In Book III, Ptolemy is going to start using actual observations to determine how things change over long periods of time. In order to do so, we will need to explore the calendar system that is used in the Almagest, namely the Egyptian calendar. Continue reading “On Chronology and Calendars in the Almagest”
Almagest Book II: Table of Zenith Distances and Ecliptic Angles
Finally we’re at the end of Book II. In this final chapter1, Ptolemy presents a table in which a few of the calculations we’ve done in the past few chapters are repeated for all twelve of the zodiacal constellations, at different times before they reach the meridian, for seven different latitudes.
Computing this table must have been a massive undertaking. There’s close to 1,800 computed values in this table. I can’t even imagine the drudgery of having to compute these values so many times. It’s so large, I can’t even begin to reproduce it in this blog. Instead, I’ve made it into a Google Spreadsheet which can be found here.
First, let’s explore the structure.
Continue reading “Almagest Book II: Table of Zenith Distances and Ecliptic Angles”
Almagest Book II: Angle Between Ecliptic And Altitude Circle – Calculations
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, AEG2. 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”
Almagest Book II: Angle Between Ecliptic And Altitude Circle – Relationships
We’re almost to the end of book II. There’s really 2 chapters left, but the next one is almost entirely a table laying out the values we’ve been looking at here recently, so this is the last chapter in which we’ll be working out anything new.
In this chapter, we’ll tackle the angle between the ecliptic and a “circle through the poles of the horizon”. If you imagine standing outside, the zenith is directly overhead which is the pole for your local horizon. Directly opposite that, beneath you, is the nadir. If these two points are connected with a great circle, that’s the great circle we want to find the angle of with respect to the ecliptic. Because we measure upwards, from the horizon, along an arc of these great circles, to measure the altitude of a star, these are often called altitude circles.
But while we’re at it, Ptolemy promises that we’ll also determine “the size of the arc…cut off between the zenith and…the ecliptic.” In other words, because the ecliptic is tilted with respect to the horizon, the arcs between the two will be different.
To get us started, Ptolemy begins with the following diagram.
Continue reading “Almagest Book II: Angle Between Ecliptic And Altitude Circle – Relationships”
Almagest Book II: Angle Between Ecliptic And Horizon – Calculations
We’ll continue on with our goal of finding the angle the ecliptic makes with the horizon. Fortunately, this task is simplified by the symmetries we worked out in the last post meaning we’ll only need to work out the values from Aries to Libra. Unfortunately, this value will change based on latitude as well as the position on the ecliptic, but we’ll still only do this for one location. And for that location, Ptolemy again uses Rhodes.
First we’ll start with angles at the equinoxes:
Continue reading “Almagest Book II: Angle Between Ecliptic And Horizon – Calculations”
Almagest Book II: Angle Between Ecliptic And Horizon – Symmetries
In this next chapter, we’ll be looking at the angle between the ecliptic and horizon. Obviously, this one will depend on the latitude so there will be more complexity than the previous chapter. But as with before, we’ll simplify the process by starting off with some symmetries.
Continue reading “Almagest Book II: Angle Between Ecliptic And Horizon – Symmetries”
Almagest Book II: Angle Between Ecliptic and Meridian – Angle Calculations
Now that we’ve gotten a few symmetry rules developed, we can return to the main objective of calculating the angle between the ecliptic and meridian at different points along the ecliptic. Specifically, Ptolemy sets out to do this at the first point in every sign. But thanks to the previously derived symmetries, we’ll save ourselves a bit of work.
First Ptolemy does some very short proofs for these angles at the meridian and solstice, and then a slightly more complex one for the signs between them.
Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Angle Calculations”
Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries
In my last post, I mentioned that entered a paper based on the rising sign calculations presented in this post into an A&S competition. This was a very interesting piece to do because it showed how well woven the roots are, as doing so made use of almost every section we’ve gone through previously. As such, it felt like a good capstone for book II. But it doesn’t end there.
Rather, Ptolemy decides to go on for several more chapters as this book is focused on the great circles on the celestial sphere. While we’ve covered the ecliptic and celestial equator pretty extensively, we have done less with the horizon and meridian which is where Ptolemy seeks to go for the last few chapters in this book. Specifically, we’ll be covering:
- The angles between the ecliptic and meridian
- The ecliptic and horizon
- The ecliptic and an arc from horizon to the zenith (an altitude circle)
All followed by another summary chapter at various latitudes. As the title of this post may have indicated, we’ll be covering the first of these in this post3. Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries”
Almagest Book II: Applications of Rising-Time Tables
At this point we’ve spent some considerable time doing the work to develop our rising time tables. Now Ptolemy answers the question: What can we do with them?
Ptolemy provides several algorithms:
Seasonal Hours (Alternative Method)
Seasonal Hours to Equinoctial Hours
Upper Culmination (Alternative Method)
Continue reading “Almagest Book II: Applications of Rising-Time Tables”
Almagest Book II: Rising Time of the Ecliptic Tables
Over the past several posts we have worked towards an understanding of the rising times of arcs of the ecliptic at various latitudes. With this work complete, Ptolemy presents a table of these rising times in 10º intervals for select latitudes. Continue reading “Almagest Book II: Rising Time of the Ecliptic Tables”