In Secret Knowledge, David Hockney points to the various optical effects in Vermeer's paintings that have led so many commentators, over the last hundred years, to suspect Vermeer of having used the camera obscura. These include Vermeer's so-called 'photographic perspective', in which figures near the viewpoint can appear excessively large; his passages rendered out of focus; and his reproduction in paint of certain artefacts of unfocussed lenses, notably his treatment of highlights on shiny surfaces.

My own argument for Vermeer using the camera turns rather on the perspective geometry of a group of his paintings of interiors. I would like if I may to summarise my findings very briefly, especially since critics of my book, including Walter Liedtke, have directed their attention more to peripheral points, and have missed my central argument. I show how it is possible to reconstruct the three-dimensional spaces seen in ten of the pictures, using a method of inverse perspective -- roughly speaking, working the normal procedures of perspective drawing backwards.

1. 'The Music Lesson'
2. Plan of 'The Music Lesson'
3. Bird's eye view, 'The Music Lesson'

What is more, it is possible to find a common scale for all these spaces, such that three conditions are satisfied. First the dimensions of the architecture -- the sizes of the floor tiles, the windows and their spacing, the height of the room -- are closely similar (with a few anomalies) throughout. Second, where recognisable pieces of furniture occur in more than one painting, their sizes remain the same. Third, items shown by Vermeer of which the originals or copies survive in museum collections today, have their actual dimensions as measured from the objects themselves. These real items include a series of printed wall maps and globes, several paintings by other artists that Vermeer shows hanging on the walls of the rooms, the Delftware tiles used as skirtings, musical instruments including the virginals in 'The Music Lesson' recognisable as a product of the Ruckers workshop, and two distinctive designs of chair.

It is very hard to resist the conclusion that all ten paintings depict one and the same room, with furniture and models rearranged for the different compositions. One very distinctive feature of the architecture, visible in eight pictures -- and whose repetition supports this idea -- is an elaborate design of leaded panes in the casements, forming a pattern of interlocking circles and squares.

4. Casements, squares and circles pattern Here is my key finding. For each of the ten paintings in question, it is possible to locate the theoretical perspective viewpoint, and to plot a ‘visual pyramid', with its apex at the viewpoint, containing everything that is visible in the picture. If the lines forming the edges of this pyramid are carried back to meet the back wall of the room, they define a rectangle on that wall. For at least six paintings this rectangle is the exact size of Vermeer's canvas.

5. Plan of room, angles of view
6. Booth-type camera at back of room

Here is a plan of the room. The small circles mark the viewpoints of six paintings. The diagonal lines show the visual pyramids in top view. The short heavy lines show the widths of the paintings at the back wall. My explanation for this very curious geometrical property of a group of Vermeer's perspectives, considered collectively, is that it is a consequence of him using a camera obscura in the form of a closed booth or cubicle. The camera lens would have been positioned at the theoretical viewpoint for each composition, and the back wall would have served as a projection screen. The projected images are the same sizes as Vermeer's canvases, because he has traced them.

Those who are sceptical of camera theories must, it seems to me, offer some alternative explanation for this strange geometrical phenomenon, if they are to convince. In conversation this summer Walter Liedtke offered the suggestion that the result might be 'due to chance'. (Not a probability theory man, then.) Paul Taylor of the Warburg Institute consulted the mathematician Tim Gowers at Cambridge on this point. Gowers estimates the odds of the result being due to chance as at least hundreds and perhaps thousands to one against.

Nor have I, or any critics, been able to come up with explanation in terms of other techniques that Vermeer might plausibly have used to construct his extremely accurate perspectives. The most obvious of these alternatives is of course that Vermeer set up his compositions using conventional mathematical methods. Unfortunately there is no positive evidence for this view: no perspective layout drawings by Vermeer's hand, and no linear underdrawing or other traces of perspective construction marks in the canvases themselves. Much has been made by Jørgen Wadum and others of the existence of tiny pinholes at the central vanishing points of many of the pictures. But these are accounted for as easily by a camera technique as by geometrical construction, The crucial point, in any case, is that there is no way that I can see -- and no suggestions offered by critics - that a mathematical perspective method would give rise to my peculiar results for the projected images of the scenes. Meanwhile an explanation in terms of the geometrical optics of the camera obscura is simple and straightforward.

Walter Liedtke questions the accuracy and consistency of Vermeer's perspectives, and argues that it is a basic mistake even to imagine that any Delft artist would have represented, in faithful detail, the exact appearances of some contemporary Dutch interior. What we seem to have here is a serious case of the art-historical condition described by Michael Kubovy as mimetophobia -- 'the morbid fear of slavish imitation'. Now I would not want to claim by any means that Vermeer was always and everywhere a slavish copyist of what was in front of him. Indeed since the room in question no longer exists, the matter of Vermeer's truth to appearances is at least in that particular respect undecidable. There are nevertheless at least three ways in which to make an assessment of Vermeer's naturalism and truth to appearances.

The first is in his rendering of the many real items -- maps, pictures, chairs -- to be found today in libraries and museums. I have only time to show you a couple of examples.

7. Map, 'Allegory of Painting'
8. Real map, Seventeen Provinces

Vermeer's versions of the printed maps are so accurate, down to every last cartouche and little sailing ship on the ocean, that as James Welu has shown, not only can the maps' designers and publishers be identified, but in some cases the specific edition.

9. Chair, 'The Glass of Wine'
10. Real lions' head chair

Vermeer's versions of the chairs with the sculpted lions' heads are accurate down to the cut-out shapes of the cross-members and the profiles of the turned legs. What's more, my perspective reconstructions of the chairs give dimensions within one or two centimetres of the museum originals; and my calculated dimensions of the maps are all within a few per cent of those of the surviving copies.

The second kind of evidence for Vermeer's naturalism is the internal consistency with which he depicts objects, architecture and the effects of light.

11. Model of 'The Music Lesson'

As an example, Arthur Wheelock argues that the pattern of illumination in 'The Music Lesson' is artificially contrived, and that solar shadows would not strike at the varying angles shown by Vermeer. But our scale model of the room, lit by diffused lamps to simulate northlight, not sunlight, produces positions and strengths of shadows very close to those in the painting itself.

And the final kind of evidence for Vermeer following real appearances concerns the degree to which his representation of architectural detail is typical of houses and styles of interior decoration of the period, even if the very building itself is lost. Both Walter Liedtke and Jørgen Wadum have argued for example that the construction of the ceiling shown in 'The Music Lesson' and other pictures 'flies in the face of documentary and archaeological evidence'. I have consulted architect colleagues and architectural historians in Delft on this point. Willem Weve who works for the municipality as a building historian says that the form of timber contruction shown by Vermeer is by no means uncharacteristic of the period. And Wilfried van Winden, partner in an architectural firm specialising in restoration work in Delft, says that that he himself lives in a 17^th century house in the centre of town whose ceiling construction is exactly as shown by Vermeer.

Of course there are instances where Vermeer indisputably departs from real appearances. The issue is rather, where does the balance of facts lie in support of or against his naturalism; what is the accumulated weight of evidence on either side? This weight, I contend, is overwhelmingly on the side of his naturalism. All this precision, all this correct detail, all this truth to light and shadow are products of Vermeer's patient attention to images created in the camera obscura.

Philip Steadman is Professor of Urban and Built Form Studies at University College, London. He trained as an architect and, in addition to teaching at Cambridge University and the Open University, has published several books on geometry in architecture. His book Vermeer's Camera (2001) is the product of twenty years' fascination with the Dutch Painter.