Appendix One On Ancient Roots of Perspective n this appendix, I discuss how some of the ancient Greek sciences may Ihave influenced the convention of drawing in antiquity as well as the devel- opment that led to the emergence of linear perspective in the Renaissance. The sciences considered are the geometrical theory of optics, methods of making maps of the earth and sky, and scenography. The protagonists are, in chronological order, Plato, Euclid, Vitruvius, Ptolemy, and Proclus. Optics The Visual Pyramid and the Angle Axiom s mentioned in the introduction, the discipline now known as perspective Agot its name from optics (page xx). The two subjects are, however, inter- twined by more than mere etymology, most notably by the theory of vision. A basic concept within the latter theory is that of a visual cone whose apex is in the eye point of an observer and contains all the light rays connecting an object and the eye point – the rays being considered as straight lines. When introducing his model for a perspective representation, Alberti, as we have seen, took over this concept, calling it a visual pyramid (page 19). A central part of the theory of vision deals with appearances. For this the- ory Euclid introduced a fundamental axiom called the angle axiom. Although it has been discussed earlier, I will nevertheless repeat it here: Magnitudes seen within a larger angle appear larger, whereas those seen within a smaller angle appear smaller, and those seen within equal angles appear to be equal. [Translated from EuclidS Optics/1959, 1] The angle axiom implies that the size of the image of a line segment can be determined by measuring the length, seen from the eye point and placed at the location of the picture plane, that covers the line segment. Present-day draughtsmen sometimes apply this result by measuring the length of an image with a pencil. It is quite possible that something similar was done in antiquity, and that the procedure was seen as a consequence of the angle axiom. Perspective was introduced by Alberti as a section in a visual pyramid – corresponding to a central projection upon a plane – and the core problem in 723 724 Appendix One perspective was how to construct such a section. The idea of a section does not occur in Euclid’s optical theory, his aim being to account for visual appearances and not for projections. Irrespective of this, his theory may well have been used for pictorial representations, for instance, as just suggested, to determine the length of the image of a line segment. The Remoteness Theorem espite the absence of projections in Euclid’s work, the most widely used Dversion of his Optics contains a consideration that, to some degree, resem- bles a projection. This consideration is part of a proof for the following result, which I call the remoteness theorem:1 The more remote parts in planes situated below the eye appear higher. [Translated from ibid., 8] Although Euclid formulated this theorem generally about objects situated in planes below the eye point O (figure 1), he restricted his example to deal with three points, A, B, and C, lying on a horizontal line below O, proving that BC appears to be higher up than AB. The currently known proof introduces a line perpendicular to AC intersecting the visual rays OA, OB, and OC in D, E, and F, respectively. It is then argued that because F is situated higher than E, and E higher than D, and because BC is seen between the lines OC and OB, and AB is seen between the lines OB and OA, the line segment BC appears to be situated higher than AB. The line DF can be interpreted as representing a plane of projection, but in the proof its function was to get the rays OA, OB, and OC orientated with respect to above and below. The important thing in the proof was that AB is seen within the angle DOE, not that it is projected upon DE. O F E D AB C FIGURE 1. Euclid’s remoteness theorem. 1In current editions of Euclid’s Optics, this is theorem 10, but according to David C. Lindberg it was number 11 in the edition most likely used in the early fifteenth cen- tury (LindbergS 1976, 264, note 23). Ancient Roots of Perspective 725 Wilbur Knorr has convincingly argued that the intersecting line is a later addition, and that Euclid’s original proof was different (KnorrS 1991, 195–197). Knorr’s findings are, of course, an important support for the idea that Euclid did not think of projections in his Optics. Nevertheless, it is still possible that practitioners and theorists in antiquity thought that the practice of drawing more remote things higher up in a picture was an application of the remoteness theorem. The edition of Euclid’s Optics known in the Renaissance contains the proof of the remoteness theorem that I have just paraphrased. It may be, as suggested by Samuel Edgerton, that the proof, and in particular its accompanying dia- gram, influenced Alberti in creating his construction of the image of a square (EdgertonS 1966, 373), but he could as well have been inspired by something else. The Convergence Theorem he optical theory considered thus far is in agreement with perspective in Tthe sense that the optical results also apply for a perspective projection. Throughout history, however, scholars and painters have also used arguments from the theory of appearances that are not valid in the theory of perspective. There seem to be two reasons for this application, the one being that in some cases a discrepancy between the two theories was not observed. The other rea- son was that, as noted elsewhere, some perspectivists considered perspective to be a science used for reproducing a visual impression (pages 111, 559, 619). The most famous example of an optical result that is not compatible with the theory of perspective is the following result from Euclid’s Optics, which I call the convergence theorem. Parallel lengths, seen from a distance, appear not to be equally distant from each other. [Translated from EuclidS Optics/1959, 4] Euclid considered two parallel lines and looked at the cases in which the eye point is either in the plane of the parallel lines or above them. In the first case he implicitly assumed that the eye point lies between the parallel lines, and in the second that its projection upon the plane of the parallel lines falls between them (figure 2). To prove his claim, Euclid introduced line segments lying on normals to the two parallel lines and having their end points on the two lines, and then showed that the visual angles determined by the eye point O and these line segments decrease as the distance between the eye and the normals increases. The angle AOB is thus larger than the angle COD, which is in turn larger than the angle EOF, and so forth. Because the visual angles become smaller and smaller, the two parallel lines seem to converge. The convergence theorem may, at first, seem to be in accordance with the van- ishing point theorem. A closer examination shows, however, that this accor- dance is not complete because Euclid’s result is only valid for situations in which the eye point, or its projection upon the plane of the parallel lines, lies between the two parallel lines. Thus, if we apply Euclid’s method of argumentation to an eye point in another position, we end up with different results for how the 726 Appendix One AC E G O BD F H G O E C H A F D B FIGURE 2. Euclid’s convergence theorem. In the upper diagram the eye point and the parallel lines are situated in the same plane, and in the lower the eye point lies above the plane of the parallel lines. parallel lines appear and how they should be drawn in perspective. In the early eighteenth century, Humphry Ditton phrased the visual result as follows: If the Eye be seated anywhere without the Parallels, they will seem to go further from each other (or their Intervals to widen) to a certain Term of Distance; and after that continually to approach each other. [Ditton 1712, 17] In other words, some sections of the two parallel lines appear to diverge, while other parts appear to converge2 – implying that the lines appear as curves. This conclusion is obviously different from the result obtained by the rules of perspective, according to which the images of the two parallel lines (when neither of them passes through the eye point) are either two converg- ing straight lines or two parallel lines. The convergence theorem is not the only case of non-corresponding results in the theories of vision and perspective, but this one example suffices to show that the different aims of two disciplines may also lead to different results.3 Although it is conceptually important to be aware of the fact that some opti- cal results cannot be applied in perspective, it is historically less relevant because this insight may not have existed in earlier times. It is therefore quite possible that the convergence theorem inspired some drawers in antiquity and other drawers during the Renaissance to depict orthogonals as converging lines. 2 S For an argument supporting this conclusion, see Andersen 19872, note 4. 3Another often-discussed example is proposition 8 in Euclid’s Optics and its seeming con- flict with the law of inverse proportionality, which was discussed in chapter III (page 95). Ancient Roots of Perspective 727 Optics and Perspective in Harmony ome historians have seen optics and linear perspective as two conflicting dis- Sciplines.