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”
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”
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”
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.
