Tracing the history of medieval astronomy
Now that we’ve worked out the distance to the moon at the time of the observation, we can put this information back into our lunar model diagram to work out the true scale. We’ll begin with a drawing of our lunar model at the time depicted:
Continue reading “Almagest Book V: Scale of the Lunar Model”
Now that we’ve revised our lunar model to include the position of the “mean apogee” from which we’ll measure motion around the epicycle, we need to discuss how we can use this to determine the true position of the moon.
As a general statement, we know how to do this: Take the position of the mean moon, determined by adding the motion since the beginning of the epoch, and add or subtract the equation of anomaly. The problem is that our revisions in this book mean the table for the lunar equation of anomaly we built in Book IV is no longer correct.
Instead, to determine the equation of anomaly, we’ll start with the motion around the epicycle1 and need to factor in the double elongation of the moon from the sun.
To see how to do so, let’s get started on a new diagram:
In the last post, we followed along as Ptolemy determined that the position of “apogee” used for calculating the motion around the epicycle is not the continuation of the line from the center of the ecliptic or center of the eccentre through the center of the epicycle. Rather, motion should be measured from the “mean apogee” which is defined from a third point opposite the center of the ecliptic from the center of the eccentre.
Ptolemy doesn’t give a rigorous proof for this and instead relies on proof by example. So in that last post, we went through one example, but in this post, we’ll do a second one
in order to show that we get the same result at the opposite sides of the eccentre and epicycle.
Continue reading “Almagest Book V: Second Determination of Direction of Epicycle”
When we built the first lunar model, it was done using observations only at opposition, which is to say, during eclipses which only happen during the full phase. In the last few chapters, we looked at quadrature, which is to say, during first and third quarter moon and derived a second anomaly. But what happens if we consider the moon when it’s somewhere between those phases?
Ptolemy gives the answer:
[W]e find that the moon has a peculiar characteristic associated with the direction in which the epicycle points.
So what does that mean? Continue reading “Almagest Book V: The Direction of the Moon’s Epicycle”
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 hypothesis2. 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”
In the last post, we started in on the angle between the ecliptic and an altitude circle, but only in an abstract manner, relating various things, but haven’t actually looked at how this angle would be found. Which is rather important because Ptolemy is about to put together a huge table of distances of the zenith from the ecliptic for all sorts of signs and latitudes. But to do so, we’ll need to do a bit more development of these ideas. So here’s a new diagram to get us going.
Here, we have the horizon, BED. The meridian is ABGD, and the ecliptic ZEH. We’ll put in the zenith (A) and nadir (G) and connect them with an altitude circle, AEG3. Although it’s not important at this precise moment, I’ve drawn it such that AEG has E at the point where the ecliptic is just rising. Continue reading “Almagest Book II: Angle Between Ecliptic And Altitude Circle – Calculations”