Tracing the history of medieval astronomy
With our corrections for the intervals from Saturn’s oppositions in hand, we’re ready to repeat the calculation. As with before, I’m doing this in a Google Sheet to speed things along. This means that I’ll be using modern trig and that the figures I give her may be subject to some rounding as we go since the Sheet is saving higher precision behind the curtain.
We’ll begin with the same diagram as before:
Continue reading “Almagest Book XI: Second Iteration for Saturn”
Now that we’ve done our initial determination of the eccentricity for Saturn, and the position of its line of apsides, we’ll work on making the first, and only, iterative correction to these. We’ll again start by finding corrections to the angles for each opposition beginning with a diagram that should look familiar. Continue reading “Almagest Book XI: Corrections for the Equant”
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”
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”
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”
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”
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”
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”
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”
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”