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Projective Geometry JOHN PHIBBS PROJECTIVE GEOMETRY This paper shows how reliant Brownian design is on projective geometry, not just for the creation of ‘picturable’ views, but for the underlying structure of the landscape. Given initially as a lecture at the Ashridge Summer School, Hertfordshire, on 24 August 2004, the paper is designed to show that mathematics and mathematical tolerance can be brought within the reach of designers and garden historians. PROJECTIVE GEOMETRY If the aim of the English landscape movement was to produce something distinct from anything French and sensu stricto original,1 then the most obvious starting point might have been to take out the geometry that characterized the golden age of French gardening – and this is conventionally accepted as what happened: the garden as linear architecture, illustrated in the bird’s-eye views beloved of Leonard Knyff and Johannes Kip and of Charles Bridgeman, was replaced by the garden as nature, typifi ed by the landscape drawing, taken from ground level and advanced by William Kent as an instrument of landscape design.2 Hence, as convention has it, the constant association between painting and gardening peddled by eighteenth-century theorists.3 Indeed painting had become very fashionable, replacing music as ‘the sole object of fashionable care’, even to the point that ‘some of the young nobility are themselves early instructed in handling the pencil’.4 This is what Phillip Southcote meant when he told the Revd Joseph Spence c.1752 that at Stowe, Buckinghamshire, Richard Temple, 1st Viscount and Lord Cobham, had begun ‘in the Bridgeman taste’, and had then moved on: ‘’tis the Elysian Fields that is the painting part of his gardens’,5 and this was the ‘sort of gardening … which Kent and nature have brought us acquainted with; where the supreme art of the Designer consists in disposing his ground and objects in an entire landskip’.6 The wealth of evidence for this conventional account is impressive, but there is another equally constant theme in the eighteenth-century establishment of the English style and that is a sense of some intangible coherence that must inhere in the most wild and naturalistic design. This can be traced well into the seventeenth century with a struggle between the ancient idea that ‘all is number’, that all natural beauty can be, and can only be, interpreted mathematically,7 and Blaise Pascal’s discovery of the beauty in symmetry. Design was either to be based on the use of regular intervals and mensuration, or on a sense that for something to be beautiful it was enough that the viewer should not be inclined to rearrange any part of it (Pascal’s rationale for the effect of symmetry). 19 Wolvercote Green, Oxford OX2 8BD, UK 2 GARDEN HISTORY 34 : 1 It would not be too far-fetched to see a precursor for Pascal in Horace’s paradoxical ‘concordia discors’ and for the dispute in Heraclitus’ and Parmenides’ argument over the changing, or unchanging, universe. Perhaps indeed one should see the tension between order and disorder as inherent in the human perception of the world. At any rate, both these alternatives had their adherents in English philosophy. On the one hand, Bishop Thomas Burnet argued that before the fall Nature had been organized in a patently mathematical way, and that since then it had been in continuous decay – an argument that might make it the duty of the gardener or place-maker to put nature right again, and by making its mathematical structure apparent, inevitably to restore it to beauty.8 On the other hand, John Locke had established an alternative philosophical position for accepting Nature as it is as a complete work, and this thesis was set out explicitly by the botanist John Ray in 1691.9 Less rigorously logical minds still distinguished between the two ideas of beauty, and asked the question that arises from them: can there be an underlying harmony in the ‘natural error’ of unimproved countryside? At the beginning of the seventeenth century, Sir Henry Wotton had already distinguished ‘regularity’ from ‘wild regularity’,10 and a generation later the latter class was renamed the ‘Chinese’ by Sir William Temple, who distinguished ‘walks of trees in straight lines, and over against one another’ from a ‘Chinese’ style, generally critical of regularity (i.e. walks of trees), and pursuing a beauty that ‘shall be great and strike the Eye, but without any Orders or Disposition of Parts, that shall be commonly or easily observ’d’.11 This alternative derivation of beauty found its way in the eighteenth-century world of Alexander Pope and Joseph Addison,12 and seems to have been well-established among tourists at that time. So John, Viscount Perceval, claimed to fi nd something similar in Bridgeman’s Stowe, in 1724 ‘Nothing is more irregular in the whole, nothing more regular in the parts, which totally differ one from the other’.13 Sir Thomas Robinson, Bt, had noticed that despite the variety of his art, Kent’s Chinese garden design at Carlton House in Pall Mall, St James’s, still had Nature’s overall unity,14 and the distinction was maintained by Robert Castell, who from what he had read of Chinese gardening described it as ‘an artful Confusion … where, tho’ the Parts are disposed with the greatest Art, the Irregularity is still preserved’,15 and then clarifi ed by Richard Hurd: ‘the careless observer, tho’ he be taken with the symmetry of the whole, discovers no art in the combination’.16 A generation later, Thomas Hale criticized Chinese gardening for having gone ‘beyond the Laws of Nature’ and falling ‘too much into this absolute Wildness’, but his proposed remedy would have been regarded by many as itself Chinese: It is an Air of Irregularity we advise, not Irregularity itself; there requires more Art by far in this Distribution, than in any other; and there requires afterwards the great additional labour of concealing it. … Every Thing we see should be chosen for its Place, thou it seem the Result of Accident; there should be Order in every Place, though under the Aspect of wild freedom, and a certain Harmony where there is the Aspect of Confusion.17 Now it would be possible to argue from this evidence that the triumph of nature in English gardening was a long, slow one and that each of four or fi ve generations pushed the same argument further than the last in the rejection of mathematical design, and hence made different assumptions about what was ‘irregular’ and what was ‘regular’, PROJECTIVE GEOMETRY 3 but two arguments in particular point to another conclusion. First, if Wotton, or any of the other theorists, had simply sought irregularity in order to imitate nature, they might have achieved it in a single stroke. First, Nature, after all, has never been very diffi cult to imitate, nor, as Horace regretted, does she take long to undo all the tidying of the fork and take charge of arrangements herself. Second, because one cannot really partly drop mathematics: it is only up to a point true to say that 2 + 2 = 5 is less mathematical than 2 + 2 = 4, and that 2 + 2 = 6 is less mathematical again – one equation follows the rules of arithmetic, and the other two do not. Similarly, something is either random or it is not; a randomly created design cannot be ordered, no matter how much it may appear to be – in short, if the components of a design are ‘disposed with the greatest Art’, they cannot also be disposed randomly. From this observation one might infer that even ‘Chinese’ design had some underlying logic or art, and ask what this art was and how it worked. PROJECTIVE VERSUS METRICAL GEOMETRY Following on from this, this paper proposes that one should read the development of the English landscape movement as the steady advance of one way of seeing against another, of China against France, and that this can be expressed mathematically as the advance of angle-based or ‘projective’ geometry against regular or ‘metrical’. The connection with geometry has eighteenth-century precedents: ‘regularity’ at least was used fairly consistently for metrical geometry; but the alternative, projective geometry, was represented by a number of names – to ‘irregularity’, the ‘Chinese’ style and ‘design’, one should add Francis Hutcheson the Elder’s ‘uniformity’.18 These sound like very different things, but in practice I doubt they were.19 The distinction between the two kinds of geometry is sometimes expressed in landscape theory today as the distinction between the ‘ratio’ and the ‘genius loci’, but it was current in the eighteenth century as well. It may be divined in William Mason, who was hostile to the ‘Rule’ (i.e. the ruler or line), which ‘Mechanism … pursues, admires, adores’, but a modernist avant la lettre praised the cube and cone (which are essentially proportions defi ned by their angles).20 Projective is the kind of geometry that is generated by perspective (hence the kind that one might expect to fi nd in a Claudean landscape); it is generated from angles and takes advantage of harmony and balance,21 and is capable of constructions whose underlying harmony is only apparent at specifi c points within the landscape – it is a geometry of illusion, of effect, rather than that of pure form. It can be created on the drawing board with a protractor and compasses, and in the fi eld, whether surveying or setting out, with a theodolite, or by trigonometry.22 Metrical geometry, on the other hand, above all else denotes mensuration and produces regularity and congruity. Mensuration requires the building up of a design from small units (often a yard, or a 10-inch interval).
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