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Lotto's "Husband and wife," van Eyck's "Portrait of Arnolfini and his wife," and other paintings: Problems with focal lengths
In brief, Hockney and Falco claim that a concave mirror was used to project an image of the models or objects onto a canvas or other surface, which was then traced or painted over by the artist. They consider Lorenzo Lotto's "Husband and Wife" (1523-4) to be the "Rosetta stone" in their theory. Based on simple assumptions about the sizes of figures and their distances from the artist, they estimate the focal length of the concave mirror purportedly used in its creation to be f = 54 cm (p. 56). They state that other paintings were created with mirrors of a comparable focal length and settle upon f = 59 cm as a reasonable and representative value. I likewise estimated the focal length of the mirror purportedly used in the creation of van Eyck's "Portrait of Arnolfini and his wife" (1434) and came up with a value close to theirs (55 cm).
First, of course, we must note that we can calculate a focal length associated with any representational painting containing sufficiently many objects at a variety of distances, including paintings that were made with no reliance on optical devices. In short, such a focal length calculation taken alone does not imply that a concave mirror was actually used in the creation of the painting.
Where might such concave mirrors be found in the early Renaissance? Recall that the Arnolfini portrait depicts a convex mirror. Hockney (p. 82, cf artandoptics.com) gives the following suggestion: "If you were to reverse the silvering [of the convex mirror depicted in the Arnolfini portrait], and then turn it round, this would be all the optical equipment you would need for the meticulous and natural-looking detail in the picture." Likewise, Hockney and Falco (p. 55) write "What evidence is there that the fabrication technology to produce such an optical element [concave mirror] existed in the early Renaissance? Jan van Eyck gives us one answer in the 1434 portrait [of Arnolfini]... Until an opaque protective coating had been applied to the back side of the convex mirror on the wall, its obverse side would have been a mirror lens [i.e., concave mirror]." More specifically, though, how might such concave mirrors have been fabricated? Mr. Hockney (p. 229) states "[small convex mirrors] were made by blowing small globes of glass into which, while hot, was passed through the pipe a mixture of tin, antimony and resin or tar. It was cut when cooled and then formed small well-defined images."
Let me put this another way.Ê The convex mirror in the painting bulges like the belly of a woman 8 months pregnant but to be used for projection as Hockney suggests, it should instead be like the belly of a woman just two weeks pregnant.Ê It isn't the overal size of the convex mirror that is wrong, it is the 'bulginess' and hence its short focal length.Ê In sum, the convex mirror could not have been used for the projection as Mr. Hockney suggests.
Note that the suggestion made publicly at the New York symposium, that by spinning the glass sphere to flatten it, does not remove the incompatibility between the focal length of the mirror depicted in the Arnolfini portrait (f = 12 cm) and that estimated for the mirror purported used in its creation (f = 55 cm).
I searched through other contemporary paintings for depictions of mirrors or even glass spheres that might be consistent with the long focal lengths required by the Hockney/Falco theory, but found none. Here are just five. I estimated the focal length of the mirrors depicted in the Campin and Metsys paintings just as I had for the Arnolfini portrait. Estimating the focal lengths for the remaining paintings was much easier. I merely "fit" a circle (sphere) to the faces of the curved mirrors (not their circumference) depicted in the paintings and compared its size to other familiar objects in the painting (and took account of their relative distances from the painter). Estimating the focal length this way was particularly easy for the convex mirror at the lower-right of "Saint Eligius" by Petrus Christus. The Vermeer does not show a mirror but instead a glass globe; I include it because it is clearly relevant to understanding the glass mirror technology of that time.
What about other sources of mirrors? Hockney and Falco mention that "concave mirrors of polished bronze and speculum metal did exist in Medieval times and in antiquity" (p. 55), but not that there is no record that such "burning mirrors" were of sufficient quality or sufficiently long focal lengths to be used as they propose, or that we have little or no pursuasive evidence they were used for any imaging tasks (see below). In any case, polishing a mirror the size of that depicted in the Arnolfini portrait would have been an astounding challenge, nearly a quarter millennium before the invention of reflecting telescopes by James Gregory in 1663.
Creating the requisite concave mirrors presents a number of difficulties that have not been answered. For instance while a blown glass ball (even one made by spinning) that deviates from a spherical shape nevertheless yields an acceptable convex mirror (as depicted in the Arnolfini portrait), if turned around that same flawed (now concave ) mirror would yield an extremely poor and unusable image on the flat screen of a canvas. In the convex mirror usage, the light leaving a point on the object that enters your eye was reflected from a single point on the mirror. In the concave mirror usage, the light leaving a point on the object that strikes a point on the canvas was reflected from the all points on the mirror. Thus, in the concave usage all parts of the mirror must work in agreement and flaws become apparent if they do not. Here's an analogy. If one person sings sharp or flat (single point on a convex mirror), you might not notice (acceptable reflected image). If a whole chorus is made up of faulty singers some singing sharp and some flat (all points on the flawed concave mirror), their music is cacophonous (unaceptable focused image).
Summary: There seems to be no corroboratory depictions of any specific mirror from the fifteenth century that could have been used in the creation of the Arnolfini portrait, Lotto's "Husband and wife," or indeed any of the paintings for which Hockney and Falco and I computed an effective mirror focal length. In every case, particularly the mirror proposed by Mr. Hockney, the focal lengths are much too short.