Our next step in calibrating Saturn’s model, is to verify that the steps we’ve taken so far is able to reproduce the original observations. Continue reading “Almagest Book XI: Checking the Eccentricity and Apogee of Saturn”
Almagest Book XI: Second Iteration for Saturn
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”
Almagest Book XI: Saturn’s Eccentricity
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: 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 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”