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: 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”
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”