Tag: geometry

Posted on

Almagest Book III: Hypotheses for Circular Motion – Similarities in the Equation of the Anomaly

So far in this chapter, we’ve been looking at the two different hypotheses to explain the non-constant angular motion of objects in the sky. Ptolemy claimed that these were equivalent under certain circumstances and, in the last post, we showed how they do indeed produce the same results in the specific case of the object travelling 90º in apparent motion from apogee1 and that it always takes longer for the object to go from slowest motion to mean, than it does mean to fastest.

But that’s not really a full demonstration that they’re functionally the same. So in this post, we’ll show that their apparent angular position from the mean (known as the equation of the anomaly) is always the same, so long as there’s a few things that are consistent between models. Continue reading “Almagest Book III: Hypotheses for Circular Motion – Similarities in the Equation of the Anomaly”

Posted on

Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses

So far in Book III, we’ve been looking at the mean sun1. However, the sun’s motion along the ecliptic is not even. In other words, its speed along the ecliptic is not constant. This poses a problem because, to Ptolemy:

the rearward displacements of the planets2 with respect to the heavens are, in every case, just like the motion of the universe in advance, by nature uniform and circular. That is to say, if we imagine the bodies or their circles being carried around by straight lines, in absolutely every case the straight line in question describes equal angles at the center of its revolution in equal times.

Here, Ptolemy states that he accepts uniform circular motion as true. So the question becomes how can something moving at a constant speed appear to have not constant speed? Continue reading “Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses”

Posted on

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, AEG1. 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”

Posted on

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”

Posted on

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”

Posted on

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”

Posted on

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 post1. Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries”

Posted on

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:

Length of a Day

Seasonal Hours (Alternative Method)

Seasonal Hours to Equinoctial Hours

Horoscopes

Upper Culmination (Alternative Method)

Rising Point

Continue reading “Almagest Book II: Applications of Rising-Time Tables”

Posted on

Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs

With the previous theorem about the ascensional differences complete, it’s time to move on to determine how to figure out the rising time of arcs of the ecliptic for 10º segments at various latitudes using what Ptolemy promises to be a shortcut in the math. In the modern sense it really doesn’t seem to be much of a shortcut, but that’s because with the assistance of calculator’s, the equations we were using previously seem much less daunting. If it had to be done by hand, I’m sure it would be far more tedious.

Instead, Ptolemy reduces the number of calculations by going through the proof regarding ascensional differences as well as making use of some previously calculated values to avoid having to do other calculations.

To get started, Ptolemy revises the previous drawing, making it a bit simpler by removing the ecliptic and renaming a few of the points, as well as changing a few definitions.

Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs”