Message #47965

[Bianca]

                                                A.2.2. Obliquity and Latitude





The obliquity of the ecliptic can be measured by photogrammetry as the average declination of the tropic circles. The latitude of the observer will be related to the average declinations of the Arctic and Antarctic Circles as measured by photogrammetry. Appendix A.1.2 presents full details on my procedures for measuring the declinations of the circles on the Farnese Atlas.

I find that the tropics and Ant/Arctic circles are parallel to the equator to within an accuracy of one degree. By averaging together all my photogrammetric measures of the circles’ declinations I get the best measures of their declinations. Thus, the Arctic Circle is at a declination of +57.8° ± 0.5°, the Tropic of Cancer is at a declination of +26.3° ± 0.2°, the Tropic of Capricorn is at a declination of –26.0° ± 0.4°, and the Antarctic Circle is at a declination of –57.0° ± 0.5°.

The equatorial distances of the tropics should equal the obliquity adopted by the Farnese Atlas. The two tropics are at virtually identical distances from the equator, so it is reasonable to form a weighted average and as a result we get an obliquity of 26.2° ± 0.2°. Similarly, we can combine the Arctic and Antarctic Circles to get an equatorial distance of 57.5° ± 0.4°.

These photogrammetry values are substantially different from those based on the circle positions with respect to the background constellations. In fact, the photogrammetry values are larger by 2.25° ± 0.8° for the tropics (26.2° versus 23.95°) and larger by 5.8° ± 1.0° for the Ant/Arctic Circles (57.5° versus 51.7°). These differences are too large to be by random chance or measurement error. An offset of one degree in declination corresponds to a misplacement of 0.57 cm on the surface of the globe, so the tropics are off by 1.3 cm while the circles of invisibility are off by 3.3 cm.

How can we reconcile these differences between the declinations from the stars-on-circles method and the declinations produced by photogrammetry? It is implausible that the sculptor correctly placed the circles on the globes and then systematically misplaced the constellations with respect to the primary coordinate grid on the globe. Also, there is no precedent for there ever being an intentional placement of the tropics for an obliquity of 26.2° ± 0.2°. However, there is excellent precedent for an obliquity of 23.95° ± 0.8°, and this argues that the sculptor was intending to place the tropics and the constellations simultaneously. So the obvious interpretation is that the sculptor placed the declination circles onto the globe with a small distortion that increases with distance from the equator, and then placed the constellations accurately with respect to the coordinate grid. Such a distortion in the placement of the grid circles could arise either from an unintentional error on the part of the sculptor or from an intentional decision on his part to improve the display of the constellations for artistic reasons.

The distortion of the globe’s coordinates is apparently independent of right ascension (because the tropics and Ant/Arctic Circles are parallel to the equator to an accuracy better than it would be if, for example, the distortion were in ecliptic latitude). The distortion is symmetric north/south. As for the magnitude of the distortion, we only have two values: 2.25° for the tropics and 5.8° for the Ant/Arctic Circles. Some model of the distortion is required for photogrammetry of positions away from the circles. While the distortion might well be smooth, many correction functions are plausible, given that we know the curve at only two points. I have therefore adopted the simple piecewise linear function, and this surely produces an additional uncertainty that is small compared to other sources of error.

These two ways of measuring the declination of the circles also accounts for a large disagreement between authorities on the derived obliquity and latitude. Gialanella and Valerio place the tropics at 25.5° and the Arctic Circle at 58°, so it is clear that they used photogrammetry. Fiorini gives an obliquity of 23° and a declination of the Ant/Arctic Circles of 50°, so it is clear that he was deriving the declinations of the circles from their positions with respect to the constellation fi gures.

In all, I conclude that the obliquity used in constructing the Farnese Atlas was 23.95° ± 0.8° while the Ant/Arctic Circles were 51.7° ± 0.9° from the equator. The globe has a small distortion in the declination which increases with equatorial distance, and this must be accounted for by the photogrammetry.