Ancient Perspective and Euclid's Optics

Ancient Perspective and Euclid's OpticsAuthor(s): Richard TobinReviewed work(s):Source: Journal of the Warburg and Courtauld Institutes, Vol. 53 (1990), pp. 14-41Published by: The Warburg InstituteStable URL: http://www.jstor.org/stable/751337 .Accessed: 25/08/2012 06:12

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ANCIENT PERSPECTIVE AND EUCLID'S OPTICS* Richard Tobin

E RWIN PANOFSKY'S ESSAY, Die Perspektive alssymbolischeForm, appeared in 1927.1 In the sixty years since its publication it has become a landmark for subsequent studies on perspective theory and practice in Antiquity and in the

Renaissance. In the essay Panofsky claimed that ancient painting had failed to develop a 'correct' or central-point projection because such a system 'conflicted', as he wrote in 1960 in a brief restatement of the argument, 'with one of the basic tenets of classical optics', which he identified as the proportionality of apparent magnitudes and their visual angles-what he termed the 'angle axiom'. For Panofsky, this axiom was implicitly affirmed in Theorem 8 of Euclid's Optics, which he translated as: 'The apparent difference between equal magnitudes seen from unequal distances is by no means proportional to these distances.'2 Any construction derived from ancient optics as a 'perspectiva naturalis in usum artificium'-for painters, that is-would have been based upon this axiom.3 Such a projection would thus be merely an approximation of that optical theory, as arc magnitudes cannot be measured exactly for a flat projection plane (e.g. a wall surface), and any method for projecting these arcs-or their chords-would yield a convergence of parallel lines along a central axis, not to a single vanishing point (Figs 1, 2). Linear perspective, however, with its flat picture plane, employs what he termed the 'distance axiom', which maintains that equal magnitudes appear inversely proportional to their distances from the eye. In linear or plane projection then, parallel lines converge to a single vanishing point directly opposite the eye: central-point projection. The difference between the two systems-one determined directly by the visual angle, the other inversely by the magnitude's distance from the eye, is shown in Figs 3-5.4

* For their encouragement and counsel I must thank Professor Brunilde S. Ridgway, Professor Armando Petrucci, Dr Franca Nardelli and Professor Eugene Dwyer.

1 Erwin Panofsky, Die Perspektive als 'symbolische Form', Vortrige der Bibliothek Warburg, xxii, 1924-25, Leipzig and Berlin 1927, pp. 258-330. A brief restatement of the article is found in his Renaissance and Renascences, Uppsala 1960, edn 1972, pp. 121-29. See also his book, The Codex Huygens and Leonardo da Vinci's Art Theory, London 1940, pp. 62ff, 99ff.

2 Panofsky 1972 (as in n. 1), p. 128. 3 Panofsky 1940 (as in n. 1), p. 106. 4 Fig. 3: from Panofsky 1927 (as in n. 1), p. 264, text

fig. 4. Figs 1, 2, 4, 5: in John White, 'Developments in Renaissance Perspective', this Journal, xii, 1949, pp. 58-60. As White notes (p. 59 n. 2), these diagrams repeat or elaborate those used by Panofsky, Little and Hauck. Fig. 2, the diagram of A. M. Little ('Perspective and Scene Painting', Art Bulletin, xix, 1937, pp. 487-95, fig. 11) is an improvement on the construction provided by Panofsky (fig. 1) to illustrate his Antique

solution, as it includes both vertical and horizontal diminution. Little notes (p. 492 n. 14) that the process behind it is purely arbitrary. Fig. 5 ('Synthetic Perspec- tive') visualizes the mathematically correct projection of are magnitudes on a plane surface. As for actual representations of angular perspective (Figs 1, 2), they are compromise solutions, lacking a systematic and practical set of rules. They do not result in a central- point construction, but rather depict a concurrence of receding lines upon the median with a herringbone effect' (Little, as above, p. 492). To quote White (as above), p. 60: 'All these compromises have two points in common. The first is that they disrupt the unity of space representation. The second is that although they may be relatively easy to abstract as the mathematical skeleton of finished works of art, it is impossible to reduce them to a simple and complete set of instructions for the working artist. They are therefore inherently empirical in practice. The only alternatives are rules so simple that they only partially cover the system, or so complicated that they require the use of advanced mathematics.'

14

Journal of the Warburg and Courtauld Institutes, Volume 53, 1990

ANCIENT PERSPECTIVE AND EUCLID 15

FIG. 1: Vanishing axis perspective (after White, as in n. 4)

FIG. 2: Vanishing axis perspective (after White, as in n. 4)

C E

J F

B D G

FIG. 3a: The 'distance axiom' (after Panofsky, as in n. 4)

b b b

FIG. 3b: The 'angle axiom' (after Panofsky, as in n. 4)

FIG. 4a, b: Artificial perspective (after White, as in n. 4)

16 RICHARD TOBIN

FIG. 5a, b: Synthetic perspective (after White, as in n. 4)

Panofsky concluded that classical optics, with its spherical vision and the (consequently) curved, inexact perspective derived from it, is antithetical to plane, central-point projection. Any ancient route to plane perspective and central-point convergence was thus barred by the spherical nature of optical theory and by the flat-surface limitations of any perspective construction derived from it. This, for Panofsky, was the injunction of Theorem 8.5

This optical argument against an ancient central-point projection system became an axiom in its own right for subsequent studies on perspective. Yet any inquiry into the nature of ancient perspective is confronted with fragmentary and contradictory evidence. A case for inexact, curvilinear perspective conforms to Panofsky's use of Euclidean optical theory and can indicate partial vanishing-axis projections in some surviving Roman architectural frescoes (P1. la-b).6 Opposing arguments for central-point projection have drawn from literary testimony (including Euclid)7 and from other Roman wall paintings which appear to follow (again partially) single-point convergence (P1. 1c-d).8 Whatever its conclusions, any attempt to chart the course of perspective theory and practice has had to take into

5 Panofsky 1972 (as in n. 1), p. 129: 'Since even the simplest curved surface cannot be developed on a plane, no exact perspective construction could be evolved or even envisaged until the urge for such a construction had become stronger than the spell of the 'angle axiom' which is the foundation of classical optics.'

6 Extant frescoes of the Second Style in Pompeii which appear to employ an axial system as a partial solution (for projection schemes see Little, as in n. 4): P1. la-b, a wall from the Boscoreale Villa of Publius Fannius Sinistor (Little, fig. 3); a wall from the Boscoreale cubiculum (Little, fig. 4); a wall from the Villa of Diomedes (Villa ofJulia Felix), Pompeii (Little, fig .5).

John White argues ('Antique Perspective Theory and Pompeian Practice', Journal of Hellenic Studies, suppl.

7, 1956, pp. 43ff) that 'neither Euclid nor Lucretius reveals any awareness of the problems of pictorial representation ...' as their arguments are 'confined to the purely optical problems that arise in natural perspective.'

8 Illustrations of architectural frescoes of the Second

Style in Pompeii which are claimed to indicate the use, again partial, of a single vanishing point, can be found in White (as in n. 7), and H. G. Beyen, Die Pompejanische Wanddekoration vom 2. bis 4. Stil, i, 1938, ii, 1960, and 'Die Antike Zentralperspektive', Jahrbuch des deutschen archieologischen Instituts, liv, 1939: Corinthian oecus, Casa del Labirinto, Pompeii (P1. Ic-d; White, as above,

pl. 7a); west wall of triclinium, Boscoreale (White, pl. 7b; Beyen 1939, as above, figs 11, 12, and 1938, figs 22, 22b); alcove fresco of cubiculum, Boscoreale (White, pl. 8a; Beyen 1939, fig. 13).


account the ancient injunction, according to Panofsky, against central-point projection made explicit in Theorem 8 of Euclid's Optics.

Are ancient optical theory and central-point perspective antithetical? Could ancient artists have in fact devised a systematic method of central-point projection, proceeding from and without violation to the curvilinear premise of classical optics?

Based upon my analysis of Euclid's text, and upon application of its theorems and diagrams to a corresponding curvilinear projection, I shall address the following propositions in this article:

1. that the optical argument initially advanced by Panofsky is fundamentally flawed by its misinterpretation of Euclid's Optics, due in large part to Panofsky's recourse to a truncated version of Theorem 8 introduced into the MS tradition of the Optics by Theon of Alexandria;

2. that analysis of the 'genuine Optics', represented by Heiberg's critical edition, suggests a distinction in the text between image reception-what is recorded or 'is seen' (6p&zmt) in the eye-and appearance, or image perception-what actually appears (xaivwzat) to the observer;

3. that a reading of Theorem 8 within the context of the genuine Optics does not rule out central-point projection in ancient perspective practice: it suggests it, in fact;

4. that a correct (i.e. mathematically-determined) and practical method of curvilinear central-point projection can be reconstructed with graphic data derived from several theorems and diagrams of the Optics. Its debt to optical theory for its principles and procedures argues strongly for the availability of such a system to ancient artists and scenographers.

A future on-site comparison of variants or abbreviated versions of this construction with extant Roman frescoes could yield significant results which might point to the survival of this system in late Hellenistic painting, in the guise of a workshop inventory of graphic formulae for Roman wall design.

I. EUCLID, THEON AND THEOREM 8

As we have seen, Panofsky proposed his 'angle axiom' as a basic tenet of classical optics, supporting this claim with the opening statement from Theorem 8 of Euclid's Optics.9 According to Panofsky, this statement explicitly rejects the 'distance axiom' of plane perspective (hence classical optics is 'anti-planperspektive') ,? implicitly asserts the angle axiom, and assumes the identification of apparent magnitudes as arc images, since arcs alone have this exact relation with the visual angles which subtend them. " Panofsky cites two sources in his recourse to Theorem 8, both from the 1895 Heiberg edition containing that scholar's critical text of the

Optics.'22 Panofsky's second citation is to the Greek text of Theorem 8 of Heiberg's

9 Panofsky 1972 (as in n. 1), p. 128. o10 Panofsky 1927 (as in n. 1), p. 264. " ibid., p. 264, and Panofsky 1972 (as in n. 1), p. 129. 12 Panofsky 1927 (as in n. 1), n. 15, p. 264, and p. 299:

'Euklid, Theorem 8, bzw., (a.a.O.S. 164 bzw. S. 14): 'Tx

'ioa CIEyOe1 &vtoov E8teorlKtc( O)K tXvac(XyOg roig

&nooaira'tv 6p9tax ... Der Satz wird damit bewiesen,

dass der Unterschied der Abstfinde betrichtlicher ist, als der der Winkel, und dass nur diese (nach den in voriger

18 RICHARD TOBIN

edition of E6KXki6ot oitttd;'3 Panofsky quotes his first citation in the footnotes in the original Greek, paraphrases it in the essay, and will translate it into English in the 1960 restatement.'4 This first citation is taken from Heiberg's text of Theon's recension of the Optics, which follows the critical text of Euclid's work in the same edition.'5

The MS tradition of Opticorum recensio Theonis originated with Theon of Alexandria, a mathematician writing in the fourth century, some six hundred years after Euclid. In an earlier (1892) edition of Euclid's Optics,'6 Heiberg characterized the work of Theon as a compilation for the instruction of students, marked in its treatment of details by significant departure from the genuine Optics, to the extent that at times all the clarity and precision of Euclid is lost in the Theonine recension.'7 Thomas Heath, in his English translation of and commentary on Euclid's Elements,'8 reviewed Heiberg's findings regarding Theon's interpolated renditions of Euclid's works-their alterations, emendations, additions and omissions-noting that Theon at times omitted the substance or phrasing of Euclid because apparently 'he regarded Euclid's language as being too careful and precise','9 hence posing difficulty for students. Heath also noted Heiberg's discussion of Theon's alterations, which 'show carelessness in the use of technical terms, as when he uses &irtcoeat (to meet) for i dtteTOu (to touch) although the ancients carefully distinguished the two words. The desire of keeping to a standard phraseology also led Theon to omit or add words in a number of cases, and also, sometimes, to change the lettering of figures.'2" All these practices of Theon's recension of the Elements are at work in his recension of Euclid's Optics.2'

I have included this brief description of the Theonine handling of Euclid's works to underscore the truncated and often careless character of Theon's fourth- century recension of Euclid's Optics. Concern for the ipsissima verba of Euclid is justified even more given the fact that the manuscript which Heiberg found to be the most reliable representative of the tradition of the 'genuine Optics' (V = cod. Vindobonensis XXI, 13 = philos. Gr. 103 Lambecius) is itself often flawed. As

Anmerkung genannten Axiomen) fiir die Sehgr6sse massgebend seien.' 13 Panofsky 1927 (as in n. 1), p. 14. 14 Panofsky 1927 (as in n. 1), pp. 164, 299 and 264:

'(Das 8 Theorem Euklids vewahrt sich sogar ganz ausdrficklich gegen eine gegenteilige Ansicht, indem es feststellt), dass der scheinbare Unterschied zweier

gleicher, aber aus ungleicher Entfernung erblickter Grossen nicht etwa durch das Verhaltnis dieser

Entfernungen ...'; in Panofsky 1972 (as in n. 1), p. 128: '... the Eighth Theorem of Euclid's Optics, explicitly

stating that "the apparent difference between equal magnitudes seen from unequal distances is by no means proportional to these distances..."' 15 I. L. Heiberg ed., Euclidis Optica, vii, 1895 (I. L.

Heiberg, H. Menge eds, Euclidis Opera Omnia, Leipzig), p. 164 (Theorem 1', Opticorum recensio Theonis): 'T 'i•ca p1Eyenh iVLcOV

6v v tEa tc6a o1jK &vac,6yto

tot; &noonjapaotv 6p&zrc ...'

The opening statement of Theorem 1' (8) of the Heiberg edition of Euclid's Optics, which precedes the Theonine version (above) in vol. 7, and which Panofsky cites (p. 14) but does not quote, reads as follows: 'T 'isol

pyOn: KaXi

TtopdC(lXXCo vLiVaov 6tEatrjK6zot ob to 6ippcarog

o)K &vcX•6yOg 'roi; 1tXafiCL•Catv 6p&rct ...' It is clear from

Panofsky's choice of the Theonine version (p. 164) of Theorem 8 over that of the Euclidis Optica (p. 14), and from his interpretation of its meaning and significance for classical optics, that his optical argument is primarily based upon recourse to the Theonine recension of the Optics of Euclid, particularly to Theon's version of Theorem 8.

16 I. L. Heiberg, Litterargeschichtliche Studien iiber Euclid, Leipzig 1882.

17 ibid.: 'Und wirklich stimmt diese Auffassung sehr gut zum Charakter der Bearbeitung ... Im einzelnen ist aber in der Regel grosse Verschiedenheit ... manchmal auch alle Schirfe und Genauigkeit verloren gegangen ist...' (p. 146).

18 The Thirteen Books of Euclid's Elements, 1956, a republication of the Cambridge 2nd edn (1926), with introduction and commentary by T. L. Heath.

19 ibid., pp. 54ff (quotation from p. 56). 20 ibid., p. 55. 21 Consult Heiberg (as in n. 16), pp. 139ff; Theon (as

in n. 15), pp. xxix ff.


Heiberg states it: 'Still the text of the Vindobonensis is in no way completely satisfactory or even tolerable. Quite apart from smaller writing errors, as happens in all manuscripts, one finds a pointedly large number of totally wrong or unintelli- gible passages, which can be the fault only of the transcriber or redactor, in any case not Euclid's.'22 With Heiberg's caveat for his own critical edition of the genuine Optics, recourse to the far less reliable Theonine recension, written six centuries later, can lead to significant misreading of Euclid. Such is the case with Theorem 8. For comparison I have quoted key passages of Heiberg's Greek texts for Theorem 8 of Euclid's Optics23 (which Panofsky only cited) and Theon's Recension24 (which Panofsky cited, and from which he quoted and translated in his essay; see Fig. 6a-b):

A P

B ~H E

Be a E

FIG. 6a: Euclid's Theorem 8 (after diagram, MS Vat. gr. 1039)

B b

E

H Z P

p H Z K

FIG. 6b: Theon's Theorem 8 (after diagram, MS Vat. gr. 204)

Euclidis Optica, pp. 14, 16: A) Tb 'ior tEy90rl

Kai tnapd&•Xhrai &vtiov

6tfozrlK6rzn obob toi5 61aji•oo oK &vozX6Tyo; zog St;otiCLaxmv 6parat ... ('Equal and parallel magnitudes set out unequally from the eye are not seen proportionately with the distances ...')

B) hXdyo, bxt o1)K •oztv G; ,aivEzat Exov 6); tb FA tpbg tb AB, oito;g tb BE tpbg tb EA ('I say, that it is not (the case)/is not held, as it is held for appearance, that GD is to AB, as BE is to ED ...')

C) o1)K &wvdoyov 6pzat &nooztiCLamv 6parat tx 'iox a j.EYerl ('So equal magnitudes are not seen proportionate to the distances').

Opticorum recensio Theonis, pp. 164, 166: A') T 'ioa ~

tEy0rl &vItov &StEacr6-K a otas

&vaoX6yo;S toi &iXooni•Laaotv 6pat ... ('Equal magnitudes unequally set out are not seen according to a proportion with the distances ...')

B') pijtt 86i, 6ti o•jK &voX6yo;g Oxvfia•xt bx& BE, AZ jwy0rl torig FK, KZ touximCLamv ... ('I say therefore, that the magnitudes BG, DZ, do not appear proportionately with the distances GK, KZ ...')

C') o1K d&vdXoyov &pa -oig &noonticaa tx 'ica gE•eh 6p&xat ('Thus equal magnitudes are not seen proportionate to the distances').

There is more than a formal distinction between Euclid's Theorem 8 and Theon's recension of it. Euclid does not simply deny, as does Theon, that the ratio between the sizes of seen magnitudes is inversely the same as the ratio of their distances from the eye. In stating that this is not the case for image reception (o•ijK ... 6p&atx), Euclid also notes the fact that it is held to be the case for perception, or appearance: 6;g xaiv•vat

'Xov. Theon's wording, however, suggests no awareness of

22 Heiberg (as in n. 16), p. 133. 23 Euclid (as in n. 15), pp. 14, 16.

24 Theon (as in n. 15), pp. 164, 166.

20 RICHARD TOBIN

any distinction between what is recorded or seen (6pa&at) in the eye, and what actually appears (caivEzat) to the observer. Inverse proportionality is denied to images indiscriminately referred to as seen magnitudes (o-iK ...

6p9&at) and

apparent magnitudes (ol5K ... ~vav cGat). What is absent in the Recension's rejection

of inverse proportionality is the key clause in the Optics referring to apparent magnitudes, against which clause Euclid opposes his denial of inverse proportionality for seen magnitudes: (hCym, 6n

oU• ronv), 6•x caivEczt

v Xov ... The fourteenth-century Latin translation that accompanies Heiberg's Greek text

of the Optics renders this clause as: '(dico, quod non est), sicut apparet habens, (gd ad ab, ita be ad ed)'-' (I say, that it is not the case), as it is held for appearance, ([that as] gd is to ab, so be is to ed).'25

Perhaps the clearest insight into what Euclid is-and is not-saying here is gained through comparison of his thesis (B) with the corresponding expression in the later, Theonine recension (B'). Note in the first place that in Euclid's statement (B) the verb that is understood in connection with 6) tb FA ... oiitt tb BE

rtpbS tb EA

is 6pa&at used explicitly in the earlier and later statements (A, C). Theon also uses

6p&xat in his antecedent and subsequent statements (A', C'), but mistakenly copies a form of Qxivw ( avioExat) in the corresponding statement (B').

Secondly, the different syntax in Theon, together with his misapplication of

cxivw, radically alters the meaning of Euclid's thesis:

Euclid: Myo / Ott ot Ko tv, (0 (aivtEat W ov / 6) tb FA (6paoat). Theon: 4n91t 8i / 6ti o1')K &xvao6yco ;avilezt ...

Euclid is stating that it is not the case, as it is held for appearance (G; Qxivevzt ~Xov) that magnitudes are seen (6pa&at) in inverse proportion to their intervals from the eye. Theon states that magnitudes do not appear (xavaexzat) proportionately26 to their intervals from the eye. There is no equivalent in the Theonine recension to 6)

aivetaCt Exov, because Theon is not aware of any distinction in Euclid's Optics between vision (6pdi) and perception (0aivw).27

25 Euclid (as in n. 15), p. 15. On the meaning of

'(paivErat EXov', see below, Appendix I. 26 Theon's expression of inverse proportionality

between magnitudes and their intervals from the eye (i.e. BF:AZ = KZ:FK) actually suggests the opposite: ztx BE, AZ Cy•irL Zoig FK, KZ &tccoroolav (p. 164, line 8). Taken together with his abbreviations of Theorem 8, his careless, mechanical copying of the diagram (the letters of Euclid's diagram in the genuine Optics [p. 14] correspond to the steps of its construction outlined in the theorem, while Theon's lettering is arbitrary), and his divergence from the sequence and substance of Euclid's proof-all these factors suggest that Theon's interest in Theorem 8, and his regard for its import for Euclid, was not especially high. 27 This distinction between the two versions provides a

more convincing explanation than that which Panofsky offered for the radically 'amended' Theorem 8 of 16th- century Renaissance translators of the Optics, i.e. an attempt to 'eliminate a flagrant contradiction between two equally respected authorities', Brunelleschi and Euclid (Panofsky 1972, as in n. 1, p. 128 and n. 1; consult Panofsky 1927, as in n. 1, n. 17, p. 301). Heiberg

had determined by 1895 that the relevant translators -Zamberto in 1505, Pena in 1557-relied upon Theonine codices of the text for their translations

(Euclid, as in n. 15, pp. xli-xliii, xvi-xviii). It is worth noting here, as well, another relevant finding of Heiberg, regarding the availability of the two traditions of the Optics. Heiberg established that throughout the Middle Ages only the 'genuine Optics' was at hand, while in the Renaissance the Theonine recension became

very common or widespread ('renascentibus vero litteris recensio Theonis pervulgata est': Euclid, as in n. 15, pp. xl, xli). I would submit that any felt contradiction in the 16th-century translations of the Optics between plane- perspective and Euclidean optics was due largely to the distinctions between Euclid's text and Theon's recension of it.

No such 'contradiction' seems to have troubled Leon Battista Alberti, much less deterred his use of the Optics in his efforts to formulate a theory and practical method of perspective for painters. Even had he relied upon a Theonine codex (less likely in 1430 than after 1500), the author of De pictura was well versed in Euclidean geometry. In his use of it in the treatise,


Our understanding of Theorem 8 should not rely upon an interpretation of its language and syntax apart from the Optics as a whole, however. Euclid's distinction in Theorem 8 between the process of image reception (6p&at) in the eye and perception or appearance (aiveatx) for the observer is maintained throughout the text by the author's consistent and separate application of the verbs 6pd0 and xaiv0. The claim, then, for this distinction in Theorem 8 finds broad support from the context of the treatise.

II. EUCLID's OPTICS: 6pd9o AND WtVo

In my analysis of the Heiberg edition of the Optics I found that Euclid's use of 6pd& and ~aivw can be classified under four headings or categories; categories iii and iv depend upon the fundamental distinction between

6pd9 and $uivw represented by

categories i and ii.

Category i: 6pdo: Forms of the verb 6pd& used by itself in a theorem (i.e. not accompanied by $aiv0) denote the image of the object as it is recorded or is seen (6pa&at, from 6pd0) in the eye by the rays of the subtending visual angle; vision, or reception.28 The theorems under category i illustrate the principles enunciated in the Definitions (6pot) which introduce the Optics and precede the theorems;

Category ii: gaiv0: Forms of the verb xaiv0, used alone or accompanied by 6pd& in a theorem, always denote how the object, recorded or seen (6pthpEvov, from 6pd0) in the eye, actually appears (xaivEtt, from Qaiv0)

to the observer; appearance, or perception.29

Alberti was quite aware of the nature of its application: 'sed in omni nostra oratione spectari illud vehementer peto non me ut mathematicum sed veluti pictorem hisce de rebus loqui': De pictura, I, 1: see Leon Battista Alberti (On Painting and On Sculpture), ed. C. Grayson, New York 1972, p. 36). I hope to examine key MSS of the Optics to determine which tradition-the Euclidean or Theonine-Alberti relied upon in the formulation of his own perspective. 28 'Op&o is found alone (i.e. apart from 4xxivo) in

theorems which simply measure or state the relation of size between the image recorded in the eye and the object seen by the eye under given circumstances. For example (cE'): 'With a sphere seen (6poCinvii) by two eyes, if the diameter of the sphere is equal to the straight line by which the eyes are set apart from each other, the whole hemisphere will be seen (6p6firEztr) ...' (Euclid, as in n. 15, p. 40). This use of 6pdr by itself is found in Theorems y', n', iE',

K•', K•', K7r', X', hl', XO'. What is common to all these theorems involving diverse objects, intervals, and eye locations is an expression of relation of size between the object and its image recorded in the eye by the visual rays which subtend or define it within its visual angle. 29 Euclid always uses a form of the verb 4)aivo in

theorems or parts of theorems which primarily address the actual appearance, to the observer, of the image

recorded or seen in the eye by the subtending rays of the visual angle. These theorems do more than state the relation of size between recorded image and the object seen: they address the perceived difference or 'distortion', for the observer between object and image, i.e. the altered relation of the objects as they appear, compared to the actual relation of the objects themselves, in respect to distance and shape (Definition 3), size (Definition 4, e.g. E', [P'), and location (Definitions 5 and 6, e.g. t', t6'). For example, Theorem 0 illustrates the consequence for perception of the principle affirmed in Definition 3, which states that 'those things are seen (6poa0t) upon which the rays fall, and (those things) are not seen (lil 6p&owa) upon which the rays do not fall': (0') 'Rectangular magnitudes, seen

(6pc••vca) from a distance, appear (PaivEzl) round ...

since therefore each of the things seen (6po•0wvoy)

has a certain length of distance, having arrived at which it is no longer seen (6paat), then the angle G is not seen (oijK 6pZat), but only the points D and Z appear

(c•aivezat). In the same way also in each of the

remaining angles this will happen. So that the whole thing will appear (pavio~zat) round (paivEzat,

cavfioernz , from privo)'. The theorems of category ii, in

which Euclid applies 4xxivo in order to address the perceived 'distortion' for the observer between objects

22 RICHARD TOBIN

Theorems in this second category further elaborate the Definitions or first principles by addressing the consequences of the process of vision (6pdw) for perception ($aivo).30

Category iii: Qcivw/6p9w: 6p9w

is used by Euclid in place of caivw

in several theorems to underscore the geometric relation between what appears (4aivetat) to the observer and what is recorded or seen (6paat) in the eye, i.e. the relation determined by the common visual angle subtending both the image as it is seen by the eye and the image as it appears to the observer.31

While there are no instances in the treatise in which Euclid applies gxivw to denote the image recorded or seen in the eye, i.e. apart from its appearance to the observer, there are several theorems in which Euclid uses 6pd9 in conjunction with

caivw in addressing perception. These occasions underscore the relation between vision and perception based upon the principles of the visual angle enunciated in Definitions 1-7.32

Category iv: 8o0 K: Euclid employs forms of the verb 860oK ('seems') with an infinitive-often involving 6pd& (6pt&•oat)

and gaivw (caivEGO0t)

-to denote what only seems to be seen in the eye and perceived by the observer, not what actually is recorded and actually appears; illusion, misperception. Euclid applies a form of 80OKx with an infinitive to denote an illusory state, action, size, shape, or relation of an image for an observer.

The theorems of category iv deal with the object as it only seems to be seen, and seems to appear to the observer-in other words, with instances of 'deception.' By 'deception' I mean an apparent image that is misleading to the observer, is misread by the viewer as something other than what was actually recorded in the eye and is actually represented for perception. For example, in Theorem va' (110), what is actually apparent is an object (G) moving at the same speed as the eye, one (B) moving more slowly, and one (D) moving more quickly. What will seem (866~Et) to appear to the viewer, however, is: object G will seem to stand still (odtvxat S6oEt), object B will seem to move

(~epE0czt [66&Et]) in the opposite direction, and object D will seem to advance (•peoot 66 Et) to the lead.

There are situations in which the observer might not be aware of the misperception; these instances involve the perception of spheres, cylinders, and cones.

and their appearance, are: e', 5', 0', t', ta', y', tS', te', t', KP', X6', Xr', h1', pYP, y', gi', gE', •',

ui', and vi'.

30 A third group of theorems could also be included under this heading but for one additional feature whose reference to the discussion merits the classification of this group in a separate category (iii, below). I refer to those theorems in which Euclid employs 6pd~o in conjunction with 4aivo to address the apparent distortion of objects for the observer, as described above. This would seem to blur any proposed distinction in the Optics between image reception in the eye (6pd~o) and image perception by the observer (lxxivo). Yet Euclid's practice here, intended to stress the common ground of both operations, serves at the same time to distinguish each of them.

31 I shall propose below (pp. 26ff) that in the geometry of vision, the arc magnitude is the image of what is recorded or seen in the eye, and its axial chord is the apparent image, the recorded image as it appears to the observer. 32 For the theorems which use 6p&o and 4caivo see

Appendix II. 33 For example, Theorem K6' (Euclid, as in n. 15, pp.

38, 40), ie' (ibid., pp. 50, 52), Xa' (ibid., p. 54). The observer, then, confronting a sphere, a cylinder, and a cone from a distance, and then at close range, perceives a smaller part of the object at close range than the part perceived at a distance; yet what actually is recorded and apparent at close range as the smaller part, will seem to be recorded (86&t 6pforcnt) and will seem to


In summary, Euclid's language in the Optics makes a strong case for: 1) a distinction in optical theory between the process of image reception-what is recorded or is seen (6prat) in the eye-and its consequence for perception-what appears (xaivevat) to the observer; and 2) a proportional relation between both recorded image and apparent image on the one hand and their common visual angle with its subtending rays on the other. Given the principle of the visual angle common to both seen and apparent image, Euclid was free to alternate

6pd9 for

Oaivw throughout the treatise with far greater frequency than he actually did. Yet on the whole the author of the Optics was very careful to confine his use of each of these verbs to the separate but related stages of vision

(6p90) and perception

(~aivw) .4 In all of the Optics, forms of the verb gaivw are found in some one hundred

thirty locations-over twice the number in which forms of 6pdw are found. Apart from the three instances cited in Appendix III, gaivw is never used by Euclid in a statement to denote the recording of the object's image in the eye, in either a general or specific way (i.e. with explicit reference to the agency of the eye, to the visual angle, or to the rays).

This is the context in which any theorem should be interpreted-and especially, for our purposes, Theorem r' or 8 (14).

III. EUCLID's THEOREM 8

A reading of Theorem 8 within the context of the 'genuine Optics' represented by Heiberg's critical text does not rule out the possibility of central-point projection in ancient perspective practice; it suggests it, in fact. According to the analysis of the text, above, Theorem 8 falls under category i, involving theorems primarily addressing the recording of the object's image in the eye. The fundamental distinction in the treatise between vision-the reference for Euclid's use of 6paC- and perception-the reference for his use of quivw-is found here in Theorem 8 as elsewhere in the text. While it is valid to say that this theorem affirms 'one of the fundamental tenets of classical optics',3" we should understand that the tenet involved is a principle of the visual angle which embraces both the image that appears (xaivezat) to the observer as well as the image that is recorded or seen

(6p&zat) in the eye. To equate the proportional property of the visual angle and the seen image with the principle of Definition 4 is to narrow and thus misinterpret the broader scope of that definition:

... things seen (ztx ... 6pc6vxv) under a larger angle appear (4~aivExzm) larger, (things seen) under a smaller (angle appear) smaller, things seen (tax ... 6pcbCEvx) under equal angles (appear) equal.

appear (066wt caivE0ea ) as a larger part than the part seen at a distance. Even if the observer knows geometry and the action of angles on curved surfaces, the illusion in these instances is convincing. Note that even in this group of theorems dealing with illusory appearance or misperception, Euclid uses 6p~aCo in conjunction with 4aivo to underscore the link between vision and perception. And the use of 6piao is always in the context of a clear reference to the recorded image encompassed

(xiptCpoav6jCevov: Euclid, as in n. 15, p. 50, line 20) by the eye, to the image that is seen (pX~Eneat, ibid., p. 40, line 6) under the rays of the visual angle. The theorems of category iv, in which the author uses o60K0 with the infinitive to denote 'deception', or misperception, are: a', K6', ie', ha', hy', tP', tr', v', va', vp', vy', v6', ve', vS', vS'. 34 See Appendix III. 35 Panofsky 1972 (as in n. 1), p. 128.

24 RICHARD TOBIN

This definition covers the apparent images as well as the corresponding seen images of their common subtending angles.

It is also valid to state that Theorem 8 primarily addresses image reception

(6p&co) in the eye; but note that the thrust of the proposition is not to affirm direct

proportionality between seen objects (arcs) and subtending angles, but to deny their inverse proportionality to their distances from the eye: 'Equal and parallel magnitudes unequally set apart from the eye are not seen proportionately to the intervals ... I say, that it is not the case, as it is held to be for appearance, that (magnitude) GD (is seen) in relation to (magnitude) AB, as (interval) BE is in relation to (interval) ED.'36

Euclid moves through this argument in Theorem 8 in a crisp and coherent progression of twelve steps and concludes with a final restatement of it."• Read in this way, Theorem 8 says more (to us) about perception theory and practice current in Euclid's time (c. 300 BC) than about image reception. At its strongest, Theorem 8 is an attack, at its mildest, a caveat, against a principle 'held to be the case for appearance'-namely, inverse proportionality between images and their intervals from the eye. Whether taken literally ('as it is held/ for appearance/'), or figuratively to express either subjective belief, opinion ('as it seems held') or objective certainty, positive declaration ('as it is manifestly held/ very much apparent'), the same question arises in each rendering of 6g 4aivct~t ~Xov: by whom is the principle of inverse proportionality 'held for appearance', 'seemingly held', or 'manifestly held'? Surely not by mathematicians; who else then but (some) artists and scenographers of Euclid's day? I have proposed that the interpretation of ig caive~zt EXov along the lines of 'as it is held for appearance' is the correct one, and

finds support from analysis of Theorem 8 in the genuine Optics. I suggest, moreover, that any other rendition (above) points to the same conclusion: Euclid is acknowledging here that the 'principle' of inverse proportionality-whose validity he denies for both the seen image and the apparent image-is held in some quarters of perspective theory and practice current in his own time, contrary to a basic tenet of optical theory. Normally one does not need to deny the veracity of what is not advanced or does not exist. If late fourth-century perspective practice- or some proponents of it-did not hold to this view, then Euclid 'dot protest too much'. The point of Theorem 8 is not to prove the direct proportionality of seen magnitudes and their subtending visual angles (though it does), but rather to disprove-and disapprove of-inverse proportionality between seen magnitudes and their intervals from the eye."8

36 See above, p. 19.

37 See Appendix IV. 38 Were we to consider how ancient scenographers

arrived at the principle of inverse proportionality in perspective, one route is found within curvilinear optics itself (for another, see n. 39 below). Artists habitually constructing the correct 'curved-screen' diagram to determine the perspective size of equal magnitudes (such as the diagram of Theorem 8) would in time discover (see Fig. 7) that there is an inverse proportion between the apparent size (axial chord) of equal magnitudes and their diagonal distances from the eye, i.e. the interval to the eye measured from the extreme

end of the magnitude (its width in plan view, its height in side-view), not from its base: (GD) HI~ (AB) ZD =

AE:EG. The desire to put this discovery to practical use would have reinforced various adaptations of the arc- screen system (see text, p. 26, and Fig. 16b-f on p. 33), and could have finally led to the 'correct' concept of inverse proportionality (the groundline 'distance axiom') at work in plane perspective. Hence we have the intriguing possibility of the artists' awareness of a principle of plane perspective before their discovery of the flat-screen process which should logically precede and ground it.


A G

B T D K

FIG. 7: After Euclid's Theorem 8 (see diagrams, FIGS 17a, b, p. 39)

Euclid is careful, as Theon is not, to denote the reception of the image in the eye by 6pc&at and by p[X•1tEat. The only instance in the proposition in which gaivEtat is used is clearly referring to image perception, the object's appearance to the observer. Given the principle of Definition 4 common to both the seen image and the apparent image, Euclid's denial of inverse proportionality of what is seen, would apply equally to what appears. His 0g 4aivEtat kxov acknowledges that precept at work in current perspective practice but does not concede its validity as a sound principle for perception any more than he will affirm its truth for image reception. Theorem 8, and in particular its og 4aivetrat Axov, read against the fundamental distinction between 6pam and gaivw in the Optics, suggests this theory:

1. Classical optics, if accurately transmitted in Euclid's compendium, acknowl- edged a distinction between image reception in the eye and image perception for the observer, while maintaining the principles of the visual angle which governed both operations;

2. By the late fourth century BC, practitioners of perspective-or a significant group of them-held to a method of central-point projection markedly divorced from a key principle of classical optics: it ignored direct proportionality of seen images and their subtending angles, in favour of an inverse proportionality between apparent images and their intervals from the eye. In other words, they substituted a flat screen for the curved screen of vision. 39

39 If we were to speculate how this radical shift might have occurred within an optical tradition premised upon curvilinear vision, Euclid could provide a clue. Within his proof for Theorem 8 are two steps-omitted in Theon's recension-which serve to illustrate the ease of transition in perspective from the curved to a flat screen: step 7 (GD = AB) and step 8 (AB:DZ = BE:ED). While GD = AB is a given, its reassertion here in step 7 by Euclid is integral to what follows (Theon's argument diverges from Euclid after step 6). A glance at the diagram (Fig. 7; see n. 38 above) accompanying the proposition graphically suggests the latent force of the latter equation (AB:DZ = BE:ED) for perspective. If the curved screen, at D, of arc LET is replaced by the flat screen of line GD, then DZ becomes the image of magnitude AB recorded on GD. Given that GD = AB, then as AB:DZ = BE:ED (step 8), and with AB recorded as DZ, so GD:DZ = BE:ED, then GD:AB = BE:ED, or

magnitudes GD and AB are 'seen' in inverse proportion to their distances (BE, ED) from the eye. It is worth noting that this equation, GD:DZ = BE:ED, which is implicit in Euclid's argument, is explicitly used in the much later recension of Theon, and its connection to

step 8 is confirmed by a Theonine scholiast (Opticorum recensio Theonis, as in n. 15, p. 164, lines 24, 25; p. 261, Ad prop. VIII, line 19ff). As the scholiast of the recension noted (p. 261), Theon's statement of this equality of ratio between magnitudes GD,DZ and intervals BE,ED (in Theon's diagram, AZ, ZO and UK, KZ) is derived from the fact that AB:DZ = BE:ED (in Theon's diagram, BT:ZO = KF:KZ)--which Theon omits. Euclid explicitly states it, however, as step 8). This procedure clearly violates optical theory, and in substi- tuting a flat screen for the curved one, its practitioners exchanged a theoretical tenet for a practical tool.

26 RICHARD TOBIN

If Theorem 8 does indeed suggest a central-point perspective at work in Euclid's time, and in violation of a key, curvilinear premise of classical optics, it might well be a radical departure from an earlier, optically 'correct' tradition: i.e. a central- point perspective that derived from and conformed to, curvilinear theory. If the Optics as a whole maintains a distinction in classical theory of vision between the seen image of vision and the apparent image for perception, with their mutual dependence upon a common visual angle, then such an optically correct method of projection would have been possible, in theory, for classical artists.40

IV. A RECONSTRUCTION

My analysis of the Optics argues for the existence of an 'optically correct' perspective system available to ancient Greek painters. I propose the projection method below as a reconstruction of that perspective system. The method and its diagrams have been derived from the definitions, theorems and diagrams of Euclid's Optics. Its theory is premised on the Definitions or 'first principles' which ground the tract's geometric theory of vision. These Definitions (1-7) articulate the basic property of visual angles, namely their direct relation of size with the images of the objects that they subtend. This property of visual angles governs the relative size and location of images for both vision and perception. For the arc and corresponding axial chord of plane geometry, dependent upon their common visual angle, become the seen magnitude and corresponding apparent magnitude, respectively, in the applied geometry of the Optics. The arc magnitude is the image of what is recorded or seen in the eye (vision, 6p&m), and its axial chord magnitude is the image of what actually appears to the observer (perception, paiv) ).41

We begin with the fact that in each theorem Euclid represents the locations of both eye and object from a third, fixed position perpendicular to the plane of both eye and object. From this position, either above or to the side of the visual angles between eye and object, Euclid depicts the eye and object on a horizontal plane in a plan diagram, and in a vertical plane by means of a side-view diagram. The curved screen upon which arc and axial chord magnitudes are subtended by the visual angles is either indicated or assumed in the diagrams (Figs 8-10).

In the Optics, these plan and side-view diagrams illustrate propositions about the relation of eye and object from a third, fixed position above or to the side of both. A perspective construction corresponding to a given plan and/or side-view diagram would illustrate the same relation of eye and object from the subjective position of the eye of the observer. By pursuing the progression of theorems and diagrams in the treatise, we can reconstruct the steps by which this frontal or perspective construction would have been developed with the data 'transferred' from the plan and side-view diagrams (Appendix V and Fig. 11).

40 It is possible in the first place because apparent images are not arc magnitudes. 41 An arc is measured here by an axial chord, i.e. by

the straight line joining the two end points of an arc perpendicular to the eye's central axis or horizontal 'line of sight.' While inclined or 'acute' arcs and their chords (e.g. Fig. 14 on p. 32) produce a projection similar to the reconstruction below, the designation of

the axial chord as the apparent image would have been justified by the rapid and continuous movement of the rays of the eye which form or constitute that apparent image ('visibus velociter transportatis'; see Theorem ac'). The composite side-view and plan diagrams of steps 2 and 3, respectively (Appendix V below) also reflect the 'composite' character of such an image.


A P K A

B

FIG. 8a: Theorem a' (1), diagram

A C K F D

aC a d

FIG. 8b: Plan diagram of Theorem a'

r A

A B

E

FIG. 9a: Theorems P', e' (2, 5), diagrams

C D

A B

E

FIG. 9b: Plan diagram of Theorems 3', e'

B A E P

B E A

B E A 1'

FIG. O10a: Theorems t', ta' (10, 11), combined diagrams

P+ E+ Ct

FI+~#,

C+

e- d4

O- E- C-

FIG. 10b: Side-view diagram of combined Theorems t', ta'

28 RICHARD TOBIN

T L K

Z M H/ \

D N\D !\ /2\ -'.\

\\ \\

FIG. 1la: Theorem 5' (6), plan diagram

FIG. 1 ilb: Transfer of horizontal distances to side-view diagram

N+ D" H L" K+

ND H LK

The combination of data derived from the theorem diagrams (read as plan, composite side, and composite plan views) will yield a corresponding frontal projection or perspective. That construction will illustrate, from the subjective position of the observer, the relation of eye and object depicted in the given diagram from a position above (plan) or to the side of (side-view) both object and observer. The three steps comprise a single operation with three stages, illustrated in Fig. 13a.42

42 Let the object be a rectangular solid with height, width, and depth in the ratios of 2:2:1 (Fig. 12g), with the observer at O, at a given distance OF from the solid, with the observer's centric line of sight perpendicular to line AB and striking the solid at F, the horizontal and vertical centre of the face of the solid. The solid's plan and side-view diagrams (Fig. 12d-f) conform to the corresponding diagrams for Theorems 1 and 2 (Fig. 12a, b; Figs 8b, 9b) and Theorem 8 (Fig. 12c). Figure 13a-c, depicts the three steps of the operation and the step-by-step formation of the frontal or perspective construction. This chord construction (Fig. 13d) is a

frontal or perspective view, on a flat surface, of an object projected through a curved screen. The fully curvilinear projection, depicting the perspective of all points on the vertical segments of the solid (rather than only those points along the horizon and at the extremities) is easily obtained through bisection of each inclined line in the projection with a compass, and extension of the dividing line back to the horizon line AG (Fig. 14); the are of the inclined line taken from its point of intersection with the line AB constitutes the curvilinear perspective of that vertical segment.


ff

/

7-

/ \ 1/ /\ / / / \ I

/ Nt\ D•,

HtL K

I /!

I I

E N D H L K

I+ N m I- N ap

FIG. 11 c: Transfer of 'diagonal' distances to composite-plan diagram

T L K

Z M H

B N D

, I \ N? D+ H) L? K?

E N D H L K

FIG. 11d: Perspective/chord magnitudes of plan, side-view and composite-plan diagram

30 RICHARD TOBIN

A PI K &

B

FIG. 12a: Theorem a' (1), (plan) diagram

K A

B

FIG. 12b: Theorem 3' (2), (plan) diagram

A

FIG. 12c: Theorem r (8), (side-view) diagram

C H I J D

A E F B

O

FIG. 12d: Plan diagram of object g C H I J D

A E F B

O

FIG. 12e: Plan and side-view diagrams (d, f) of object g

B+ D

(0)------ -.. ...D

B" D"

FIG. 12f: Side-view diagram of object g


A?

C"E G+ B

C A E F G_

B-

C-

AF A'

FIG. 12g: Object g, rectangle constructed from diagrams for Theorems u', 3', n'

FIG. 12h: Perspective construction of object g

C DJ /D

FIG. 13a

A F _ B 4, ' d /

I / / ""

/ ' -'b

FIG1t B1t" I''1 D~w I I/ /

/ -II -

I'-" .,0

I l

/

I

- - -

.

i

,.,,,j - ,''-,-,, .- i.1 _

____. _--,

I +~ 1I / -?

I ,

I/ - I\ - // \

I I

FIG. 13b FIGc. 13d FIG. 13c

FIG. 13a: Three-stage operation for perspective construction of object in FIG. 12 FIG. 13b: Perspective height, side-view diagram FIG. 13c: Perspective widths, composite-plan diagram FIG. 13d: Perspective constructions (a-c)

32 RICHARD TOBIN

A' B

FIG. 14: Conversion of perspective segment to its corresponding circumference (arc)

FIG. 15: Fully curvilinear perspective construction

Any projection constructed by the above method, using the diagrams of the Optics, will exhibit the following traits (Fig. 13d):

1. Entasis and diminution: all vertical magnitudes on the plane perpendicular to the eye's centric line of sight will appear curved (entasis) and tapering inward (diminution) as they rise toward the vertical axis opposite the eye's line of sight;43

2. Horizontal curvature: all lateral magnitudes on the same plane will exhibit horizontal curvature, convex above the horizon line, concave below;

3. Magnitudes seen or recorded under larger visual angles will appear larger, those seen under lesser angles will appear smaller, and those seen under equal angles will appear equal (Definition 4);

4. Central-point convergence: all connecting orthogonals recede to a central vanishing point.44

The three separable stages of this optically correct projection system lend themselves easily to abbreviation in actual perspective practice. Several variants of the perspective construction are produced by an omission or modification at a given stage, as illustrated in Fig. 16b-f. These variants could point to the route that ancient perspective might have followed in the transmission of this doctrinaire

43 A trait common to this perspective construction (Fig. 13d) and to all its variants except Variant V (Fig. 16f) is diminution of lateral spacing along the horizon line, with the widest interval flanking the central vertical axis and the narrowest at the ends. The presence of lateral diminution in a given architectural fresco would discount the possibility that procedures such as those applied in Variant V (identical to Renaissance plane perspective constructions) were used. The presence of this trait on peristyles of Greek temples suggests its use in architecture as an optical refinement, but this motive could well have coincided with other considerations,

either practical or theoretical (e.g. the problem of the corner triglyph; see my article, 'The Doric Groundplan', American Journal of Archaeology, lxxxv, 1981, pp. 379-427). 44 A key difference between the perspective construc-

tion proposed in this article (Fig. 15) and the idealized construction that illustrates conversion of arc magnitudes to a plane surface ('Synthetic Perspective', Fig. 5 and n. 4 above): in the construction proposed here, orthogonals recede to the vanishing point in straight lines, while in the arc diagram they recede in curves convex to the eye.


FIG. 16a: Stage. Perspective construction FIG. 16b: Variant I: adjustment of inclined lines to vertical

FIG. 16c: Variant II: reduction to steps 1 and 2

FIG. 16d: Variant III: designation of uniform height

FIG. 16e: Variant IV: substitution of first perspective chord

FIG. 16f: Variant V: uniform height (III) and substitution (IV)

34 RICHARD TOBIN

system-grounded on the curvilinear premise of classical optics-through later workshop practice.

In the first century BC, wall painters at Roman Pompeii, Boscoreale, and Oplontis, as artisan heirs of this workshop legacy, would have far less awareness of, or interest in, the geometric pedigree of their graphic tricks-of-the-trade. They would have looked for their inspiration more likely to Eros than to Euclid.

UNIVERSITY OF ROME, FULBRIGHT SCHOLAR

APPENDIX I

The meaning is perhaps clearer if Exov in bg xaiverat Exov is also understood after 6•tn olK itriv (~oiiv EXov = Exet, Greek-English Lexicon, eds H. G. Liddell and R. Scott, Oxford 1968, 7th

edn), so that we have, in effect: 6it o00K &ltV (?xov), 6g quivezat lxov; or 6tin o5K XyEt, G; qivezat Exov

f•otv, where ?xo has the force of the Latin teneo or habeo, to hold, to maintain (Liddell

and Scott, 341, A.I. 8; A Latin Dictionary, ed. C. Lewis and C. Short, Oxford 1879, 1966 impression, 1854, I.B. 2.b,c; 835, II.D.1). Thus: 'dico, quod non habet/tenet, sicut apparet habens ...'-'I say, that it is not held/maintained, as it is held for appearance, (that ...)' For 6g 4aiverat Exov, Euclid could have intended 6g #aiverat Exov: 'as it is manifestly held/maintained' or ;g #aiverat Exetv: 'as it seems to be held/maintained' (Liddell and Scott, 854.B.II.1, Qxivu, Pass.: 'to appear to be "so and so"', with infin., fintg &pikozrl Qiverza

Ctvat. Od.; to'0t6 got 0BEoiaztov Qxive~ta yevioeat. Hdt.: -infin. omitted, 6otng aivezat &putroog.

Od., etc.: -also with part., but #aiveoiat with inf. indicates that a thing appears to be so and so, 4aiv~oeat c. part. states the fact that it manifestly is so and so, 4toi oi &i 0ovuitv qaivetat you appear to me to be rich, Hdt.; but, etvoog ixaivezo ?bv, he was manifestly well-inclined, id.; Qaive~at 6 v6gog j3dXritv the law manifestly harms, but #aive~zatx 6 v6iog lCgag 1; jdv\3 yv, it appears likely to harm us,, Dem). Yet the use of #aivo in these figurative senses ('manifestly is', 'seems/appears to be') would contradict Euclid's restriction of that verb to its technical meaning ('is apparent') throughout the treatise. For the author of the Optics found himself in the awkward situation wherein both the literal or technical, and the figurative or derivative sense of a key verb (4aivo) were called upon in the same context, hence could easily be confused. While most theorems that deal with perception concern what actually appears ('is apparent') to the observer, some theorems (e.g. those involving perception of parts of spheres, cones, and cylinders) address the observer's misreading of what actually appears, i.e., what only 'seems to appear' or 'appears to appear' (which the Latin translation renders as videtur apparere). In those instances Euclid avoided the potential for confusion by using, instead of Qxivu, a form of

86•0Km with the infinitive: 866~t tporljo0ut (videbitur

praecedere), 860Et #aivE0iae (videtur apparere; Theorems v', Xy' respectively, Euclid, as in n. 15; see also text above, p. 22. On occasion [e.g. Theorem

t'] Euclid does use the unequivocal

pavep6v, 'evident.'). Hence in this instance it is highly unlikely that Euclid would use Qaivu in the figurative senses of 'is evidently held' (4aive~at Exov) or 'seems/appears to be held' (Quivrat ?E~xv). The parallel in Latin to the distinction in Greek between Quivta~x (with participle)-'it is manifest/apparent/clear'-and 866Kce (with infinitive)-'it seems/appears to be'-is that between apparere (apparet/adparet) and videre(videtur), respectively: 'res apparet and far more freq. impers. apparet with acc. and infin. or rel. clause, 'the thing (or 'it') is evident, clear, manifest, certain, 8i4dv ozt, xaiveat, (objective certainty), while videtur, 8OKei,


designates subjective belief.' (See Lewis and Short, as above, 140, II; also 1988, II.B. 7.) Happily for Latin the distinction between apparet (objective certainty, positive declaration) and videtur (subjective belief, guarded opinion (Lewis and Short, as above, 1989, II.B. 7.b) does not seem (sic) to blur as does that between gaive~a and 8oKEi, where #aivEatx can, as noted above, indicate either certainty (with participle: aivexat ... Xdurttov or opinion (with infinitive: #aive~at ... 1Xk&etv). Less happily for the 14th-century Latin interpretatio which Heiberg used to accompany his edition of Euclid's 'OrttIKd&, the careful Latin translator felt constrained to render instances in the Optics of 866~t 6p&0oia as 'videtur videri' ('it seems to be seen', Theorems 1(6', pp. 38, 39; 1(', 50, 51; ka', 53, 55), where perhaps a less exacting translator would have used 'videtur (videbitur) cerni.' But the choice of the literal rendition of videtur videri would suggest the same care in his approach to Theorem 8's ;g qaive~tx ~xov, which he translates as sicut apparet habens. This would indicate perhaps that the phrase was understood to express a figurative sense of Qaivo: 'it is manifestly held'-though it does not exclude the possibility that the translator simply transferred a puzzling Greek syntax into its awkward Latin equivalent. What it would exclude is the interpretation of xivwTxt as 'it seems (i.e. videtur).' That would suggest a rendering of the clause as, as 'it is quite manifest/very much apparent.' The only English translation of the Optics (H. E. Burton, 'The Optics of Euclid', Journal of the Optical Society of America, xxxv, 5, 1945, pp. 357-72); Professor Burton's translation relies upon the Heiberg text) renders the full statement (Cyom, 6it o••K fotv, 4g quivezti Xov, bg tb FA itpbg tb AB, ototg tb BE itpbg tb EA) in this way (p. 358): 'I say that, as it appears, BE is not in the same relation to ED as GD is to AB.' In the first place, however 'appears' is intended here (literally, 'is visible', or figuratively, 'it seems'), the position of otKi in the Greek (before bg xaiveatx ~xov) should render that translation as: 'I say that it is not, as it appears, (namely) as BE is in relation to ED so (otto;) GD is in relation to AB' (oitog: Liddell and Scott, 580 [ottog], I.1. and 907 [6g], A.I. 2. 6g [tb FA] ... oitgo [tb BE] as in Latin ut ... sic', although 'properly, otrog is antec. to 6g ...'). Secondly, if we accept the revised translation (above) as a more accurate rendition, interpreting qaiverta to mean (literally) 'is/are apparent', leads to one or the other of two conclusions. Either it yields a patent contradiction with what follows (6g tb FA tpbg tb AB, otto"g tb BE tpbg tb EA, p. 14, lines 6, 7; 6g [line 6] is omitted in several MSS) if that is understood to refer to the relative size of apparent images (in effect: 'I say that it is not, as it is/they are apparent/is the case for appearance, that as [apparent image] GD is to [apparent image] AB, so [interval] BE is to [interval] ED)'; or it infers that what follows is restricted in its reference to the seen image-what is recorded in the eye (6p&ta)-as opposed to the apparent image-what appears to the observer(aivEatw). In this latter interpretation, Euclid would be saying in effect that GD:AB = BE:ED is not the case for what is seen

(Stt oiKl ~otv), as it is for what actually appears to the observer (6g gaive~at ~xov). I conclude that Euclid assumes or makes a distinction between the recorded image in the eye and the apparent image for the observer, and I have shown that Theorem 8 primarily addresses what is seen (6p9&ta), and the invalidity of inverse proportionality (GD:AB = BE:ED) for the recorded image (tb 6p6)gEvov). But while the clause 6g tb FA xpbo tb AB, oijtog tb BE tpbg tb EA (lines 6, 7) does indeed refer to the seen or recorded image, the interpretation as 'I say that it is not the case (for vision), as it is the case for appearance, that as GD:AB, so BE:ED' is incorrect in one key area. For while Euclid does distinguish throughout the Optics between what is seen (6p&rat) in the eye and what appears (xaiveat)) to the observer, at the same time he underscores their mutual dependence upon their common visual angle. Hence Euclid's denial of inverse proportionality for seen images and their intervals from the eye applies equally to apparent images, and so a literal translation of #aivetat in &; #aivetat

~xov should render it 'as it is held for appearance' rather than 'as it is the case for appearance.

36 RICHARD TOBIN

APPENDIX II

For example, Theorem (8'): 'Of intervals equal and on the same straight line those seen

(6pcbthva) from a greater distance appear (quiveat) smaller ... I say, that AB will appear (•aviocEat) larger than BG, and BG (larger) than GD ... Therefore the angle ZEB is larger than the angle ZBE; and (the angle) ZBE is equal to the angle BEG; and the angle ZEB is then larger than (angle) GEB. Therefore, AB will be seen (60ilo•tat) larger than BG. Again in the same way, if through the point G a line parallel to DE is drawn, BE will be seen

(d60zlatcn) larger than GD' (d6-ioea~t from 6pd9o):

Euclid (as in n. 15), pp. 6, 8. Midway through this theorem addressing the unequal appearance to the observer of equal intervals on the same straight line, Euclid abruptly shifts from Qaivu to 6pdo: he substitutes d6-~letax ('will be seen') for xav~lceat ('will appear'). This switch occurs immediately after his indirect reference to the principle of Definition 4 ('... the angle ZEB is larger than the angle ZBE; and [the angle] ZBE is equal to the angle BEG; and the angle ZEB is then larger than [angle] GEB. Therefore ...'). The Definition itself enunciates the fundamental relation between the object and the visual angle which subtends it under its visual rays. The image of the object that appears to the observer is dependent upon the visual angle and its subtending rays which initially record it in the eye. What is recorded in the eye (6p&tat) and what appears to the observer (xaivewat) are mutually grounded in the properties of the visual angle affirmed in this Definition (4) and in all the Definitions which precede the theorems. Euclid's recourse here to the principle of Definition 4, to explain the unequal appearance to the observer of equal intervals, acknowledges the visual angle as the common source of the image that is seen in the eye (6p&tat) and of the image that appears to the observer (xaive~at). His sudden shift from the apparent image (xavioetat) to the seen image (60ri~etat) after allusion to a principle of the Definition, serves to underscore what the allusion points to: the relation between reception and perception based upon the visual angle and its rays governing both operations (see also, e.g. Theorem Xe', ga': Euclid, as in n. 15, pp. 64, 88ff). In all such theorems in which Euclid suddenly shifts from Qaivu to 6pdo, the transition is invariably accompanied by his recourse, in the proof, to the properties of the visual angle and its subtending rays. In all there are ten theorems involving the author's use of 6p&o in conjunction with Qxivo (Theorems P3', 8', g', y', Xe', )g', g', pa', plg', p0'). In two of these theorems (P', p') dealing primarily with appearance, the shift is from 6pdo to xaivo; in the remaining eight theorems the transition is from Qxivu to 6pdo. In seven of those theorems (8', g', XE', ;g', ta', tg', 1t0'), in which Euclid abruptly shifts from a form of #aivo in the opening statement to a form of 6pdo in its restatement within the argument, the change is occasioned by that part of the proof in which he invokes, but does not explicitly cite, the principle of the visual angle and its subtending rays enunciated in Definition 4 ('[a] and things seen under a larger angle appear larger, [b] things seen under a smaller angle appear smaller, [c] things seen under equal angles appear equal': Theorems 8' (4a); g' (4a); X' (4a);

)g' (4c), pa' (4c), pt' (4c), ge' (4a, b). Definition 4 provides the fundamental property of the visual angle for reception and perception: a direct relation between the size of the magnitudes seen and the size of the visual angles which subtend them. Euclid's attention to the common tie of the visual angle to both what is seen by the eye and what appears to the observer is evident also in the two theorems involving a shift from

6pd9 to Quivo. In

Theorem P' Euclid cites Definition 7 as the basis for the relation between the number of subtending rays of what is seen (6p&at) and the clarity of what appears (aiveztx); and the greater number of rays in the visual angle common to both depend upon the larger size of that angle-the principle of Definition 4. In the second theorem,

t', the argument deals

with appearance (cxvil~jem) until Euclid alludes to the principle of Definition 4: once the

principle is invoked, the author shifts to d64O~iemtw (Euclid, as in n. 15, p. 88). This consistent


practice of Euclid points to the author's tendency to underscore-wherever it is not explicitly stated as a definition-the geometric link between the apparent image for the observer and the recorded or seen image in the eye, grounded in their mutual compliance with the principles of the visual angle fundamentally stated in Definition 4. In claiming this concern as the reason why Euclid reinforces pciva with 6pd&a in these theorems (category iii) dealing primarily with appearance, it is relevant here to review briefly the comparable theorems of category ii in which he does not. As stated earlier, Euclid uses pciva alone in twenty theorems which address the apparent image (category ii). Three theorems in this group, which come toward the close of the treatise (1X', pp. 80f;

t•', gW', pp. 104f) involve

straight-line magnitudes subtended by right angles and inscribed within semicircles. The equality of right-angles and arcs subtended by them, together with the visual clarity of the accompanying constructions, clearly convey to the reader the governing role of visual angles in the resulting apparent images. A fourth theorem in this group, the last in the treatise (varl', pp. 118f), is an adaptation to a square of theorems previously applied to circles. Of the remaining sixteen theorems of category ii, twelve explicitly summon the appropriate definition-rather than simply allude to its principle-in support of the thesis (0', Def. 3;

t', Def. 5; E', Def. 4; X6', Def. 4; XAl', Def. 4; g•',

Def. 4, ge', Def. 4) or refer back to a previous theorem which has already done so (ta', ty', t6', e', it',

refer back to Theorem t',

Def. 5); and four theorems (c', KP', PI3', gy) invoke the principles of Definition 4 without citing the definition itself-as do theorems in category iii (uxiva/6p&a) as well as explicitly referring to the related seen image, that is recorded or is seen (3phertez ) by the visual angle or its subtending rays-as theorems in category iii do not. For example, in the explicit reference in the proof of Theorem r' (Euclid, as in n. 15, p. 12) to the seen images of AB and GD recorded in the eye by the visual angles ('But under the angle AEZ, AB is seen [~h~ezat] and under [the angle] LED, GD [is seen] ...'), Euclid underscores the relation of the seen image to the image that appears to the observer: gEiSOv &pa

i1 FA ti; AB Quive~•i-in

other words, the proportionality of the visual angle common to both the seen and the apparent image: Ki zt& gtyv ri5b gteiovo; ywvix 6p9bevac

... (Definition 4). Euclid tends to shift to the process of reception (6p&zat, P3XFiact) whenever he does not have occasion in the proof to affirm its role for perception otherwise (i.e. by citing the appropriate definition or by invoking its principle together with explicit reference to the related image that is seen

[pkhrctwz] under the rays of the visual angle). For Euclid, the image that is seen or recorded in the eye (6p&aat) is logically prior to the image that appears to the viewer (qxivext)), just as the arc of the visual angle is logically prior to the axial chord of the same subtending angle. The author's abrupt shifts from perception to reception, and vice versa, reveal his concern to convey the relation between the two processes and their common grounding in the proportional property of the visual angle.

APPENDIX III

Beginning with the definitions and throughout the Optics, Euclid's references to the recording of the object's image in the eye are dominated by his use of forms of the verb

6p&a (6pi01vov, 6p&uat, 6pQ~Oo~am, 6p&oam) and its cognates (tb i6ggCla, axi 6byt). The Definitions (1-7) enunciate the principles of vision and their consequence for perception. They state that the object or magnitude is seen (6p&oam, from 6p&a) within straight lines of sight carried from the eye (&tob tof5

6u•Ctaoq: tb 6tlxa, from ggCw1a, pf. pass. of 6pd&o; Definition

1). Thus a cone of vision, made up of visual angles, is contained (ptCEX6e1vov) within these vision lines of the eyes ("rnb -0v 6emov: axi 6bWt, from 6ylog•at, fut. of 6pd&; Definition 2). And these vision lines or rays fall upon or strike the object or magnitude (Definition 3). Euclid applies other verbs besides 6pd&a to designate this stage of image-reception. These verbs are

38 RICHARD TOBIN

often called upon to address the process of vision in one or another of the three ways in which it is described in the Definitions: i.e., an object is seen 1) by the eye: riOb to 6bggcaxtog or Tzv

6x?&mv (Defs 1-3); 2) by the visual angle:

+rib tri; irnb tfi; yivlig (Defs 4, 7); and 3) by the rays of the visual angle: rib tdv Xtziv(v (Defs 5, 6). In applying these synonymous verbs Euclid follows the same restrictive procedure which governed his application of 6pda. This practice can be confirmed by a tally of the forty locations in the Optics where the author addresses the recording of the image in the eye with explicit reference to the eye, the angle, or the visual rays. There are twenty-two places in the Optics where a form of 3phXCo is written or understood with explicit connection to the agency: 1) of the eye (e.g. K6': p3Xneratt Orob tof 13 6tc1xoto; tb FA gtpog trig aoxipo;:

Euclid, as in 15, p. 38; (e', K6' [3] KIg', K•');

2) of its visual angle (e.g. K6': 1

K[3 rib orig T

nb KAB ycavix;g phn?eat: ibid., p. 26; 5' [2], i' [2], Kr3' [2], plt' [2],

1y' [2], g6S' [3]) or of its subtending rays (e.g. icy': ... rinb

T)v (htldvOv tbv BT, BA ph~ercat; tcx [2], cy', Icrl', p3'). The only instance in which p3Xio is found apart from explicit connection

with the eye, with the visual angle, or with the subtending rays, follows a statement which already describes the encompassing of the angles (and hence their magnitudes) by the subtending rays: (vS') ~tEi yxp ac AZ, ZA bv ZB, ZE

O18ooova yvciav TEPt•XoU(T, j~wiov

&9pa tb BE To AA p~EXneat: p. 113, line 27, p. 114, lines 1, 2 (&X1a;h). Heiberg states that such alternate proofs were not advanced by Euclid, but rather were interpolations (see xxix). P3XiECO is clearly equivalent to 6pda for Euclid, and in three theorems (iy', K', •p') he applies both verbs to describe the same operation in different parts of the theorem-e.g. (iy'):

•i 6pTact rnob TOv BT, BA (hXlv(v tob a?Tob TTg oeoipog jitpog

... KWi tfOob

... TOWv ~-rTivcO)v Tdv BT, BA p31c••Eat

: ibid., p. 39, lines 18-21). Indeed ph[~o is interchangeable with 6pda in denoting the recording of the object's image in the eye, and has, as a geometric term, more force for Euclid than does 6pda with its more applied (i.e. optical) associations: 6pda is found together with explicit connection to the eye, the visual angle, or the subtending rays only one fourth as often (y', K;g', cr', ?p', vy' [2]) as P3XIto is under the same conditions. Appearing less often in the Optics, but equally evident as alternates to 6pda in denoting the recording of the image in the eye, are the verbs

&rtoXcqatpd3vo, epiXthatPdv.O and omp~om. The

first two verbs denote the 'sectioning off or encompassing of the object by the eye (Ke': ronb To) ogc(XaTo;: ibid., p. 39, line 20), by the visual rays (Kr': [EOitck ac ZE, ZB] 6roXa4pd3vovtv &pa LaXTtov pito•aiptov ...: ibid., p. 46, line 10; (X'): Tb &pa •inepthaXp3cav6gevov riOb TOv AB, AT

E0t~EeV ...: ibid., p. 52, lines 25, 26), or by the eye and its lines of sight mentioned together (Xa': rtl8tv oiv

•prepthlXpd•v~etT •ncb To) E

6otb(aTog KWi tv EZ, EH 6Oyv tb ZEH pipog tOO KCbvoW:

ibid., p. 39, lines 21-23). In the seven instances where these verbs occur (tg' [2] Kg', K•',

X',

Xx', [2]), as well as in the three places where Eomp~o is found (ir' [2], Ke'), the meaning intended by the author is clearly that of 6pda. What of

pcivo and perception? While there

are some forty instances in which Euclid opts for the verb 6pdo, or one of its synonymous verbs noted above, to denote the recording of the object's image in the eye with explicit reference to the agency of the eye, the visual angle, or its subtending rays, the author applies aXivo under the same conditions on three occasions only, and even then indirectly and in conjunction with P[3tO in one of them. They occur early in the Treatise: (i'): ... 8tx 6i T&v AT, AA fl AF cQave~at , 6tx 6is Tv AA, AE i1 AE, l FA &pXa T~ig AE LgE empolipa qcivrtt

... (Euclid, as in n. 15, p. 27, lines 9-11); (c'): &Xc1a 8th tiv TOv FA, AE Tb FE ph3•r•aT, 8t6 1i T&v AA, AB Tb AB XaiveTm ... (ibid., p. 18, lines 28, 29, p. 20, line 1); (te'): IC•t oiv iOob ToO

6Ot1caXTog Ki Tig EZ

xKzivog T(x ZB, FA QXivT~tat, tb AB &pa to) FA ixrepe~0v QuivEat Ttit AZ

gOt&;0; ... (ibid., p. 22, lines

26, 27, p. 24, line 1). Note, however, that in the case of Theorems i' and ta', the use of 8t&, in connection with the visual rays, instead of ntcb-used in addressing the recording of the seen image by means of the eye, the angle, or the rays-suggests that Euclid here is addressing the emergence of the apparent image through the general agency of the rays which are the specific and direct instruments for encompassing the 'prior' seen image. This interpretation of 8tx in reference to location rather than to instrumentality or agency, is reinforced in


Theorem te', dealing with magnitudes unequal in length below the level of the eye. Here, where Oxnb is used, Euclid is clearly denoting location, not agency (i.e. apparent magnitudes will emerge below the level of both the eye and the ray EZ);

de•t oiv Orob zof 61waxzo; Ki til

EZ dhwivo z~x ZB, FA Qaivezat ...: in the Latin translation, 'quoniam ergo sub oculo et ez radio zb et gd apparent ...': ibid., p. 23.

APPENDIX IV (Fig. 17)

A P H

B O E

FIG. 17a: Euclid's Theorem 8 (after diagram, MS Vat. gr. 1039)

A G

B T L

D B T

FIG. 17b: Diagram with Roman letters (after Latin translation)

(r'): (1) ?Inci o0v tb EFZ ZpiyWvov toi5 EZH zoCingw g•i6v aoztv, (2) zb 6~ EZA zpiywvov zoi5 EZO top03g ?ctzz6v

outv, (3) tb EFZ &pca ptiyov tpbg tbv EZH topCa ~Eiiovca Xd6yov lXEv i'tEp tb EZA tpiCOvov Tpbog to EZO topCa, (4) Kai evaXlxk b EZE tpiywvov tpbg tb EZA tpiyovov geigova Xd6yov EXet ietp 6 EZH toybig pbog tbv EZO topia. (5) KAi ovOivt tb ETA tpiywvov tpbg tb EZA

tpi•ovov gEiSova 6Xdyov EXet i~xp 6 EHO tog~ibg pbog bv EZO zopia. (6) 6XX'

dr zb EAFT pbg ztb EZA zpiywvov, oiitzg 7i FA tpbg zilv AZ. (7) i 6i8 FA zti AB dotv ior, (8) Kai Ag J AB tpbog ziv AZ, il BE tpbog tiv EA. (9) 1 BE &pa tpbg tilv EA ~i•Eova X6yov EXet itup 6 EHO togebg

tpbo zbv EZO zopCa. (10) 6S &t 6 togEb,

tpbo Zbv zojia, ottzo,

4 Gxb HEO yovia spbo tiiv 6xb ZEG yoviav. (11) fj BE &pa cpbo zilv EA

g•eiova X6yov EXEt i•Ep

fj xTb HEO ywvia tpbo ziAv xtb ZEG. (12) K•ai

K iv tifi iixbo HEG

yoviac pXral tob TFA, ?K 8& tifg 6xb ZEO zb AB. oXK aXvd6loyov &pa zoig &xoonfigcactv 6p&zal z& '~oa gC0EYi.

(Euclid, as in n. 15, p. 14, 1. 14 f).

For the sake of brevity I have 'devised' the symbol 4 to represent 'is seen under/by' (6p&-at/P[3 erXrat);

here in Theorem 8 Euclid uses ph3•Xrcat (step 12).

1) Since triangle EZG is greater than section EZL, and A EZG > EZL 2) triangle EZD is less than section EZT, A EZD < EZT 3) therefore triangle EZG has a greater ratio to section

EZL than triangle EZG to section EZT; A EZG/EZL > A EZG/EZT 4) alternately, triangle EZG has a greater ratio to triangle

EZD than section EZL to section EZT, and A EZG/A EZD > EZL/EZT 5) added together, triangle EGD has a greater ratio to

triangle EZD than section ELT to section EZT; A EGD/A EZD > ELT/EZT 6) but as (triangle) EGD is to (triangle) EZD, so (line)

GD is to (line) DZ A EGD/A EZD = GD/DZ 7) but GD is equal to AB, and GD = AB 8) as (line) AB is to (line) DZ, so is (line) BE to (line) ED, AB/DZ = BE/ED 9) therefore BE has a greater ratio to ED than section ELT

to section EZT; BE/ED > ELT/EZT 10) and as section (ELT) is to section (EZT), so the angle

LET is to the angle ZET; ET/E -

LET/ ZET 11) therefore (line) BE has a greater ratio to (line) ED than

angle LET to angle ZET; BE/ED > Z LET/Z ZET

40 RICHARD TOBIN

12) and (magnitude) GD is seen from angle LET, (magnitude) AB from angle ZET. GD L LET / AB L ZET

Therefore equal magnitudes (GD, AB) are not seen proportionately to the intervals (BE, ED). GD/AB ? BE/ED

Euclid's argument can be summarized as follows. From the sequence developed in steps 1 to 4 the author states, in step 5, that the composite triangle EGD (nearer the eye and bounded by side GD) has a greater ratio to the lower of its two constituent triangles, EZD (bounded by DZ or side GD), than the entire arc section ELT (cut off or subtended by rays EG and ET) to its corresponding arc section EZT (subtended by rays AE, EB). In steps 6 to 8 Euclid shows that this same ratio of triangle EGD to triangle EZD is equal to the ratio of the interval or distance BE (of side AB from the eye at E) to that of ED (of side GD from E); therefore (step 9) the ratio of BE to ED is likewise (step 5) greater than the ratio of arc section ELT to arc section EZT. As arc sections are directly proportionate to (i.e. have the same ratio as) their subtending angles (step 10), the interval ratio BE:ED exceeds also the ratio of angle LET to angle ZET (step 11). 'And as (magnitude) GD is seen from angle LET

(K•l ?K gpv tflq ~nb HEO yozving phi3nEat tb FA) and (magnitude) AB from angle ZET (EI &~i i

6no ZEO tb AB-step 12)', the interval or distance ratio BE:ED is likewise greater than the ratio of magnitude GD (seen under angle LET) to magnitude AB (seen under angle ZET), or BE:ED>GD:AB, thus proving that 'equal magnitudes (GD, AB) are not seen proportionately to their distances (BE, ED), o0C &vdXloyov t&pa zoi; &nooztigucatv 6p&zat ix 'ioa gederl', or GD:AB # BE:ED.

APPENDIX V

Step 1: Plan Diagram. The diagrams of the initial theorems of the tract (Theorems a', P', y', 6',

e', S', 5', [1-7]) provide data for recording the apparent or perspective widths of an object at the level of the eye. Transverse lines of a rectilinear object, opposite the eye and on the same horizontal plane, are recorded by the visual angles on the curved screen as the arc magnitudes subtended by those angles, while the corresponding axial chord magnitudes of those arcs are recorded on the plan diagram as the apparent widths of those lines. For example, in Theorem e' (5, Fig. 9a), transverse lines AB and CD are recorded on the curved screen of its plan diagram (Fig. 9b) as arc magnitudes ab and cd (a•b, cd) and as the corresponding axial chords ab, cd (ab, cd). Chords ab, cd (ab, cd) are the apparent or perspective widths of transverse lines AB, CD.

Step 2: (Composite) Side View. These diagrams illustrate propositions involving lines or surfaces above or below the level of the eye at the horizon, and to the left or right of the eye (e.g. Theorems 8 [rl'], 10 [t'], 11 [t~a'], 13 [ty'], 14 [tS'], 15 [te'], 16 [ti'], etc.). Visual angles subtend the heights (or depths)of these surfaces or line segments, located on or above, and perpendicular to the horizon line at the point which marks their actual distance from the eye, measured along the horizon line. This horizontal distance of each line segment from the eye is obtained from the plan diagram of step 1. For example, in a side-view diagram for Theorem ; (6, Fig. 1l b) the horizontal distances from the eye of vertical segments N+N, D+D, H+H, L+L, and K+K have been transferred from the 'corresponding' plan diagram for Theorem 6 (Fig. 1la). This same procedure would locate segments N-N, D-D, H-H, L-L, and K-K. The result (Fig. 1lb) would resemble the combined side-view diagram of Theorems 10 and 11 (Fig. 10b). The images of the vertical segments are recorded as arc magnitudes and their axial chords on the curved screen. The axial chords (n+n, m+m, etc.) are the apparent or perspective lengths of those vertical segments.


Step 3: Composite Plan. The side-view diagram of step 2 is a graphic composite view which depicts all vertical segments above and below the eye, intersecting the plane of the horizon at their actual (horizontal) distances from the eye. In that side-view diagram, visual angles in vertical planes subtend vertical line-segments. The composite plan of step 3, however, locates the extreme points of those vertical segments on a plan diagram which depicts their actual (diagonal) distance from the eye and their lateral distance from the eye's central (and vertical) plane of sight (Fig. 1l b). This lateral interval between each extreme point of a segment and the centric line of sight is the same here as on the plan diagram for step 1 (i.e. N+D+ in Fig. lc = ND in Fig. 1la). The actual distances between these extreme points and the eye-depicted on the side-view diagram as the diagonal lines between the eye and the vertical segments at their maximum height (or depth)-are transferred from the side-view diagram to the composite plan (Fig. 1 lc). This graphic composite of the plan view for each vertical segment at maximum height provides the apparent or perspective width of each segment at that height, just as the plan view in step 1 provided the perspective width of each segment recorded at the level of the eye. Visual angles in each plane record the intervals between vertical segments (at their extremities) as arc magnitudes. The axial chords of these arc magnitudes constitute the perspective intervals between those segments at maximum height or depth (e.g. Fig. 1ld: axial chords n+d+, m+h+, l+k+).

a--Boscoreale, Villa of Publius Fannius Sinistor, summer b--Projection scheme for P1. l a (p. 16) triclinium, south wall (p. 16)

c--Pompeii, Casa del Labirinto, Corinthian oecus (p. 16)d-Poetnshmefr1.l(p16

z

z Hd

Hr

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Ancient Perspective and Euclid's Optics

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Transcript of Ancient Perspective and Euclid's Optics