Now that we’ve ironed out Mars’ position for the time of the Babylonian observation we discussed in the last post, and we’ve revised the periodic motions, it’s time to calculate the epoch position. Continue reading “Almagest Book X: Mars’ Epoch Positions”
Almagest Book X: Correction to the Periodic Motions
Back in IX.3, Ptolemy noted that the planetary periods
have been corrected by us, on the basis of the comparison of their positions which became possible after we had demonstrated their anomalies, as we shall explain at that point.
As a reminder, what he was telling us then, was that he had made corrections to the periods provided to him by Hipparchus. We’ll now go over those corrections for Mars. Continue reading “Almagest Book X: Correction to the Periodic Motions”
Almagest Book X: The Size of Mars’ Epicycle
Having determined the eccentricity of Mars’ model as well as the line of apsides,
Our next task is to demonstrate the ratio of the size of the epicycle.
Continue reading “Almagest Book X: The Size of Mars’ Epicycle”
Almagest Book X: Checking the Eccentricity and Apogee
Now that we’ve sorted out the angles around the circle of mean motion, Ptolemy sets about checking to ensure that they return the observed angles from apogee from the point of view of the observer. Continue reading “Almagest Book X: Checking the Eccentricity and Apogee”
Almagest Book X: Third Iteration for Mars
Finally we’re ready to calculate our final eccentricities and line of apsides. And instead of breaking it into two posts, I’m going to do it all at once so we can be done with it1.
Here we go! Continue reading “Almagest Book X: Third Iteration for Mars”
Almagest Book X: Second Iteration Correction for Equant – Third Opposition
We’re now ready to make the correction for the third opposition in this second iteration. And the good news is that this is the last time we’ll need to calculate a correction.
Ptolemy still does a third iteration which relies on these corrections, but since he won’t be doing a fourth iteration, further corrections aren’t necessary. So let’s get to it. Continue reading “Almagest Book X: Second Iteration Correction for Equant – Third Opposition”
Almagest Book X: Second Iteration Correction for Equant – Second Opposition
Next, we’ll work on the correction for the second opposition in our second iteration. Continue reading “Almagest Book X: Second Iteration Correction for Equant – Second Opposition”
Almagest Book X: Second Iteration Correction for Equant – First Opposition
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”