Marcus FringsDarmstadt University of TechnologyArt History El-Lissitzky-Str. 1 D-64287 Darmstadt GERMANY
For Vitruvius the educated architect has to know arithmetic and geometry [I,1,4]. Thus his own rules of design combine both methods: all plans of temples are completely developed by geometrical partitions and relations [IV,4,1; IV,7], with the round temple even the elevation [IV,8,1-3]. Arithmetic modular operations using the lower diameter of the column as the base module regulate the dimensions and proportions of the columns and intercolumniations [IV,1]. Here Vitruvius uses geometrical methods too - the height of the entablature and its decoration [III,8-12] and the tapering of the shafts [III,3,12] are bound to the columns' dimensions. Even the modulus is derived from a division of the temple's width, dependent on each temple-type [III,3,7; IV,3,3-4; IV,3,7]. This interrelationship of modularity and geometry is also found in the detailed analysis of the proportions of the human body [III,1]. Vitruvius presents his canon, the famous figure of a man in circle and square, in support of his claim that "no Temple can have a rational composition without symmetry and proportion, that is, if it has not an exact calculation of members like a well-shaped man" [III,1,1].[2] So the body is a model by virtue of its perfection of symmetrical shaping first and foremost and not in its inherent proportions, which is often misunderstood; cf. [Frings 1998]. The human body, as an example of modular creation from nature, is chosen by Vitruvius as a paradigm for the required rules of proportion. The Roman architect and author is far away from the development
of a convincing system of proportions. However, his precepts
of design are a notable attempt to unify different methods apparently
drawn from the Greek and Roman traditions. His knowledge of musical
theory seems to exceed his skills in geometry, in which he takes
less interest. The only sophisticated geometry is ad quadratum,
which organizes the capital of the Corinthian order [IV,1,11].
We do not find the Golden Section in the In the Middle Ages architectural writing may be categorized according to the source: by scholars in encyclopedias referring to Latin authors like Vitruvius or Pliny, by theologians who explain the building of the church (ecclesia materialis) in allegoric ways, pointing to the ecclesia spiritualis, and lastly by the practitioners of architecture themselves. In this third category we have some books of simple advice (e.g., the mappa claviculae, 8th century [Mortet 1911: 225-267]) and of patterns and building designs (e.g., Villard d'Honnecourt, ca. 1220/25-1230/35 [Hahnloser 1972]). In all three categories the Golden Section is lacking.
Luca Pacioli, however, was a great admirer of the Golden Section,
as evidenced by the name of his treatise,
Here we clearly recognize Pacioli's metaphysical character, Platonic and Christian. In the rest of the first book, Pacioli describes in detail the geometrical attributes of the "proportion having a middle and two extreme ends", in medieval words used also by Pacioli. He does not advance the field of research in his discipline, does not give any advice or thought about the practical application in the pictorial arts. In fact Pacioli's achievement consists more in diligent compilation than in scientific originality.[5] Harsh critique has also been uttered, since Pacioli declares much but proves nothing.[6] Of course the What he offers would scarcely have satisfied the architects
- his Pacioli understands the Vitruvian figure of the man in circle
and square with Filarete in the sense that the geometrical figures
have their origin in the human body (ch. I,1, f. 25r). He is
the first who describes the so-called This shows the eminent role Pacioli gives to geometry. In fact, he harshly criticizes the architects who make use of geometry - and other craftsmen as well - "even though they don't know it". They do not even know of the world's order in measure, number and weight, so that often their buildings fall down. [12] In this chapter he recommends a special three-dimensional
form often used in architecture, the body with 72 planes, which
he treats also in the Another chapter of his architectural treatise recommends the Platonic regular bodies in its title.[14] So buildings were "not to criticize", he writes in this chapter, if for details like bases or capitals here and there were used these mathematic bodies, although Vitruvius did not mention them at all. They not only "adorn buildings, but also give the scholars and wise opportunity to speculate, in so far as they are always constructed by that sacred and divine proportion", which we call the Golden Section.[15] Then he tells the anecdote of Phidias who gave a statue of Ceres in Rome an icosahedron for the element of water, earning highest praise. Hence it seems to be clear that Pacioli does not recommend at all to replace traditional parts, whose "correct" formation has been the architects' and sculptors' task for decades, by the Platonic polyhedra. Pacioli himself describes the orders in detail, as already stated, in Vitruvian manner. On the contrary he thinks of a selected insertion as the case may be, like we know it from other examples in the history of architecture (for instance Vitruvius's explanation of caryatids; [Vitruvius I,1,5-6]), which attest to the architect's historical knowledge. There is no general recommendation of the Golden Section for use in architecture, and it is only implicit in the Platonic solids. On the contrary Pacioli recommends to architects - besides the Vitruvian rules - the forms of circle and square, whereas the formerly praised triangle is lacking. He also recommends simple ratios of integral numbers like 1/2, 1/3, 3/4, 2/3 etc. If irrational proportions like those based on Ö2 must be used, the architects should draw them separately and not calculate them.[16] So Pacioli advises the practitioners on commensurable proportions, which are easy to construct on drawing board or building site - he apparently thinks of practicability and is aware of the limited mathematical skills of his readers. In this context the Golden Section does not play any role. The three illustrations at the end of Pacioli's text only
explain the technical terms of some parts of the building (column
and pedestal, entablature) and present - without relation to
the text - an ideal architecture, the façade of Solomon's
temple. The second version of the profiled head is titled Neither in the text nor in the illustrations is the Golden Ratio recommended for practical use. This is astonishing, given the great valuation as divina proportione, but has its reasons: Pacioli does not want to give concrete proportional precepts but to show the meaning of calculated proportions in architecture in general, similar to Vitruvius. Pacioli's role in architectural theory has been overestimated for a long time.[17] He presents himself as mathematician, who certainly deals with Vitruvius, but who does not develop his own propositions on how to design a building. In stark constrast, this is exactly the achievement of Sebastiano Serlio (1475-1554). He dedicates the first volume of his great architectural treatise to the orders [Serlio 1537]. His consistent system of the five orders with simple proportional construction is the lesson book the practitioners had been waiting for. With a juxtaposition of Italian text and instructive figures, Serlio advisedly addresses the builders. With this treatise he becomes the creator of the modern orders. Following the same traditions as had Pacioli, Serlio uses
simple, commensurable proportions. The columns are constructed
in a module system, the entablature and its parts use divisions
of the module, and also the pedestal is designed by geometrical
operations. The dado is an ashlar, which is slendered from Tuscan
to Composite order - in the Tuscan order it is a cube, in the
Doric order its height is equal to the square root of its width,
in the Ionic one and a half its width, in the Corinthian one
and two thirds, and in the Composite at last the cube is doubled.[18] These
are some of the seven most important proportions which Serlio
[1545: 21r] later mentions with their classical terms ( This surely cannot be explained only by Serlio's pragmatic attitude. There is no need for such a complicated proportion. The same is true for the other great treatises from Vignola and Palladio, Scamozzi and the Italian classicists to the theory of the 19th century. Nowhere occurs the Golden Ratio; in the overwhelming majority of these treatises, the simple, commensurable ratios are recommended, and the only irrational proportion is the well-known Ö2. In addition to the human body as relative figuration, Venetian authors of the Cinquecento (e.g., Francesco Giorgi, Barbaro) introduce musical harmonies. When Palladio deals with the heights of rooms, the Golden Ratio could have been employed, but Palladio sets the length, width, and height using other mathematical relations [Palladio 1570: ch. I, 23]. Let us look at two persons at the edge of architectural theory,
who were active as artists and theorists of proportions. Even
a master of the applied perspective such as Leonardo, Pacioli's
collaborator, did not take interest in the Golden Ratio but gives
only approximations to the regular pentagon.[19] Albrecht Dürer indeed knew - perhaps
by Pacioli himself on the second journey to Italy 1506/7 - a
correct construction of the regular pentagon by means of the
Golden Section, but he did not seem to make up his mind about
their connection [Dürer 1525: II,15]. His practical instrument
called
Zeising's work remains largely unstudied: his sources and
inspirations, his writings and their impact.[21] After having retired from active service
as a secondary-school professor in 1853, he devoted himself to
his studies and lyrics. Even the first of Zeising's publications
[Zeising 1854] declares in its title his program: Having extensively discussed actual theories Zeising develops his own aesthetics, born from a romantic, idealistic tradition. In this theory the Golden Section plays an important role as the perfect balance between absolute unity and absolute variety. Zeising is convinced that in the Golden Section "is contained the fundamental principle of all formation striving to beauty and totality in the realm of nature and in the field of the pictorial arts, and that it from the very first beginning was the highest aim and ideal of all figurations and formal relations, whether cosmic or individualizing, organic or inorganic, acoustic or optical, which had found its most perfect realization however only in the human figure" [Zeising 1854: V]. Thereto he presents his own proportional analyses of the human
body, comprising vertical and horizontal measures, but illustrates
it separately ( Less intensively he looks at works of art, at the Parthenon - he is the first to publish an analysis in which he finds the Golden Section - at two important churches in Germany (Cologne cathedral, St. Elizabeth in Marburg/Hesse), and at Raphael's Sistine Madonna. To find the Golden Section in music gives him trouble, but he embraces poetry, philosophy and religion. In an appendix he gives artists practical advice on the construction of the human figure and on the measuring of proportions. Only some years before Zeising published his One should not underestimate Zeising's impact. It is he who introduces the Golden Section into the literature on art and so placed it at the artists' disposal. From the above historical discussion we can conclude that Zeising "discovered" the Golden Section for architecture and the pictorial arts in general.[25] In a strange combination of idealistic aesthetics and pretended exact-scientific analysis, he proclaims a simple truth about the rather complex manifestations of nature and about the scarcely less divergent forms of art, which in this time did not show a stylistic unity. In an age of deep uncertainty Zeising returns to an anthropocentric, normative aesthetics, which seemed to have become out of favor since the French Rationalism and British Empirism of the 18th century. He rehabilitates the man as the crown of creation in the actual Darwin debate, and he rehabilitates the concept of the inherent harmony of the cosmos, which he sees as intelligible and in keeping with the artistic production of mankind. It is probably this unity that caused the immense popularity
of Zeising's convincingly formulated theory. Numerous "disciples"
adopted his axiom and widened the scope to other arts and further
parts of nature.[26]
Of all attempts to find norming proportional systems in the history
of art (e.g. The arising psychology tried to prove the effect of the Golden
Section on people, which had become the paradigm of experimental
psychology. Gustav Theodor Fechner in 1864 observed that in a
series of rectangles between square and double-square, his probationers
preferred the Golden Ratio rectangle.[28] Since he could not confirm this result
with ellipses or in the simple division of lines he said that
"Zeising thought too highly of the aesthetic value of the
Golden Section" [Fechner 1865: 162], which Fechner [1913]
then even ridicules. After the
Within this context the Golden Section attracted two architects
whose attitude to Proportions could not be more different: Ernst
Neufert and Le Corbusier. More efficacious for architecture probably
was Neufert's decision to embrace the Golden Section in his famous
Neufert explains historical systems of proportion and the
Golden Mean more extensively in his On the other hand we see planning in proportions caused by
aesthetic reasons in the other great system of the 20th century,
Le Corbusier's (1887-1965) Only in his most important tract, The Modulor in Le Corbusier's story combines square and Golden
Section, but as a result it does not offer anything else than
a modular system. From a blue series of numbers (Golden Section
of the total height) and a red series (height of the navel) results
a sequence of measures from 27 cm to 226 cm (and then much more)
in steps of 27 and 16 ( The Modulor has some deficiencies, however. First, although
Le Corbusier meant for it to be used for all dimensions, vertical
and horizontal, he bases it solely on the vertical dimension.
Further it is based on approximations to the Fibonacci numbers.
Since the blue and the red series can be combined, the system
becomes so elastic that the Golden Section is hard to detect.
Neufert himself criticizes, in a newer edition of the It is not our task to examine where Le Corbusier has built the Golden Section and the Modulor, so it should do for our purpose to say that he applies his system less than thoroughly. Le Corbusier's use of the Golden Section begins by 1927 at the Villa Stein in Garches, whose rectangular proportion in ground plan and elevation, as also the inner structure of the ground plan, approximately show the Golden Section [Le Corbusier 1948: 34 ff.].[33] Le Corbusier himself calls his Unité d'Habitation in Marseille (1945-52) a demonstration of his Modulor system. Indeed the real measures considerably differ from the theory: real length 140 m, instead of Modulor 139,01 m; width 24 m, Modulor 25,07 m; height 56 m, Modulor 53,10 m [Le Corbusier 1948: 134, cf. the series of measures p. 84]. To address these differences, attempts have been made to trace back the building's dimensions not to the Modulor but to the exact relations of the Golden Mean, but with little success [Padovan 1999: 332 ff.]. Le Corbusier's importance and his great impact seem to originate in his normative aesthetics, which propagates the combination of abstract geometry and anthropomorphic measures. So nearly all modern architectural lesson books extensively describe the Golden Section, e.g., Ching [1996] or Krier [1988]. Here and there it serves as a theoretical reflection that can be given - post festum? - to architectural designs [Bofill 1985].
in
extenso. After early experiments Le Corbusier uses the Golden
Section to develop his catalogue of measures, which has - due
to roundings and combinations - not much in common either with
the Golden Mean or with the Fibonacci series. In fact, Neufert
and Le Corbusier seem to use the Golden Section as a way to embellish
their own subjective artistic creation by theory and ratio. In
any case, the Golden Section certainly does play a role in the
writings of these architectural theorists. Prior to the 19th
century, however, the Golden Section is simply absent in written
architectural theory.
[2] Translated by the present author from [Vitruvius
1990: 6]: [3] Anthropos and mimesis: a mimetic imitation of the
body. [4] Two manuscripts dated 1498 contain 60 drawings of
Leonardo da Vinci: Geneva, Bibliothèque Publique et Universitaire,
Cod. 250, and Milan, Biblioteca Ambrosiana, Ms. E 170 sup., the
last one published as édition de luxe on hand-made paper
fully illustrated 1956 in Milan (Fontes Ambrosiani, 31). [5] Examples given in [Davis 1977: 98, n. 2]. [6] So his [7] I, f. 23r: [8] I, f. 23r: [9] I,8, f. 29v. However he mentions Alberti only to
rebuke his Tuscan companion to speak of an order "Italiche"
rather than "Tuscane". [10] I,7, f. 29r: [11] Paris, Bibliothèque Nationale, fonds français,
1903, f. 18v; the plate cited is still the best illustration.
[12] Divina proportione I,54, f. 16r: [13] Rackusin [1977] thinks too highly of this chapter
as "in actuality a miniature architectural treatise"
(p. 479) and cannot prove here remarks on the number-symbolism
of 72 for Pacioli. [14] Divina proportione I,18, f. 32v: [15] Ib.: [16] Divina proportione I,19, f. 32v.: [17] So still Peter Kidson [1996: 345]. [18] [19] In the Parisian Codex, between 1487 and 1497 (Ms.
A f. 13v, Ms. B f. 13v, Ms. B f. 27v), and in the Codex Atlanticus,
ca.1508 (f. 362 r-b), see [Meckseper]. [20] E.g. the London leaves c. 1513 [Rupprich 1966: n.
12, n. 16, n. 17]; see [Fredel: 1998]. [21] On the philosophical backgound concentrates Albert
van der Schoot [1997: 179-200]. [22] This is not to be confused with the small, posthumuously
compiled treatise [23] [Zeising 1854: 389]; Darwin's theory of evolution
had not been published yet, but was already - with others - in
discussion. [24] [Schmidt 1849]; following him Zeising [1854] designed
his fig. 2, p. 85. Furthermore he illustrates [Hay 1843; cf.
Zeising 1854: fig. 1, p. 63] and [Carus 1853; cf. Zeising 1854:
fig. 3, p. 95]. [25] Carefully considered already in [Scholfield 1958:
98]: "A fairly good case could be made out for the view
that the nineteenth century actually discovered the golden section
as an instrument of architectural proportion." [26] Still in Zeising's lifetime [Bochenek 1875], then
e.g. [Goeringer 1893] or [Pfeifer 1885]. [27] See, for example, [Lund 1921; Hambidge 1920; Funck-Hellet
1951; up to Doczi 1996]. [28] [Fechner 1865] and with varying exposition and reflection
[Fechner 1876]. [29] On the other hand Dessoir [1906: 124-127], and Borissavlievitch
[1954] discuss the Golden Section critically. [30] Most famously [Cook 1922], and more recently [Ostwald
2000]. [31] Here the influence of Otto Bartning is probable,
who advised Neufert in this part. [32] In 1921 Le Corbusier had published (under the name
Le Corbusier-Saugnier) [33] Le Corbusier himself presents the façades
with rectangular
De re aedificatoria.
Giovanni Orlandi and Paolo Portoghesi, eds. 2 vols. Milan: Edizioni
Il Polifilo (Trattati di Architettura, 1).Barbaro, Daniele. 1556. Bochenek, Johannes. 1875. Bofill, Ricardo. 1985. Borissavlievitch, Miloutine. 1954. Brooks, H. Allen. 1997. Bühler, Walther. 1996. Carus, Carl Gustav. 1853. Ching, Francis D. K. 1996. Choisy, Auguste A. 1899. Cook, Theodore A. 1922. A new disease in architecture.
Daly Davis, Margaret. 1977. Dessoir, Max. 1906. Doczi, György. 1996. Dürer, Albrecht. 1525. Dürer, Albrecht. 1527. Dürer, Albrecht. 1956/1966/1969: see Bernhard Rupprich. Fechner, Gustav Theodor. 1865. Über die
Frage des goldnen Schnitts. Fechner, Gustav Theodor. 1876. Fechner, Gustav Theodor. 1913. Warum wird
die Wurst schief durchgeschnitten? In Filarete (Antonio Averlino). 1972. Filarete (Antonio Averlino). 1965. Frankl, Paul. 1938. Fredel, Jürgen. 1992. Dürer und
der Goldene Schnitt. Pp. 174-180 in Fredel, Jürgen. 1998. Frings, Marcus. 1998. Funck-Hellet, Charles. 1951. Goeringer, Adalbert. 1893. Gros, Pierre, ed. 1990. Ghyka, Matila C. 1998. Hahnloser, Hans R. (ed). 1935. Hambidge, Jay. 1920. Hay, David Ramsay. 1843. Herz-Fischler, Roger. 1984. Le Corbusier's
'Regulating Lines' for the Villa at Garches (1927) and Other
Early Works. Herz-Fischler, Roger. 1997. Le Nombre d'or
en France de 1886 à 1927. Herz-Fischler, Roger. 1998. Hilpert, Thilo. 1999. Menschenzeichen. Ernst
Neufert und Le Corbusier. Pp. 131-143 in Walter Prigge, ed. Kidson, Peter. 1996. Architectural Proportion,
§ I: Before c. 1450. Pp. 343-353 in Jane Turner, ed. Krier, Rob. 1992. Kroll, Johann F. 1839. Le Corbusier-Saugnier. 1921. Les tracés
régulateurs. Le Corbusier. 1923. Le Corbusier. 1948. Le Corbusier. 1955. Lund, Fredrik Macody. 1921. Martini, Francesco di Giorgio. 1967. Martini, Francesco di Giorgio. 1989. Meckseper, Cord. 1983. Über die Fünfeckkonstruktion
bei Villard de Honnecourt und im späten Mittelalter. Mortet, Victor. 1911. Neufert, Ernst. 1936. Neufert, Ernst. 1941. Neufert, Ernst. 1943. Ohm, Martin. 1835. Ostwald, Michael J.
2000. Under Siege: The Golden Mean in Architecture. Pacioli, Luca. 1509. Pacioli, Luca. 1889. Padovan,
Richard. 1999. Palladio, Andrea. 1570. Pfeifer, Franz Xaver. 1885. Philandrier, Guillaume. 1544. Rackusin, Byrna. 1977. The architectural theory
of Luca Pacioli: De divina proportione, chapter 54. Rupprich, Bernhard, ed. 1956/1966/1969. Schmidt, Carl. 1849. Scholfield, Paul H. 1958. van der Schoot, Albert. 1997. Serlio, Sebastiano. 1537. Serlio, Sebastiano. 1545. Serlio, Sebastiano. 1619. Serlio, Sebastiano. 1996/2001. Villard d'Honnecourt: see Hans R. Hahnloser. Vitruvius. 1969-1992. Wesenberg, Burkhardt. 1983. Wiegand, August. 1849. Wolff, Ferdinand. 1833. Zeising, Adolf. 1854. Zeising, Adolf. 1884.
Copyright ©2002 Kim Williams top of
page |
NNJ HomepageNNJ Winter 2002 Index Related
Sites on the WWWAbout
the AuthorComment on this articleOrder
books!Research
ArticlesThe
Geometer's AngleDidacticsBook
ReviewsConference and Exhibit ReportsThe Virtual LibrarySubmission GuidelinesTop
of Page |