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Does Euclid's Optics Correct False Appearances?

Colin Webster

The purpose of Euclid’s Optics is not primarily to explain false appearances or to provide an account of sight, as is often claimed. Rather, the text has a far less defined or unified goal; it simply inscribes vision and a few related phenomena into a set of mathematical practices.

Scholars such as Berryman commonly maintain that the Optics was constructed in order to explain certain perceptual misjudgments, perhaps even as a response to a growing skeptical tradition that used such false appearances to deny that any knowledge could be based on the senses. Some ancient commentators support this interpretation. For instance, Geminus (Frag. Opt. 22) states that the purpose of optics is to articulate why a row of columns appear to converge, why towers seem to be rounded at a distance, or why unequal things can appear equal. Similarly, Proclus (In Eucl. Elem. 40) also claims that optics concerns itself with these same few illusions. Many modern scholars, including Andersen, Panofsky and Jones join these ancient commentators and maintain that Euclid’s Optics was written to explicate visual illusions. Conversely, Simon argues the Optics was concerned with the propagation sight and not, as scholars before him proposed, the propagation of light or perspective. A close examination of several key propositions within Euclid’s text shows that neither formulation is sufficient.

Although the Optics does include an account of why a square can appear rounded at a distance (prop. 9, B), several other propositions do not correct any perceptual misjudgments. For instance, propositions 18-21, B all use similar triangles to calculate the size of distant objects, and proposition 18, B even fails to mention the eye entirely, dealing solely with the shadow cast by an object. Thus, in no way do these propositions deal with false appearances. Moreover, as Knorr has shown, proposition 35, B explains why circles continue to appear circular when viewed from any point along their hemisphere, even though this ‘appearance’ is in fact a phantom of Euclid’s geometry.

Proposition 1, B presents an even more complicated cluster of issues. It attempts to demonstrate that we never see the entirety of an image all at once; rather, because the visual rays extending from the eyes are discrete (def. 1, B), we only see those points onto which the rays fall (def. 3, B). Euclid then argues that objects only seem to look whole because the visual rays projected from our eyes move quickly back and forth to cover the gaps. Thus, the proposition explains a false appearance in no ordinary sense—that is, Euclid first seeks to explain why objects do not actually appear whole, only to immediately elucidate why this ‘pointillist’ illusion of the world does not itself appear to our eyes. In this way, Euclid’s proposition does not explain away a perceptual misjudgment as much as it constructs a visual illusion according to a geometrical scheme.

In general, then, Euclid’s Optics presents a number of different goals, depending on the individual proposition. Thus, the work operates more as a palimpsest, drawing on multiple predecessors and traditions, and, rather than presenting a sustained effort to explain perceptual errors, the text often includes examples based solely on whether geometry can successfully be applied to an issue concerning vision. Recognizing the diversity of concerns within Euclid’s text helps us understand both how applied mathematics formed and how it relates to certain philosophical traditions, although without absorbing the goals of the former into those of the latter. 

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Scientific Modes of Perception and Expression

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