John White writes in a similar vein:

The advantages of [using several viewpoints in a single composition] - sometimes even the necessity for it, are shown most obviously in Uccello's Hawkwood ... A fairly high degree of realism was desirable in frescoes which were substitutes for more expensive marble monuments, and this element of illusion is supplied by the steep foreshortening of the architectural [base]. On the other hand a worm's eye panorama of a horse's belly and a general's feet can be at best a dubious tribute to his memory. The realism of the low-set viewpoint is therefore restricted to the architecture. In Uccello's fresco there is no foreshortening of the horse or rider ... (1967, p. 197)

Peter and Linda Murray attribute the effect to Uccello's incompetence:

During the 1430s [Uccello] became fascinated by the new ideas in perspective and foreshortening, although he never really mastered the full implications of the system, which became for him, eventually, no more than another form of elaborate pattern making. Even when the impact of the new ideas was fresh, his treatment of them was quite arbitrary, as can be seen in the [fresco of] Sir John Hawkwood... This has two separate viewpoints, one for the base and another for the horseman...; a similarly irrational approach was also used in his Four Heads of Prophets of 1443 in the roundels in the corners of the clock of Florence Cathedral. (1963, pp. 113-4)

In view of our analysis of marginal distortions, I believe that Hartt and White are only partially correct in their analysis of why Uccello chose two inconsistent centers of projection, and, a fortiori, I believe that Murray and Murray err in their attempt to debunk Uccello's mastery of perspective. Hartt and White are mistaken in thinking that, as Hartt puts it, if Uccello "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." Hartt's and White's analyses are based on a failure to appreciate the importance of the distinction between the central projection of a scene (in our case, the monument) from a low vantage point onto a vertical picture plane, and its projection onto a titled picture plane. As long as the picture plane is, on the whole, parallel to the important surfaces of the objects represented, such as the side of the horse, none of the features of these important surfaces is lost by moving the center of projection. To better understand this point, let us ask the question in a slightly different way: How would the appearance of the horse and rider have changed had they been depicted in a manner consistent with the projection of the base, that is, from a low vantage point onto a vertical picture plane? It is true that more of the horse's underbelly would be visible in the picture, and that the soles of the figure's boots would be seen, but that is true of any equestrian monument erected on a tall pedestal. But Hartt and White are wrong to think that the horse's underbelly and the figure's soles would be visible to the exclusion of the side of the horse and the side of the rider. That would happen only if the picture plane were tilted, which would not be consistent with Uccello's representation of the base of the statue. I do not think that a representation of the horse and rider that would be consistent with the representation of the base would have been "a dubious tribute" to the general's memory and therefore do not believe that the officials of the Opera del Duomo who viewed the first version of Uccello's fresco were angered by having been the butt of a practical joke (an unlikely action on the part of an aspiring young artist, dependent on further commissions). What is at stake here is marginal distortion: I believe that Uccello's first attempt was a correct central projection of the pedestal, the horse, and the rider, which suffered from extreme marginal distortion; that his second attempt was a partial compromise, which was still afflicted with too much distortion; and that his third attempt - which is the masterpiece we know so well - was perceptually acceptable. Leonardo elevated Uccello's procedure to the level of principle:

In drawing from the round the draughtsman should so place himself that the eye of the figure he is drawing is on a level with his own. This should be done with any head he may have to represent from nature because, without exception, the figures or persons you meet in the streets have their eyes on the same level as your own; and if you place them higher or lower you will see that your drawing will not bear resemblance. (Leonardo da Vinci, 1970, 541, p. 325)

In conclusion, we have seen that non-rectangular bodies that are not on the principal axis of a central projection cause problems for the would-be orthodox user of this sort of projection. In general, such bodies - including humans and animals - are not drawn in accordance with the geometry of central projection. Instead, each body is drawn from a center of projection on a line perpendicular to the picture plane intersecting the picture at a point inside the contour of the body. Only the size of the non-rectangular objects and their position in the two-dimensional space of the picture are subject to the rules of central projection. We have argued in this chapter that this convention of painting reflects the perspectivists' acceptance of the primacy of perception and that central projection is applied principally to architectural settings of scenes. So perspective, as it was practiced by artists, was far from being an inflexible system. Because it was subordinated to perception and because different kinds of objects were made to obey the laws of central projection to different extents, a unifying concept such as Alberti's window cannot do justice to the subtleties and complexities of Renaissance perspective.

Some artists and scholars, who did not recognize the richness and elaborateness of perspective, have thought of it as an awesome monster unleashed on the art of the Renaissance, a geometric system so truculent that it confined the imagination of artists to an inescapable four-square grid. Here, for instance, is how Carlo Carrà wrote in his 1913 manifesto of Futurism, The Painting of Sounds, Noises, and Odors:

The old running perspective and trompe l'il, a game worthy at most of an academic mind such as Leonardo's, or of a designer of sets for realist melodramas.⁷

The Gestalt psychologist Rudolph Arnheim expresses a similar disdain for perspective in his classic Art and Visual Perception:

[Perspective] must distort sizes, shapes, and spatial distances and angles in order to convey depth, thus doing considerable violence not only to the character of the two-dimensional medium but also to the objects in the picture. We understand why the film critic André Bazin has called perspective "the original sin of Western painting." In manipulating objects to foster the illusion of depth, picture-making relinquishes its innocence ... The discovery of central perspective bespoke a dangerous development in Western thought. It marked a scientifically oriented preference for mechanical reproduction and geometrical constructs in place of creative imagery. William Ivins [1973, p. 9] has pointed out that, by no mere coincidence, central perspective was discovered only a few years after the first woodcuts had been printed in Europe. The woodcut established for the European mind the almost completely new principle of mechanical reproduction. It is to the credit of Western artists and their public that despite the lure of mechanical reproduction, imagery has survived as a creation of the human spirit ... Nevertheless, the lure of mechanical faithfulness has ever since the Renaissance tempted European art, especially in the mediocre standard output for mass consumption. The old notion of "illusion" as an artistic ideal became a menace to popular taste with the beginnings of the industrial revolution. (1974, pp. 258, 284-5)

Perhaps it is this mimetophobia, the morbid fear of slavish imitation, that impelled scholars like Herbert Read, Nelson Goodman, and Rudolph Arnheim, to name a few, to look for flaws in central projection as a method for the representation of space. Let us consider the most sustained critique, by Nelson Goodman in his important book Languages of Art. One line of Goodman's attack concentrates on what has been called the projective surrogate⁸ conception of perspective, namely

that pictorial perspective obeys laws of geometrical optics, and that a picture drawn according to the standard pictorial rules will, under the very abnormal conditions outlined above [viewed with one eye only, through a peephole] deliver a bundle of light rays matching that delivered by the scene portrayed. Only this assumption gives any plausibility at all to the argument from perspective; but the assumption is plainly false. By the pictorial rules, railroad tracks running outward from the eye are drawn converging, but telephone poles (or the edges of a facade) running upward from the eye are drawn parallel. By the `laws of geometry' the poles should also be drawn converging. (1976, pp. 15-6)

Although criticism along these lines is fairly widespread,⁹ it rests on a misunderstanding of the basis of perspective. Goodman erroneously assumes that when one talks about the "laws of geometry" one is referring to a law according to which the further an object is from the viewer the smaller the visual angle it subtends, which is correct, but is not the basis of perspective. According to the geometric rules of central projection, the projection of any two lines that are parallel to the picture plane, such as two telephone poles, or the edges of an appropriately oriented facade, will be two parallel lines.