Loschmidt and Venn

Symbolic Logic in Chemistry and Mathematics
  • Ian D. Rae

Abstract

The circle is a universal symbol of wholeness or perfection. It may represent the cycle of minutes and hours, as in a clock, or of the seasons as depicted in a calendar. Inevitably the cyclic nature of the representation conveys a sense of timelessness — time without end. In religious symbolism the circle is variously the earth, the heavens or a heavenly body like the sun or the moon. In the Christian tradition the circle is most often associated with the cross, although three interlocking circles may be used to denote the trinity. These are powerful symbols of closure or enclosure and they find their way into everyday language when we speak of someone who is ‘part of our circle’ or ‘moves in the right circles.’ We may even speak of an ‘inner circle’ as denoting special status or privilege, such as that attending the admission of magicians to fellowship of the Inner Magic Circle. Stonehenge reminds us of pre-Christian rituals about which we know very little, but many circles of standing stones are found in western Europe and their significance has been widely discussed.

Keywords

Acetone Europe Benzene Radium Smoke 

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Notes and References

  1. 1.
    C.R. Noe and A. Bader, Josef Loschmidt, in J.H. Wotiz, ed., The Kekulé Riddle, Glenview Press, Carbondale, USA, and Cache River Press, Vienna, USA, 1993, p.221.Google Scholar
  2. 2.
    A. Bader, Out of the Shadow, Chem. Ind. (London) 1993, 367.Google Scholar
  3. 3.
    G.P. Schiemenz, Goodbye, Kekulé? Josef Loschmidt und die monocyclische Struktur des Benzols, Natur-wiss.Rundsch. 46(3), 85 (1993).Google Scholar
  4. 4.
    F. Kirchhof, Josef Loschmidt und die Benzolformel, Chem. Ztg. 91(2), 48 (1967).Google Scholar
  5. 5.
    J. Loschmidt, Chemische Studien, Carl Gerold’s Sohn, Wien, 1861; reprinted by the Aldrich Chemical Company, Milwaukee, Wisconsin, 1989 (cat. no. Z18576-0).Google Scholar
  6. 6.
    J. Loschmidt, Konstitutions-Formeln der organischen Chemie in graphischer Darstellung, herausgegeben von R. Anschütz, in Ostwald’s Klassiker der exacten Wissenschaft, No 190, Verlag Wilhelm Engelmann, Leipzig, 1913; reprinted by the Aldrich Chemical Company, Milwaukee, Wisconsin, 1989 (cat. no. Zl 8577-9).Google Scholar
  7. 7.
    CA. Russell, The History of Valency, Leicester University Press, Leicester, 1971, p.97.Google Scholar
  8. 8.
    R. Anschütz, note 6, p.110; Anschütz’s endnote 3, relating to p.5 of the text.Google Scholar
  9. 9.
    Russell gives the following references: A. Crum Brown, The Theory of Chemical Combination, M.D. Thesis, University of Edinburgh, 1861; D.F. Larder, Ambix 14, 112 (1967).CrossRefGoogle Scholar
  10. A. Crum Brown, Trans. R. Soc. Edinburgh 23, 707 (1864).CrossRefGoogle Scholar
  11. 10.
    C.A. Coulson, Valence, Clarendon Press, Oxford, 1952.Google Scholar
  12. 11.
    Coulson’s manuscript was read by his then-coworker, R.D. Brown, now Emeritus Professor of Chemistry at Monash University in Melbourne. In correspondence with the present writer, Brown drew attention to the following paper: L. Pauling, The Nature of the Chemical Bond. Application of Results Obtained from the Quantum Mechanics and from a Theory of Paramagnetic Susceptibility to the Structure of Molecules, J. Am. Chem. Soc. 53, 1367 (1931), and specifically the statement on p. 1370 that ‘...the resonance integral defined qualitatively on what may be called the overlapping of the single-electron eigenfunction involved...’ Coulson seems to have been the first to express this concept diagramatically.CrossRefGoogle Scholar
  13. 12.
    Professor Brown also drew my attention to the work of G.N. Lewis, Valence and the Structure of Atoms and Molecules, The Chemical Catalog Company, New York, 1923, p.74.Google Scholar
  14. 13.
    J.L. Heilbron, Thomson, Joseph John, Dictionary of Scientific Biography, ed. C.C. Gillispie, Charles Scribner’s Sons, NewYork, 1976, Vol XIII, p.362.Google Scholar
  15. 14.
    J.J. Thomson, The Corpuscular Theory of Matter, Archibald Constable & Co. Ltd., London, 1907, p. 121. For the most recent writing on Thomson, see:.Google Scholar
  16. N. Robotti, J.J. Thomson at the Cavendish Laboratory: The History of an Electric Charge Measurement, Ann. Sci. 52, 265 (1995).CrossRefGoogle Scholar
  17. 15.
    Lewis (note 12).Google Scholar
  18. 16.
    R.H. Silliman, William Thomson: Smoke Rings and Nineteenth Century Atomism, Isis 54, 461 (1963).CrossRefGoogle Scholar
  19. 17.
    Lord Kelvin, Plan of a Combination of Atoms to have the Properties of Polonium or Radium, London, Edinburgh, Dublin Philos. Mag. J. Sci. (third series) 8, 528 (1904). See also the precursor of this paper.CrossRefGoogle Scholar
  20. Lord Kelvin, Aepinus Atomized, London, Edinburgh, Dublin Philos. Mag. J. Sci. (third series) 3, 257 (1902).CrossRefGoogle Scholar
  21. 18.
    W. Böhm, Loschmidt, Johann Joseph, Dictionary of Scientific Biography, ed. C.C. Gillispie, Charles Scribner’s Sons, New York, 1973, Vol VIII, p.507.Google Scholar
  22. M. Kohn, Josef Loschmidt (1821–1895), J. Chem. Educ. 22, 381 (1945).CrossRefGoogle Scholar
  23. 19.
    See Noe and Bader (note 1).Google Scholar
  24. 20.
    Heilbron (see note 13) cites Lord Rayleigh, The Life of Sir J.J. Thomson, O.M., Cambridge, 1943.Google Scholar
  25. 21.
    T.A.A. Broadbent, Venn, John, Dictionary of Scientific Biography, ed. C.C. Gillispie, Charles Scribners Sons, New York, 1974, Vol X, p.611.Google Scholar
  26. W.B.H., John Venn—1834–1923, Proc. R. Soc. London, Ser. A 110, p.x (1926).Google Scholar
  27. J.A. Venn, Venn, John (1834–1923), Dictionary of National Biography, 1922–1930, ed. J.R.H. Weaver, Oxford University Press, London, 1937, p.869.Google Scholar
  28. 22.
    H. Midonick, John Venn (1834–1923), in The Treasury of Mathematics, revised by M. Vesselo and R. Vesselo, Penguin Books, Harmondsworth, UK, 1968, p.262.Google Scholar
  29. 23.
    J. Venn, On the employment of geometrical diagrams for the sensible representation of logical propositions, Proc. Cambridge Philos. Soc. Math. Phys. Sciences 1, 47 (1880).Google Scholar
  30. 24.
    J. Venn, On the Diagrammatic and mechanical Representation of Propositions and Reasonings, London, Edinburgh, Dublin Philos. Mag. J. Sci. (5th series) 10, 1 (1880).CrossRefGoogle Scholar
  31. 25.
    J. Venn, Symbolic Logic, Chelsea Publishing Company, New York, second edition, revised and rewritten, 1894.Google Scholar
  32. 26.
    Venn (note 25), Chapter 10, Historic Notes, p.477.Google Scholar
  33. 27.
    E.T. Bell, Men of Mathematics, Simon and Schuster, New York, 1986, p. 152, notes that Euler’s Letters to a German Princess, composed as lessons in mechanics, physical optics, astronomy and sound for the Princess.Google Scholar
  34. of Anhalt-Dessau, ‘were immensely popular and circulated in book form in seven languages.’ The copy examined in the present work was: Lettres á une Princesse d’Allemagne sur divers sujets de Physique & de Philosophie (Academie Impériale des Sciences, Saint Petersbourg, 1868). The pertinent letters are Nos 102 (14 February 1861) to 105 (27 February 1861) which appear on pp.95-125 of Vol 2.Google Scholar
  35. 28.
    L. Couturat, Opuscules et Fragments inédits de Leibniz. Extraits des manuscrits de la Bibliothèque royale de Hanovre, Georg 01ms Verlagsbuchhandlung, Hildesheim, 1961, p.292 (Facsimile reprinting of the original 1903 publication). L. Couturat, La Logique de Leibniz. D’apres des documents inédits, Félix Alcan, Paris, 1901, p.21.Google Scholar
  36. 29.
    A.N. Kolmogorov and A.P. Yushkevich, Mathematics of the 19th Century. Mathematical Logic, Algebra, Number Theory, Probability Theory, Birkhäuser Verlag, Basel/Boston/Berlin, 1992, p. 1.Google Scholar
  37. 30.
    J. Lüroth, Ernst Schröder, in E. Schröder, Vorlesungen über die Algebra der Logik (Exakte Logik), 2nd edition, ed. E. Müller, Chelsea Publishing Company, New York, 1966, p.iii.Google Scholar
  38. 31.
    R. McWeeny, Coulson’s Valence, third edition, Oxford University Press, Oxford, 1979.Google Scholar
  39. 32.
    H.C. Longuet-Higgins, in N.H. March (ed.), Orbital Theories of Molecules and Solids, Clarendon Press, Oxford, 1974, p.vii. See also.Google Scholar
  40. N.H. March. Coulson, Charles Alfred (1910–1974), Dictionary of National Biography, 1971–1980, ed. Lord Blake and C.S. Nicholls, Oxford University Press, Oxford, 1986, p. 182.Google Scholar
  41. Anon. Coulson, Charles Alfred (1910–1974), in The Biographical Dictionary of Scientists-Chemists, ed. D. Abbott, Frederick Müller Limited, London, 1983, p.31.Google Scholar
  42. 33.
    McWeeny (note 29) p. 130.Google Scholar
  43. 34.
    Many’ serious’ mathematicians have no time for symbolic logic, topology or even (in some cases) for geometry, preferring to define mathematics in terms of more algebraic and numerical topics but admitting graph theory (topological, not Cartesian graphs) to the canon. Geometers were heartened when recent efforts to prove fermat’s last theorem rested to some extent on geometric reasoning.Google Scholar
  44. 35.
    Midonick (note 22).Google Scholar
  45. 36.
    L. Hogben, Mathematics for the Million, Allen and Unwin, London, 1960, 3rd ed; Bell (note 27).Google Scholar
  46. 37.
    R.B. Woodward and R. Hoffmann, The Conservation of Orbital Symmetry, Academic Press, New York, 1970.Google Scholar
  47. 38.
    J.M. Tedder and A. Nechvatal, Pictorial Orbital Theory, Pitman, Massachusetts and London, 1985.Google Scholar
  48. 39.
    Few modern theoretical chemists employ orbital diagrams as basic tools of trade, as opposed to those who produce diagrams to help explain their mathematics. An exception is Dr R.D. Harcourt of Melbourne, whose recent work makes extensive use of orbital depictions: R.D. Harcourt, Bohr Circular orbit descriptions of Chemical Bonding via a 2n × n Factorization of 2n2, J. Mol. Struct.; accepted for publication in special issue (1995) to commemorate van’t Hoff and le Bel. McWeeny (note 29), pp. 121-4, comments on such valence bond work as having been ‘largely superseded by MO theory and its variants’ but leaving ‘a substantial legacy of well-established concepts which are still widely used in describing chemical bonds.’ A no doubt unintentional example of the tendency to make concrete even the abstraction of a covalent bond, the index entry in McWeeny reads ‘Valence bone (sic) (VB) theory’!.Google Scholar
  49. 40.
    J.W. Kennedy and L.V. Quintas, Applications of Graphs in Chemistry and Physics, North-Holland, Amsterdam, 1988, p. 1.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Ian D. Rae
    • 1
  1. 1.Victoria University of TechnologyMelbourneAustralia

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