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The Geometry of a Piece of String

  • David Yeomans


line, n. […] 4. cord for measuring, levelling etc. […] 6. long narrow mark traced on surface. O.E.D.

Between the design and the realization of a building there are a number of ‘drawing’ processes, either on the building site or in the workshop, which range from the setting out of the plan to the production of ‘shop drawings’ from which details are derived. Unfortunately, while the latter have survived in sufficient numbers to have attracted scholarly attention, the setting out of the ground plan leaves no trace. Nevertheless, it is that process that determines the building's basic geometry. While much scholarly effort has gone into attempting to divine the geometrical principles behind designs from Antiquity to the Gothic period, it has not always been informed by an understanding of the setting-out process. Without taking the constraints of that process into account, one is reduced to looking for geometrical relationships within the building, and of course one will find some. Clearly, there were geometrical principles behind almost all buildings, if only that they should be rectangular or symmetrical, but the difficulty is that a few simple rules can easily result in a large number of geometrical relationships within the building that were not used, and possibly not even recognized, by their designers. At the very least, therefore, we should consider how to distinguish those relationships that were actually used by the designers and builders from those that are merely epiphenomenal.



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1 Examples of shop drawings for Gothic architecture are described by Harvey, John, The Tracing Floor of York Minster, 40th annual report to the Friends of York Minster (York, 1968); and Davis, Michael, ‘On the Drawing Board: Plans of the Clermont Cathedral Terrace’, Ad Quadratum: The Practical Application of Geometry in Medieval Architecture, ed. Nancy, Y.Wu (Aldershot, 2003), pp. 183204 .

2 Stuart's surviving notebook (see n. 11) has all the indications that it was the first to be used on the expedition with its notes on dialling, possibly an intellectual exercise during his journey to Greece, and its comparisons of the various lengths of a foot. This does not seem to have led him to any useful conclusions because no use was made of the data.

3 Movement of the building as a result of poor foundations, timber shrinkage and possible decay and insect attack need to be allowed for in making measurements.

4 Stuart, James and Revett, Nicholas, The Antiquities of Athens, 5 vols (London, 176294).

5 Vitruvius, , De architectura, I, 6, 67 .

6 Vitruvius, , Ten Books of Architecture, trans. Rowland, Ingrid D., with a commentary and illustrations by Howe, Thomas Noble (Cambridge, 1999), p. 168 .

7 Adam, Jean-Pierre, Roman Building: Materials and Techniques (Bloomigton, Ia., 1994). The method of its use is also illustrated in Rowland and Howe's edition of Vitruvius (p. 170). The modern equivalent is the optical square which can be seen in nineteenth-century instrument makers' catalogues and is still available today as a simple tool for determining right angles on site.

8 Engravings of this building are in Stuart, and Revett, , The Antiquities of Athens, I, ch. III, pt II .

9 For a study of the geometry of Notre Dame, see van Leifferinge, Stefan, ‘The Hemicycle of Notre-Dame of Paris: Gothic Design and Geometrical Knowledge in the Twelfth Century’, Journal of the Society of Architectural Historians, 69 (2010), pp. 491507 . Van Leifferinge makes a convincing case for a scheme based on an octagon. However, the columns are noticeably out of position in relation to the ideal plan, which is probably a result of the difficulty in setting out. To plumb down accurately into a foundation trench from a line stretched over it would not be a simple matter, but is solved by the method suggested in Figure 6. This method could not have been used for a foundation out of sight in a narrow pit and masons would have needed to resort to less accurate means, resulting in the misalignment noted by Van Leifferinge.

10 For recent measurements, see Rottlander, Rolf C. A. et al., ‘Untersuchungen am Turm der Winde in Athen’, Jahreshefet des Österreichischen Archäologischen Instituts in Wien, 59 (1989), pp. 5592 . These, however, do not correspond closely with the ones given by Stuart and Revett, or allow their data to be checked.

11 Stuart's only surviving notebook from their expedition is in Edinburgh University Library (Laing MSS, La.III.581). Calculations in this suggest that he was looking for rules for proportion within the measurements made.

12 Morgan, Bernard George, Canonic Design in English Medieval Architecture: the Origins and Nature of Systematic Architectural Design in England, 12151515 (Liverpool, 1961).

13 London, British Library, Add. MS 22153. Note also that this is not a dimension that would have been made by the builders.

14 Stuart, and Revett, , The Antiquities of Athens, II, p. 24 .

15 Perrault, Claude, Les Dix Livres d'architecture de Vitruve (Paris, 1673), pp. 16063 and 170–71.

16 Because Vitruvius is not very clear, Stuart says, ‘If we do not mistake Vitruvius’; Stuart, and Revett, , The Antiquities of Athens, II, p. 24 . At this point Stuart was relying heavily on Perrault whom he quotes and translates in his notebook. His manuscript comments are found in f. 165V of his notebook (see above, n. 11).

17 Jones, Mark Wilson, ‘Principles of Design in Roman Architecture: the Setting Out of Centralised Buildings’, Papers of the British School at Rome, 57 (1989), pp. 10651 .

18 Jones, Mark Wilson, Principles of Roman Architecture (New Haven and London, 2000), pp. 6061 .

19 Most of this outer ring of piers was destroyed in a twelfth-century earthquake, so the small proportion remaining exacerbates problems of measuring.

20 Wilson Jones, Principles of Roman Architecture, p. 88.

21 I have not had the opportunity to make measurements myself. I have simply reworked the figures provided by Wilson Jones to produce a hypothetical explanation.

22 The Engineering of Medieval Cathedrals, ed. Courtenay, Lynn T. (Aldershot, 1997) is a collection of previously published papers. She would certainly have included something on early works on structural design had there been any.

23 See Shelby, Lon R. and Mark, Robert, ‘Late Gothic Structural Design in the “Instructions” of Loeenz Lechler’, Architectura, 9 (1979), pp. 11331 ; and Shelby, Lon R., ‘The Geometrical Knowledge of Mediaeval Master Masons’, Speculum, 47 (1972), pp. 395421 . Both of these are reproduced in The Engineering of Medieval Cathedrals, ed. Courtenay, pp. 87–105 and 27–53 respectively.

24 Ackerman, James S., ‘“Ars Sine Scientia Nihil Est”: Gothic Theory of Architecture at the Cathedral of Milan’, The Art Bulletin, 31 (1949), pp. 84111 .

25 Fitchen, John, The Construction of Gothic Cathedrals (Oxford, 1961).

26 Coldstream, Nicola, Masons and Sculptors (London, 1991), p. 37 .

27 Coldstream, Nicola, Medieval Architecture (Oxford, 2002), p. 69 .

28 Fernie, Eric, ‘The Ground Plan of Norwich Cathedral and the Square Root of Two’, Journal of the British Archaeological Association, 129 (1976), pp. 7786 .

29 Ibid., p. 86.

30 Note also that the lines can be extended to setting out points that lie outside the lines of the foundations.

31 Murray, Stephen, Beauvais Cathedral: Architecture of Transcendence (Princeton, 1989), p. 14 and fig. 9. Murray points out that the error results in the hemicycle being swung in a clockwise direction.

32 Zenner, Marie-Thérèse, ‘A Proposal For Constructing the Plan and Elevation of a Romanesque Church Using Three Measures’, Ad Quadratum, ed. Wu, , pp. 2555 .

33 Ibid., p. 25.

34 One can only speculate about this, but it is possible that some site constraint, perhaps the same as the one producing the slight misalignment of the south transept, necessitated the switch between the two dimensions.

35 Zenner, ‘A Proposal For Constructing the Plan’, p. 30.

36 Zenner gets carried away by a five-pointed star that a carpenter of the Compagnon des Devoir Unis de Tour de France drew on the basis of her circles.

37 Davis, Michael and Senechal, Lester, ‘Scenes From a Design: the Plan of Saint-Urbain, Troyes’, Journal of The Association Villard de Honnecourt For the Interdisciplinary Study of Medieval Technology, Science and Art, 10/1 (1996-97), pp. 1521 .

38 See Hiscock, Nigel, The Wise Master Builder: Platonic Geometry in Plans of Medieval Abbeys and Cathedrals (Aldershot, 2000); and The Symbol at Your Door: Number and Geometry in Religious Architecture of the Greek and Latin Middle Ages (Aldershot, 2007).

39 Hiscock, The Wise Master Builder, pp. 236, fig. 23.

40 Nigel Hiscock, ‘A Schematic Plan for Norwich Cathedral’, in Ad Quadratum, ed. Wu, pp. 83–121.

41 Mainstone, Roland, ‘Brunelleschi's Dome’, Architectural Review, 162 (1977), pp. 15666 .

42 Jones, Barry, Sereni, Andrea and Ricci, Massimo, ‘Building Brunelleschi's Dome: a Practical Methodology Verified by Experiment’, Construction History, 23 (2008), pp. 331 .

43 To make a straight line on a piece of stone, a mason (or a carpenter) uses a line covered with a colouring material. Chalk or ink-black are both commonly used and the line is kept in a box together with the colouring specifically for this purpose. The line is stretched tight between the end points, lifted and then allowed to snap back onto the surface, thus leaving a mark.

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Architectural History
  • ISSN: 0066-622X
  • EISSN: 2059-5670
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