This post a collection of my class materials which will be posted below the fold. In general, this is a Powerpoint presentation and the script I follow is included in the file. Continue reading “Class Materials”
Almagest Book V: Components of Parallax – Corrections
At the end of the last post, we noted that Ptolemy wasn’t quite satisfied with what we did previously because we used some rather faulty assumptions.
As Ptolemy states it:
For lunar parallaxes, we considered it sufficient to use the arcs and angles formed by the great circle through the poles of the horizon [i.e., an altitude circle] at the ecliptic, instead of those at the moon’s inclined circle. For we saw that the difference which would result at syzygies in which eclipses occur is imperceptible, and to set out the latter would have been complicated to demonstrate and laborious to calculate; for the distance of the moon from the node is not fixed for a given position of the moon on the ecliptic, but undergoes multiple changes in both the amount and relative position.
The key phrase here is the “at the ecliptic, instead of the moon’s inclined circle.” This got swept under the rug in that post because Ptolemy didn’t really explain why the algorithm he gave us should work. So to understand, let’s start by taking a harder look at what’s actually going on.
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Almagest Book V: Lunar Distance Adjustments for Eccentre
So far in this chapter, we’ve reviewed how to calculate the lunar parallax for certain limits of the lunar position and looked at what’s necessary to estimate the effects for lunar positions away from those limits due to the epicycle. Now, we need to discuss the impact of the eccentre and how that we can estimate the effect on parallax due to it bringing the moon closer and further.
So let’s set up a generic diagram of our eccentric model, ignoring the epicycle and only concerning ourselves with the mean moon:
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Almagest Book V: Lunar Distance Adjustments for Epicycle
In the last post, we explored how to calculate parallax if the distance to an object is known and its distance from the zenith. This was done for the sun and the moon at four different distances. However, because the moon varies so widely in distance in Ptolemy’s model, we need a way to estimate between those positions and we’ll begin by looking at the effect the epicycle has on distance for various points throughout its cycle. To help us, we’ll start with a new diagram:
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Almagest Book V: Scale of the Lunar Model
Now that we’ve worked out the distance to the moon at the time of the observation, we can put this information back into our lunar model diagram to work out the true scale. We’ll begin with a drawing of our lunar model at the time depicted:
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Almagest Book V: Calculation of Lunar Distance
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:
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Almagest Book V: Calculation of Lunar Parallax
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 $50 \frac{11}{12}º$.
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Almagest Book V: Lunar Parallactic Observations
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:
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Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee
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:
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Almagest Book V: The Difference at Syzygies – Maximum Lunar Anomaly
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 bit3. 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.
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