 66.

    vol. II, pp. 3–4. Our translation.

  67. 67.

    vol. II, p. 99.

  68. 68.

    Recalling many of his predecessors, among whom Varignon, Venturoli, Clebsch, Mossotti, Belanger, Ritter, he also remarked that the law of composition of forces could have been deduced from that of the motions if one had put dynamics before statics, and also provided an interesting historical resume on the subject ([72, vol. II, pp. 12–14]). Similar remarks had also been provided by Gabrio Piola, who had started precisely from this point of view.

  69. 69.

    In the first article of the second volume Saviotti declared how force was a primitive element for him:

    A body cannot shift itself by its own if it is at rest, nor can modify the movement it has without the intervention of a cause exterior to it. [...] We do not investigate its origin; we only evaluate its effect [...] [72, vol. II, p. 5.] Our translation (A.5.26)

    This vision was substantiated by the model of matter:

    In Statics we consider ideal bodies, that have all their dimensions infinitesimal, without having a determined shape, and that are called elements or material points. In addition, we consider ideal forces applied to them, with finite magnitude, which, being concentrated onto a point, are called concentrated forces. [...]

    Several points of application are called rigidly connected when they be linked in such a way that their relative distances always remain unchanged, or when they are part of a non-deformable body.

    Even though we consider bodies as material in Statics, still at first we will make abstraction of their weight, that is, we will consider them as geometrical bodies, or rigid joints, infinitely resistant, of the points of application [of outer and inner forces] that concur to form a system with invariable shape [72, vol. II, pp. 6–7]. Our translation. (A.5.27)

    The mechanical model we already saw in Maxwell and Menabrea is apparent.

  70. 70.

    Giuseppe Jung was born in Milan in 1845 and died there in 1926. In 1867 he graduated in Naples and soon after, he went back to Milan, where he became Cremona’s assistant. In 1876, when at the Higher technical institute in Milan also the two-year period preparatory for the studies in engineering was introduced, he was appointed professor of Projective geometry and Graphical statics, but he became full professor only in 1890. He was fellow of the Regio istituto Lombardo.

  71. 71.

    Zucchetti would have later on be an assistant to the chair of Steam engines and railways in Turin.

  72. 72.

    Italics is ours.

  73. 73.

    p. 6. Our translation.

  74. 74.

    Here is the theorem to which Zucchetti refers: “The figures that may be seen as plane projections of polyhedra always admit reciprocal figures”; [84, p. 46].

  75. 75.

    pp. 6–7. Our translation.

  76. 76.

    Camillo Guidi was born in Rome on July 24th, 1853. He graduated at the School of Application for Engineers in Rome in 1873. Guidi was professor of Graphical statics from 1882 at the School of Application for Engineers in Turin. He was appointed with the chair of Scienza delle costruzioni (Structural mechanics) from 1887, and in 1893 he became director of the Cabinet of Structural mechanics and theory of bridges. He died in Rome on October 30th, 1941. His investigations on reinforced concrete have remained famous.

  77. 77.

    We remark that, besides the textbooks by Saviotti and Zucchetti, two other books were particularly considered in the Schools of application for engineers, the already quoted one by Bauschinger [1] and that by Maurice Lévy [59].

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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). Computations by Means of Drawings. 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_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-05523-7

  • Online ISBN: 978-3-319-05524-4

  • eBook Packages: EngineeringEngineering (R0)

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