Tracing the history of medieval astronomy
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 $\angle{\theta}$.
Continue reading “Almagest Book I: Angle Between Celestial Equator and Ecliptic”
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”
Here displayed in all its glory is Ptolemy’s Table of Chords. Notes to follow.
Continue reading “Almagest Book I: Ptolemy’s Table of Chords”
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:
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:
Continue reading “The Almagest – Book I: Corollaries to Ptolemy’s Theorem”
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:
Continue reading “Instrumentation – The Great Quadrant: Day 4”
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.
Previously, we rough cut the beams for the lattice. Today’s goal was to get them notched and assembled. To do so, we used a special blade for the table saw that cut wide chunks out instead of a narrow line.
Continue reading “Instrumentation – The Great Quadrant: Day 3”
From the full scale plans we started cutting out pieces today.
First we cut out the arcs which will hold the scale. As noted previously, the idea was to have several to choose from so we could see which had the smoothest arc. If nothing else, it just reduced the amount of plywood we had laid out.
Continue reading “Instrumentation – The Great Quadrant: Day 2”
Now that we’ve introduced a bit about Ptolemy’s math we can take a look at his derivation of the chord tables. Exactly what those are we’ll get into later, once we have played around with some geometry that will hopefully clarify that issue.
To start, Ptolemy considers the following figure.
Continue reading “The Almagest – Book I: Special Angle Chords”
