It’s been awhile since I’ve posted anything on the Almagest. That’s mostly because I was at a nice stopping point and with the quadrant getting finished up and observations beginning, that stole my focus for awhile. But it’s finally time to jump into Book II. The first chapter in this book is just text. No math. So what does Ptolemy have to say? Continue reading “Almagest Book II: Introduction”
Almagest Book I: Rising Times at Sphaera Recta
We’ve finally hit the last chapter in Book I. In this chapter our objective is to “compute the size of an arc of the equator”. At first pass, that doesn’t seem to have much to do with the title. Arcs of equator vs rising times?
However, Earth is a clock, rotating once every 24 hours. Thus, if we know the length of an arc, we know something about when an object following that arc through the sky will rise and set because it’s a certain proportion of 360º per 24h. Notice that if you actually complete that division, it comes out to an even 15º/hr. That’s not a coincidence.
Fortunately, to work on this problem, we won’t even need a new diagram. We can recycle the one from last chapter. Again this time we’ll be wanting to determine all sorts of arc lengths, but we’ll start with the one where $arc \; EH = 30$º.
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Almagest Book I: Table of Inclinations
In the last post, we derived a method to calculate the angle between the celestial equator and ecliptic and generalized it to be applicable for any angle. Here, I present the table derived from those calculations.
Almagest Book I: The Arcs Between the Equator and Ecliptic
Before going on the several post detour in developing several theorems and making extra sure we understood a few, we stated the goal was to determine the length of an arc at various points between the celestial equator and ecliptic1.
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Exploring Menelaus’ Theorem
In the last post, we used several theorems we’d developed to arrive at Menelaus’ theorem. However, at the very end Ptolemy simply mentions another version of the theorem, but does not derive it. I simply took his word that it worked, but as that alternative form is used first thing in the next chapter, I want to make sure at the very use, we know how to use it, even if we don’t go through how it’s derived.
First, let’s set up a generic Menelaus configuration on a sphere which is the intersection of the arcs of four great circles:
Almagest Book I: Menelaus’ Theorem
So far in these preliminary theorems, we’ve looked at some that were based on triangles and some that were based on circles. We’ll be going one step further with this next one and work with spheres. Thus far, we’ve briefly touched on spheres in this post discussing the celestial sphere. If great circles and spherical triangles aren’t familiar to you, I suggest reading over that post.
But since this is the first time we’ve encountered math in 3D if you’ve been following along, I want to build this up more slowly2 and will be trying to add some 3D elements to make the visualization a bit easier.
[L]et us draw the following arcs of great circles on a sphere: BE and GD are drawn to meet AB and AG, and cut each other at Z. Let each of them be less than a semi-circle.
Almagest Book I: Circular Lemmas for Spherical Trigonometry
As noted in the previous Almagest post, I wanted to break this next set of lemma off because they use a new mathematical term: Crd arc.
The term is very simple. In fact, the name’s on the tin. It refers to the chord subtended by a specific arc. So let’s dive right in to the next proof and see how it’s used:
Continue reading “Almagest Book I: Circular Lemmas for Spherical Trigonometry”
Almagest Book I: Triangular Lemmas for Spherical Trigonometry
Our next task is to demonstrate the sizes of the individual arcs cut off between the equator and the ecliptic along a great circle through the poles of the equator. As a preliminary we shall set out some short and useful theorems which will enable us to carry out most demonstrations involving spherical theorems in the simplest and most methodical way possible.
In opening the next chapter in Book 1, Ptolemy again gives us a goal to work towards, namely, the length of the chord shown in solid red below3.
But before we do that, we’re going to have to lay out some lemma4 to get us there. There’s going to be several, but for this post, I’m only going to address the first two which come from triangles, whereas the remaining involve circles and a bit of new notation that I’ll want to introduce before getting into them. Continue reading “Almagest Book I: Triangular Lemmas for Spherical Trigonometry”
Almagest Book I: Angle Between Celestial Equator and Ecliptic
As Ptolemy begins building his model of the celestial sphere, there are a few fundamental values that will need to be known. One of them is the angle between the celestial equator and the ecliptic (denoted in the below diagram as $\angle{\theta}$.
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Almagest Book I: Ptolemy’s Table of Chords
Here displayed in all its glory is Ptolemy’s Table of Chords. Notes to follow.
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