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”
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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The Almagest – Book I: Aristarchus’ Inequality and the chords of 1º & $\frac{1}{2}$º
To get at the chord length for an angle of $\frac{1}{2}$º, Ptolemy makes use of a proof from Aristarchus. It starts with this diagram:
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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Instrumentation – The Great Quadrant: Day 4
With the lattice finished the goal today was to put the edging and scale on and clean things up.
To start, we inserted some additional horizontal supports along the outer arms:
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The Almagest – Book I: Ptolemy’s Theorem
We shall next show how the remaining individual chords can be derived from the above, first of all setting out a theorem which is extremely useful for the matter at hand.
Having derived a handful of special angle-chord relationships, Ptolemy next set out to derive a more general theorem to get the rest. So Ptolemy constructs a new diagram from which to start his calculations.
