Tracing the history of medieval astronomy
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”
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”
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”
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.
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”
