Tracing the history of medieval astronomy
Now that we’ve demonstrated a parallax of $1;07º$ for the moon, we can use that do determine a distance to the moon. As a forewarning, some of the math may seem suspect here, so I’ll do my best to explain it.
To being, let’s start off with a new drawing:
Continue reading “Almagest Book V: Calculation of Lunar Distance”
So far in this book, we’ve refined our lunar model, shown how to use it to calculate the lunar position1, discussed a new instrument suitable for determining lunar parallax, as well as an example of the sort of observation necessary to make the calculation.
Now, Ptolemy walks us through an example of how to calculate the lunar distance using an example entirely unrelated to the one we saw in the last post.
In the twentieth year of Hadrian, Athyr [III] $13$ in the Egyptian calendar [135 CE, Oct. $1$2], $5 \frac{5}{6}$ equinoctial hours after noon, just before sunset, we observed the moon when it was on the meridian. The apparent distance of its center from the zenith, according to the instrument, was $50 \frac{11}{12}º$. For the distance [measured] on the thin rod was $51 \frac{7}{12}$ of the $60$ subdivisions into which the radius of revolution had been divided, and a chord of that size subtends an arc of $50 \frac{11}{12}º$.
Continue reading “Almagest Book V: Calculation of Lunar Parallax”
In the last post we followed along as Ptolemy discussed the construction and use of his parallactic instrument, which he would use to measure the lunar parallax. To do so, Ptolemy waited for the moon to
be located on the meridian, and near the solstices on the ecliptic, since at such situations, the great circle through the poles of the horizon and the center of the moon very nearly coincides with the great circle through the poles of the ecliptic, along which the moon’s latitude is taken.
That’s pretty dense, so let’s break it down with some pictures, First, let’s draw exactly what Ptolemy has described above:
Continue reading “Almagest Book V: Lunar Parallactic Observations”
The primary instrument I’ve used for my observing is an astronomical quadrant. That instrument is designed primarily to measure the angular distance of an object above the horizon3, otherwise known as its altitude. However, this isn’t the only instrument good for this sort of thing. Brahe’s Astronomiae Instauratae Mechanica is filled with instruments that essentially fill this same purpose, but in different ways.
One design, he describes as a “parallactic instrument” but it was also known as a triquetrum in period. This design dates back to Ptolemy and is described in Chapter $12$. Here’s a drawing of it from Toomer:
Continue reading “Almagest Book V: The Parallactic Instrument”
Chapter 11 of Book 5 is one of those rare chapters that’s blessedly free of any actual math. Instead, Ptolemy gives an overview of the problem of lunar parallax, stating that it will need to be considered because “the earth does not bear the ratio of a point to the distance of the moon’s sphere.” In other words, the ratio of the diameter of the earth to the distance of the moon isn’t zero.
However, this does pose an interesting question. We’ve previously given the radius of the eccentre as $49;41^p$, but we haven’t given the radius of the Earth in the same units. Thus, how can this ratio even be taken to know this? Continue reading “Almagest Book V: Lunar Parallax”
In the last post, we looked at how much the total equation of anomaly would change during syzygy due to the eccentre we added to the lunar model in this book, when the moon was at its greatest base equation of anomaly. As Ptolemy told us, it wasn’t much. However, there was a second effect that can also change the equation of anomaly, which was based on where we measure the movement around the epicycle from. Namely, the mean apogee instead of the true apogee. This has its maximum effect when the moon is near apogee or perigee so in this post, we’ll again quantify how much.
Let’s start off by building our diagram:
Continue reading “Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee”
Syzygy is one of those words that has popped up very little in the Almagest so every time it does, I’m always thrown off a bit4. Especially when Ptolemy is going to spend an entire chapter discussing a topic that has scarcely even come up. But here we have Ptolemy spending the entirety of chapter $10$, to demonstrate that these modifications we’ve made to the lunar model have a negligible effect because he fears readers might think it does since
the centre of the epicycle does not always … stand exactly at the apogee at those times, but can be removed from the apogee by an arc [of the eccentre] of considerable size, because location precisely at the apogee occurs at the mean syzygies, whereas the determination of true conjunction and opposition requires taking the anomalies of both luminaries into account.
Continue reading “Almagest Book V: The Difference at Syzygies – Maximum Lunar Anomaly”
It’s been awhile since I’ve done an Almagest post so quick recap. In Book IV, we worked out a first model for the moon, which was a simple epicycle model inclined to the plane of the sun’s sphere. In this book, Ptolemy showed us that this model was insufficient as the moon’s speed varies more than should predict and so we added an eccentric as well as having the center of the eccentre rotate around the observer. Finally, we introduced the concept of a “mean apogee” which is the position on the epicycle we’ll need to measure from in order to do calculations.
That’s been a lot, but with all of this completed, we should now be able to use it to calculate lunar positions which Ptolemy walks us through in this chapter. Unfortunately, he does this in the form of a generic prescription of steps instead of a concrete example. Fortunately, Toomer provides an example5 that I’ll use to supplement Ptolemy’s narration. Continue reading “Almagest Book V: Calculation of the Lunar Position”
After all our work, here’s the final table of lunar anomaly
Since this table gets a bit wide, I’m sharing it as a publicly visible Google Sheet. Continue reading “Almagest Book V: Complete Table of Lunar Anomaly”
In our last post, we showed how it is possible to determine the equation of anomaly by knowing the motion around the epicycle and the double elongation. This, combined with the position of the mean moon6 gives the true position of the moon. As usual, Ptolemy is going to give us a new table to make this relatively easy to look up. But before doing so, Ptolemy wants to explain what this table is going to look like. Continue reading “Almagest Book V: Constructing the Lunar Anomaly Table”