Tag: geometry

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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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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:

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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.

So let’s get started.

[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.

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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”

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Introduction to the Celestial Sphere & Astronomical Coordinates

The goal in the next chapter in the Almagest, Ptolemy’s goal is to is to find the angle between the celestial equator and ecliptic. These are both features on the celestial sphere which, while fundamental to astronomy, are not terms we’ve yet explored (aside from a brief mention in the first chapter of Astronomia Nova). So before continuing, we’ll explore the celestial sphere a bit. In addition, if we’re to start measuring angles on that sphere, we will need to understand the coordinate systems by which we do so. Continue reading “Introduction to the Celestial Sphere & Astronomical Coordinates”

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The Almagest – Book I: Corollaries to Ptolemy’s Theorem

If you’ve been following the Almagest posts, you’ll recall that we’ve done some work to derive the chord lengths of various angles. But Ptolemy’s goal is to derive the chord length for every angle between 0-180º in $\frac{1}{2}$º intervals. To do that, we’re going to have to develop some new tools using Ptolemy’s theorem on the angles we already know in order to add, subtract, and divide them. These new tools are referred to as corollaries since they come from applications of Ptolemy’s theorem.

The first one comes from the following diagram:

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