In order to determine the relationship between the true distance to the sun and moon, one of the key pieces of information we’ll need is their apparent angular diameters at syzygy. I’ve mentioned several time that both are around half a degree. For this to work, we’ll need to be more accurate than that, but measuring small angles like this is especially tricky. Ptolemy mentions a few ways astronomers prior to him tried to tackle the issue which include
measuring [the flow of] water or by the time [the sun and moon] take to rise at the equinox.
However, he rejects these stating that they are not sufficiently accurate. Instead, he states he used a dioptra which is a surveying instrument. While he doesn’t get into the detail of its construction or use, he does give a summary of some of his key findings:
First, he states that the sun’s apparent diameter does not appear to change, but the moon’s apparent diameter does. According to Ptolemy, it has the same angular distance as the sun only when it is at its maximum distance which he states disagrees with his predecessors who claimed that the moon’s diameter matched that of the sun only at mean distance. Ptolemy’s position is easily refuted as the existence of annular eclipses by necessitates a smaller angular diameter of the moon than the sun.
The second conclusion Ptolemy gives is that the angular diameters he has determined are “considerably smaller than those traditionally accepted.” Although he doesn’t give any of those previous values, Toomer notes that Hipparchus had a value of “a six hundred and fiftieth of its circle” which is about $0;33,14º$. Ptolemy also doesn’t give away his figure just yet either, but does indicate that it didn’t actually come from use of the dioptra he just mentioned, but “on certain lunar eclipses.”
So in this post, we’ll explore that method, but as a warning, while Toomer praises this section as “elegant and theoretically correct” this section has many problems. Continue reading “Almagest Book V: Angular Diameter of the Moon and Earth’s Shadow at Apogee During Syzygy”