The last planet we’ll need to determine the eccentricity and line of apsides for is Saturn. To do so, we’ll follow exactly the procedures we developed previously. And again, we’ll begin with three observations. Continue reading “Almagest Book XI: Saturn’s Eccentricity”
Almagest Book XI: Jupiter’s Epoch Positions
The last thing to do before finishing up with Jupiter for now, is to determine its epoch positions.
Ptolemy does this by taking the intervals from the ancient observation we explored in the last post. Continue reading “Almagest Book XI: Jupiter’s Epoch Positions”
Almagest Book XI: Jupiter’s Periodic Motions
Before moving on, Ptolemy describes how he arrived at the periodic motions of Jupiter, beginning with an ancient observation. Continue reading “Almagest Book XI: Jupiter’s Periodic Motions”
Almagest Book XI: Determining the Size of Jupiter’s Epicycle
Next, we’ll determine the size of Jupiter’s epicycle. To do so, we’ll use another observation that Ptolemy has made. Continue reading “Almagest Book XI: Determining the Size of Jupiter’s Epicycle”
Almagest Book XI: Verification of Jupiter Derivations
Now that we’ve gotten an understanding of the eccentricity and line of apsides for Jupiter, Ptolemy wants to demonstrate that they are correct. He plans to demonstrate this, as he did for Mars,
from the fact that the apparent intervals [in longitude] of the planet derived from the ratios we have thus found turn out to be the same as those observed.
Continue reading “Almagest Book XI: Verification of Jupiter Derivations”
Almagest Book XI: Second Iteration for Jupiter
Now that we’ve worked out corrections for the change in angles caused by the equant, we’ll apply them and recalculate the eccentricity and line of apsides. Continue reading “Almagest Book XI: Second Iteration for Jupiter”
Almagest Book XI: Correction for the Equant – Third Opposition
Continuing on with our corrections for the equant, we’ll work on the third opposition for Jupiter. Continue reading “Almagest Book XI: Correction for the Equant – Third Opposition”
Almagest Book XI: Correction for the Equant – Second Opposition
Continuing on with finding the corrections necessary for the presence of the equant, we’ll produce a diagram similar to the last but with the first opposition switched out for the second at $B$. Continue reading “Almagest Book XI: Correction for the Equant – Second Opposition”
Almagest Book XI: Correction for the Equant – First Opposition
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”
Almagest Book XI: Jupiter’s Eccentricity
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”