Tag: eccentre

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Almagest Book X: Second Iteration for Mars – Part 1

Chapter $7$ of Book X has been a lengthy one. We already have six posts behind us and we’re really just getting started because now we have to repeat all of those posts again, iterating through to better approximate an angle we didn’t know at the outset.

To do a quick review, in the first post in this chapter, we needed $\angle ZNH$, the apparent change along the equant circle from the point of view of the observer in the diagram below.

However, what we had was $\angle LNG$ which was the apparent change along the ecliptic from the point of view of the observer.

As you can tell, they’re close, but not quite the same. However, we could get to what we needed from what we had by subtracting $\angle LNT$ and $\angle YNM$.

We found both of these angles to be $0;33º$ in this post and $0;50º$ in this post, respectively.

Unfortunately, both of those values were calculated starting from an incorrect value for $\angle ZNH$. But, the good news is that applying them anyway will (Ptolemy hopes) still get us closer to the correct value! And then we’ll be able to iterate through again to reduce the discrepancy even further.

So let’s get started. Continue reading “Almagest Book X: Second Iteration for Mars – Part 1”

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Almagest Book X: Preliminary determination of Mars’ Apogee & Eccentricity – Part 2

In the last post, we began the process of determining Mars’ eccentricity and the position of apogee. The technique is fundamentally similar to the three eclipse method used for the moon, but the presence of the equant is throwing a wrench in things.

Specifically, right at the beginning of the last post we had to make an assumption about $\angle BDG$ that we know was incorrect. We’re not quite ready to address that, but for now we’ll work on finishing finding the line of apsides as well as the eccentricity as if that assumption were correct. Then, in future posts, we’ll go about trying to iteratively correct that assumption. Continue reading “Almagest Book X: Preliminary determination of Mars’ Apogee & Eccentricity – Part 2”

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Almagest Book V: Parallax Table

In the previous chapter, we worked out the parallax for the sun and the moon at 4 different positions. Next, we determined how we might make some corrections for the moon if it were between the positions we described, breaking the corrections into ones for the epicycle and for the eccentre. From that work, Chapter $18$ presents the results in tabular format. Since this one is pretty wide, I’ve again made the decision to put it into a Google Sheet.

Continue reading “Almagest Book V: Parallax Table”

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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:

Continue reading “Almagest Book V: Lunar Distance Adjustments for Epicycle”

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Almagest Book V: Parallaxes of the Sun and Moon

Now that Ptolemy has worked out his model for the sun and moon in earth radii, we can use this result to calculate parallax for any position in our model. To begin, we will calculate

the parallaxes with respect to the great circle drawn through the zenith and body.

This is essentially the reverse calculation of what we did in this post, so we can reuse the same diagram:

Continue reading “Almagest Book V: Parallaxes of the Sun and Moon”

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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:

Continue reading “Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee”

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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 bit1. 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”

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Almagest Book V: Constructing the Lunar Anomaly Table

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 moon2 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”

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Almagest Book V: Second Determination of Direction of Epicycle

In the last post, we followed along as Ptolemy determined that the position of “apogee” used for calculating the motion around the epicycle is not the continuation of the line from the center of the ecliptic or center of the eccentre through the center of the epicycle. Rather, motion should be measured from the “mean apogee” which is defined from a third point opposite the center of the ecliptic from the center of the eccentre.

Ptolemy doesn’t give a rigorous proof for this and instead relies on proof by example. So in that last post, we went through one example, but in this post, we’ll do a second one

in order to show that we get the same result at the opposite sides of the eccentre and epicycle.

Continue reading “Almagest Book V: Second Determination of Direction of Epicycle”