Tracing the history of medieval astronomy
We’re almost finished with chapter 6. All that’s left is to determine the position of the mean moon during one of the eclipses which will tell us the equation of anomaly at that point. To do so, we’ll add a few more points to the image we ended the last post with:
Continuing on with Ptolemy’s check on the radius of the epicycle, we’ll produce a new diagram based on the positions of the Alexandrian eclipses. However, instead of doing it piece-by-piece as I did when we explored the Babylonian eclipses, I’ll drop everything into a single diagram since we already have some experience and the configuration for this triple is a bit more for forgiving on the spacing:
Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle”
In the last post, we were able to determine the radius of the epicycle when the radius of the deferent is $60^p$. It took a lot of switching between demi-degrees contexts, but in the wake of all that math, we’re left with a mess of lines and arcs that we’ve already determined. So in this post, we’ll use that starting point to go just a little further and determine the position of the mean moon, specifically for the second eclipse. To do so, we’ll need to add a bit more to the configuration we ended with last time:
In the last post, we introduced three eclipses from Babylonian times which we used to build a couple intervals: The first eclipse to the second, and the second to the third. Using those, we used lunar and solar mean motion tables to figure out the solar position, as well as its change. Since the moon must be opposite the sun in ecliptic longitude for an eclipse to occur, we used the change in solar position to determine the true change in lunar position in these intervals. From that, we could compare that to the mean motion to determine how much of it must be caused by the lunar anomaly. But while we’ve determined this component, we haven’t done anything with them yet.
So in this post, we’ll start using these to answer several questions that will build out the details of the model. Specifically, we want answers to questions like what is the radius of the epicycle? Where, in relation to the ecliptic was the mean moon during these eclipses and what was the equation of anomaly? That’s a lot of information to extract so I’m going to try to break it up a bit and in this post, we’ll only tackle the radius of the epicycle1
To begin, let’s sketch out the epicyclic lunar model2 with the three eclipses drawn on it.
Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle”
In Book III of the Almagest, Ptolemy developed a methodology by which the solar position could be predicted. This method had a few key components. This included determining the mean motion, the equation of anomaly, and calculating a start date/position to which those could be applied.
Unfortunately, trying to use Ptolemy’s solution would no longer work in the present day. While the mean motion and equation of anomaly are still fine, precession of the equinoxes and other small effects over the past ~1,900 years mean too much has changed. So while we can’t use the exact results of Ptolemy’s it should be possible to recreate his methodology with a more recent starting date again allowing for reasonably accurate predictions of the solar position.
So for my entry to this year’s Virtual Kingdom Arts & Science competition, I did. Continue reading “The SCAdian Astronomical Epoch – Solar Position”
As the final post for this chapter, we’ll examine the equation of anomaly for angles measured from perigee in the epicyclic model. Again, we’ll start with a basic diagram for the epicyclic model and this time, we’ll drop a perpendicular onto $\overline{DA}$ from H to form $\overline{HK}$.
Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Epicyclic Hypothesis”
In the past couple Almagest posts, we’ve demonstrated that you can derive the equation of anomaly if you know the angle from apogee of either the mean or apparent motion using either the eccentric or epicyclic hypothesis3. Now, we’ll do the same using the angular distance from perigee again using 30º as our example. Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Eccentric Hypothesis”
Despite Ptolemy demonstrating the equivalence of the eccentric and epicyclic hypotheses many times now, he sets out to prove that we can arrive at the same equation of anomaly using the epicyclic hypothesis as we do the eccentric. As we did for the eccentric, we’ll again use the case when the mean motion is 30º from apogee. Continue reading “Almagest Book III: Equation of Anomaly from Apogee using Epicyclic Hypothesis”
Our goal in this chapter will be to determine a systematic way to the equation of anomaly for the sun at 6º intervals from apogee (and perigee) along the eccentre. In this post, we’ll explore the method for finding this value from apogee for a single value of 30º. Continue reading “Almagest Book III: Equation of Anomaly from Apogee using Eccentric Hypothesis”
In the last post, we determined the basic properties of the eccentric model to predict the motion of the sun. Now, we’ll use these properties in conjunction with the model itself to be able to predict “the greatest difference between mean and anomalistic motions” by referring back to our original model. Continue reading “Almagest Book III: On the Anomaly of the Sun – Difference Between Mean and Anomalistic Motion”