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:
Continue reading “The Almagest – Book I: Corollaries to Ptolemy’s Theorem”
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.
Instrumentation – The Great Quadrant: Day 3
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.
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Instrumentation – The Great Quadrant: Day 2
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.
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The Almagest – Book I: Special Angle Chords
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.
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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”
Instrumentation – The Great Quadrant: Day 1
Work began today on the prototype of the great quadrant. I’d drawn up a set of plans for the quadrant arm last week (each box = 1″) so that’s where we started today (we being my father and I since he does construction and contracting professionally and has a workshop we can use).
To begin, we used a sheet of plywood to draw out a full sized set of blueprints. Mathematically, the most important parts of the entire structure is the arc which will display the scale, and that the pivot around which it rotates and from which the plum bob will hang is exactly in the center of the circle the arc lies on.
So to start, we measured the box in the upper left corner on the plywood, found the center, and screwed a long wood arm in the center. Holes were drilled in that arm at 36″ and 34″ from the center to draw the arc 2″ wide.
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The Almagest – Book I: Introduction
Ptolemy begins by drawing a distinction between practical and theoretical philosophy. The distinction is not defined here although Ptolemy gives “moral virtues” as an example of the former and “understanding of the universe” as an example of the latter. Likely, he is referring to Aristotle’s work, Metaphysics (Ptolemy almost never cited sources and Aristotle is the one name which is later cited which indicates how much Ptolemy drew from him), which defined three types of philosophy: The theoretical (knowledge for its own sake), the practical (morality), and the productive (works of utility and beauty). Continue reading “The Almagest – Book I: Introduction”