Up to this point, we have been discussing the marginal distortion of right angles. Figure 8.8 (the panel labeled Fig. 243) shows that the correct central projection of a sphere that is not centered on the principal ray is an ellipse. Nevertheless, if the projectively correct ellipses were substituted for the circles with which Raphael represented the spheres in his School of Athens³ (Figure 8.9 and the detail in Figure 8.10), they would not look like spheres (unless the fresco were viewed through a peephole at the center of projection). This misperception of the correct projection of a sphere is a marginal distortion very much like the misperception of projectively correct representations of the vertices of cubes when they are outside the area of normal perspective (because they are likely to violate Perkins's laws). There is, however, one major difference: A cube ran be anywhere within the area of normal perspective and Still look like a cube; a sphere that is not on the principal ray will look distorted. The visual system, so tolerant of variations in the representations of vertices of cubes, is completely intolerant of variations in the representations of spheres. As discussed earlier in this chapter, the link between perspective exagérée and Perkins's laws is that the latter are a convenient rule of thumb that separate pictures that could represent objects in our normal field of view from those that could not. Vertices of cubes that are on the principal ray vary in their appearance depending on the distance of the center of projection from the picture plane; the projection of a sphere whose center is on the principal ray is always a circle. Furthermore, there is no convenient, easy to perceive, rule of thumb (analogous to Perkins's laws) to separate the unlikely projections of spheres from the likely ones: The difference between the projection of a sphere that falls just within the area of normal perspective and one that falls just outside is a purely quantitative difference in the ratio of the long dimension of an ellipse major axis) to its shorter dimension (minor axis). As a result, only circles are considered acceptable projections of spheres. And because artists have always accepted the primacy of perception over geometry, whenever they represented spheres in their paintings (which was not often), they always represented them as circles. In other words, it is as if whenever a sphere had to be represented, an ad hoc center of projection and a new principal ray (which passed through the center of the sphere) was created.

Fig.8.9 Raphael, The School of Athens (1510-1) Fresco. Stanza della Segnatura, Vatican, Rome.		Fig.8.10 Detail of Figure 8. 9 showing Ptolemy, Euclid, and others.

		Fig.8.11 Plan of four cylindrical columns, C, C1, C2, C3, projected onto picture plane AB using O as center of projection. Although frontal chords of circular cross sections of the columns (mn, m1n1, m2n2) project as constants (MN, M1N1, M2N2), diameters of columns project wider images the further away they are from principal ray.

Just as the correct central projection of a sphere becomes a more elongated ellipse the further the center of the sphere is from the principal ray, the wider the correct central projection of a cylindrical column becomes under these circumstances⁴ (see Figure 8.11). This marginal distortion is mostly academic, because I have not found any Renaissance paintings of colonnades that could have been subject to this sort of distortion acid were corrected to accord with perception. Nevertheless, Leonardo was aware of the problem, and he correctly pointed out that, although the progressive thickening of the pictures of columns the further they are from the principal ray (and the concomitant narrowing of the spaces between them) is implied by central projection, this "good" method (as he puts it) is "satisfactory" only if the picture is viewed through a peephole located at the center of projection. He concludes that when the picture "is to be seen by several persons" the only perceptually acceptable solution (which is "the lesser fault," i.e., not as good as using a peephole) is analogous to what Raphael did with the sphere: to ignore the rules of geometry and to represent the columns in the colonnade "in their proper size," that is, with equally wide projections (Leonardo da Vinci, 1970, §544, pp. 326-7).

		Fig.8.12 Paolo Uccello, Sir John Hawkwood (1436). Fresco, transferred to canvas. Cathedral of Santa Maria delle Fiore, Florence.

If spheres and cylinders are treated in a special way by the practice of perspective, it should not come as a surprise that the same is true of human bodies. If we think of the human body as a flattened sphere on top of a flattened cylinder, we can appreciate the distortions its picture undergoes as it is displaced away from the principal axis of the projection. In Figure 8.8, the panels labeled Fig. 246, 246^bis , 246^ter , Olmer shows three versions of the figure of Aristotle from Raphael's School of Athens, successively displaced to the right from the principal ray. Needless to say, artists never complied with this implication of geometry. Let us examine the famous fresco by Paolo Uccello Sir John Hawkwood to illustrate this most interesting violation of the geometric rules of central projection (Figure 8.12). Here is Hartt's description of the work:

[Uccello's] earliest dated painting is the colossal fresco in the Cathedral of Florence, painted in 1436 on commission from the officials of the Opera del Duomo, an equestrian monument to the English condottiere Sir John Hawkwood, known to the Italians as Giovanni Acuto, to whom a monument in marble had been promised just before his death in 1394...

The pedestal rests on a base that is supported by three consoles... The simulated architecture is projected in perspective from a point of view far below the lower border of the fresco, at about eye level of a person standing in the side-aisle^5. But the horse and rider are seen from a second point of view, at about the middle of the horse's legs. One is tempted to speculate as to why Uccello changed the perspective system. If he had projected the horse and rider from below, in conformity with the pedestal, the observer would have looked up to the horse's belly, and have seen little of the rider but his projecting feet and knees and the underside of his face. But might not Uccello, a lifelong practical joker, have done exactly that? Perhaps at first he did. The officials of the Opera objected to his painting of the horse and rider and compelled him to destroy that section of the fresco and do it over again. The explanation of this oft-noted circumstance⁶ may well have been Uccello's view of the great man from below. (1969, pp. 212-13)

³ An experiment credited by Pirenne (1970, p. 122) to La Gournerie (1859, p. 170). The second edition of La Gournerie's treatise (1884) does not mention the experiment.

⁴ This kind of marginal distortion was first discussed by Uccello and analyzed extensively by Leonardo. For a review, see White Chapter (1967, Chapter XIV). A more detailed analysis was published in 1774 by Thomas Malton; see Plate 144 (Figure 34) in Descargues (1977). La Gournerie (1884) also discusses it in detail.

⁵ Actually, the present viewing level is near the floor. Hartt is describing the original state of affairs as if it were current.

⁶ Described by Pope-Hennessy (1969, p. 7) as follows:

On 30 May [1436] Uccello was ordered to replace [Agnolo] Gaddi's fresco [of Hawkwood, commissioned in 1395] with a new fresco in terra verde [meaning green earth, a natural earth color], on 28 June he was instructed to efface the horse and rider he had executed on the wall `because it was not painted as it should be,' on 6 July he was told to make a new attempt, and by the end of August the fresco was complete. The erasure of the first version was due probably to some technical defect in the preparation of the ground, and not, as is often implied, to dissatisfaction with Uccello's cartoon [full-size drawing used for transfer to a wall on which a fresco is to be painted].

I find Pope-Hennessy's attribution of the erasure of the first version to a "technical defect in the preparation of the ground" implausible: Why would there be such a defect in the preparation of the ground of the horse and rider and not a similar defect in the ground of the base? Furthermore, what prompted the erasure of the second version? I think, as I explain later in this chapter, that Uccello had discovered that strict adherence to the laws of perspective made for unacceptable paintings and that he had to compromise twice before the result was acceptable to viewers. I also think that the tolerance of his employers was due to the avant-garde nature of Uccello's application of perspective.