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Martianus Capella’s calculation of the size of the moon

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Abstract

The eighth book of Martianus Capella’s famous De Nuptiis Philologiae et Mercurii deserves a prominent place in the history of astronomy because it is the oldest source that came down to us unambiguously postulating the heliocentrism of the inner planets. Just after the paragraph in which Capella asserts that Mercury and Venus revolve around the Sun, he describes a method for calculating the size of the Moon, as well as the proportion between the size of its orbit and the size of the Earth. It is possible to find some descriptions of the argument in general histories of astronomy or in books dedicated to Capella’s work, but usually they do not try to make sense of the argument. Rather, they limit themselves to describe or paraphrase what Capella says. As far as I know, there is no single study of the argument itself. The explanation for this absence is simple: the calculation offers many difficulties in its interpretation, for it shows obvious inconsistencies in the steps of the argument and apparent arbitrariness in the selection of the data used. In this article, I offer an interpretation that tries to discover, behind Capella’s confusing presentation, a well-sound argument for calculating the Moon’s absolute size. Interestingly, we have no records of this argument in other sources, at least in the form described by Capella.

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Notes

  1. For the differences between the 250,000 stadia reported by Cleomedes and the 252,000 stadia informed in many other sources see (Carman and Evans 2015). Because Capella knew that the value of Eratosthenes was 252,000 stadia, it remains a mystery why he attributes to Eratosthenes (and Archimedes) the value of 406,010 stadia that he will use in the calculation of the size of the Earth.

  2. It is interesting to note that Cleomedes, after describing Eratosthenes’ method that used the summer solstice, asserts that others used the same method but with the winter solstice (I;7;111; Bowen & Todd: 84). Therefore, it is not unlikely that Capella’s source used the equinox. Interestingly, the shadow cast by the gnomon at the equinox at Alexandria is around 30\({^{\circ }}\) (equal to its longitude) and 30\({^{\circ }}\) multiplied by 24 is equal to 720\({^{\circ }}\), i.e., to a double circle. So, maybe Capella is referring to the size of the shadow of Alexandria, but he says that the calculation has been made using Syene and Meroë!

  3. It is interesting to note that while the penumbra of the Moon’s shadow projected over the Earth is equal to two Moons, the umbra of the Earth projected over the Moon also equals about two Moons (exactly 2 Moons for Aristarchus, 2,5 to Hipparchus and 2,6 to Ptolemy).

  4. I want to thank Anibal Szapiro, Diego Pelegrin and Gonzalo Recio for their invaluable help in interpreting the details of this diagram.

  5. Value taken from Nasa. See: http://nssdc.gsfc.nasa.gov/planetary/factsheet/Moonfact.html.

  6. Thanks to Dennis Duke for suggesting and preparing this graph for me.

  7. This table has been calculated using Eclipse Predictions by Fred Espenak and Chris O’Byrne (NASA’s GSFC), at http://eclipse.gsfc.nasa.gov/JSEX/JSEX-index.html. Data used for Meroë: lat: 16.9382\({^{\circ }}\)N, long \(33.7488{^{\circ }}\hbox {E}\)

  8. Something similar could be said about Cleomedes’ description of Eratosthenes’s argument. It could also be understood as a simplification of what Eratosthenes actually could have done. But in this case it is a good description of the simplified argument. See (Carman and Evans 2015).

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Acknowledgements

I would like to thank Dennis Duke, Anibal Szapiro, Diego Pelegrin and Gonzalo Recio for discussing previous versions of this paper; Juan Torbidoni, who sent me bibliography I could not find and Juan Francisco Franck, who corrected the English.

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Correspondence to Christián C. Carman.

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Communicated by: Noel Swerdlow.

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Carman, C.C. Martianus Capella’s calculation of the size of the moon. Arch. Hist. Exact Sci. 71, 193–210 (2017). https://doi.org/10.1007/s00407-016-0185-0

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