With the upper portion of the base ready to hold the center column, today’s focus was on getting a socket for the center column to rest in. To start, we created a set of cross beams.
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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 slowly1 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:
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Data: Summer Solstice Solar Altitude
Happy Summer Solstice everyone!
Since one of the things Ptolemy was concerned with was the angle between the ecliptic and celestial equator, I built a solar angle dial to help determine this. Given that the measurements needed to be taken on the solstices (or a solstice and equinox), I set up the instrument today to take the measurement.
Instrumentation – The Great Quadrant: Day 6
Today’s focus was on starting to build the discs that would allow the central column to rotate.
The central column is 4.75″ square (a roughly 6.75″ diagonal), so we decided to cut an 8″ disc as well as a 9″ disc which would lay over it.
Continue reading “Instrumentation – The Great Quadrant: Day 6”
Instrumentation – Ptolemy’s Solar Angle Dial
Since the solstice is only a few weeks away, I decided it might be fun to make Ptolemy’s solar angle dial we recently saw.
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Instrumentation – The Great Quadrant: Day 5
With the quadrant itself pretty much complete (aside from the scale), it was time to start working on something to mount it on. So today we started with the central column which was assembled from plywood to make a 5″ box that was 8′ tall.
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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 below1.
But before we do that, we’re going to have to lay out some lemma2 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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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”