Tag: mathematics

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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”

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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:

Continue reading “The Almagest – Book I: Corollaries to Ptolemy’s Theorem”

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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.

Continue reading “The Almagest – Book I: Ptolemy’s Theorem”

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The Almagest – Book I: Sexagesimal & Ptolemy’s Math

Before continuing into the math portion of this book, a brief interlude is necessary to explore how Ptolemy does his math. Chiefly, he uses the sexagesimal system which is a base 60 (as opposed to our base 10). The reason for this is that 60 has a large number of factors, which means it’s ideal for quick math since it you can make lots of fractions out of it.

This may sound odd at first, but consider that in some respects, it’s one we already use for telling time. There are 60 minutes to an hour and 60 seconds to a minute. In fact, that’s where the word “second” for measuring time comes from as it was the second division of the whole number. Continue reading “The Almagest – Book I: Sexagesimal & Ptolemy’s Math”