Tracing the history of medieval astronomy
Having worked out a preliminary eccentricity and position for the line of apsides, we now turn to finding a better approximation iteratively. Fortunately, Ptolemy only finds one iteration necessary for Jupiter.
To get started, we’ll produce a new diagram: Continue reading “Almagest Book XI: Correction for the Equant – First Opposition”
Now that we have established the periodic motion, anomalies, and epochs of the planet Mars, we shall next deal with those of Jupiter in the same way.
It certainly strikes me that splitting up the planets between books is an odd choice on Ptolemy’s part, but at least the methods will be familiar. As Ptolemy tells us:
Once again, we first take, to demonstrate [the position on] the apogee and [the ratio of] the eccentricity, three oppositions [in which Jupiter is] directly opposite the mean sun.
This will be directly paralleling the work we did in X.7-10 for Mars. So bring on the observations! Continue reading “Almagest Book XI: Jupiter’s Eccentricity”
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”
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”
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”
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”
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”
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”
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”
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”