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'It makes no difference': optics and natural philosophy in late antiquity.

Whatever the strengths of Aristotle's account of vision, it would be difficult to reconcile it with later work on geometrical optics. Nonetheless, some Aristotelian commentators in late antiquity--especially John Philoponus in the 6th century CE--sought both to acknowledge contemporary work in optics and to defend Aristotle's theory of light and vision. It is perhaps not surprising that Aristotelians believed that such a reconciliation was possible: Aristotle uses ideas from the optics of his day in Meteorology, and classified optics as one of four mathematically grounded 'intermediate sciences'. But commentaries on Aristotle's Meteorology faced a more complex task in attempting to reconcile Aristotelian natural philosophy with the work of the succeeding centuries.

I shall argue that Philoponus' integration is unsuccessful, and that Aristotle's account of vision is seriously challenged by new theories of the geometry of perspective and the physics of deflection. But a larger theme is also apparent in this discussion. The commentaries offer evidence of a strategy used in late antiquity to insulate work on optics from the dominant theory of vision, in effect asserting the independence of the mathematical models of the intermediate sciences from conflicts with doctrines in natural philosophy. In claiming that 'it makes no difference' to the geometry of optics what theory of vision is adopted, the commentators employ a strategy that is explicitly paired with the idea of 'saving the appearances' in astronomy. This discussion is therefore crucial in understanding changing perceptions of the relationship of the 'intermediate sciences' to natural philosophy.

1 Compatibility between Aristotle's Texts

Aristotle classifies optics, along with astronomy, harmonics and mechanics, as one of the four disciplines intermediate between mathematics and natural philosophy. These disciplines investigate the mathematical relationships found in the natural world, without offering full-fledged physical explanations. (1) While the objects studied by optics cannot exist apart from bodies, Aristotle says that optics considers its objects qua lines, not as physical bodies. (2) Unsurprisingly, the parts of Aristotle's work most closely concerned with geometrical optics fall short of presenting a theory of vision. Meteorologica 1.5-8 and especially 3.2-6 of Meteorology introduces ideas about mirror-reflections to help explain puzzling effects like rainbows and haloes. These passages account for what appears to us, rather than attempting to explain the nature of light, color, vision or perception itself. (3)

Although the account does not lay out its central principles systematically, it does seem to be based on a coherent view of the geometry of perspective. The account is organized around the idea of an opsis that is bent or deflected from certain smooth surfaces and to the visible object. (4) Opsis is used in the singular and is usually depicted diagrammatically by a line connecting eye to object. The term has several possible meanings: the eye; (5) sight; (6) the physical process required in vision; (7) or more specifically a 'visual ray,' i.e. some kind of linear physical emanation from the eye. (8) I shall later note evidence that opsis might be used simply as a name for a straight-line relationship, such as the lines drawn in optical diagrams to represent a line of sight between eye and object. To avoid question-begging, the term will be left untranslated.

The focus of Meteorology is on the geometrical relationships between perceiver, object perceived and its apparent location. Some remarks have physical implications, (9) but there is little explicit theory of vision. No explanation is offered of how we actually see, nor how images get back to the eye, nor how shapes or colors appear in mirrors. (10) Aristotle's use of optical theory in Meteorology, then, fits the programmatic suggestions he makes elsewhere about the proper focus of geometrical optics: it focuses on the changes brought about by the geometry of perspective, without delving into the physical relationships involved. It shows how the deflection of the opsis at smooth surfaces alters the geometrical relationship between observer and the points at which objects appear, or from which they are visible. Apparent phenomena of meteorology are thus accounted for by indicating how they could be produced: the causal processes involved in vision are not in question, only the linear relationships that complicate the picture and--so it is claimed--are responsible for the unusual appearances.

While the subordination relationships Aristotle posits between intermediate sciences and natural philosophy exempts the former from the need to engage with physical theory, it implies that their claims are compatible with accepted theories of the natural world. Their subject matter is the mathematical objects that can be considered in abstraction from the natural bodies, but which are nonetheless instantiated in those objects and so must represent their relevant features accurately. This formula needs a little interpretation to see how it would apply to a theory of vision, because what is at issue is not natural bodies, but a series of processes that occur in a medium. What is abstracted in the case of optics would be the mathematical structure of processes, not of things. Nonetheless, the implication of Aristotle's classificatory remarks in Posterior Analytics is clearly that the intermediate sciences should be compatible with the causal accounts offered in natural philosophy.

Even between Aristotle's own texts, then, there is a potential conflict. If the Meteorology is understood as asserting that a 'visual ray' literally travels out from the eye, this would therefore be incompatible with the account of vision in de Anima and de Sensu. Many have thought that this is the implication of the use of opsis in Meteorology, and suppose that Aristotle simply changed his view in the natural philosophical treatises. The strongest evidence for this is de Anima 3.12, where Aristotle seems to be explicitly rejecting the Meteorology view, saying that deflection should not be explained by an opsis going out from the eye. (11) There is thus a prima facie problem of reconciliation, either if we take opsis unambiguously to refer to a 'visual ray' theory, or if we take geometrical optics to be true only if there is some physical emanation from the eye.

And yet the commentators take the theories to be compatible. There are two possible ways that the discussion of the effects of deflection might be reconciled with Aristotle's theory of vision, depending on how we interpret the account of vision. In de Anima 2.7, color is said to be kinetikon, capable of changing the transparent medium; the effect it causes in the medium in turn affects the sense organ. (12) Color is visible through a transparent medium, if we are looking in the right direction, but is not apparent in it, such that it would be visible from any angle. (13) This is sometimes taken to imply that there are actual physical impulses through the medium, i.e. a kind of local motion that could have effects in one direction and not another; (14) others deny this. (15)

One line of reconciliation depends on supposing Aristotle thinks that vision works by actual physical impulses. If color causes physical impulses in the medium, these impulses could be rectilinear and admit deflection. They would be localized in the medium: impulses from particular colored objects of differentiated shapes would reach only specific locations, and it would be from these that objects are visible. The difficulty for this line of reconciliation is that in Meteorology the causal directionality between eye and object seems to be reversed.

A second line of reconciliation avoids this problem, and is available to those who do not interpret the color effects on the medium as localized changes. The term in question, energeiai, can mean either 'activities' physical processes--or 'actualities', states brought into fruition. On this interpretation, Aristotle means to say that the presence of colored objects 'actualizes' the potential of the transparent medium to convey its appearance, but without undergoing any physical process. The visibility of objects is a property that comes to be instantiated throughout the medium. To take account of the phenomenon of perspective, however, would require adding some new assumptions that could account for the localization of perception. Aristotle's account of vision would need to take note of a perceiver with the right kind of sense-organ, and moreover one that is oriented in the right way towards the visible object. With this in mind, a geometrical notion of an opsis--a line between the eye and a visible object --might be thought to represent a factor implicit, if never discussed, in the de Anima account of vision.

Consider 'line of sight' as an abstract notion of the line along which objects are visible. It must be distinct from the direction of gaze if it is to undergo deflection. One way to conceive of a line of sight abstracted from any causal implications might be to think of tracing the points from which objects can be seen. The fact that points in line with one another relative to the eye are blocked from view provides a basis for thinking of the visual field in terms of diverging lines. (16) Note that this notion of a line of sight is descriptive rather than explanatory: it is a way to characterize what is represented when we draw a line from the eye to the mirror to the object bent at equal angles, but does not say why this characterization obtains.

Either of these possibilities might suggest a line of reconciliation between the Aristotelian texts. On the first option, something needs to be said about the difference of directionality, perhaps interpreting the Meteorology account in a way that does not require actual travel. The second, static option avoids this problem, achieving the localization of the objects of sight by introducing the observer's position and orientation. Both reconciliation strategies are imperfect, since some indications in the Meteorology regarding the physics of vision would be harder to accommodate. (17) Optics may concern itself with the geometry, but it does have some physical implications. These became more problematic, I will argue, in the postAristotelian period.

2 The Reconciliation

Whatever reconciliation might have seemed feasible in Aristotle's day, the compatibility of optics and natural philosophy became vastly complicated in ensuing years. While Meteorology considers the deflection of a unitary opsis, it is a central tenet of geometrical optics after Euclid that multiple opseis diverge from the eye in a cone-shaped array. (18) These two features the multiplicity of opseis from each eye and the conical presentation--are not features of Aristotle's understanding of optics. (19) The idea that opsis could be taken to represent a single 'line of sight' localizing the line of visibility was thus greatly complicated.

The explanatory advantage of the Euclidean model was considerable, because it showed how more distant objects appear smaller and less distinct, allowing some problems in the geometry of perspective to be handled by geometrical demonstration. Along the lines of Aristotle's taxonomy, the Euclidean model of optics seems to forge an intermediate science 'between' geometry and theory of vision, i.e. assuming the demonstrations of the former and providing results to the latter. (20) It presupposes as background the proof techniques in Euclid's geometry, adds some extra axioms about the geometry of what is seen, and produces demonstrations about the mathematical structure of perspective. The cone-shaped visual field--represented as a flattened two-dimensional 'slice'-is a central feature of the proofs.

While some aspects of this account of optics are anticipated in Aristotle's Meteorology, it was only articulated in its classic form a generation or so after Aristotle's death. (21) The Euclidean model of optics became standard, and influences the reading of Aristotle's Meteorology by his followers some centuries later. That is, they do not seem to be aware of distinctions between the theory of Meteorology and the optical theories they were trying to accommodate. Alexander remarks that Aristotle's discussion in Meteorology should not be taken to mean that we see by rays--aktines flowing from our eyes; (22) Philoponus suggests that Aristotle uses the hypothesis of opseis merely because it is clearer, not because he endorses it. (23) Both read the use of optics in Meteorology as implying the plural lines or 'rays' of the Euclidean account.

Philoponus explains how he thinks we can understand geometrical optics in Aristotelian terms. He tells us to understand discussions of opseis from the eye, in the Meteorology passages, as equivalent to the energeiai, 'activities' or 'actualities' that come from objects in the de Anima account:
 And quite simply, whatever they say concerning opseis, we say the
 same things in precisely similar terms concerning actualities and
 save the appearances. For what difference does it make whether
 straight lines come to the mirror proceeding from the eye, or are
 deflected from the mirror to the eye? (in DA 331, 26-29). (24)


Especially with the talk about emission in straight lines and deflecting from smooth surfaces, it would be easy to read this passage as taking the energeiai to be movements in the medium, along the lines of the first reconciliation I sketched above. Sambursky indeed concludes that Philoponus must be understanding energeia here as an activity rather than an actuality, something that travels through the medium. (25) Sorabji shows that this cannot be Philoponus' considered view, however, given the context of the crucial passage, which distinguishes between direction of causal influence and direction of travel. Philoponus claims that Aristotle does not need to think that colors literally travel in order to have directionality: (26) causal influence is transferred to successive bodies or media, so that the change happens to each substance as a whole and instantaneously. Nothing travels, yet there is a clear sequence of causation. Philoponus applies this idea of direction of influence without travel both to light and color. (27) This static conception of energeiai, 'actualities,' is meant to do what the opseis of the geometers do. The difficulty should be evident: the 'actuality' view claims that the medium as a whole actualizes its capacity to transmit a particular visual experience: the actuality exists equally throughout the medium. How can it account for the evidence that points to a process with local properties?

Sorabji supposes that Philoponus takes over from Aristotle the distinction between travel and direction of causal effect, and merely extends a sense of directionality possessed by color effects to light. He points out that Philoponus goes beyond Aristotle when he says that a ray of sunlight passing through stained glass can convey the color and shape of the glass onto a solid body it encounters. (28) The effects of color and shape can be 'ferried' through a medium that is unaffected, and yet still produce localized effects on the solid surface. Philoponus explicitly denies that this effect can be explained by the deflection of vision. (29) Light as well as color can be present through a medium as an actuality and yet have unidirectional effects.

When Philoponus talks of the actualities as present in the air so as not to affect it, he insists that this must be so if many different colors, and opposite ones at that, are to be visible throughout the same medium simultaneously. (30) The difficulty is also noted by Alexander. (31) This may not provide an independent argument against taking the color effects to be actual impulses: the notion of contraries present in a medium already supposes that the color effects are present in the medium as a whole and are not localized impulses passing through. It may be the attempt to integrate the account of color with that of light that nudges aside the view that vision involves localized impulses. Later work on optics seems to suggest a parallel between the behavior of light and vision, whereas this is not assumed at the time of the Euclidean text. (32) It may be that writers on optics found the parallel compelling under the increasing weight of evidence that light undergoes the same kinds of deflections that are needed to explain visual effects.

For Philoponus, both light and color are thought to be present through the medium as incorporeal actualities, and yet to have localized effects. Because of this, the direction of causation alone is not sufficiently fine-grained to provide the localization needed for an account of vision, as Philoponus recognizes.

For the actualities are always in the air without affecting it, but they do not act on us, except when the perceptible objects are on a straight line with the eyes, as neither do the opseis, according to them, contact the visible object, unless they are on straight lines with it (in DA 331,13-5).

The direction of causal influence is from a visible object to a medium and thence to the eye. But if that were the whole story, then actualities present throughout the medium should be visible from any angle, whereas in fact they are not. (33) The eye needs to be not only causally downwind, but also oriented in the right direction. This is needed to show why visibility is localized: an object is visible from one point and not another, from one angle and not from another. While it looked initially as if actualities are going to be cast as direct analogues to opseis but going in the opposite direction, it seems that the opsis--as line of sight--takes over the role of fixing the direction and localization of visibility. An observer looking in the right direction is needed to fix the points and angles from which the object is seen.

So far, the attempt at reconciliation is not wildly implausible. However, there are complications. I want to suggest that Philoponus faces two difficulties in his attempt to integrate Aristotle and optics: the geometry of perspective and the physics of deflection.

3 The Failure of the Reconciliation

The commentators need to accommodate a model of optics based on a cone-shaped array diverging from the eye, not a single line connecting eye to object. The problem of explaining why a cone-shaped visual field converges on the eye is a real one for theories of vision based on something traveling to the eye. There is no apparent reason why localized effects instigated by the visible object on a medium should narrow to a point, and moreover converge at any of the possible locations from which an eye can see the object. Philoponus does not have a problem explaining how we could see them at every point, but he needs some analog to the diverging lines of perspective if he wants to retain the explanatory power of optics.

The point of considering vision as a cone diverging from the eye is, again, to account for problems in perspective: why an object looks smaller if seen from further away. (34) In Euclidean terms, we judge this by the angle at the eye of the lines connecting the eye to the object. (35) Philoponus does recognize that the hypothesis of actualities needs to accommodate the task of judging relative sizes of objects seen:

But as many things as are established by the hypotheses <of opseis>, the same things also shall be established by means of <actualities>. For as we said there that cones go out from the eyes from a certain centre, then widen as they go forward, evidently it will be said thus also in the case of actualities. For since the actualities of visible objects are discriminated in a certain narrow centre of the crystalline <humour>, they must originate from the width of the visible object itself and terminate in the narrow eye (in DA 333,25-31).

The reference to the width of the object only makes sense if the problem here is that of judging the size of the object, a problem he refers to again later (in DA 340,22ff). If this capacity is to be retained in an account involving effects from the object, these would not only need to converge, but to converge at every possible point in the medium from which the object is visible. The difficulty is not that we need to be able to see objects from many points, but that they must somehow convey the relationships between size and distance involved in perspective. Philoponus gestures towards the difficulty: it may be that the eye looking at the object is somehow thought to explain how effects in the medium converge on the eye. (36) In discussing how the deflection of light can occur at acute or obtuse angles, Philoponus acknowledges that features implying size seem to be corporeal, not immaterial (in DA 333,36-334,1). In the passage above, Philoponus seems to be conceding that the actualities are localized and possess magnitude, rather than states of the medium present throughout.

The problem of explaining deflection also complicates this account. In the Meteorology passages, deflection merely redirects the opsis: we see the object much as in direct vision, but the line of sight merely happens to be bent. A minimal requirement for deflection is that eye, mirror and object stand in a certain spatial relation. (37) Alexander and Philoponus use the spatial relationship to define deflection. (38) But standing in the right relation cannot be the whole story in deflection. Integrating the optics into a causal account needs to show not only what relationship hold between the objects, but also how the line of sight comes to be bent. If there are actual impulses, these could be said to have resistance and therefore to be deflected, but Philoponus' incorporeal actualities cannot do this. Nor could an abstract notion of the observer's direction of gaze explain why the line of sight is bent and not simply blocked on encountering a mirror. If neither the actualities in the medium nor the line of sight can account for a change of direction within the medium, it seems that the mirror must be involved in a causal story that explains why there are changes in the points and directions from which objects are visible.

The idea introduced to solve this problem is that mirrors do not merely act as resisting bodies, but receive the actualities from the medium and transmit it in turn back to the medium, themselves becoming sources of actualities. Alexander suggests this, (39) and the idea is taken up by Philoponus. The mirror needs to be involved in the causal account to show how mirrors can intensify the effects already present, as well as change their direction. This is particularly evident in the case of burning glasses. (40) Aristotle had talked about rays from the sun being deflected and heating the air, (41) but need not be thinking of the rays as light. When Philoponus discusses burning glasses, he makes clear that he is thinking that light is deflected. Light is intensified because mirrors do not merely deflect light, but become light sources themselves:

Whenever it is said, then, that actuality is reflected from smooth bodies, nothing else is said than that such bodies, when they receive the actualities of both light and visible objects, do not reflect the numerically same actualities which they received, but that they are themselves naturally inclined to produce such actualities in return (in DA 334,21-25).

The mirror's capacities enable it to receive one actuality and transmit another. This account is not limited to what happens when light is deflected: he adds that the same account applies to the opseis. Presumably Philoponus is claiming that images of visible objects, as well as light effects, are first received by the mirror and then passed on.

Aristotle in Meteorology said rather little about what happens in mirrors: his idea seems to be that we see the object to which the opsis is deflected, but seem to see it at the point of deflection. Color effects like rainbows somehow appear in the mirrors when the deflected opsis reaches something bright; we see the color at the point where sight is deflected (3.4, 373b26; 3.6, 377b3ff). (42) A difficult passage in de Sensu supports this idea that mirrors are mere conduits and not causally implicated in vision. (43) Aristotle seems to be claiming that there is no observer-independent image in the mirror: what we see by deflection is the object, not an image. Despite Aristotle, the commentators think that we see color effects produced by the sun acting on the mirror, not that we see the sun dimly in the mirror. Philoponus stresses the insubstantial nature of the rainbow: (44) we are looking at the effect produced by the sun and not looking by deflection at the sun; the rainbow 'looks' like a real colored object, whereas there is no colored object, only an effect produced by light.

There is a real advantage to thinking of deflection, not a mere bending of the line of sight, but as two distinct causal processes. Later work on catoptrics shows that there is a precise geometry to the angles and apparent distances at which objects appear 'behind' the mirror, a phenomenon that Aristotle's account ignores. (45) However, Philoponus and Alexander forfeit a key aspect of Aristotle's account, which is the idea that images in mirrors are merely the observer looking at the object. If there are two distinct causal processes, the mirror receiving the effect and then transmitting it again, it would be hard to claim that this only happens in the presence of a perceiver. In the case of burning glasses, the deflection of light was shown to be observer independent; if the accounts of light and color are parallel, this would challenge Aristotle's account. Moreover, in the case of color actualities, there needs to be something to provide the occasion of perspective and the means of judging what size an object appears from a particular point of gaze. It would seem odd to suggest that mirrors inherit our capacity to 'look' out at objects and to form the actualities from an object into an image of the appropriate size.

Philoponus not only takes on board the Euclidean optics, but he also refers to late antique ideas about binocular vision and the importance of the axis of the visual cone, apparently from Ptolemy. (46) Ptolemy's account of refraction seems to depend on the medium impeding progress to different degrees, whereas incorporeal energeiai would not encounter resistance. There may be further difficulties afoot for this line of reconciliation. (47) Where Aristotle distinguishes the role of light from that of the color activities that produce vision, Philoponus wants to give an account wherein light and color actualities are much more closely analogous. Light, like the actualities of colors, was seen to have localized effects. An explanation of vision must take account not only of the lines of visibility that would fan outward from the object--the points from which the object is visible but also the lines converging on the eye used to trace relationships of perspective. Despite thinking that actualities are present throughout the medium, Philoponus needed to show how color actualities are not only visible along straight lines, but could also convey the size of the object relative to every point at which it can be seen. This becomes even more difficult if mirrors are thought to receive and retransmit actualities.

4 'It Makes No Difference': Finding Room for the Intermediate Sciences

It is, I suggest, against a background in which the implications of geometrical optics were increasingly at odds with accepted theory in natural philosophy that the commentators develop a new strategy to insist on their compatibility. The texts discussed above contain a particular phrase that deserves attention, because it seems to be used rather programmatically. When the commentaries justify Aristotle's use of geometrical optics in the face of its apparent incompatibility with the natural philosophical account of vision, they frequently employ the expression, 'it makes no difference'. This phrase, ouden diaphorei, is often invoked in a manner suggestive of a party line. The phrase has apparent canonical authority, since it comes from Aristotle himself. However, the use to which the commentators put it is quite different.

Aristotle uses the phrase three times in the Meteorology. Two concern the brightness of an object seen through a dense medium or via a mirror. (48) Weakening of the opsis makes the object appear colored, but since deflection is only one cause of weakening, Aristotle tells us that 'it makes no difference' how the effect is produced. In observing that a cloud will seem darker either if the cloud itself is moving further away or if the opsis changes, Aristotle claims that 'it makes no difference' where the change occurs. (49) In both cases he is advancing only a very limited neutrality claim. Aristotle does use the phrase in Generation of Animals to claim neutrality between two theories of vision, but only to exempt an anatomical point from a controversy about the direction of causation. (50)

Alexander of Aphrodisias and John Philoponus use the phrase in a way that far outstrips these precedents, to claim the neutrality of the geometry of perspective and deflection between physical interpretations. (51) After asking what difference it makes whether we talk about something from the eye or something from the object, Philoponus says that Aristotle's view 'saves the phenomena'. This, in effect, suggests a parallel between the relationship of optics to natural philosophy and attempts to reconcile the mathematics of astronomy with natural philosophy. (52) What is at stake here is not merely a theory of vision, but the compatibility of Aristotelian natural philosophy with the intermediate sciences. (53)

The defensive posture that 'it makes no difference' seems not only to have been official doctrine among the Neoplatonist Aristotelians, but may reflect practice among optical theorists. The claim that optics does not offer physical explanations is explicit in a text from the 1st century BCE, (54) and may lie behind the method of presentation of the Euclidean optics. Despite some suggestions of direction to the lines connecting object to eye, (55) there is no mention of direction in the introductory statements, and the account does not depend on there being any directionality or travel. (56) The account applies geometry to the relationships between objects and observed effects, issuing predictions about what will appear to us in certain conditions. (57) This all may have been part of a general strategy among optical theorists of claiming indifference between competing theories of vision.

A. Mark Smith argues that the Platonic tradition would have seen a deep-seated parallel between the need to reaffirm the status of astronomy and that of optics, because our ability to access mathematical realities by contemplating the regularities of astronomy depends, epistemologically speaking, on our ability to understand and trust the processes involved in vision. (58) Euclidean optics implicitly addresses challenges to the possibility of knowledge from scepticism about the senses. (59) But the problem of reconciliation in late antiquity may not necessarily be addressed in terms of the relationship between data and theory. In the specific context of the Aristotelian architecture of the sciences, the problem of reconciliation may have been complicated by the successes of the mathematical disciplines on their own terms, and the degree to which they seemed to flourish by avoiding natural philosophical questions. Even geometers also found it useful to avoid embroilment in philosophical disputes about definitions of terms or about the existence of mathematical entities. (60) The difficulty may not merely have been that natural philosophers need to account for the errant data, but also that natural philosophy needed to assert its own relevance to disciplines that were flourishing in spite of it.

Aristotle's account of the subordination relationships faced a number of challenges, as each of the intermediate sciences presented some tensions with accepted views in natural philosophy. Astronomy had a long history, but as natural philosophical opinion was coming to coalesce around a widely accepted picture of the natural world, this was increasingly in conflict with a developing body of observational data about the motions of the 'wandering stars' or planets in particular. (61) The much-discussed need to 'save the phenomena' has generally been considered in the light of Duhem's claim that an instrumentalist interpretation of the observational data prevailed in antiquity. Lloyd has shown that the evidence does not support this reading, and that the idea of 'saving' the phenomena referred to the idea of seeking theories that accounted for the data. (62) It is nonetheless clear that there was a serious perceived gap between the physical models compatible with deeply held natural philosophical assumptions about the centrality of the earth and the simplicity of celestial motions, and the pathways that mathematical astronomy described. Rather than seeing the conflict in modern terms--as a question of a fit between data and theory--the question might equally be considered as one of the relationship between the purely mathematical and the natural philosophical disciplines. When Philoponus talks about 'saving the appearances' in the context of optics, what he is 'saving' is not raw data, but the way of modeling the effects that is used in geometrical optics.

A similar problem existed in harmonics, over the reconciliation of harmonic theory with the natural philosophy of sound. Since the former requires a quantifiable aspect, the attempts to reconcile mathematical harmonics with theories of sound needed to explain how such an apparently qualitative phenomenon could be subject to such precise mathematical relationships. One school of thought tried do accomplish this by regarding sound as produced by a number of collisions, but the reconciliation remained an open problem without a satisfying response. (63)

Mechanics was another intermediate discipline that was only beginning to establish itself in Aristotle's day: it is barely mentioned in Plato, and the story of Plato's associate Archytas' involvement may be only a reference to Archytas' mathematical achievement in finding a method for solving cube roots, the notorious problem of 'duplicating the cube' that was vital in producing ballistic devices to scale. (64) Nonetheless, there was considerable work in various branches of mechanics in the Hellenistic period particularly, of a kind that late antique philosophers would generally be aware of and feel the need to respond to. As with optics, the fact that a work on mechanics existed in Aristotle's corpus and was commonly taken to be by him helped underline the problem of reconciliation. One obvious problem with reconciliation of mechanics concerned the distinction between natural and nonnatural motion asserted in the Aristotelian Mechanica, since the latter text offers a different analysis of the distinction than does Aristotle himself. However, the practice of the discipline of mechanics seems not to have depended on this distinction; although many texts are lost, there does not seem to be much reliance on, or use of, the theoretical analysis at the beginning of the Mechanica in later mechanics.

I have suggested elsewhere that a perceived barrier to reconciling mechanical theory with natural philosophy stemmed from the evidence for chemical change and for nonlinear relationships between the properties of composites and the properties of their parts. (65) A passage in Simplicius in particular highlights the way in which Archimedes is taken to have asserted the generality and indefinite extendability of the 'law of the lever' in his claim that he could move the earth, given a place to stand. This claim took as central the idea that certain quantitative relationships existed between various parameters involved in the causes of motion, and that these could be extended indefinitely. Simplicius responded by asserting the existence of distinct limits to such linear relationships. While mechanics asserted that the properties of wholes were a product of the properties of parts, Aristotle's theory was taken to imply that the matter merely provided suitable basis for the supervenience of properties of the composite, but did not produce them. Philoponus provides additional evidence that chemical changes do not display the linearity assumed by mechanics. Although the connection to the critique of mechanics is not drawn in this context, other Neoplatonists and teleologists object that the mechanics' attempt to show how all effects are produced by structural arrangements ignores the explanatory need for qualitative transformations. (66)

It seems, then, that in late antiquity, all four of Aristotle's key examples of intermediate sciences were in some kind of tension with received natural philosophy. Aristotelians could hardly dismiss the results of the intermediate sciences, yet Aristotle's program for reconciliation had only grown less convincing in the intervening centuries. Where the mathematical models of the intermediate sciences conflicted with prevailing models in natural philosophy, this 'insulation strategy' had real application. I suggest that it is no accident that the programmatic use of ouden diaphorei occurs in the philosopher whose work shows most evidence of a problem of reconciliation. Nor, I suggest, is it accidental that Philoponus invokes comparison to the idea of 'saving the phenomena' in exactly this context.

The question why the ancient sciences did not have more impact, or why natural philosophers resisted or ignored work in the intermediate sciences, cannot be detached from the real conceptual difficulties involved in integrating the disciplines into a general mathematical physics. (67) Changing views about the perceived relationship of the intermediate sciences to natural philosophy was evidently a major factor in the emergence of a new, unified mathematized physics in the 17th century. (68) A full account of impact of the Greek sciences, or of the pre-modern background to the New Science, must take account of the increasing tension between the implications of the mathematized branches of the sciences and the natural philosophical views that bought their comprehensive systematicity at the cost of adaptability to new evidence. It may make a difference after all. (69)

DOI: 10.1515/apeiron-2012-0001

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(1) APo 1.13, 78b35ff; Ph 2.2, 194a6.

(2) Metaph 3.2, 997b20; 13.3, 1078a14; Ph 1.13, 194a8. On the different ways of conceiving ofabstraction, see Mueller (1990); de Groot (1991a) and (1991b).

(3) Aristotle indeed refers further discussion of one point to the study of perception: Mete 3.4, 374b15-16. For an in-depth study of the organization of the Meteorology, see Wilson (2009).

(4) It looks as though this characterization of deflection is ascribed to Hippocrates of Chios at Mete 1.6, 342b36-343a5. As the Greek distinguishes between the idea that something is deflected at a mirror--anaklasthai--and something appearing in a mirror--emphainesthai--it could be misleading to use the one word 'reflecting' to translate both. See Lindberg (1976), p. 3; Simon (1988), p. 45.

(5) Mete 3.3, 373a18. Here it is represented as a point, not a line.

(6) This could mean either the capacity or the activity: Caston (2002), p. 762.

(7) Preus (1968).

(8) e.g. Simon (1988), p. 30; Lejeune (1948), p. 38.

(9) The opsis is said to 'stretch out', apoteinomene: Mete 3.4, 375a33; 3.6, 377b33; cf. ekteinomene, 3.4, 373b12; directionality is implicit in the claim that the opsis deflects from one surface and towards another. The opsis is described as 'striking against' a surface, and by verbs that suggest travel (diikneisthai, Mete 3.4, 374b15; aphikneisthai, Mete 1.6 343a20; 3.4, 375b15; 3.6, 378a11; pheresthai, Mete 3.6, 378a11). Poorer vision of more distant objects is attributed to the fact that the opsis weakens, olige (Mete 3.4, 373b3). Weakening is used to account not only for changes in brightness, size, and shape of what is seen, but also for changes in hue: different colors seen in a rainbow are explained as successively waning brightness when the opsis is deflected to the sun: Mete 3.4, 374b31-3.

(10) The reference to seeing objects by mirrors in Mete 3.2, 372a33-4 contrasts mirrors in which shapes appear to those in which color appear: the phenomena discussed are cases of the latter.

(11) de An 3.12, 435a6-7. On the deflection of light, cf. also de An 2.8, 419b30-3. The claim that the opsis 'stretches out' is also explicitly rejected: Sens 2, 438a27.

(12) de An 2.7, 418b1; 419a11. There is of course a huge controversy over the interpretation of de Anima and whether the sense organ is thought to undergo physical change: it should be clear that no attempt is made here to do justice to the difficulties involved in interpreting these texts; the present discussion is moreover limited to what happens in the medium outside the eye. I merely canvas interpretative strategies that would allow compatibility with the Meteorology texts.

(13) Burnyeat (1996).

(14) Two passages particularly support this idea that an actual impulse or impact is transmitted through the medium: de An 3.12, 435a1-10 and de An 2.2, 423b12-17.

(15) Among contemporary interpretations of de Anima, the controversy is expressed most clearly in the debate between Richard Sorabji and Myles Burnyeat: see especially the collection of essays in Nussbaum and Rorty (1996).

(16) On the use of this way to characterize an opsis in the Euclidean Catoptrica prop. 1, see Lejeune (1957), p. 56-7, who stresses that the points on the line are taken to be blocking each other from view. He notes parallels in Parmenides 137E, Top 2.11, 148b28; cf. PA 2.10, 656b29. See also Burnyeat (2005). Smith's comparison to the pin-point traces of astronomy is interesting: Smith (1981), p. 83. The Catoptrica is thought to be a compilation and has been assigned variously to Euclid, Archimedes, Hero and Theon: see Smith (1996), p. 15; Simon (1988), p. 58; Eecke (1938), p. xxix; Lejeune (1957), p. 5;.

(17) For example, the opsis is said to undergo physical effects like weakening. In Generation of Animals, Aristotle suggests that the weakening effect of distance are equivalent whether we think something goes out from or comes to the eye (GA 5.1, 781a3-12). This does assume that there are actual physical motions instigated by objects, of such a kind that would weaken (dielueto, GA 5.1, 781a10) incrementally over distance. There are also problems in reconciling the explanation of color effects like rainbows-Mete 3.4, 374a4-30; 3.4, 377b21; 3.6, 377b9-11; Sens 3, 440a11-12--with the de Anima view.

(18) On the singular use of visus in Ptolemy, see Lejeune (1948), p. 20. On Euclid's originality, cf. Eecke (1938), p. xi.

(19) pace Burnyeat (2005), p. 37. While some of the examples in Meteorology are diagrammed using multiple lines, this is not said to be a feature of our sight, and the cone-shape found in one diagram (Mete 3.5, 375b22) arises as a result of the configuration of bodies being views, not from the configuration of the visual field per se.

(20) Barnes (1975), p. 152 writes that it 'no doubt is a sophisticated version of the sort of thing Aristotle had in mind'.

(21) The date of the Euclidean text, as well as its authorship, are uncertain.

(22) in Mete 141,3-11. Sambursky thinks Alexander regards it as 'an embarrassing lapse': Sambursky (1958), p. 116.

(23) in de An 333,22-5; cf. Alexander in Mete 147,19.

(24) Translations of in de Anima passages are by de Groot (1991a), slightly modified.

(25) Sambursky (1958); (1962), p. 115.

(26) Sorabji (1987), p. 29. Philoponus glosses Aristotle's statements that color is kinetikon of the actually transparent rather freely, claiming that kinetikon is used instead of 'completive', anti tou teleiotikon, in de An 322,2; 349,31.

(27) in de An 329,35-330,15; in de An 329,5-16.

(28) in de An 335,30-336,4.

(29) in de An 335,22.

(30) Philoponus in de An 329,18.

(31) de An 61.30ff; Sorabji (1981).

(32) A point well made by Simon (1988), pp. 23ff, 64; cf.Lejeune (1948), pp. 22-31.

(33) Philoponus in de An 331,13-16.

(34) Brownson (1981).

(35) Euclid Optica, definition 4.

(36) Philoponus recognizes a distinct problem about judging distance: if something goes out from the eye and is still somehow connected to it, the eye has a basis for judging how far away the visible object is, whereas if something comes to the eye, there is no way to know the distance traversed (334,40-335,8). Philoponus replies that it is quite possible in the case of sounds, so why not with colors? (in de An 340,22-341,10). Our ability to judge distances must depend on the fact that things perceived from further away are seen less clearly, ie. that the energeiai weaken.

(37) cf. Mete 3.4, 373b31.

(38) Alexander in Mete 141,23; Philoponus in de An 331,20. Alexander uses the idea of a change of relation elsewhere to differentiate the actualization of light from an alteration: in Sens 134,11-19; Mantissa 143,10-18. Burnyeat praises Alexander's account here as understanding Aristotle's view that no physical change is involved in illuminating: 'Remarks on De Anima 2.7-8,' p. 424. On the authorship of Mantissa and its relationship to Alexander's in Sens, see Sharples (1997).

(39) in Mete 141,20-30.

(40) This effect appears as the last proof in the Catoptrics attributed to Euclid.

(41) Mete 1.3, 340a26-32.

(42) The Meteorology account of the Milky Way seems even to suggest that mirror appearances cannot further appear in other deflecting surfaces (Mete 1.8, 345b28). It is not quite clear here what the objection is, but he may be doubting that the appearance in mirrors is a real change that would be reproduced in further reflection.

(43) Aristotle objects to Democritus' theory that vision involves an emphasis in the eye, explaining that it only looks this way to a third party because to the third party, a seeing eye acts as a mirror (Sens 2, 438a5-15). Aristotle is often understood as resisting the idea that seeing is reflection: e.g. Hett (1935), p. 217; Ross (1955), p. 191; Stratton (1917), pp. 97, 109; Johansen (1997), pp. 44-51. See note 1 above. In his criticism of Democritus, Aristotle seems to be rejecting the view that seeing requires an observer-independent image in the eye and saying that this would be insufficient anyway as an account of vision (Sens 438a10-13).

(44) eg. Philoponus in Mete 69,4-15; 75, 11-14.

(45) Philoponus in Mete 28,2; 106, 28-32. It is in discussing the precise placing of images in concave mirrors that the Euclidean Catoptrics turns from discussing the visible object to discussing an eidolon.

(46) in Mete 85, 24-8; in de An 339,35-340,11; cf. Olympiodorus in Mete 261,34; 238,21ff.Lejeune (1948), pp. 35-7. Sambursky noted that the commentators made conscious or unconscious concessions to geometrical optics: Sambursky (1962), p. 112.

(47) cf. Smith (1996), p. 37. Burnyeat notes that Ptolemy's optics attributes a physical property of relative strength to the opsis: Burnyeat (2005), p. 36.

(48) Mete 3.6, 377b11; 3.4, 374a5.

(49) Mete 3.4, 374b23. Aristotle talks of appearances that can be produced by different means in other contexts, e.g. Mete 1.3, 339b34-6; Cael 2.8, 290a24.

(50) GA 5.1, 781a2-7. This passage is noted in Sorabji (1987), p. 27n175.

(51) Alexander in Mete 141,17-8; 147,20-21; Philoponus in de An 331,27. Olympiodorus uses the tag line 'it makes no difference' to make only a modest point, echoing Aristotle's statement that a change of appearance could be produced by different means: in Mete 241,30; cf. 237,28-31.

(52) Duhem (1969); Lloyd (1991). Alan Bowen and Bernard Goldstein challenge Simplicius' account of early astronomy: Goldstein and Bowen (1983); Bowen (2002).

(53) As Malcolm Wilson noted in private correspondence, this cannot be Philoponus' entire answer, since he stresses the problems with the opsis account.

(54) Attributed to Geminus, 1st century BC: Schone (1897), p. 24; Hahm (1978), p. 61.

(55) They are described indifferently as grammai, opseis, aktines, i.e. lines or rays.

(56) Euclid Optica, definitions 1,2,6.

(57) Lejeune (1948), p. 22; Lindberg (1976), p. 12-4; Hahm, 'Early Hellenistic Theories of Vision,' p. 90.

(58) Smith (1981).

(59) Berryman (1998).

(60) See Mueller (1982); cf. Lloyd (1991).

(61) cf. Smith (1981), pp. 81-2.

(62) Lloyd (1991).

(63) I thanks David Creese for pointing me to this issue: see Creese (2010).

(64) Huffman (2005), esp. pp. 379-84; Berryman (2009), 87-97. Engineers began by tinkering with small prototypes of ballistic devices to find proportions of maximum efficiency, and then increased range by scaling up the size, a method that required solving cube roots.

(65) Berryman (2009), pp. 184-90.

(66) Berryman (2005); (2009).

(67) On the view that ancient natural philosophers regarded mechanics as working 'against nature,' see Schiefsky (2007); Berryman (2009), pp. 43-53.

(68) cf. Kuhn (1977), p. 41.

(69) I am particularly grateful for advice and discussion of many points to Han Baltussen, Myles Burnyeat, Verity Harte, John Milton, R. W. Sharples, A. Mark Smith, Richard Sorabji, and three anonymous reviewers. Many points were clarified during discussion in seminars lead by Richard Sorabji, and Michael Martin and Martin Stone's London series on problems in perception and vision. All errors are of course my own.

SYLVIA BERRYMAN

Department of Philosophy

The University of British Columbia

1866 Main Mall E-370

Vancouver BC Canada V6T 1Z1

sylvia.berryman@ubc.ca
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