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Hipparchus' Celestial Globe

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Bianca
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« Reply #15 on: December 08, 2007, 09:17:20 am »








Many uncertainties contribute to the error bars. First, there is my measurement errors, which are 1° – 2° as based on the repeatability of positions as measured from picture-to-picture. Second, there is the uncertainty as to my placement of the dot on the constellation figures. For example, does the star α Her correspond to the top or the middle of the head of Hercules? Third, the sculptor will not have placed the constellation figure perfectly with respect to the position of the star, for example because the sculptor has a high priority in not making the constellation fi gures look wrongly elongated. Fourth, the original observations on which the sculptor is working will not be perfectly accurate. The star catalogue in the Almagest has positional accuracies of a little better than a degree, whereas the verbal descriptions in Aratus are accurate only to around 4° for placing parts of constellations onto the celestial circles. Fifth, the Roman sculptor did not make a perfect reproduction of the original Greek statue, and this introduces yet more errors.

A.1.4. A Worked Example I will here present a detailed example, with all intermediate values presented. This will allow researchers to test my procedures and to see typical values, and will provide a known example to check later applications. I will take for my example the fi rst of my photographs, which is a typical case with neither large nor small error bars.

The photograph was taken with the camera 6 feet from the edge of the globe (Dcamera = 183 cm). Recall that Rglobe = 32.5 cm. The angular radius of the globe as viewed from the camera has ζedge = 0.151 rad. The image of the globe was expanded and printed onto paper such that the radius of the image was ρedge = 10.3 cm. The centre of the image was found by repeated trials with a compass until all edges were within

0.1 cm of a circle drawn around this centre (except for a 45° arc to the north caused by the hole in the globe). I constructed a rectangular coordinate with an origin at this centre and with the Y axis roughly towards the north. I next placed red dots where the vernal equinox colure intersected the two tropics and the Arctic Circle. I also placed green dots at 8 positions in constellations that can be identified with distinct stars in the modern sky. Then, with a millimetre ruler, I measured the distance of each dot from the two axes (see Table 1). With these positions, I then made calculations (in EXCEL) as presented in Appendix A.1.1 so as to convert to spherical coordinates on the globe. Intermediary and final values are presented in Table 1.
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