Now it’s time to determine the small angles that get applied as corrections. As with before, we’ll start with the first opposition, again using the same diagram. Continue reading “Almagest Book X: Second Iteration Correction for Equant – First Opposition”
Almagest Book X: Second Iteration for Mars – Part 2
Continuing on with our second iteration in which we calculate the revised line of apsides and eccentricity of Mars’ model.
As with before, I’m doing my calculations in a Google Sheet to make it easier to correct should I make an error. This also means that the Sheet will be keeping much higher precision to prevent rounding errors. However, as I’m only showing rounded values, things may look a bit off. So if something seems funny, I encourage checking out the aforementioned Sheet. Continue reading “Almagest Book X: Second Iteration for Mars – Part 2”
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”
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”
The Almagest Manuscripts – State Library of Victoria: RARESF 091 P95A
I have finished adding another Almagest manuscript to the collection in the Google Sheet. This time, it’s the RARESF 091 P95A manuscript1 at the State of Victoria Library in Australia and it has some interesting features. Continue reading “The Almagest Manuscripts – State Library of Victoria: RARESF 091 P95A”
Almagest Book X: Correction for Equant – Second Opposition
Having determined a correction for the equant for the first opposition, we’ll now repeat the procedure for the second. The diagram to do so is identical to the previous one, but is flipped over the line of apsides. Continue reading “Almagest Book X: Correction for Equant – Second Opposition”
Almagest Book X: Correction for Equant – First Opposition
Having gotten our preliminary estimate for the eccentricity and position of apogee
let us investigate the differences which can be derived from them in the ecliptic arcs which we week to determine at each of the oppositions [in turn].
In this post, we’ll cover Ptolemy’s method for the first opposition. Continue reading “Almagest Book X: Correction for Equant – First Opposition”
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”
Almagest Book X: First Iteration of Mars’ Apogee & Eccentricity – Part 1
Now that we’ve laid out what we’re going to be attempting to do, let’s get started on finding the line of apsides and eccentricity. Continue reading “Almagest Book X: First Iteration of Mars’ Apogee & Eccentricity – Part 1”
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”