Organized by Istituto e Museo di Storia della Scienza

J. V. FIELD

What mathematical analysis can tell us about a fifteenth-century picture

Masaccio's Trinity fresco is the only picture that I have ever measured in detail. The investigation, carried out in collaboration with Dr Lunardi and Professor Settle, was directed to understanding the perspective scheme. It led to various unexpected conclusions, and has caused me to reflect upon the limitations of mathematical investigation.

We clearly had to start from the actual picture, because it is large, which means that in an A4 photograph the very finest line one could draw would correspond to a stripe several centimetres wide across the actual picture. Accordingly, writers of textbooks can show a simple and apparently mathematically correct perspective scheme, in which, for instance, the lines of the ribs of the barrel vault can be extended to converge neatly at a point in the step on which the donors are kneeling. However, such coarse lines are not likely to be very good guides to what the painter actually did when laying out his picture on the wall. To understand the possible techniques used in practice, one needs to take account of the actual sizes of things, and to bear in mind what instruments were likely to have been used. I do not mean specialised perspective instruments but mundane items such as compasses, nails and string.

The problem of scale is clearly fundamental. One has to put oneself in a position to see and then measure the results of the painter's operations to the same degree of accuracy as was available to him when he carried out the work. However, what we see is the finished work. Sometimes, as in the Trinity fresco, there are substantial traces of the lines drawn to guide the painter as he applied colours, and these are (of course) good evidence regarding construction of the final image, but they are not necessarily the lines that were used to establish the overall design, and specifically the perspective scheme.

Moreover, if a picture is in correct perspective, certain elements in it will display certain mathematical properties. For instance, lines which in the 'real scene' were at right angles to the picture plane ('orthogonals') will appear in the picture as lines converging to a point that is the foot of the perpendicular from the eye of the ideal observer to the picture plane. Thus finding this property in a picture simply tells one it is correct. The property can of course be found in a photograph. It does not tell you that the person who made the picture knew the property in question, and still less does it tell you that he used this property in constructing the picture. However, one can check up on such properties to see whether a picture is correct. All the same, one needs to bear in mind that some such properties can arise simply from the symmetry of a design. Beware reading too much from the fact that when you extend the two receding edges of the dais under the Madonna's throne you find that they meet on the centre line of the picture. This kind of mistake seems to have led to the idea that medieval painters practised a kind of 'fish-bone' perspective, in which corresponding lines at either side of the picture meet, two by two, on the centre line. This is of course, not to deny that the painters of such pictures may have intended to convey a sense of depth.

A convincing example of these snags in action is provided by the Trinity. The centre line is marked by a sinopia line in the central rib (not exactly in its centre!) and the lines of the mouldings at the base of the vault meet exactly on it - where 'exactly' means that it looks exact to me when running string (slightly finer than Masaccio's) over the picture. But the lines of the edges of the ribs do not meet so neatly. In fact, I rather suspect that Masaccio got them by dividing the upper and lower (front and back) semicircular edges of he vault into appropriate parts and then joining up the points. (Dividing front and back edges in the same pattern may well be the basis for the construction of Ambrogio Lorenzetti's apparently correct pavimento in his Presentation now in the Uffizi.)

So the ribs do not really provide strong evidence for Masaccio having understood the properties of what Alberti, writing about ten years later, was to call the 'centric point'. However, there are two more lines which also go through the putative 'centric point'. These are the lines established by the inner edges of the abaci on the columns. The actual edges of the abaci are short, but the two on each side are exactly in line and when extended the two lines pass exactly through the centric point. This looks encouraging. Better still, the two engraved lines marking the receding edges of the lower right abacus both pass exactly through the centric point. Unfortunately we have not got the relevant part of the ariccio with a nail hole in it, preferably marked with a sinopia circle to avoid mistakes when a nail is used again. In the absence of the hole, we cannot go further than concluding that the evidence suggests rather strongly that Masaccio actually used the convergence property in constructing the images of some orthogonals and was aware it held for the rest. The fact that the point of convergence is in a position that makes it a reasonable height for the eye of an observer further confirms that Masaccio understood what he was working with.

In fact, although a large number of fifteenth-century pictures convey a strong sense of depth, it turns out - even when one merely plays around with a transparent ruler laid over a small photograph - that almost all of them show noticeable departures from mathematical correctness. This is a saving grace for an art historian. If the picture is not absolutely correct then the things that are right are presumably done deliberately, rather than merely being the consequences of some other construction. Moreover, certain types of error may be revealing in themselves. For instance, the incorrect shapes of the abaci suggest very strongly that in the Trinity fresco Masaccio did not use a perspective construction like that described by Alberti.

Masaccio faces one with archaeology; to understand his use of perspective, one has to work from the lines left on the picture. But surviving texts tell us a fair amount about the kind of mathematics he was likely to have been taught. The written accounts of perspective construction - Alberti's sketch in the 1430s and Piero della Francesca's detailed treatise about thirty years later - provide some guidance for analysing later pictures. However, the guidance does not prove to be overwhelmingly useful, because only the very simplest constructions seem to have been employed in most pictures. Moreover, even when one has grounds for suspecting the use of more elaborate methods, there is no guarantee that details can be recovered from the picture. There are some instances of this in the Trinity fresco also, so I make the usual mathematician's claim of not having sustained any loss of generality in using it as my example.