euclid's optics

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Colin Webster Euclids Optics and Geometrical Astronomy Abstract: This paper seeks to demonstrate that propositions 2327 of the Eucli- dian Optics originated in the context of geometrical astronomy. These entries, which deal with the geometry of spheres and rays, present material that over- laps considerably with propositions 13 of Aristarchus of SamosOn the Sizes and Distances of the Sun and the Moon. While all these theorems deal with ma- terial that could conceivably be native to celestial illumination, the proofs do not work for binocular vision. It therefore seems probable that the proofs were borrowed from Aristarchus or, more likely, some common astronomical prede- cessor. As an extension of this observation, this paper argues that the Optics displays a far less unified purpose than has typically been assumed. Instead, it reflects a conglomerate of concerns, goals and dependencies, insofar as it pre- sents multiple proofs that emerge from a set of geometrical concerns not di- rectly related to vision as a physical phenomenon. These entries complicate the idea that the Optics is about explaining light, sight or false appearances in any straightforward sense. Instead, the text is oriented toward a somewhat more diffuse goal: simply to articulate the geometry of vision and rays. Optics as a discipline should should therefore be understood not as a purely mathematical explanation of vision, but as a target-field of explanation constituted by fea- tures and boundaries derived from extra-visual geometric concerns. Keywords: Euclid, Optics, Ancient Greek Mathematics, Astronomy, Vision Colin Webster: Columbia University Classics, 1130 Amsterdam Ave. 617 Hamilton Hall MC 2861, New York, New York 10027, United States, E-Mail: [email protected] Introduction When ancient Greek philosophical authors talk about visual perception, they most often focus their attention on its physical mechanisms. For instance, such authors as Plato, Aristotle and Theophrastus all examine various arguments for and against opposing theories, addressing the physical aspects of eyes, light and sight. They also mention the accounts of Empedocles, Anaxagoras and De- DOI 10.1515/apeiron-2014-0007 apeiron 2014; 47(4): 526 551 Brought to you by | Harvard University Authenticated Download Date | 3/31/15 8:50 AM

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  • Colin Webster

    Euclids Optics and Geometrical AstronomyAbstract: This paper seeks to demonstrate that propositions 2327 of the Eucli-dian Optics originated in the context of geometrical astronomy. These entries,which deal with the geometry of spheres and rays, present material that over-laps considerably with propositions 13 of Aristarchus of Samos On the Sizesand Distances of the Sun and the Moon. While all these theorems deal with ma-terial that could conceivably be native to celestial illumination, the proofs donot work for binocular vision. It therefore seems probable that the proofs wereborrowed from Aristarchus or, more likely, some common astronomical prede-cessor. As an extension of this observation, this paper argues that the Opticsdisplays a far less unified purpose than has typically been assumed. Instead, itreflects a conglomerate of concerns, goals and dependencies, insofar as it pre-sents multiple proofs that emerge from a set of geometrical concerns not di-rectly related to vision as a physical phenomenon. These entries complicate theidea that the Optics is about explaining light, sight or false appearances in anystraightforward sense. Instead, the text is oriented toward a somewhat morediffuse goal: simply to articulate the geometry of vision and rays. Optics as adiscipline should should therefore be understood not as a purely mathematicalexplanation of vision, but as a target-field of explanation constituted by fea-tures and boundaries derived from extra-visual geometric concerns.

    Keywords: Euclid, Optics, Ancient Greek Mathematics, Astronomy, Vision

    Colin Webster: Columbia University Classics, 1130 Amsterdam Ave. 617 Hamilton Hall MC2861, New York, New York 10027, United States, E-Mail: [email protected]

    Introduction

    When ancient Greek philosophical authors talk about visual perception, theymost often focus their attention on its physical mechanisms. For instance, suchauthors as Plato, Aristotle and Theophrastus all examine various arguments forand against opposing theories, addressing the physical aspects of eyes, lightand sight. They also mention the accounts of Empedocles, Anaxagoras and De-

    DOI 10.1515/apeiron-2014-0007 apeiron 2014; 47(4): 526551

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  • mocritus, who do the same.1 In stark contrast, the Optics attributed to Euclid isa fully geometrical treatment of vision, composed of fifty-seven mathematicalproofs.2 The Euclidian author weighs no alternative theories. He makes no phy-

    1 See especially Pl. Tim. 45b2c4, 67c469a4, Arist. De sens. and Theophr. De sens.2 Before we begin an examination of the Optics, we need to determine 1) which text we arediscussing; 2) whether Euclid is its author; and 3) when such a text was composed. In regardsto the first question, there are two different versions of the Optics preserved in the manuscripttradition, now referred to as version A and version B. Since the publication of Heiberg (1882),edition A has been considered the authentic Euclidian text, while B was thought to have beena later edition produced by Theon of Alexandria (ca. 335 CEca. 400 CE). Version B was thusreferred to as the Recensio Theonis. Heibergs conclusion was based on the fact that B containsmany difficult and unclear passages and he considered these unworthy of Euclid. In more re-cent years, however, both Knorr (1991) and Jones (1994) have pushed back against Heibergsargument and proposed that it is more typical for an editor to add clarifications and additionalproofs, rather than muddy long, clear syntax. Moreover, B employs a highly idiosyncratic letter-ing scheme, unique to the Optics, while A displays the more standardized alphabetical letteringof later geometrical texts. Jones and Knorr argue that lettering was more likely standardizedthan made more idiosyncratic and therefore A is more likely the recension, while B is an ear-lier, more authentic version of the text. That being said, both scholars recognize that B stillcontains many stylistic oddities and geometric errors uncharacteristic of Euclid in his Elements.Thus, in regards to our second question above, then, not only do these oddities suggest thatthe text contains multiple interpolations, but it also casts doubt on whether Euclid is in fact itsauthor.To Knorr and Jones arguments, I will add that Geminus, Astron., Frag. Opt. 22.1415 appearsalmost certainly to have had version B in the first century BCE, since he quotes its first lines:optics establishes that visual rays from the eyes move along straight lines [ - ]. This is identical to the firstdefinition found in B, Let us establish that visual rays from the eyes move along straightlines [ ], (Eucl. Opt. def.1, B), while the A edition reads Let us establish that straight lines leading from the eyes movea distance of great magnitude [ ] (Eucl. Opt. def. 1, A). Although Heiberg (1882), p. 134,seems to acknowledge that Bs first definition is more correct, he nevertheless supplies Asversion in both his 1882 and his 1895 editions of the Optics. In light of this evidence and recentarguments, I will follow Knorr and Jones and use manuscript B as the main text for my investi-gation.Lastly, it remains difficult to assess precisely when the Optics was composed or compiled, espe-cially since Euclids own dates are uncertain (although most scholars, contra Schneider (1979),accept the date of fl. c. 300 BCE). Nevertheless, while the text uses both and torefer to the visual rays, Netz (1999), pp. 103113, argues that mathematical language tends to-wards economy, shedding superfluous terms in order to denote each concept by a single mon-iker (many thanks to my anonymous reader for this observation). To my mind, this suggests anearly provenance, probably sometime in the third century BCE, even if the text neverthelesscontains interpolations added later. Moreover, Bs idiosyncratic lettering system suggests that itwas composed before the practices of the Elements became fully standardized, which further

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  • sical arguments in support of his views. He provides no justification for histreatment of vision aside from the success of his own propositions. These arebuilt on his seven definitions []:

    . .. , .. , , , .. , , .. , .. , .. .

    1) Let it be established that visual rays move along straight lines from the eyes andproduce some distance between one another;

    2) and that the shape inscribed by the visual rays is a cone that has its vertex at the eyeand its base at the limits of the things being seen;

    3) and that those things are seen against which the visual rays fall, while those thingsare not seen against which the visual rays do not fall;

    4) and that those things which are seen [subtended] by a greater angle appear larger,those things which are seen [subtended] by a smaller angle appear smaller, and those[subtended] by equal angles appear equal;

    5) and that those things which are seen by higher rays appear higher, while those whichare seen by lower rays appear lower;

    6) and likewise that those things which are seen by rays more to the right appear moreto the right, while those things which are seen by rays more to the left appear moreto the left;

    7) and that those things which are seen [subtended] by greater angles appear more shar-ply (Eucl. Opt. def. 17, B).3

    In its structure, the text displays far greater similarity to contemporary mathe-matical treatises, such as Euclids Elements, than it does to earlier philosophicalworks concerning vision. Yet, while the Optics remains distinctly mathematical

    supports this (for the emergence of standard geometrical lettering practices, see Netz (1999),pp. 1288). An inability to date the text securely, however, has been accounted for in my argu-ment, as well as potential multiple authorship; see below.3 Netz (1999), pp. 89103, analyzes the practice of starting geometrical texts with definitions ofthis type which he points out are actually second-order statements, since they make assertionsabout entities rather than providing foundational descriptions. All translations are my own.

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  • in form, it includes many statements that rely on certain assumptions about thephysical process of sight. For instance, even within the above definitions, theEuclidian author commits himself to the claims that 1) vision results from therectilinear propagation of discrete rays [/] proceeding from theeyes; 2) that these rays fall in the shape of a cone; and 3) that only those pointson which the rays fall are actually visible.4 As a result, objects can actually fallin between the visual rays (prop. 3), which must therefore move side to side tofill in the gaps (prop. 1). To be sure, if visual rays can move back and forth inthis manner, they must be real physical entities, rather than simple mathema-tical abstractions. Still, the text never fully explicates its implicit physicalclaims and instead concerns itself solely with using geometry to articulate cer-tain aspects of vision, including how objects of the same size can appear to beof different magnitudes, how height and movement affect the appearance ofthese magnitudes, how various geometrical shapes appear at different angles,etc.

    The sheer multitude of propositions presented in a single treatise has ledscholars to suspect that the Euclidian author must have drawn some of hisproofs from earlier works, especially since, at least a generation prior, Aristotlementions as though it were already a firmly established geometricalscience. Even if no extant examples survive, it is likely that optical treatises ex-isted on which the Euclidian author could have relied.5 Scholars have proposedseveral potential candidates, including Democritus lost text, the ,6

    4 For a discussion of the physical assumptions implicit in Euclids definitions, see Jones(1994).5 Arist. Metaph. 997b20, 1077a5, 1078a14; Phys. 194a812; An. Post. 75b16, 76a24, 77b2, 78b37,79a1120. Plato, by contrast, does not mention optics at all and instead considers the art ofmeasuring [ ] to be the tool used to decide whether objects that appear to besmaller when viewed from a distance actually are smaller (cf. Pl. Prot. 356c4e4). Hahm (1978),p. 62, suggests that the author of the Optics relied on earlier source texts, but makes no claimsas to what they might be.6 Rudolph (2011), p. 74, suggests that this lost treatise of Democritus was a source text forEuclid. Yet, despite her claims, as well as those of LeJeune (1957), p. 4, Keuls (1978), p. 66, andSider (2004), pp. 1519, it seems unlikely that the was actually an account ofvision or perspective. Thrasyllus (D. L. 9.48 = DK 68A33) lists it as a mathematical work in atetralogy otherwise devoted to celestial concerns, which includes Diagrams of the Heavens[], Diagrams of the Earth [] and Diagrams of the Poles [].Moreover, in his extant fragments Democritus uses only to refer to rays from the sun(DK 68A91) and never mentions them in the context of eyesight presumably because (paceRudolph) it would conflict with his theory of vision, which involves atoms and impressions inthe air, not rays. Instead, Democritus displays considerable interest in general celestial matters(cf. DK 68A85, 67B1, 68A87, 68A89, 59A81), in geometrical astronomy (cf. DK 68A86, 59A78,

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  • and two of Philip of Opus lost works, the and .7 Yet, sinceonly titles remain, what specific information this Euclidian author could havederived from these texts if any remains a matter of speculation. This paperseeks to demonstrate that in being focused exclusively on these lost opticalsources, scholars have neglected other potential precursors. In fact, several pro-positions in the Optics display features that are not endogenous to the opticaltradition at all. Rather, as I shall demonstrate, multiple proofs and several keyideas have been appropriated from the tradition of geometrical astronomy which, like optics, concerns itself with the geometry of rays. In turn, recognizingthat the Optics presents proofs borrowed from other subject fields provides in-sight into the construction of the text as a whole, revealing that the work dis-plays a less unified purpose than has typically been assumed. Instead, the Op-tics reflects a diversity of sources and goals, presenting multiple propositionsthat are neither directly related to vision as it is experienced in the world, nor tothe Euclidian authors implicit physical theory, but only to sight as a geometri-cal phenomenon, transplanted and translated into diagrammatic space. Opticsshould therefore be understood not as a purely mathematical explanation ofvision, but as a target-field of explanation constituted by features and bound-aries derived from extra-visual geometric concerns.

    Geometrical Astronomy and Optics

    It is difficult to determine when geometrical astronomy first appeared in ancientGreece.8 It may have been practiced as early as the sixth century BCE, sinceHerodotus claims that Thales was the first to predict an eclipse, and it is concei-vable that this entailed the application of some geometry to celestial matters.Still, as Heath argues, Thales could have relied on Babylonian observational

    68A87, 68A88, 59A77, 59A80, 68B14.1) and in explaining eclipses by means of the shadow castby the rays of the sun (cf. DK 67A1, 68A89a). Thus, it is more likely that the provided an account of celestial illumination, not optics.7 Suida s.v. ; cf. D.L. 3.37. See Tarn (1975). The Suida lists the title of the secondwork as , which, if correct, is a hapax legomenon. Later commentators report that Aris-totle himself wrote an optical treatise, called the or (Frag. var.7.43, n. 12). This, however, could be a reference to Mete. 3.372b12377a28, which deals withthe halo and rainbow, books 15 and 31 of the pseudo-Aristotelian Problemata, which deal withoptical problems, or even a later misattributed text, such as the pseudo-Ptolemaic De speculis.8 For various general accounts of early astronomy, see Heath (1913), (1932); Lloyd (1970), pp.8098, (1973), pp. 5374, Pedersen (1974), Neugebauer (1975, v. 2), pp. 573776, Evans (1998)and Graham (2013).

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  • records instead of geometrical methods if he did in fact predict an eclipse atall, which is itself doubtful.9 More reliable evidence suggests that some form ofgeometrical astronomy was being practiced by the early fifth century BCE, sincenot only is Anaxagoras reportedly the first person to demonstrate that a lunareclipse is caused by the shadow of the earth,10 but he is also purportedly thefirst person to have included a diagram [] in his text.11 Even though itremains ultimately impossible to determine what such a diagram looked likeand what it actually depicted, it is certainly plausible that his explanation ofcelestial illumination included some basic geometrical image of the earth inter-posed between the sun and the moon. In fact, Democritus seems to have emu-lated his predecessor, insofar as he adopted Anaxagoras theory of the eclipse12

    and reportedly discovered that the volume of a cone was 1/3 that of a cylinder an innovation that corresponds closely to the geometry of rays, cones andspheres involved in celestial illumination.13 Thus, while it is difficult to discernwhat geometrical astronomy looked like in the late fifth century BCE, it seems

    9 Herodot. 1.74 = DK 11A5; cf. Aristoph. Birds 9921009. Heath (1913), pp. 1223 casts doubt onthe idea that Thales knew the mechanism of eclipses and instead insists that he more likelyfollowed the observational records of the Babylonians. Dicks (1959) takes a skeptical positiontowards Thales astronomical teachings in general; cf. Burkert (1972), pp. 413417 and Netz(2004), p. 246. Most recently, Whrle (2009) has collected all the extant testimony for Thales,but for a discussion of the prediction of eclipses in particular, see Bowen (2002).10 Plut. De fac. in orb. lun. = DK 59B18 mentions demonstrating the Anaxagorean proof thatthe sun gives its light to the moon [ , - ]. Similarly, Plut. Nic. 23 = DK 59A18 states that The first to establish an ac-count of the illuminations and shadow of the moon in an image, most clearly and boldly of all,was Anaxagoras ( - ); cf. 59A5, 59A42 (8), 59A76.Anaxagoras displays an overriding interest in astronomy, having being condemned for impietyafter calling the sun a red-hot stone; cf. DK 59A1, 59A2, 59A3, 59A11, 59A19, 59A35, 59A72,59A73. He also addresses other celestial matters to which geometry was generally applied, in-cluding rays, orbits and the size of the earth which he considered to be drum-shaped; cf. DK59A30, 59A47, 59A72, 59A74, 59A80, 59A81, 59A87, 59A88, 59A89.11 D.L. 2.11 = DK 59A1. Similarly, Clement, Misc. 1.78 = DK 59A36 claims that Anaxagoras wasreportedly the first to publish a book through an image/writing [ ]; cf. Plut. Nic. 23= DK 59A18 quoted above in n. 10.12 DK 67A1; cf. DK 68A89a.13 Archimedes, Quadrature of the Parabola 262.12 claims that although Democritus did not ac-tually prove the proposition, he was the first to discover that the volume of a cone was 1/3 of acylinder with the same height and diameter; cf. DK 68B155; Archim. Sphere and Cylinder I 4.9;Method 430.1. For a discussion of Archimedes reliability, see Cuomo (2001), p. 50. AlthoughRudolph (2011), p. 7, suggests that Democritus geometrical discovery may have been the resultof envisioning a cone of visual rays, it is far more likely that it arose from the discussion ofluminous rays and celestial shadows; cf. n. 5 above.

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  • likely that diagrams of some sort were circulating in astronomical texts by thistime even if they did not yet take the axiomatic-deductive form characteristicof later proofs.14 In any case, by the fourth century BCE both Plato and Aristotlemention astronomy as a geometrical practice,15 and it flourished during the Hel-lenistic period in the works of Autolycus, Eudoxus, Euclid, Aristarchus, Hip-parchus and others.

    Some authors in antiquity seem to have noticed the resemblance that theOptics bears to this astronomical tradition, since the treatise appears as part ofa compendium called the Little Astronomy, a set of texts that included Eu-clids Phaenomena, Autolycus On the Moving Sphere and On Risings and Set-tings, Theodosius Sphaerica, On Days and Nights and On Habitations, Aris-tarchus of Samos On the Sizes and Distances of the Sun and the Moon andHypsicles Risings.16 They must have been circulating together by at least thebeginning of the fourth century CE, since Pappus of Alexandria refers to thosewho teach the Treasury of Astronomy [ ],17 andHeath notes that a third hand in the margin of the oldest manuscript has iden-tified this collection as the Little Astronomy [ ].18

    Still, while Heath recognizes the close relationship that several of the textswithin the group bear to one another, he remains confused as to why the Op-tics was included, suggesting that at most it might have been used to inoculatestudents against the Epicurean argument that objects including the sun must be the size that they appear.19 At first glance, his suggestion seems rea-sonable, since this text is the only member of the group that does not dealdirectly with celestial astronomy. Yet, rather than looking for an explanationbased on the potential utility of this optical text for budding astronomers, weshould first note that there are many parts of the Optics that appear to have

    14 Netz (1999), pp. 272275, argues that lettered diagrams and axiomatic-deductive proofs didnot become part of Greek mathematics before 440 BCE. Similarly, Netz (2004) argues that Hip-pocrates of Chios quadrature of the lunules is one of the earliest such texts.15 See Pl. Rep. 528d56, 530b6c1, Theaet. 145d12, Phaedr. 274c7d2, Euthyd. 290b7c6, etal.; cf. Arist. Cael. 291a29b11, 1723, et al. Aristotle uses the term , not ,although Lloyd (1992), p. 570, discusses the interchangeability of these terms in early Greekastronomy.16 Heath (1913), p. 317; cf. Heiberg (1882), p. 152. An Arabic version of the manuscript alsocontains the Sphaerica of Menelaus, on which Pappus relies; see Bjrnbo (1902), p. 4, 51 and55.17 Pappus, Sunagoge 6.474; see Heath (1913), p. 317.18 Heath (1913), p. 317; cf. Heiberg (1882), p. 152.19 Heath (1913), p. 320. For this type of Epicurean argument, see At. 2.21.5; cf. Arist. De anim.428b24. See also n. 37 below for other visual illusions against which Euclids Optics couldpotentially inoculate the reader.

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  • been born within the astronomical tradition itself. First and foremost, the Op-tics, just like geometrical astronomy, depicts rays interacting with objects. Inthis way, it bears both a conceptual and visual similarity to the Little Astrono-my as a whole, even if its rays happen to be projecting from eyes and fallingon hypothetical round objects, rather than proceeding from the sun and illumi-nating particular celestial spheres. Yet, more than simply having a general re-semblance, the Optics contains several clues suggesting that it has derivedsome of its geometrical content from this parallel astronomical tradition, mostnotably the set of propositions 2327, which concern spheres, their magnitudesand their distances.

    Proposition 23 is the first in this particular series. It demonstrates that asingle eye will never see a full hemisphere, since regardless of the spheres size,a single vantage point will never provide an unimpeded view of any full crosssection:

    , .

    In whichever way a sphere is seen by a single eye, it is always the case that less than ahemisphere is visible, but the part of the sphere that is visible appears circumscribed bya circle (Eucl. Opt. prop. 23, B).

    The Euclidian authors proof is simple and starts by cutting the sphere along itshorizontal plane to give a two-dimensional circle, (fig. 1):

    Fig. 1: Eucl. Opt. prop. 23, B.20

    20 Except otherwise indicated, all figures are my own, based on firsthand examination ofmanuscripts Parisianus gr. 2107, 2347, 2350 and 2390.

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  • , , , , , , . , [] - , , , , , . , , , , , . , , , , , , . , . .

    Let there be a sphere, whose center is , and let be the eye; and let line be joined;and let line be extended orthogonally to it through ; and let the plane beproduced through ; it will make a circle in the sphere. Produce ; and let a circlebe drawn with as its diameter; and let lines , , , and be joined. Andso, since the angles and are right angles (because they are each on a singlehemisphere and and are radii), the lines and touch the sphere at a singlepoint [i.e. are tangents]. Therefore, the rays proceeding from the eye at fall along lines and . And since each of the angles at are orthogonal (since is parallel to ,and line is equal to ), if, then, with remaining stationary, the triangle isrotated and returns back to its original spot from whence it began to move, line ,being rotated, will touch the surface of the sphere at a single point aligned with , anda circle will be drawn through the points and . Thus, the visible part of the spherewould be circumscribed by a circle, and it is less than the hemisphere. For the shape is less than a semicircle. Thus, the shape circumscribed by the eye is less than a hemi-sphere (Eucl. Opt. prop. 23, B).

    This proof begins by demonstrating that when rays project from a single eye,they will always run tangent to the circumference of a circle at two points infront of its full diameter. That is, the rays projecting from a single eye will nevercomprehend the circle across its full cross section, since some part of the circlewill always get in the way. In order to extend this conclusion to the three di-mensions of a sphere, the Euclidian author simply rotates the proof aroundthe circles center axis. Since the rays will always run tangent to the sphere atthe same point of the circumference regardless of which circle we take as ourplane, rotating these points will inscribe a vertically oriented circle sittingslightly in front of the diameter. Thus, we will see a figure that looks like a fullhemisphere, but is in reality slightly smaller.

    Proposition 24 seeks to extend this discovery to examine how distance affectsthis relationship. It demonstrates that as an eye moves closer to a sphere, it cansee progressively less of it, although the portion that is visible will occupy a pro-gressively larger section of the visual field or, as the Euclidian author states:

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  • , .

    When the eye approaches, less of the sphere will be visible, but it will seem to appearlarger (Eucl. Opt. prop. 24, B).

    The proof that follows simply repeats the geometry of proposition 23, exceptthat it hypothesizes a second eye, , located slightly closer to the sphere thanthe first, (fig. 2).

    , , - , , , , , , . , , . , . , , , , , . , . , . , . , . , .

    Let there be a sphere, whose center is , and from the eye at let the line join at thecenter; and let line be extended through at a right angle; and let a circle be drawnwith as its diameter; and let lines , , and be joined. And so, the anglesat points and will be right angles, since they are each on a single hemisphere.Therefore, the lines and touch the sphere at a single point. Therefore, the raysfrom the eye at will fall along the lines and . In turn, let the eye at be movedto ; and let a circle be drawn with as its diameter; and let lines , , and be joined. And so lines and touch the sphere at a single point. And the rays pro-ceeding from the eye at will fall along the lines and . Thus, is seen by theangle at , but is seen by the angle at . But is larger than , although itappears smaller. For angle is larger than angle , and the things seen by a larger angleappear larger. Therefore the shape appears larger than , but is smaller (Eucl. Opt.prop. 24, B).

    Fig. 2: Eucl. Opt. prop. 24, B.

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  • As Heath notes, although the diagram makes it obvious that the proposition iscorrect, the Euclidian author does not actually prove his assertions, since hedoes not move through the requisite steps to show how the two visual conesrelate to one another.21 Nevertheless, both theorem 23 and 24 are true and po-tentially applicable to vision.

    Propositions 2527 of the Optics further expand on these conclusions, rely-ing on the previous proofs, but now addressing how binocular vision affects ourperception of spheres. In particular, these theorems address the relationship be-tween a spheres magnitude and the portion of it that remains visible to twoeyes collectively. Proposition 25 states:

    , , .

    When a sphere is seen through two eyes, if the diameter of the sphere is equal to thestraight distance between the eyes, a hemisphere will be seen (Eucl. Opt. prop. 25, B).

    In turn, proposition 26 seeks to demonstrate what happens when the sphere isslightly smaller:

    , - .

    If the distance between the eyes is greater than the diameter of the sphere, a visible partof the sphere greater than a hemisphere will be seen (Eucl. Opt. prop. 26, B).

    Proposition 27 seeks to prove the converse assertion:

    , - .

    If the distance between the eyes is less than the diameter of the sphere, a visible part ofthe sphere less than a hemisphere will be seen (Eucl. Opt. prop. 27, B).

    In one way, insofar as these are the sole theorems in the Optics to considerbinocular vision, they seem to display a unique concern with sight as it actuallytakes places in the world. That is, by examining how binocular vision effectsthe geometry of spheres, the Euclidian author may at first seem to be demon-strating sensitivity to the fact that none of his previous propositions account fora very basic aspect of human sight, namely, that we each (generally) have twoeyes. Yet, we may ask whether these particular propositions would conceivablyever have arisen from a direct meditation on eyesight as it takes place in theworld, especially since it is uncertain whether the Euclidian author would have

    21 Heath (1913), p. 363, n. 1.

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  • witnessed the problems to which his proofs provide ostensible solutions. To besure, it is certainly possible that the author noticed at some point that he couldnever see an entire hemisphere when looking at a large ball (prop. 27), but see-ing more than a hemisphere (prop. 26) would have required that he placed asmall sphere at the end of his nose and then went cross-eyed. Although theseparticular experiences could have led the author to include these proofs, itseems somewhat unlikely. Instead, the text does not seem concerned with ex-plaining appearances that the author himself witnessed and wondered about,but with constructing how objects ought to appear within a strict set of geome-trical circumstances. Moreover, a closer examination of the proofs suggests thatthese geometrical circumstances are not endogenous to optics at all, but arederived from the context of celestial illumination especially since the Eucli-dian proof bears considerable similarity to propositions found in another textfrom the Little Astronomy, Aristarchus On the Sizes and Distances of the Sunand the Moon (hereafter Sizes and Distances).

    Aristarchus of Samos (ca. 310-230 BCE) is most famous for advocating a he-liocentric view of the universe.22 Aside from this detail, however, little is knownabout him. Atius calls him a pupil of Strato,23 and Ptolemy says that he ob-served a summer solstice in 281/280 BCE.24 He was therefore most likely ayounger contemporary of Euclid and indeed, Sizes and Distances takes manyof the discoveries of the Elements for granted.25 Aristarchus treatise proceeds ina manner very similar to that of both the Optics and the Elements. He begins byestablishing 6 primary hypotheses, which he then explicates, demonstrates andutilizes in 18 subsequent propositions.26 Because he relies on celestial illumina-tion to calculate the relative magnitudes and distances of the sun and themoon, his first step is to assert: the moon receives its light from the sun [ ].27 He then establishes geometricalproofs about that will be crucial for his later calculations. It is these first threepropositions that bear considerable similarity to what we have discussed fromthe Optics:

    22 Cf. Archimedes, Arenarius 1.4-7 (Heiberg, v. 2, p. 244). No trace of this theory appears inSizes and Distances, Aristarchus only extant text.23 At. 1.15.5.24 Ptolem. Syntax. 3.2.25 At. 4.13.9 mentions that Aristarchus wrote works concerned with optics, which providesadditional justification for considering the two authors together.26 For a discussion of how Euclids Optics, Phaenomena and Elements use definitions and hy-potheses, see Berggen and Thomas (1996), pp. 910.27 Aristarchus, Sizes and Distances, hypoth. 1.

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  • , , - .

    The same cylinder circumscribes two equal spheres; the same cone circumscribes un-equal spheres with the vertex on the side of the smaller sphere; and the straight lineextended through their centers is at a right angle to each of the circles, where the sur-face of either the cylinder or the cone touches the spheres (Aristarchus, Sizes and Dis-tances, prop. 1).

    Although Aristarchus formulation does not correspond directly to the wordingof propositions 2527, he proceeds just as the Optics does, insofar as bothauthors move from equality to inequality that is, the Euclidian author startswith eyes separated by a distance equal to the diameter of a given sphere, thena distance larger than the diameter, then smaller; similarly, Aristarchus startswith spheres of equal size and then moves to spheres of unequal size, one lar-ger and one smaller. Moreover, Aristarchus proposition 2 is certainly related topropositions 2327 in the Optics, even if the same geometrical insight is used toslightly different ends (fig. 3).28

    , . , , , , , . , , - , . , , . , , , , -

    Fig. 3: Aristarchus, Sizes and Distances, prop. 2.28

    28 Diagram based on Heath: (1913), p. 356, fig. 17.

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  • , .

    If a sphere is illuminated by a sphere larger than itself, more than a hemisphere will beilluminated. For let a sphere, whose center is B, be illuminated by a sphere larger thanitself, whose center is A. I say that the illuminated part of the sphere whose center is Bis larger than a hemisphere. For since the same cone inscribes two unequal spheres withits apex on the side of the smaller sphere, let there be a cone inscribing the spheres;and let a plane be produced through their [common] axis. It will make (two) circles ascross sections inside the (two) spheres, and a triangle in the cone. Thus, make circles and in the spheres, and triangle in the cone. It is clear that the segment ofthe sphere along the arc , whose base is the circle with the diameter , is a partilluminated by the segment along the arc , whose base is the circle with the diameter, since it is orthogonal to the straight line . For the arc is illuminated by thearc . For the rays and are the most extreme. And the center of the sphere inthe segment is ; thus the illuminated part of the sphere is greater than a hemi-sphere (Aristarchus, Sizes and Distances, prop. 2).

    The similarity of Aristarchus proof to those in the Optics did not go unnoticedin antiquity. In his commentary on Ptolemys Almagest, Theon of Alexandria,editor of Euclids Optics and Elements, claims that both Aristarchus and Euclidhad demonstrated the same facts about celestial illumination.

    - , . - .

    On every side, more than a hemisphere [of the moon] is illuminated by the rays of thesun, since the sun is larger than the moon; and because of this, the light and dark por-tions of the moon are not divided by its full circumference, but by a smaller one. Forthese things have been demonstrated by Aristarchus and Euclid (Theon, Comment. inPtolem. Syntax. Math. 957).

    In his note on this passage, Heiberg identifies the two proofs to which Theon isreferring as Aristarchus proposition 2 and proposition 27 of the Optics (which ismore or less the inversion of proposition 23 above).29 In most respects, it wouldnot be surprising that both authors demonstrate a common geometrical fact,especially since the geometry of spheres and cones remains the same whetherapplied to eyes or to the sun. That being said, Theon neither claims that Euclidhas demonstrated a general geometrical truth, nor some particular aspect of vi-sion, but that Euclid has proved an essential feature of celestial illumination. Asit turns out, Theon is at least in some sense correct, since one important

    29 Heiberg (1882), p. 131, n. 1.

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  • aspect of propositions 2527 betrays the fact that these proofs have been bor-rowed from an astronomical context: while their geometrical arguments remainfully applicable and valid for the illumination of celestial spheres, they do notfunction for binocular eyesight. Whereas Aristarchus proofs are true, the Eucli-dian proofs are false.

    Optics 2527 all commit the same error, but it is easiest to see in proposition25 (fig. 4):

    , , , , , , , , . , , , . , .

    For let there be a sphere whose diameter is the line ; and from the points and letlines and be extended at right angles; and from point let a line be extendedparallel to ; and let one eye be placed at and the other at ; and from the center, ,let a line be extended parallel to . And so if, while remains stationary, the rec-tangle is rotated and returns again to the same spot whence it began to be moved, theshape circumscribed by will be a circle, which runs through the center of the sphere.Thus, only half of the sphere will be seen by the eyes and (Eucl. Opt. prop. 25, B,emphasis mine).

    At first glance the diagram may intuitively suggest that the proposition is cor-rect: since our eyes are far enough apart, no part of the circle protrudes farenough to prevent visual rays from striking it across its full diameter. Similarly,the eyes are also not far enough apart to allow any rays to land behind thecircles diameter either. Thus, the rays projecting straight from our eyes will falltangent at the exact diameter the circle. Up to this point, the Euclidian author iscorrect: a full semicircle will indeed be visible when the circles diameter is thesame magnitude as the distance between the two eyes. Nevertheless, the propo-sition does not stop there. Rather, proposition 25 seeks to demonstrate thatthese two eyes also comprehend a perfect hemisphere. In order to demonstrate

    Fig. 4: Eucl. Opt. prop. 25, B.

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  • this, the author suggests that we simply rotate the proof in this case therectangle around the circles center axis, . Indeed, it may seem that,just as with proposition 24 above, the rotation translates the two dimensions ofa circle into the three dimensions of a sphere. Yet, the physical parameters ofvision make this impossible. Put simply, in the physical world, our eyes cannotrotate; they must remain stationary along a single horizontal plane in our head.Rotating our eyes around a center axis (by doing a cartwheel?) would at mostproduce a whole series of sequential vantage points, and even these would notallow us to see a full hemisphere all at once. Proposition 25 and for the samereason, propositions 26 and 27 proves incorrect.

    We can also demonstrate that this proof is false by splitting it into two, witheach eye producing its own unique visual cone. This way we can see what eacheye comprehends individually before combining their vantages together. Let usagain consider an overhead view of a sphere with a horizontal circular planeproduced through its middle; and let that circle have a center and a diameter (fig. 5):

    Let a line be extended from to at a right angle to ; and let a line beextended from to at a right angle to ; and let a line connect and andbe parallel to ; and let one eye be placed at and the other at . We knowfrom proposition 23 that neither eye will comprehend a full semi-circle on itsown. Thus, let lines representing visual rays be projected from the eye at notonly to its tangent on the circle at point B, but also to its tangent at point A;and let a line be extended from to the center orthogonal to BA, which itcrosses at point . And so if, while remains stationary, the triangle isrotated and returns again to the same spot whence it began to be moved, it willproduce a cone with its vertex at the eye and its base circumference resting onthe surface of the sphere (see prop. 23). We can then repeat the same process toproduce a second visual cone extending from the other eye (fig. 6). Let linesrepresenting visual rays be projected from the eye at not only to its tangenton the circle at point , but also at point ; and let a line be extended from to

    Fig. 5: Eucl. Opt. prop. 25, my reconstruction 1. Fig. 6: Eucl. Opt. prop. 25, my reconstruction 2.

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  • the center orthogonal to , which it intersects at point . And so if, while remains stationary, the triangle is rotated and returns again to thesame spot whence it began to be moved, it will produce a cone with its vertexat the eye and its base circumference resting on the surface of the sphere.

    From proposition 23, we know that neither cone will comprehend an entirehemisphere individually. Moreover, since neither cone will include point , theywill also fail to do so collectively. Some portions of the sphere those corre-sponding to the area within the triangle on the top and bottom of thesphere will always remain unseen. Therefore, contrary to the Euclidian proof,a full hemisphere is never visible when the eyes are spaced apart by the samedistance as the spheres diameter. Some sections will always remain hidden.The proof is false.

    This demonstration is not undertaken merely to point out that the Euclidianauthor has made a mistake. Rather, if the proof does not work, we may ask howit made its way into the Optics in the first place. It could be a simple case ofgeometrical error, whereby the author did not notice that his proofs were incor-rect. That is, the mistake could be native to the Euclidian authors own meth-odologies, insofar as he first finds a theorem that applies to a circle and thencarelessly translates this into three dimensions by applying the definition of asphere. In fact, the vocabulary found in proposition 25 (italicized above)matches the definition of a sphere found in the Elements precisely.

    , , , .

    A sphere is whenever, while the diameter of a semicircle remains in place, the semicircleis rotated and returns again to the same place whence it began to move (Eucl. Elem. XI,def. 4).30

    Even this formulation in the Elements, however, seems to have had its origins incelestial geometry, since the language may have been borrowed from Autolycusof Pitane (fl. 310 BCE), whose treatise On the Moving Sphere, provides severalelementary geometrical proofs concerning the rotations required for many astro-nomical calculations. The first proof, dealing with the simple rotation of asphere about an axis, contains vocabulary almost identical to the Optics:

    30 Euclid uses the same grammatical formulation to define a cone (Eucl. Elem. XI, def. 18) anda cylinder (Eucl. Elem. XI, def. 21).

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  • If, while line remains stationary, the semicircle is rotated and returns again into thesame spot whence it began to move (Autolycus, On Moving Spheres 2.2130)

    Autolycus was a contemporary of Aristotle and an older contemporary of Euclid.His works On the Moving Sphere and On the Rising and Setting of the Stars arethe oldest Greek geometrical treatises remaining in their entirety.31 Therefore,given the debt that the vocabulary of the Elements owes to Autolycus, we mightconsider it even more unlikely that propositions 2327 of the Optics are alsoadapted from an astronomical context. This becomes even more assuredly thecase when we realize that proposition 25 even if it fails for binocular vision does work if the source of rays is a spherical sun. In fact, Aristarchus begins histext by demonstrating just this, proving that the same cylinder will inscribe twoequal spheres (fig. 7):32

    , , , , . , , , , , . , - , , . , , , . , - , , , , , , , , . , , .

    Let there be equal spheres; and let their centers be the points and ; and let line be extended and connect them; and let a plane be extended through . Indeed, thisplane will make great circles as cross sections in the spheres; and let these circles be and ; and let the lines and be drawn at right angles to line from

    Fig. 7: Aristarchus, Sizes and Distances, prop. 1.32

    31 Aujac (1979), p. 7.32 Diagram based on Heath (1913), p. 354, fig. 16.

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  • the points and ; and let the line be joined. And since lines and are equaland parallel, therefore lines and are also equal and parallel. Therefore, is aparallelogram and the angles at points and will be right angles. Thus, will touchthe circles and (as a tangent). Indeed, if, while line remains stationary, theparallelogram and the semicircles and are rotated and return again into thesame spot whence they began to move,33 the semicircles and will be moved in thespheres and the parallelogram will make a cylinder whose bases will be the circlesaround the diameters and , which are at right angles to the line AB, since angles and remain orthogonal to AB throughout the entire motion. And it is clear that its surfacetouches the spheres, since the line will touch the semicircles and throughoutthe entire motion (Aristarchus, Sizes and Distances, prop. 1, emphasis mine).

    Since the source of the rays in Aristarchus proposition is itself spherical, whenhe rotates the parallelogram and the semicircles and to translatethe two-dimensional proof into three dimensions, the rotation inscribes both thehemisphere of the visible object (the moon) and the edges of the illuminatingobject (the sun). Since light rays project from entirety of the spherical sun, Aris-tarchus geometry actually works. The Euclidian author, by contrast, misappliesthis geometry and thereby produces a false proof. Even though the uncertaindate of the Optics makes it difficult to discern whether the Euclidian author bor-rows directly from Aristarchus, or whether both authors rely on a commonsource (even though Theon mentions no such predecessor), the success of theproposition within the context of celestial illumination makes it likely that theseproofs about rays and spheres originated out of a concern with the sun, themoon and the earth, rather than out of a meditation on either eyes or perspec-tive. Even though they have been reformulated, propositions 25-27 still remainapplicable to astronomy, not optics. The Euclidian author (or some predecessor)has refashioned them based on a simple and apparent resemblance to the geo-metry of sight.

    The Purpose of the Optics

    Along with being of interest for a general history of Greek mathematics, ac-knowledging that the Euclidian author includes propositions borrowed fromother subject fields can alter our assumptions about purpose of the Optics as awhole. This can be particularly useful, since scholars read the treatise in a vari-ety of ways. Because of the influential work of Erwin Panofsky, many commen-

    33 The same formulation appears again in Archimedes, De sphaera et cylindro v. 1, p. 88, ln.17; v. 1 p. 93, ln. 2; cf. Hero, Mechan. 7.1.9, 76.1.9, 83.1.7.

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  • tators especially those from the discipline of art history investigate almostexclusively whether the Optics is compatible with modern ideas about perspec-tive; they thus treat the work as though it were fundamentally constructed inorder to explain how to produce two-dimensional images of a three-dimensionalworld.34 Others, such as Grard Simon, have pushed back against this reading,arguing that such an interpretation takes for granted that the text deals with thepropagation of light, rather than, as seems more accurate, the propagation ofsight.35 Others still, such as Thomas Heath and Silvia Berryman, suggest thatthe Optics was written to combat false appearances and thereby to solidify vi-sual information as a valid access point to knowledge or, in other words,these scholars argue that the text is rooted in a set of philosophical concerns.36

    All these interpretations, however, implicitly assume that the Euclidian authorpossesses a more or less unified goal whether a phenomenon to explain or aphilosophical point to demonstrate that he then uses his mathematical exper-tise to achieve. Or, to use slightly broader terms, the Euclidian author first has aset of questions and then seeks to answer them. While propositions 23-27 maybe modest adaptations, they cloud these typical assumptions about the priorityof question (primary) over answer (secondary). Instead, the Euclidian authorwould have had a considerable number of proofs at his disposal, which couldact as ready-made tools to be applied to new (physical) questions. Yet, by recog-nizing the indebtedness that the Optics bears to earlier astronomical texts, wecan ask whether such questions always existed especially since the appear-ances that the Euclidian author sometimes attempts to explain do not actuallyappear in any normal sense of the word. They are phenomena created by thevery explanations that the Euclidian author employs.

    The problem of false false appearances is not unique to the propositionsborrowed from celestial geometry. C.D. Brownson and Wilburt Knorr have de-

    34 Panofsky (1991), p. 66; cf. Andersen (1987). For the most recent discussion in this vein, seeSinisgalli (2012).35 Simon (1988).36 Heath (1913), p. 320. Berryman (1998); cf. Jones (1994), p. 47. This idea appears as early asGeminus, Frag. Opt. 22, who lists the apparent convergence of columns, the rounding oftowers and unequal things appearing equal as phenomena that optics explains; cf. Procl. inEuclidis Elementiis 40, who also claims that optics has as its subject the illusions created byobjects at a distance, and he also cites the convergence of parallel lines and the round tower asexamples. Plato presents different examples of the paradigmatic visual illusions, includingstraight things appearing bent in water, painting with shadows to make flat surfaces appearconcave or convex (which he calls ) and things of the same magnitude appearing tobe of different sizes at various differences; see Pl. Rep. 602c78; cf. Rep. 523b5c3, Prot. 356c4e4.

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  • monstrated that difficulties also surround proposition 35 as well, which arguesthat if you look at a circle from any point on the hemisphere that circumscribesit, the diameters of that circle will appear equal.37 As both scholars have shown,the diameters of circle will indeed appear equal; nevertheless, the diameterswill no longer appear to be diameters.38 In other words, this proposition is alsoincorrect precisely because it explicates a false appearance that seems to makegeometrical sense, but never actually appears. In these cases, we can say thatin the Optics proofs and geometrical insights occasionally come first, and expla-nanda are constructed to fit them. In fact, there may be no real explananda atall.

    These are not the only proofs in the Optics that show what could be calleda priority of tool over task. For instance, propositions 18-21 display it in anotherway. These four propositions use similar triangles to demonstrate how to calcu-late the size of a given object 1) by using the size of its shadow; 2) by using itsreflection in a mirror; 3) by aligning the eye with the edge of a drop-off (such asa cliff); or 4) by aligning objects in the visual field. None of these propositionstruly deal with vision or address false appearances; they deal with measurementat a distance. Proposition 18 even fails to include any mention of the eye.39 Wemight then wonder why these are included in a work about optics proper.40 Onemight suspect that the author is exploring the practical implications of geome-trical perspective, but I suggest that proposition 18 offers another potential clue.In it, the author (rather uncharacteristically) uses the first person to report howhe enacted his proof (fig. 8):

    37 Eucl. Opt. prop. 35, B: And if a line is extended not at right angles to the plane (of thecircle), but it is equal to the diameter of that circle, the diameters will appear equal [ , , ].38 Brownson (1981); Knorr (1992).39 Unlike Eucl. Opt. prop. 18, B, the A manuscript includes a very awkward reference to theeye, claiming that it would be at the very edge of the objects shadow (e.g. on the ground atpoint ). This is a completely unnecessary inclusion if one is simply measuring the triangles(as well as being physically quite difficult). To my mind, this suggests that the inclusion of theeye has been added in the later edition; see n. 42 below.40 Even though no theoretical concerns seem to explain the inclusion of these proofs dealingwith similar triangles, the measurement of an object by its shadow becomes crucial for the useof the gnomon in celestial astronomy and for estimating the height of a distant wall (for in-stance, when designing siege engines) in dioptrics. Therefore, the propositions could conceiva-bly have been included for their practical use. Yet, in the Optics, the Euclidian author does notpresent these propositions as if they have these specific purposes in mind, so arguing thatthese are the real reason the proofs were included would be misleading, since it is simply inthe nature of geometry as elemental as this that it can be applied to multiple situations.

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  • , . , , , , . . - . , , . . .

    To know how big a given height is. For let there be a height, whose size it is neces-sary to know, and let a ray of the sun, , fall though point . Therefore, will be itsshadow. Indeed, I took some known magnitude, , and I fit it so that it subtended theangle and was parallel to . And so, just as is to , so too is to . And theratio of to is known. Therefore, the ratio of to is also known. And theshadow is known. Therefore the height is also known (Eucl. Opt. prop. 18, B,emphasis mine).41

    The first person verb form seems to allude to the fact that the author himselfdiscovered the application of this proof perhaps as he wandered around inthe sun with a stick. This suggests that this particular group of propositionsmay have been incorporated, not out of a deep concern with articulating visionper se, but because the Euclidian author had found a way to measure the heightof objects by using their shadows and thought it was relevant or at least re-lated to the subject at hand. In fact, the remaining propositions in this groupseem to be a simple extension of this basic insight, insofar as they utilize thegeometry of similar triangles to describe three comparable examples. At thesame time, however, these solutions are somewhat difficult to enact in practice.All four rely on already knowing some of the magnitudes involved in the calcu-lation, and it is not always clear how this would be the case in most practicalsituations. For instance, Optics proposition 20 suggests that to calculate a givendepth, we could simply stand above it, near the depths edge, and measure thedistances from the edge to our feet and from our feet to our eye. Since thiswould produce a right-angled triangle with a known ratio, we could extend the

    Fig. 8: Eucl. Opt. prop. 18, B.

    41 The first person verb does not appear in the A version of the text. To my mind, this furthersuggests that B is older, and the proofs have been streamlined and polished in A.

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  • visual ray from our eye along the path that aligns with the edge and mark theplace where it strikes the ground on the floor of the depth. We could then mea-sure back from that point to the place directly under the edge itself, creatinganother right-angle triangle with the same ratio of sides. We could then use theratio from the first triangle to calculate the missing height of the second. Ofcourse, this could only work if the depth 1) had a completely vertical side (forexample, a cliff or well); and 2) it was possible to measure from the base of thatwall to the point where our visual rays falls. This would be extremely difficultto do in almost all circumstances where we could not just as easily measure thedepth itself. The proposition may be true, but its practical application seemsquite minimal. Therefore, we can conclude that the primary justification for in-cluding these proofs is that they 1) utilize the geometry of rays and 2) seem towork mathematically. Any theoretical concern or actual practical applicationseems secondary, derivative and perhaps even somewhat irrelevant.

    The fact that the Optics relies on earlier astronomical propositions, incorpo-rates erroneous proofs and attempts to measure heights complicates the ideathat the treatise is about explaining light, sight or false appearances in anystraightforward sense. Even though the text has certain physical implicationsand relies on certain physical assumptions, vision is not encountered solely oreven primarily as a phenomenon outside the diagrams; it is instead constructedas a geometrical entity within them. In fact, in several moments within the Op-tics, the geometry even leaves vision behind. In short, the text reflects a con-glomerate of concerns and dependencies, oriented toward a somewhat diffusegoal: simply to articulate the geometry of vision and rays. In this way, the textcould be said to be as much about diagrams as it is about sight.

    The use of the first person verb in the above quotation also helps draw ourattention to a problem that we have not yet addressed, namely, whether we cantreat this mathematical work as though it is the product of a single author.42

    The Optics itself survives in multiple editions and, as a survey-type text, likelydrew from multiple sources. As such, we cannot simply assume that the Eucli-dian author of the Optics was the first to re-fashion the pre-existing proofs fromcelestial astronomy to fit an optical context. Rather, the Euclidian author could

    42 Netz (2009), pp. 107109 classifies three different types of mathematical works: 1) ludic, 2)survey; and 3) pedagogical. Ludic works are constructed around a single mathematical purpose(such as calculating a given value) and will incorporate surprise by delaying the justificationbehind intermediary steps in order to dazzle the reader. Pedagogical works privilege lucidityand lead the reader to an explicit goal (or set of goals). Survey works, like the Optics, attemptan exhaustive account of a given subject (e.g. Eucl. Elem. X, which provides an account ofirrationals). Netzs discussion of survey works, however, focuses more on theoretical geometryand does not deal with applied mathematics in particular.

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  • merely be replicating some earlier adaptation (although the first-person verbmakes this unlikely in the case of proposition 18). Nevertheless, even if therewere some predecessor who originally borrowed these proofs, the same problemwould simply be displaced; the author who first appropriated these propositionswould still be demonstrating the fact that applied mathematics can sometimesbe just as much about engaging with mathematical concerns and earlier textsas with addressing physical phenomena themselves.

    Conclusion

    As a consequence of recognizing the various motivations behind the Optics pro-positions, it is harder to maintain that the text is wholly about addressing asingle philosophical or mathematical goal. It reflects different priorities at dif-ferent times. Sometimes geometry is used to articulate physical ideas aboutsight. Sometimes preexisting mathematical propositions help construct visionas a geometrical phenomenon. Sometimes geometry is used for more mundanetricks. Indeed, in recent years, scholars have begun to recognize the diversity ofmotivations behind the construction of ancient geometrical treatises. RevielNetz has shown that Hellenistic geometers responded to literary as well asmathematics traditions, demonstrating aesthetic sensibility by incorporatingelements of surprise and delight into their arrangement of proofs.43 Similarly,Serafina Cuomo has argued that Pappus Sunagoge is designed to engage with anumber of different traditions, including mathematical, scientific and philoso-phical debates for which reason she refers to the text as a palimpsest.44 In acomparable way, we can shift the assumption that what Aristotle calls morephysical mathematics [ ]45 is involved in thepure explanation of natural phenomena by geometrical means. As the abovepropositions within the Optics help demonstrate, authors can adopt pre-existinggeometrical proofs, modifying them for subject matter at hand. In many ways,this remains the fundamental strength of theoretical geometry, since it is pre-cisely by relying on earlier discoveries that knowledge can advance. In the caseof applied geometry, however, this occasionally creates inconsistencies insofaras repurposed proofs can fail to account for the restrictions imposed by the phy-sical world. In some instances, the phenomena explained by these texts can

    43 Netz (2009); cf. Lloyd (1992), pp. 569570.44 Cuomo (2000).45 Arist. Phys. 194a78; cf. Asper (2009).

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  • even be phantoms of geometry, appearing only within the confines of the dia-gram. Tracking these inconsistencies allows us insight into how these treatiseswere composed and helps us understand the complex relationship between con-struction and explanation, as well as between mathematical entities and theobjects they supposedly represent.

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    Bjrnbo, A. A. Studien ber Menelaos Sphrik. Beitrge zur Geschichte der Sphrik undTrigonometrie der Griechen, in Abhandlungen zur Geschichte der mathematischen Wis-senschaften, Heft 14. Leipzig: Teubner, 1902.

    Brownson, C. D. Euclids Optics and its Compatibility with Linear Perspective, Archive forHistory of Exact Sciences 24, (1981): 165194.

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