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The Life and Work of Godfrey Clive Miller, 1893 - 1964

PhD thesis by Ann Wookey




Chapter 5. Miller's application of mathematics in painting, ca 1929 - 1942

notes

  1. 1.See letter from Miller dated 25 September 1935 in Appendix 1, and this chapter, fn 60.

  2. 2.Matrix traces in Warrandyte [Plate 16] were identified by Macnaughton when Conservator at the National Gallery of Australia, and then examined and the grid more fully mapped by myself. Evidence of the structure is subtle. What are most probably graphite marks are visible around the work's perimeter, most often as points only, some of which have been engulfed with a lacquer-like substance, while through the body of the picture plane, traces of straight linear indentations are recognisable below the pitting and cragging of the oil paint surface. Possibly there are lines inscribed in pencil or incised into the picture's priming layer although some incisions would reflect Miller's paint surface scrapings using the plush mat. Pictures that were similarly naturalistic in style to Warrandyte [Plate 16] were the most likely prospects for confirming this analysis. Conveniently, the two canvases Still life, London (Table group, London) [Plate 12] and Jug and egg [Plate 14] were in Victorian public collections. Their initial examination by myself was facilitated through the National Gallery of Victoria, and similar conditions recognised. Since then, Jug and egg has been professionally assessed by the Gallery's Conservation Department; Gillian Leahy, Painting Condition Report, Melbourne, National Gallery of Victoria, December 1993. The Conservator's report has led me to partly revise my earliest conclusions. The resulting matrix extractions of plates 13, 15 and 17 were done by myself, their dotted line grids estimated by matching up peripheral graphite points. Because of directional correlations between these estimations, when I derived the diagram of Plate 17 for Warrandyte most horizontals and two verticals isolated on first examining the picture were omitted. The outcome of the recent conservation assessment of Jug and egg [which led to the diagram of Plate 15] warrants further examination of Warrandyte [Plate 16] - a better identification as to which marks relate to the matrix and which to Miller's use of the plush mat is probable [time considerations disallow this step now]. Miller's plush mat technique was outlined in Chapter 1.

  3. 3.Laughton [1986], op cit, p 35. Such variations of transfer grids are not uncommon English art then [see Claude Rogers, Drawing for 'The Coates Family', 1929, in Laughton - Rogers left the Slade in 1929, just prior to Miller's arrival there, and heard Sickert's London lecture a year later];  ibid, fig 38.

  4. 4.The Listener early in 1935 reviewed the National Gallery's recent acquisition of the seven-panel Sassetta altarpiece featuring the life of St Francis [to which this panel belongs], the work thus becoming well-known to Londoners and so, probably, to Miller; 'Radio news-reel', The Listener, 13 March 1935, found in Bibliography under 'The Listener, London', and see Saint Francis and the wolf of Gubbio panel illustrated [underdrawing not apparent] in John Pope-Hennessy, Sassetta, London, Chatto & Windus, 1939, pl XIX. The Italian art shown at the Royal Academy in 1930 would have been another stimulus. Miller particularly admired Botticelli's painting, studied closely around Europe over these years, for its "underlying Rational Principles" or mathematical or geometrical form.

  5. 5.Morris [1985], op cit, p 37.

  6. 6.Sweet [1992], op cit, and Matila Ghyka, Esthétique des Proportions dans la Nature et dans les Arts, 10th edn, Paris, Gallimard, nd [1st edn, 1927], pp 220-251; see also Appendix 2, entry for Matila Costiesen Ghyka. Miller later credited Sweet with talking to him about geometry and 'golden proportions etc'. He did not say when but the possibilities are limited to 1929-31 and after mid-1935; letter from Miller to George Sweet, Sydney, 3 May 1953, Miller Papers VII, op cit.

  7. 7.The Diagonal was reprinted as The Elements of Dynamic Symmetry; Hambidge, The Elements of Dynamic Symmetry [1948 and 1959 reprints], op cit, and Jay Hambidge, Dynamic Symmetry: the Greek Vase, New Haven, Connecticut, Yale University Press, 1920, pp 23, 24.

  8. 8.Ghyka notes in 1927 that Hambidge had not dealt with the √Ø ratio; Ghyka, Esthétique des Proportions [1927], op cit, p 251 and pl 53.

  9. 9.Ghyka, Le Nombre d'Or [1931], op cit, vol 1, pl XXVIII.

  10. 10.Matila Costiesen Ghyka, Essai sur le Rythme, Paris, Gallimard, 1938, pl XXV.

  11. 11.Osborne [1981], op cit, pp 373, 374.

  12. 12.Letters from Miller to Miss Freedman, London, 26 September 1938, and to C V Allen, London, 28 September 1938 [brackets mine, partly reproduced in Appendix 1], Miller Papers I, op cit, vol 13, pp 276, 283, 284.

  13. 13.Letter from Miller to Miss Freedman, London, 26 September 1938, ibid, vol 13, p 277; Miller noted the 1938 paintings in progress while talking of Hambidge. Dr Margaret Garlake from London has commented that revealed structure was a belief of the International School in architecture, by way of Bauhaus teaching.

  14. 14.Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art [1983], op cit, pp 12-14.

  15. 15.Henry's Éléments d'une Théorie Générale de la Dynamogenie makes several references to Helmholtz on geometrical descriptions of movement in space, and light and colour, etc [from the Optique physiologique]; see Charles Henry, Éléments d'une Théorie Générale de la Dynamogenie autrement dit du Contraste, du Rythme et de la Mesure, avec Applications Spéciales aux Sensations Visuelle et Auditive, Paris, Charles Verdin, 1888, pp 11, 13, 28, 29, 35, and fns to pp 6, 29, 52.

  16. 16.Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art [1983], op cit, pp 15-17. Poincaré, whose mathematics were also known to the English speaking world, was a friend of Miller's "intellectual" hero Paul Valéry; see H Poincaré, 'On the foundations of geometry', The Monist, vol IX, no 1, October 1898, pp 1-43, and Valéry [1932], op cit, translator's preface.

  17. 17.Wilhelm Worringer, 'Abstraction and empathy' [Munich, 1908], trans by M Bullock [Routledge and Kegan Paul, 1963], in Francis Frascina, Charles Harrison and Deidre Paul, eds, Modern Art and Modernism: a Critical Anthology, London, Harper and Row, Publishers, in association with The Open University, 1982, pp 162, 163.

  18. 18.See the very fine analysis by William Camfield of Gris' use of the golden section from around 1912; William A Camfield, 'Juan Gris and the golden section', The Art Bulletin, vol XLVII no 1, March 1965, pp 128-134. The figures of rotated planes, etc, in Power's Éléments de la Construction Picturale apercu des Méthodes des Maitres Anciens et des Maitres Moderne of 1933 support my opinion as to the intuitive nature of the systems devised by artists then rather than being strictly, as seen by Brien, a recourse to 'classical geometry'. Classical geometry enters Power's approach to the extent that he deals only with flat Euclidean space and the golden section is one of the many proportional devices present [the others seem derived from some inherently personal perspective]; J W Power, Éléments de la Construction Picturale apercu des Méthodes des Maitres Anciens et des Maitres Moderne, Fr and Eng edns, Paris, Aux Editions Antoine Roche, 1933, and Brien and Spate [1991], op cit, p 22; see also Appendix 2, entry for John Joseph Wardell Power.

  19. 19.Murray [1987], op cit, pp 309, 310 ['perspective'].

  20. 20.Macdonald identifies the Egyptian system discovered by the nineteenth century German Karl Lepsuis which divides the standing human body, base of feet to top of head, into nineteen equal parts and, to top of head-dress, into 211/4 equal parts [interestingly, the golden section break coincides with the figure's genital organs], the module being the longest finger. Another he notes was the Greek Canon of Polykleitos, known through Vetruvius [and so to the Renaissance] where the body with limbs extended occupies a square, comprises sixteen equal squares, the face and hands occupying a/b each of the vertical and horizontal dimension, head 1/8, head and chest to nipple 1/4, hands and arm to elbow 1/4, etc;  Macdonald [1970], op cit, pp 41-46. In contrast John Marshall, Professor of Anatomy at the Royal Academy of Arts, scaled the female and male bodies into 71/2 equal units; John Marshall, The Rules of Proportion for the Human Figure, London, Smith, Elder & Co, 1879. These differences indicate the wide inconsistencies then between texts and, it can thus be anticipated, in art teaching.

  21. 21.Hay, Fock and Gougy are such volumes; David Hay, Proportion, or the Geometrical Principles of Beauty Analysed, Edinburgh, William Blackwood and Sons, 1843, H C A L Fock, Popular Æsthetic Considerations on the Symmetry of Pleasing Proportions, Cambridge, Hall and Sons, 1877, and Charles Gougy, L'Harmonie des Proportionset des Forms dans l'Architecture d'après les Lois de l'Harmonie des Sons, Paris, Librairie Génénerale de l'Architecture et des Arts Decoratif, 1925.

  22. 22.Macdonald [1970], op cit, p 44.

  23. 23.A T Porter, The Principles of Perspective and their Application to the Representation of the Circle and the Sphere, London, University of London Press, 1927 [dedicated to Tonks because instigated by discussions between them]; see also Appendix 2, entry for the Slade School of Fine Arts.

  24. 24.Incomplete draft from Miller to Lewis Miller, London, 28 May 1935 [second date of 1 June, p 240;  partly reproduced in Appendix 1], and letter from Miller to Lewis Miller, London, 18 June 1935, Miller Papers I, op cit, vol 16, pp 242, 245, 255.

  25. 25.Calender: Session MCMXXX-MCMXXXI, London, University College London, 1930, p 117; see also Appendix 2, entry for Walter Bayes.

  26. 26.Bayes [1927], op cit, p 130 [brackets my addition]. Dictionary definitions of isometric include 'of equal measure or dimensions' and [from 1840] 'applied to a method of projection or perspective, in which the plane of the projection is equally inclined to the three principal axes of the object, so that all dimensions parallel to these axes are presented in there actual proportions;  used in drawing figures of machines, etc'; Little, Fowler, Coulson [1978], op cit, vol 1, p 1118 ['isometric'].

  27. 27.Bayes [1927], op cit, p 133. Bayes' term "point of sight" in relation to isometric drawing seems an anachronism, although his sight plane could be seen as a "point" in extension to left and right. In his example of a boating scene, as distance from the horizon increases the cubic dimensions of clouds and waves "lengthen" on the sight plane; ibid, illustration opposite p 144.

  28. 28.Ibid, pp 144, 145.

  29. 29.Ibid, p 107.

  30. 30.Ibid, pp 261-263.

  31. 31.Squaring-up is known from as long ago an ancient Egyptian art; see Macdonald [1970], op cit, p 42. For the nineteenth century Jones' propositions on the decoration of surfaces and proportion read:

  32. Proposition 8.

  33. All ornament should be based upon a geometrical construction.

  34. Proposition 9.

  35. As in every perfect work of Architecture a true proportion will be found to reign between all the members which compose it, so throughout the Decorative Arts every assemblage of forms should be arranged on certain definite proportions; the whole and each particular member should be a multiple of some simple unit.

  36. Those proportions will be the most beautiful which it will be most difficult for the eye to detect.

  37. Thus the proportion of a double square, or 4 to 8, will be less beautiful than the more subtle ratio of 5 to 8; 3 to 6, than 3 to 7; 3 to 9, than 3 to 8; 3 to 4 than 3 to 5.

  38. Jones [1856], op cit, pp 5, 6. The first combination in each pair of numbers here gives a regular numeric ratio while the second reduces to an approximation of the irregular ratio known as the golden section [0.617] or its inversion [0.383].

  39. 32.Excepting the Purists in Paris from 1918-25 and the stream that links the Dutch de Stijl movement and Mondrian's Neo-Plasticism, ca 1917-28, to the German Bauhaus and International Constructivism of the 1920s and 1930s. Other than Mondrian, many of these artists continued to express a New Age spirit aesthetically weighted in part towards functionalism, a conception at the very heart of mechanical achievement, and so used the forms of planar and solid geometry in an associated sense.

  40. 33.See Christopher Green, Cubism and its Enemies: Modern Movements and Reaction in French Art, 1916-28, Newhaven and London, Yale University Press, 1987, pp 25-36, 44. Green introduced the term "Crystal Cubism" in 1976 because of the regularity with which "crystal" appeared in reviews after 1918 with reference to recent Cubist art; ibid, p 303 fn 62]. Both Metzinger and Severini were shown to Londoners in 1930, with every likelihood that their work incorporating the golden section was seen, while Herbin also showed there during the 1930s; see Exhibition of Paintings by Léger, Metzinger, Severini and Viollier; also Paintings in Space by Kotchar [1930], op cit, also Appendix 2, entry for Abstraction-Création, Paris.

  41. 34.Green [1987], op cit, p 44.

  42. 35.Ibid, p 36.

  43. 36.Ibid, p 26.

  44. 37.See Brooks, 'On abstract art' [1936], op cit, pp 195, 196. Brooks began this article [already noted in relation to Miller's comprehension of Modernist concerns] by noting the 'reawakening of a sense of the necessity of constructive qualities in painting'. Following his observation of the search by modern artists for a 'defined trigonometrical basis' for their constructions, he notes that "formulae" being adopted then, the mid-1930s, are idiosyncratic, as I argue. Brooks infers from this situation that 'Construction then . . . is still dependent upon instinct'. He goes on to claim that the chaos of the known world yet contained 'the law of divine order', and that it is the latter's rhythms that the artist seeks. Other theoricians of the time such as R H Wilenski tend to argue likewise; see also Green [1987], op cit, p 37.

  45. 38.The relevant proposition in The Grammar of Ornament is cited in this chapter, fn 31. Macdonald documents the history of the golden section ratio from Greek times through to Ghyka; Macdonald [1970], op cit, pp 47-49.

  46. 39.Ibid, pp 247, 269. Miller is likely to have known Jones' book since the 1911 examinations he passed in New Zealand were set by the South Kensington Science and Art Department, who sponsored the volume.

  47. 40.Paul Sérusier, A B C de la Peinture, repr, Paris, Librairie Floury, 1950 [1st edn 1921], pp 15-21.

  48. 41.Henderson cites the manifesto as published by the Revue N+1 in Paris over 1936; Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art [1983], op cit, pp 342, 343. The document was republished, its signatories amended slightly [excluding Nicholson for instance], with the magazine Plastique, a Parisian-New York journal that supported abstract art and ran to five issues between 1937-39. Which issue it accompanied was unclear from the London and Paris archival holdings that I sighted; Manifeste Dimensioniste, in Plastique, nd [5 issues, 1937-39].

  49. 42.E Beothy, La Série d'Or. Theorie et Methode Practique. Pour l'Examen Morphologique des Structures et Évolutions Naturelles.  Pour le Perfecionnement Proportional des Produits des Arts Plastiques et de l'Industrie, Paris, Chanth, nd [1939 recorded by the National Art Library, Victoria and Albert Museum, London], pp 2, 3.

  50. 43.Piero Pacini, 'Introduction' and 'Gino Severini: Verifica di un tracciato di Mondrian', in Gino Severini, Dal Cubismo al Classicismo e Altri Sagga sulla Divina Proporzioni e sul Numero d'Oro a Cura di Piero Pacini, Firenze, Marchi & Bertolli Editori, 1972, pp 7, 244.

  51. 44.Henderson cites the fifth postulate as given by Wolfe, "Through a given point can be drawn only one parallel to a given line", as its most familiar formulation [Harold Wolfe, Introduction to Non-Euclidean Geometry, 1945]; Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art [1983], op cit, p 3.

  52. 45.See ibid, pp 3-10. Non-Euclidean geometry was formulated in the 1820s and the geometry of n-dimensions in the 1840s.  Henderson provides illustrations of Lobachevsky-Bolyai and Riemannian geometries; ibid, pls 1, 2.

  53. 46.Ibid, p 96, illustrated in pl 27. Henderson's observation is greatly assisted by Albert Gleizes and Jean Metzinger in Du cubisme of 1912 making direct reference to non-Euclidean geometry and Riemann; see also ibid, pp 44, 79, 80, 93-96, 304, and Albert Gleizes and Jean Metzinger, Du Cubisme, 1912, Eng trans 1913, in Herschel B Chipp, Theories of Modern Art: a Source Book by Artists and Critics, Berkeley, Los Angeles, and London, University of California Press, 1968, p 212. Metzinger's Cubist landscape, 1911, appeared in Ozenfant's Foundations of Modern Art, and thus became a well-known image in the 1930s; Amédée Ozenfant, Foundations of Modern Art, London, John Rodker, 1931 [enl repr, New York, Dover Publications, Inc, 1952], p 50.

  54. 47.Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art [1983], op cit, p 96.

  55. 48.Miller's historical model to his deformist tendencies was not Analytic Cubism but Synthetic Cubism.

  56. 49.Miller letter dated 11 August 1934, which is more broadly cited in Chapter 3 and Appendix 1.

  57. 50.Letter from Miller to Lewis Miller, London, 14 December 1934, Miller Papers I, op cit, vol 16, p 98. These thoughts permeated Miller's philosophy and artistic expression for ever. The bead analogy, for instance, reappeared in a pamphlet he had printed many years later;  see Miller's pamphlet reproduced in Appendix 1.

  58. 51.Bragdon, The Frozen Fountain [1970 repr], op cit, p 37. I am indebted to Sydney artist Frank Hinder for bringing The Frozen Fountain to my notice; Frank Hinder Talking about Godfrey Miller, Dynamic Symmetry, etc, Sydney, 27 March 1988, interview with Ann Wookey, author's collection.

  59. 52.Letters from Miller to Lewis Miller, London, 24 April 1935 [partly reproduced in Appendix 1], Miller Papers I, op cit, vol 16, p 130; for Miller's further descriptions of his trip and Spanish architecture, see same letter, and those to Arthur Fenwick, London, 25 April 1935 [partly reproduced in Appendix 1], to C V Allen, London, 14 May 1935, to Miss Puxley, no address, undated, and to unknown correspondent, no address, undated, ibid, vol 7, pp 77-84, vol 11, pp 33, 34, vol 14, pp 195-199, 230-239, and vol 16, pp 127-135.

  60. 53.Letter from Miller to Arthur Fenwick, London, 25 April 1935 [partly reproduced in Appendix 1], ibid, vol 7, pp 80, 81; Miller continues with:

  61. [P]ools of water geometrically contained, gardens, shrubs, and borders worked all to geometry - circles, circles and wider circles to the containing squares interlaced with paths as diagonals and radii back to the gold fish in the pond.

  62. 54.Incomplete draft letter from Miller to Lewis Miller, London, 28 May 1935 [second date of 1 June, partly reproduced in Appendix 1], ibid, vol 16, p 244.

  63. 55.Letter from Miller to Lewis Miller, London, 25 July 1935 [Miller's underlining], ibid, vol 16, pp 275, 276.

  64. 56.See as an instance, letter from Miller to Lewis Miller, London, 26 August 1935, ibid, vol 17, pp 3-5.

  65. 57.Letter from Miller to Lewis Miller, London, 25 September 1935 [partly reproduced in Appendix 1], ibid, vol 17, pp 60, 61.

  66. 58.R Wilenski, The Meaning of Modern Sculpture, London, Faber and Faber, 1932, pp 87, 155, 159; form for Wilenski is not the object, but rather the shapes made by the sculptor [or the painter's marks]. In the wake of the popularisation of "God geometrizes", this definition became a favourite of Modernists to justify the mathematical æsthetic that entered their art from around 1910. As instances, it was published by the New Yorker Alfred Steiglitz in Camera Work, ca 1911 and by Amédée Ozenfant in his Purist magazine, L'Élan, during 1916 in Paris; see Henderson, The Fourth Dimension and Non-Euclidean Geometry in Modern Art [1983], op cit, pp 215 [fn], 310 [fn].

  67. 59.Brooks [1936], op cit, pp 197, 19.

  68. 60.Edward B Edwards, Pattern and Design with Dynamic Symmetry, repr, New York, Dover Publications Inc, 1967 [originally Dynamarhythmic Design, The Century Company, 1932], pp 5, 13-16, 111-113; see Appendix 3 for Miller's 1.5388 rectangular structured pictures.

  69. 61.Letters from Miller to Lewis Miller, Sydney, July 1940, and to Miss Freedman, London, 26 September 1938, Miller Papers I, op cit, vol 13, p 276, vol 17, p 65, and vol 20, p 159.

  70. 62.Hambidge, The Elements of Dynamic Symmetry [1948 repr], op cit, pp 18, 24, 65, and as an instance of other writers on Hambidge, see Ghyka, Esthétique des Proportions dans la Nature et dans les Arts [1927], op cit, pl 41.

  71. 63.The information about dynamic symmetry and these artists has come from a letter from Daniel Thomas to Ann Wookey, Adelaide, 31 May 1988, author's collection [Grace Crowley, Rah Fizelle and Weaver Hawkins], from Bruce Adams, Ralph Balson: a Retrospective, exhibition catalogue, toured Heidi Park and Art Gallery, Melbourne, 15 August - 24 September 1989, Newcastle Region Art Gallery, Newcastle, 6 October - 19 November 1989, Wollongong Art Gallery, 1 December 1989 - 28 January 1990, Art Gallery of New South Wales, Sydney, 14 February - 1 April 1990, University Art Museum, Brisbane, 12 April - 24 May 1990, Melbourne, Heidi Park and Art Gallery, 1989, Jennifer Phipps, Atyeo, exhibition catalogue, Heidi Park and Art Gallery, Melbourne, 23 November 1982 - 13 February 1983, Melbourne, Heidi Park and Art Gallery, 1982, p 27, and John Paul, Frank Hinder: in Pursuit of the Snark, exhibition catalogue, First International Lewis Carroll Conference, Christ Church, University of Oxford, Oxford, Great Britain, Carlton, Victoria, Gryphon Gallery, University of Melbourne, and Flemington, Victoria, Carroll Foundation, 1989. There is also every likelihood that Claude Flight in London around this time taught the approach to his pupils in linocut printing, among whom were the Australians Dorrit Black, Ethel Spowers and Eveline Syme, the New Zealander Frank Weitzel, and Eileen Mayo who later emigrated to Australasia; see Stephen Coppel, Claude Flight and his Followers: the Colour Linocut Movement between the Wars, exhibition catalogue, toured from National Gallery of Australia, Canberra, to Art Gallery of New South Wales, Sydney, National Gallery of Victoria, Melbourne, National Art Gallery, Wellington, and Auckland City Art Gallery, Auckland, 18 April 1992 - 18 July 1993, Canberra, National Gallery of Australia, 1992.

  72. 64.De Maistre's configuration in The drawing class [Plate 163] is more regular and less complex than Miller's tend towards [the internal rectangles of The drawing class are in the ratio 0.66, that of the 1.5388 rectangle, while the outer dimensions of the painting, as extended to the full figure suggested by the constructional lines, approximates the 0.76 ratio].

  73. 65.Richter [1932], op cit, p 6.

  74. 66.Matila Costiesen Ghyka, The Geometry of Art and Life, 3rd pr, New York, Sheed and Ward, 1966 [1st edn, 1946], p 111; Ghyka terms this principle the Hieros Logos of Pythagoras. See also Richter [1932], op cit, pp 8-12.

  75. 67.Ibid, p 6.

  76. 68.Eric Temple Bell, The Magic of Numbers, London, Whittlesey House, McGraw-Hill Book Company, Inc, 1946, p 174. The golden section has a variety of nomenclatures. It was called the "extreme and mean ratio" and the "mean proportion" by Euclid, appears in Plato's Timæus as the "mean term"; see also Richter [1932], op cit, p 17, and Roy Howat, Debussy in Proportion, Cambridge, Cambridge University Press, 1983, p 2.

  77. 69.Letter from Miller to Lewis Miller, London, 11 December 1936, Miller Papers I, op cit, vol 18, p 265 [bracket my addition, partly reproduced in Appendix 1].

  78. 70.Ibid, vol 18, pp 266, 267 [partly reproduced in Appendix 1]; further constructions do not appear through Miller's correspondence. The psychological experimentation Miller mentioned has not been identified. Experiments of a like nature involving the golden section rectangle were first undertaken by Fechner in the nineteenth century; Fechner, Zur Experimentalen Æsthetik [1871], op cit. In the 1920s this work by Fechner was acknowledged by such well-known psychologists as Thouless who devotes a full chapter of his textbook to 'The psychology of æsthetics', while continuing experimentation led to papers like Feasey's 'Some experiments on æsthetics' in The British Journal of Psychology; Robert H Thouless, General and Social Psychology: a Textbook for Students of Psychology and of the Social Sciences, 5th edn, London, University Tutorial Press Ltd, 1967 [1st edn, 1925], pp 419-432, and L Feasey, 'Some experiments on æsthetics', The British Journal of Psychology, vol XII, December 1921, pt 3, pp 253-272. General public interest continued into the 1930s, for in January 1933 The Listener conducted an experiment in æsthetic taste in which readers were asked to state their preferences for one of a pair on various items. The results were commented upon by Professor Cyril Burt from University College, who early in 1933 also contributed 'The psychology of art' which covered Fechner's work with the "golden cut" rectangle. While Miller may not have arrived back in London then, these items indicate a strong public interest in the topic and this probably carried forward to after his return. Since Burt conducted a colour-association survey among listeners as well, he presumably worked in the field, possibly even lecturing publicly at University College, and could have been heard there by Miller; 'An experiment in æsthetic taste', The Listener, 11 January 1933, p 71, and Burt, ibid, 18 January 1933, p 76, 25 January 1933, p 140, and 8 February 1933, p 217, found in Bibliography under 'The Listener, London';  see also Boring [1942], op cit, pp 275-296.

  79. 71.Richter [1932], op cit, pp 18, 19 [fig 7]; the closeness of their expression here suggests that Richter was another of Miller's sources.

  80. 72.The series was named after Filius Bonacci [Leonardo of Pisa] attributed with its rediscovering in 1202; see Ghyka, The Geometry of Art and Life [1946], op cit, p 13, and Howat [1983], op cit, p 3. Further discussions of the series are found, for instance, in Cook [1914], op cit, pp 97, 442-445, Hambidge, The Elements of Dynamic Symmetry [1948 repr], op cit, p 12, Bragdon, The Frozen Fountain [1970 repr], op cit, p 25, and Edwards, Pattern and Design with Dynamic Symmetry [1967 repr], op cit, p 44.

  81. 73.Howat [1983], op cit, p 2.

  82. 74.Assistance with this analysis has been drawn from Bouleau; Charles Bouleau, The Painter's Secret Geometry: a Study of Composition in Art, New York, Hacker Art Books, 1980, p 66 fn 26.

  83. 75.The ratio following H’B/QB is 0.97. With the series drawing ever closer to 1.00 [the square], practical inexactitudes likely at this high end of the range validates a halt at ~0.94.

  84. 76.Ghyka, The Geometry of Art and Life [1946], op cit, p 22 and fig 22, see also pp 28, 30 and pl IV. Construction diagram also given by Ghyka in Esthétique des Proportions dans la Nature et Dans les Arts [1927], op cit, pl 53, with variations in Essai sur le Rhythme [1938], op cit, pls V, XXIV and XXV; see also this thesis, Plate 171.

  85. 77.The proof is simple: the hypotenuse of a right-angled triangle is given algebraically through

a2 = b2 + c2

...   b = √a2-c2

...   for a = Ø (1.617) and c = 1

      b = √Ø2-1

         = √1.617

         = √Ø

  1. 78.See Edwards, Pattern and Design with Dynamic Symmetry [1967 repr], op cit, p viii; Edwards' research into Greek, early Mycenaean and Cretan art objects led him to claim an origin of 1600 years BC for the reversed spiral and to argue its source as natural shell forms.

  2. 79.Sérusier [1950 repr], op cit, pp 15-21.

  3. 80.Hambidge, The Elements of Dynamic Symmetry [1948 repr], op cit, pp xii, xiii.

  4. 81.The notion of duplicating the cube underlies the dynamic symmetry technique that Hambidge subsequently develops. The Delian problem according to a more recent mathematical historian, Carl Boyer, is unsolvable by 'compasses and straight edge alone', but was "solved" mathematically by Eudoxus of Cnidus (+ ca 355 BC). Boyer identifies the Delian problem as one of 'the "three famous (or classical) problems" of antiquity', the task being, 'given the edge of a cube, construct with compasses and straightedge alone the edge of a second cube having double the volume of the first'. He gives as its origin the plague of Athens, ca 428 BC, for which it is reported that on the oracle of Apollo at Delos being asked how the plague could be averted, the answer given was to double the cubical altar of Apollo. The doubling of its dimensions increased its volume eightfold, not twice, and the plague continued. Boyer relates that the problem [and the other two classic ones] was proven insoluble using compasses and straightedge alone more than 2200 years later; Carl B Boyer, A History of Mathematics, New York, John Wiley & Sons, Inc, 1968, pp 71, 103-106.

  5. 82.See Hambidge, The Elements of Dynamic Symmetry [1948 repr], op cit, pp xv-xvii [Hambidge's underlining]. The constructions of the five Platonic solids are:

  6. .  cube - formed from six squares;

  7. .  tetrahedron - formed from four equal triangles;

  8. .  octahedron - formed from eight equal and equilateral triangles;

  9. .  icosahedron - formed from twenty equal and equilateral triangles;  and

  10. .  dodecahedron - formed from twelve equal, equilateral and equiangular polygons, or pentagons.

  11. 83.Ibid, p 5.

  12. 84.See ibid, pp 3-14.

  13. 85.Ibid, pp 17, 18.

  14. 86.Ibid, p 24.

  15. 87.Ibid, pp 10, 25, 31.

  16. 88.Edwards, Pattern and Design with Dynamic Symmetry [1967 repr], op cit, pp 8, 9.

  17. 89.Jay Hambidge, Practical Applications of Dynamic Symmetry, New Haven, Connecticut, Yale University Press, 1932, p 26.

  18. 90.See Richter [1932], op cit, pp 21, 22.

  19. 91.Bell [1946], op cit, p 121.

  20. 92.Richter [1932], op cit, p 22 [fn].

  21. 93.See Edwards, Pattern and Design with Dynamic Symmetry [1967 repr], op cit, pp 103, 104.

  22. 94.Ghyka, The Geometry of Art and Life [1946], op cit, pl I, and Madonna, repro in Patrick McCaughey, essay, The Great Decades of Australian Art: Selected Masterpieces from the J.G.L. Collection, exhibition catalogue, Melbourne, National Gallery of Victoria, 1984, p 53.

  23. 95.Richter [1932], op cit, pp 24, 25.

  24. 96.These Ø and √Ø rectangle diagrams are found in Ghyka, Le Nombre d'Or [1931], op cit, pl XXVIII, Ghyka, The Geometry of Art and Life [1946], op cit, pls LXXVII, XLVII-XLIX, L, and Ghyka, Essai sur le Rhythme [1938], op cit, pl XXV.

  25. 97.See Appendix 3, especially the table 'All groups, ratio by period'. The earliest paintings [Group A of Appendix 3] noticeably spread over the full range of ratios. Various explanations can be offered for the coincidence of the external dimensions for Miller's earliest pictures from the 1920s with the ratios of proportional rhythm theory and their relatively even spread across this work. These include being due to chance, or as a result either from standard canvas and paper sizes available in Melbourne during the 1920s, or from theory and practice expounded at the National Gallery School and by the Melbourne painting milieu. On the latter, for instance, Helmer records George Bell's interest in Melbourne in dynamic symmetry and golden section proportion as a theory of composition from about 1925 but there is no evidence of Miller's involvement so early; Helmer [1985], op cit, p 10. Proportional notions acquired by Miller through his architectural training is another possibility. Landscape is the principal subject of this early group whatever ratio is present [only one still life, Early still life [Plate 8], is known besides landscape]. The artist was quite unaware, therefore, of the root-two rectangle ["porte d'harmonie"] as the traditional format for landscape as claimed by Sérusier.

  26. 98.See Appendix 3, especially the table 'All groups, ratio by period'. The early transitional works [Group B] are weighted heavily towards the square and the √2 rectangle in incidence, and late transitional paintings [Group C], towards the square and the √Ø rectangle.

  27. 99.See Appendix 3, especially Group B tables. In the early transitional phase [Group B], Miller's landscapes and cityscapes most often take the square while the high-incidence √2 rectangle, together with its neighbour, the harmonic ratio of 0.76, commonly apply for still life.

  28. 100.See Appendix 3, especially tables for groups C and D. For late transitional phase [Group C] and the mature phase [Group D] there is a concentration of still lifes as the √Ø rectangle, but the other ratios occur in lesser incidence as format for the theme [compare with fn above]. This is equally so then concerning landscape and the √Ø and √2 formats, the Madonna paintings and the 0.76 harmonic, other religious themes and the 0.66 ratio [the 1.5388 rectangle] while the reclining nudes tend to narrower dimensions approximating the Ø and √3 rectangles.

  29. 101.Miller's Figure composition is classified as late transitional because of its connections to the Cézanne bather canvases he would have seen in Paris during 1936; see also Appendix 3, sub-group C (i).

  30. 102.See also Appendix 3, sub-group C (ii).

  31. 103.Green comments of a picture of similar type, Portrait of Mme K, 1924, that Miró had 'transformed Cubist geometry with acutely subversive results by absorbing it into his erotic image of female sexuality'; Green [1987], op cit, p 278 and pl 290. The same cannot be claimed for Tête du paysan Catalan whose more "supranatural" and irrational forms seem drawn from beyond reality. The supranatural spatial quality that Miró induced into Tête du paysan Catalan reflects in other of Miller's paintings, like Space movement [Miller's matrix here is from dynamic symmetry and thus not of the same kind as Miró's]; Space movement, reproduced Henshaw, Godfrey Miller [1965], op cit, pl 67.

  32. 104.Ghyka, Esthétique des Proportions dans la Nature et dans les Arts [1927], op cit, pl 74, and Richter [1932], op cit, p 26 fig 15. Ghyka reflects through Miller's recourse to five concentric circles rather than Richter's four, and Richter by way of the complexity of the internal structure [although Miller's matrix is of still greater complexity;  see also this chapter, figs 16 and 17.

  33. 105.Ghyka, Essai sur le Rhythme [1938], op cit, p 20.

  34. 106.Ibid, pp 19, 20, and Ghyka, Le Nombre d'Or [1931], op cit, vol 1, p 51.

  35. 107.See postcard of Fra Angelico's Annunciation sent by Miller to Theni Miller, Sydney, 16 November 1945, Miller Papers I, op cit, vol 13, p 314a.

  36. 108.Richter [1932], op cit, p 25.

  37. 109.The passage in French reads:

  38. Or, dans le corps humain, qui représente, en effet, une symphonie des proportions très complexe mais (pour chaque corps) très unitaire, on sait que la section dorée et les proportions apparentées (spécialement √5) reviennent en leitmotiv constant.

  39. Le corps humain, ou microcosme, fut ainsi très tôt symbolisé par le pentagramme (pentagone étoilé), qui donne une série récurrente indéfinie et « pulsante » des puissances de la section dorée, (Ø = √5+1/2  = 1,618 . . . ), et qui avait été le signe de raillement des pythagoriciens comme symbole de la Vie et de l'harmonie dans la santé . . .

  40. Le schéma du corps (athlète viennois) figuré sur la planche I est encadré par un carré (la planche II ne fait que reproduire l'armature abstraite très remarquable de la planche I); . . . le nombril partage le corps verticalement dans le rapport de la section dorée'.

  41. Ghyka, Essai sur le Rhythme [1938], op cit, pp 17-20 [English translation and square brackets added by myself].

  42. 110.The picture acquired the title Palette design after Miller's death. While the black oval is reminiscent of the hole in an artist's palette, when turned to the left by 90o, a leaning, seated figure with knees raised and the black oval representing the head might almost be present. However, Miller's form is sufficiently obscure to allow the painting's accreditation as a full abstract; see also Appendix 5, No 72 [Palette design].

  43. 111.See this chapter, explanation to figure 8.

  44. 112.See also Appendix 3, sub-group C (iii).

  45. 113.See Appendix 3, sub-group C (iv).

  46. 114.For a description of the pantograph, see Rutherford J Gettens and George L Stout, Painting Materials: a Short Encyclopaedia, New York, Dover Publications, Inc, 1966 [originally published D Van Nostrand Company Inc, 1942], pp 304, 305; on its historical applications, see Kemp [1990], op cit, pp 180, 181. Supposing Miller used such a mechanical device from early in his career, his fitting of a pin to the downward projection intended for larger surfaces would account for the array of pin holes seen around the perimeters of various canvases and which join together as a hidden matrix to Still life, London (Table group, London) [Plate 12], etc [support for his weak arm would have been one justification for his using such a piece of equipment].

  47. 115.Other diagrams showing firstly, two pentagons inverted onto each other within a circle within a square, secondly, combinations of line within a circle and thirdly, dynamic symmetry structural breakdowns within the ï rectangle, the square, the 0.91 and 0.85 ratios of the harmonic progression are also among the Miller drawings held by Australia's National Gallery; National Gallery of Australia, Canberra, acc nos 77.347.338, 77.347.342, 79.1658.A, 79.1647.A, 79.1661.B, 77.347.337, 77.347.341.

  48. 116.Cirlot [1971], op cit, pp 281, 282 ['Seal of Solomon'].

  49. 117.Despite Miller's monetary inheritance at an early age on the death of his mother, to all appearances he lived always in dire poverty while pursuing his artistic and spiritual aspirations.

  50. 118.Quoted previously in this chapter.

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