Month: May 2018

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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”

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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.

Making the scale arc Continue reading “Instrumentation – The Great Quadrant: Day 1”

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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”