# Alberti’s construction (1435)

Source: The Invention of Infinity: Mathematics and Art in the Renaissance, Judith Veronica Field.

The view of the Palazzo della Signoria across the square certainly presented a rather more complicated drawing problem than that posed by the view of the Baptistery, but the two panels were apparently painted in fairly rapid succession and in the absence of evidence to the contrary it seems likely that much the same method was used to produce both pictures. Unfortunately, we have no direct evidence what this method was. The first written account of a method of constructing pictures in correct perspective dates from about 20 years later. It is found in the first book of a short treatise On painting, written by the learned humanist Leon Battista Alberti (1404—1472).

Alberti’s family originated from Florence, but had been exiled, so that Leon Battista first returned to his native city in the 1430s, in the train of Pope Eugcnius IV. On his arrival he found that many interesting things had been happening in painting in Florence and, eager to relate them to his own theories about art, decided to write a treatise on the subject. Being addressed to the learned, the first version of the work, finished in 1435, was written in Latin, with the title De pietura (On painting). A vernacular version (Della pittura) appeared in the following year, dedicated to Filippo Brunelleschi. Fifteenth-century conventions did not actually require formal acknowledgement of one’s predecessors, so it is not clear whether this dedication is in fact an acknowledgement that Alberti is using Brunelleschi’s work or, as it might be in the twentieth century, an elegant method of evading the charge of having appropriated it without due acknowledgement. It is, moreover, possible that Alberti dedicated the work to Brunelleschi merely as a homage, in recognition of the older man’s interest in the visual arts.

In any case, Alberti’s account of perspective is not addressed to artisans. He is merely sketching the outlines of a procedure for those who wish to understand something about it. The same is in fact true of Alberti’s treatment of other aspects of painting. For instance, his long discussion of what he considered suitable subjects for pictures is clearly addressed not to painters, who would have been told what to paint, but to their patrons, who did the telling.

As the first surviving account of a method of perspective construction, Alberti’s brief description of how to draw a picture of a chequerboard floor has often been subjected to the kind of detailed, awe-stricken analysis usually reserved for Holy Writ. To a mathematician’s eye, however, it is obvious that the description is inadequate. Either Alberti is not explaining himself very well— perhaps finding it difficult to explain mathematics in a way that makes it accessible to a readership whose real interests lie elsewhere—or he himself has not really understood the method concerned. However, since the main outline is clear, it is quite easy for a twentieth-century reader to fill in the gaps in Alberti’s account. The construction is shown in Fig. 2.4.

The problem is to construct the perspective image of a square chequerboard pavement with one edge running along the ground line of the picture, that is the line of intersection of the picture plane with the ground. The outer edge of the actual picture is shown as a square centred on the point C. This point is called the ‘centric point’ of the perspective and is the point in the picture directly opposite the viewer’s eye, that is the foot of the perpendicular from the eye to the picture plane. The line ONC has been drawn parallel to the ground line, and the distance of the eye from the picture plane is equal to ON.

The method of construction is as follows:

1. First, the lower edge of the picture is divided into the appropriate number of tiles to represent the pavement. Here we have chosen to have five tiles, and the points of division of AB are D, E, F, G.

2. The second step is to join A, B and the points of division to C. Alberti somehow fails to make it clear why this is what one has to do—which may perhaps indicate that he expected his readers to know about it already. The lines AC, DC, EC and so on represent the lines of the tile edges that run perpendicular to the plane of the picture. The present-day term for such lines is ‘orthogonals’. There seems to be no Renaissance equivalent, so for the sake of clarity the modern word will be used. (It is, of course, in general rather dangerous for a historian to depart from period vocabulary, since doing so may tacitly introduce a modern style of thought that distorts one’s view of a non-modern line of argument.)

3. The third step of the construction is to join the point O to cach of the points A, D, E, F, C.

4. The fourth step is to draw through the points where OA, OD, OE, OF and OG intersect the vertical edge of the picture, that is the vertical through B, lines parallel to the base line AB. These lines represent the lines of the tile edges that run parallel to the ground line of the picture. There seems to be no Renaissance name for them. Their modern name is ‘transversals’. The perspective image of our five by five pavement is the trapezoidal grid bounded by the lines AC, BC, AB and the uppermost transversal.

Before giving this construction, Alberti has repeatedly referred to the ‘cone’ or ‘pyramid’ of vision, that is the sheaf of rays between the eye and the scene that is viewed, and has said that the picture represents a section through this cone. His use of the words cone and pyramid to indicate solid figures whose bases arc of irregular shape is a little confusing mathematically, but was quite usual at the time. What is much more serious is that he says not a single word to indicate the nature of the connection between the cone of vision and the construction that follows. Perhaps Alberti expected his learned readers to allow such technicalities to be left to the experts. Although the general level of mathematical education was rising in the fifteenth century, it was not comparable to that prevailing in Western Europe in our own time. Alberti is addressing himself to an upper class readership. On the whole, such readers seem to have known less mathematics than their social inferiors in the artisan classes.

It is, in any case, not very difficult to provide a justification for Alberti’s construction. Let us add a point P to represent the position of the viewer’s feet. In Fig. 2.4, P will lie directly beneath O on the line AB produced. We are using O and P as the letters for the positions of eye and foot because these are the letters most generally used in perspective treatises of the following century. They are dearly derived from the Italian words for eye and foot, namely ‘occhio’ and ‘piedo’. The plan and a vertical section of the set-up are shown in Fig. 2.5. It is clear that, in the diagram of the section, the point in which the line OH cuts the vertical line through M gives the position in which H will appear as seen from O. Thus the transversal through H, that is the back line of the pavement, will appear in the picture plane as a line parallel to the ground line at the height of this point of intesection. The heights of the nearer transversals will clearly be given by the points of intersection between the vertical through M and the other lines of the pencil radiating from O. So we could think of Alberti’s construction, as shown in Fig. 2.4, as being put together by superimposing everything in the vertical plane shown in Fig. 2.5(b) on a view of the actual picture, whose significant part is contained in the triangle CAB in Fig. 2.4.

This is not, of coursc, a proper proof since we have not shown why the orthogonals should converge to C. However, there is nothing anachronistic in using superposition. It was regularly used in the Renaissance, as it had been in earlier periods, to show how the facades and internal structures of buildings were related to one another. In the fifteenth century, a learned justification

for its use could have been found in the newly recovered text of Vitruvius’ On architecture, where superposition of lines from perpendicular planes is used in the famous ‘analemma’ construction for the lines on sundials. Moreover, be it said, Vitruvius explains himself as little as Alberti.