Tracing the history of medieval astronomy
Now that we’ve demonstrated that the equation of anomaly is symmetric about the line of apsides for the general model, we’ll demonstrate the same for Mercury’s model. Or, as Ptolemy puts it,
we must prove that in this situation too the angles of the equation of ecliptic anomaly [are equal].
We’ll start by producing a diagram based on Mercury’s particular model. Continue reading “Almagest Book IX: Symmetries in Mercury’s Planetary Model”
Now that we’ve taken the time to understand the model for the four planets other than Mercury, let’s start on the model for Mercury as that’s the focus of the remainder of this book.
Let the eccentre producing the anomaly be $ABG$ about centre $D$, and let the diameter through $D$ and centre $E$ of the ecliptic be $\overline{ADEG}$, [passing] through the apogee at $A$. Continue reading “Almagest Book IX: Model for Mercury”
Having thoroughly discussed what anomalies Ptolemy wants his model to account for as well as what hypotheses1 he intends to use for each, Ptolemy is ready to start laying out the basic models. Ultimately, there will be two models. One for the four planets other than Mercury, and a special one for Mercury.
In this post, we’ll explore the first of these models. Continue reading “Almagest Book IX: Model for Planets Other than Mercury”
Now that these [mean motions] have been tabulated, our next task is to discuss the anomalies which occur in connection with the longitudinal positions of the five planets.
Having derived the mean and anomalistic motions, Ptolemy now turns to exploring the anomalies in more depth (as there’s going to be two of them), in order to derive the parameters necessary for configuring the model’s scale. Continue reading “Almagest Book IX: Preliminary Notions”
This week I completed the addition of another manuscript of the Almagest to my collection. This is the Cambridge MS 32 manuscript. And it’s very interesting.
This manuscript doesn’t have a lot of information on it. It’s one I stumbled across when doing some other research and was not included in Peters & Knobel’s work.
The Cambridge website itself gives precious little information. The manuscript itself is a composite, written by at least two hands, most likely in the early $16^{th}$ century. However, no information is given on the lineage of this manuscript, but we’ll shed some light on this here shortly.
Continue reading “The Almagest Manuscripts – Cambridge MS 32”
As with previous large tables, I’ve transcribed the Tables of Mean Motion into a set of Google Sheets.
In this, I have rearranged the tables to be in a more sensical order2 (descending periods of time).
Also, you may notice that the longitudes for Mercury and Venus are the same to that of the sun. This is not an error. As we stated in the last post, these two have the same mean motion in longitude as the sun. Continue reading “Almagest Book IX: Planetary Mean Motion Tables”
Now that we have completed the above discussion, we will first set out, for each of the five planets, the smallest period in which it makes an approximate return in both anomalies, as computed by Hipparchus.
As we’ve done with the sun and moon, we will now focus on the periods associated with the planets.
While Ptolemy cites Hipparchus, he quickly notes that:
These [periods] have been corrected by us, on the basis of the comparison of their positions which became possible after we had demonstrated their anomalies, as we shall explain at that point. However, we anticipate and put them here, so as to have the individual mean motions in longitude and anomaly set out in a convenient from for the calculations of anomalies.
This passage hard to parse at this point, but what Ptolemy is saying here is that the values Ptolemy is about to give us have been corrected from those of Hipparchus. He is presenting them, with corrections already added, even though he will not explain what those corrections are until a later chapter. This way, he doesn’t have to rewrite these large tables later. Continue reading “Almagest Book IX: Returns of the Five Planets”
Having laid out his order of the spheres, Ptolemy still has more to say as an introduction to planetary theory before moving on to a specific one. Continue reading “Almagest Book IX: Purpose of the Planetary Theory”
Having laid out “the sum total of the chief topics one may mention as having to do with the fixed stars, in so far as the phenomena [observed] up to now provide the means of progress in our understanding” Ptolemy now moves on to “the treatment of the five planets” each one getting a book to itself.
But before diving in, Ptolemy first proposes that we “avoid repetition” by “explain[ing] the theory of the [planets] by means of an exposition common [to all five], treating each of the methods [for all planets] together.”
Beginning this discussion, Ptolemy introduces us to the order of the spheres for these planets, the sun, and moon. Continue reading “Almagest Book IX: On the Order of the Spheres”
In the final chapter of Book VIII, Ptolemy discusses the first and last visibilities of stars. Before diving into the text, let’s first take a moment to think about what this means.
The description is, in some ways, self evident – we’re considering when, based on the movement of the sun, are stars first and last visible.
Recall that, when viewed from inside the celestial sphere (i.e., on earth), the sun appears to move west to east along the ecliptic. Thus, as time passes, stars that were washed out by the sun will, at some point, become visible because they have risen while the sun has moved far enough along the ecliptic that it will be sufficiently far below the horizon as to not wash the star out in the morning glow before the sun actually crosses the horizon. This point in time is known as their helical rising and describes when they are first visible in the sky throughout the year.
These stars will, of course, initially be only briefly visible as the daily rotation of the sun will quickly cause them to be lost in the morning glow, but as the sun continues to progress eastward along the ecliptic3, the star will be visible for longer.
Similarly, we can ask at what point the sun has moved sufficiently close to a star that, even after the sun has set, the star will be lost in the evening glow and thus, it is effectively invisible for the season until its next helical rising.
In case you haven’t been checking the dates on these posts, you may well notice that it’s been nearly a month and a half since my last post. It’s because this section has been quite challenging as I find Ptolemy’s descriptions here are an absolute mess. This is probably among the least sensical sections I’ve yet encountered. I think you’ll see why as we go through Ptolemy’s commentary. Continue reading “Almagest Book VIII: On the First and Last Visibilities of the Fixed Stars”