Category: Almagest

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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: Correction for Equant – Third Opposition

In the previous two posts, we’ve been looking at how the apparent position of the center of a superior planet’s epicycle shifts position along the various circles. Specifically, we’ve been looking at how the angle between oppositions differs between the eccentre of mean distance and the equant circle.

In this post, we’ll do the same for the third opposition Ptolemy gave us. Unsurprisingly, the process will be largely the same as in the past two posts. Continue reading “Almagest Book X: Correction for Equant – Third Opposition”

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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 X: Observations and Introduction to Iterative Calculations for Apogee and Eccentricity

In the last Almagest post, Ptolemy explained that for the models of the superior planets, we’ll be needing to use oppositions. So with that explained, we’ll begin by working to develop the model for Mars by looking at its eccentricity and apogee. However, this isn’t going to be nearly as straightforward as it has been previously. As we’ll see, we’re missing some of the angles and arcs that would be needed for this. Thus, Ptolemy is going to use an iterative method to close in on them. Continue reading “Almagest Book X: Observations and Introduction to Iterative Calculations for Apogee and Eccentricity”