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In the Divine Comedy, in Canto II of Paradiso, Beatrice explains to Dante why his theory to explain the dark spots of the moon is wrong. To do this she suggests conducting an experiment involving three mirrors. This is the translation by Robert Hollander of the relevant lines 97-105:

'Take three mirrors, placing two at equal distance
from you, letting the third, from farther off,
also meet your eyes, between the other two.

'Still turned to them, have someone set,
well back of you, a light that, shining out,
returns as bright reflection from all three.

'Although the light seen farthest off
seems smaller in its size, still you will observe
that it must shine with equal brightness.

This statement about light intensity, on which her explanation is based, is patently false. The intensity of light will decrease like the square of the distance, unless we are dealing with something like laser beams, not likely to be in Dante's mind in the 14th century. I would like to find a commentator who points this out. None of the comments I have read discusses it. Writes Nicola Fosca: "Boyde ('L'esegesi di Dante e la scienza,' in Dante e la scienza, ed. P. Boyde and V. Russo [Ravenna: Longo, 1995], pp. 14-16), actually performed a version of the experiment in order to test Dante's method (it passed his test).". Is there any commentator who discusses this point?

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  • The inverse square law was not formally defined until a couple of hundred years after Dante. It is not just lasers that wouldn't be in his mind. Commented Jun 27 at 9:55
  • @Chenmunka That's just quantification. People certainly noticed the quality/dimness of lights that were further away. Commented Jun 29 at 21:45

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The Italian text of lines 103–105 is

Ben che nel quanto tanto non si stenda
la vista più lontana, lì vedrai
come convien ch’igualmente risplenda.

That is,

However far may extend
the more distant image, you’ll see it
(as it should) shine equally.

Remember that Dante was writing before the science of photometry was established, and in any case Paradiso is poetry, so that we can’t expect to find precise scientific statements. It is up to us to find the most plausible interpretation.

In Beatrice’s experiment, the only instrument that Dante can use to measure the brightness of the reflected images is his eyes, so in modern terms we must take “shine equally” to refer to the perceived brightness of each light source, that is, its luminous intensity, which is defined as luminous flux per unit of solid angle, where luminuous flux is the power of the light (its radiant flux), weighted by wavelength according to the sensitivity of the human eye.

The question is correct that the luminous flux of a light source delivered to the eye decreases like the square of the distance of the source from the eye. But the solid angle subtended by the source also decreases like the square of the distance, so that this factor cancels out, and the luminous intensity is independent of distance, as Beatrice says.

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