Fig. 14. The first known diagram of the two-point perspective by Pélérin (1505), showing the pairs of vanishing points that each determine the angels of one set of parallels.

Fig. 15. Two perspective figures from Pélérin’s ‘De Artificiali Perspectiva’ A. A rare example of an oblique perspective construction. B. A design with a barrel vault.

In order to reconstruction the perspective, it is necessary to ‘read’ the intended layout of the scene, a task in which there is potential ambiguity. In view of the fact that this is the first full attempt at a painting with angular perspective, it seems reasonable to assume that Raphael intended a straightforward plan in which the chamber was rectangular but was viewed from an oblique angle. Examples of the geometry of angular perspective had recently been published in France (in 1505) and Bavaria (in 1509) in ‘De Artificiali Perspectiva’ by Jean Pélérin (Fig. 15A). A copy of this book might well be expected to have found its way to the premier artists at the center of decorative excellence, the Vatican. It is not unreasonable to suppose that Raphael would have been inspired to elaborate such a construction, implying a rectangular underlying plan. Fig. 15B from the same book bears a strong resemblance to the vaulted ceiling in the background of the painting, suggesting that Raphael may have combined the two constructions to develop his elaboration. This possibility, however, suggests the seed of his problems. While the assembly room vault is an accurate two-point construction, Raphael’s vault is in one-point perspective with a lateralized vanishing point. The two constructions cannot be validly melded together if all the corners are right angles. Strictly speaking, in fact, any shift of the principal vanishing point away from the center implies a need for a second vanishing point to appear one the opposite side. This is a requirement that was never accurately met during the Italian Renaissance. Analysis of Raphael’s ‘Coronation of Charlemagne’ provides a prime example of this lack, even if the assumption of a rectangular plan is relaxed.

Fig. 16. Reconstruction of the vanishing points in Raphael’s ‘Coronation of Charlemagne’. All vanishing points should lie on the yellow horizon line. In fact, the lines for each of the tables, the line of votive candlesticks, and the cardinals in the right foreground all deviate from this requirement.

There are a number of obviously rectangular objects in the painting that allow reconstruction of the vanishing points of each set of parallel sides. The perspective rules are simple. If all these objects are aligned with the same rectangular grid, all the edges of the rectangular objects should converge to either one of exactly two vanishing points. Both vanishing points should be at the intended level of the horizon. Any deviation from this construction anywhere in the picture implies a lack of adherence to the rectangular grid. (The more relaxed possibility is that the objects, while still rectangular, are arranged at a variety of angles on the grid. In this case, any number of vanishing points up to twice the number of objects is possible, but all vanishing points should still lie along the horizontal line of the horizon.)

The reconstruction of Fig. 16 makes it clear, however, that Raphael adhered to neither of these rules. While each object has a plausible vanishing point, the vanishing points neither cluster at two locations nor lie along the same horizontal line. To complete the set, we may assume that the oblique line of cardinals in the foreground is sufficiently homogeneous to allow construction lines across similar joints of their bodies. With this inclusion, we find that Raphael has used no less than 6 vanishing points a different levels in the scene, strongly violating the requirement that they all be at the same horizon level. Furnishings that are clearly intended to be aligned with each other also have different vanishing points. In total, there are eight different horizontal positions of the vanishing points where there should be two if the whole composition was on a uniform oblique grid. The similarity of the elements of the scene to some of the separate illustrations from Pélérin’s book (Figs. 15A & B) clearly suggests that Raphael adopted the particular construction for each part of the scene without understanding how they should be modified to form a coherent perspective projection.

The remarkable feature of angular perspective is that, although it was well-understood by geometers such as Pélérin (1505) and Vredemann de Vries (1605), it was avoided by virtually all artists until the middle of the seventeenth century. Aside from two paintings of doubtful attribution painted around 1440, the first successful use of full angular perspective was by Dutch artist Gerard Houckgeest in 1650. (There was limited use of the angular construction in floor tiling throughout the period, but this could easily be achieved by connecting the corners of a one-point perspective grid, and did not require an understanding of the rules of two-point construction.) Inspired to develop a radical design for his painting of the tomb of William the Silent, the king whose efforts united Holland in 1581, Houckgeest turned to Vredemann’s architectural representation technique of the oblique construction for the interior of the church at Delft. This dramatic shift from the unremitting one-point perspectives of the church interiors of Saenredam and Neeffs gained Houckgeest immense popularity in The Netherlands, but it was to be another half-century before the two-point construction appeared in Italy in the hands of Canaletto. The long progression from the early perspective approximations of Cavallini and Giotto around 1300, through the one-point developments of Masolino and Mantegna in the 1420s, to the two-point sophistication of Houckgeest and Canaletto was a span of 400 years. The intellectual endeavor of the full representation of space was evidently no easy matter, and was largely an effort of the artists and architects of the time.