Category: Almagest

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Almagest Book II: Symmetry of Rising Times – Arcs of the Ecliptic Equidistant from the Same Solstice

In the last post, we proved that two arcs of the ecliptic that are equidistant from the same equinox rise in the same amount of time. In this post, we’ll prove something similar for what happens with arcs of the ecliptic equidistant from the same solstice.

It’s been awhile since I’ve been able to update the blog with anything from the Almagest. As noted in the last post from the book, this section is not one of the better written ones. Indeed, it’s taken me the better part of a month to really work out how the diagram is put together.

Ultimately the trouble stemmed from the fact that it’s not a single diagram; it’s actually two pasted together1, so instead of throwing it all at you at once like Ptolemy did, let’s work through each piece in turn before pasting it together.

To begin, let’s start with a simple celestial sphere:

Continue reading “Almagest Book II: Symmetry of Rising Times – Arcs of the Ecliptic Equidistant from the Same Solstice”

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Almagest Book II: Symmetry of Rising Times – Arcs of the Ecliptic Equidistant from the Same Equinox

In this next chapter, Ptolemy’s goal is to

show how to calculate, for each latitude, the arcs of the equator… which rise together with [given] arcs of the ecliptic.

To do this, we’ll do a bit of convenient math, breaking the full ecliptic into its traditional 12 parts. However, since these signs are not of equal size, Ptolemy takes an even 30º for each sign, beginning with Aries, then Taurus, etc…

The first goal will be to prove that

arcs of the ecliptic which are equidistant from the same equinox always rise with equal arcs of the equator.

Continue reading “Almagest Book II: Symmetry of Rising Times – Arcs of the Ecliptic Equidistant from the Same Equinox”

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Almagest Book II: Exposition of Special Characteristics, Parallel by Parallel

I’ve known this chapter was coming for awhile, and have struggled to decide how to deal with it.  The reason is that, Ptolemy goes through various climate zones as defined by Hipparchus. In particular, they are defined by their latitudes, beginning with the equator, all the way up to the north pole. For that 90º, it is broken into 39 zones. 

A mid 15th century wolrd map based on Ptolemy's Geography.
A mid 15th century world map based on Ptolemy’s Geography.

As such, this chapter has much in common with the tables we’ve seen previously. But those were straightforward to reproduce. But since the information in this chapter is not particularly tabular, it would be a hefty post to cover each one. But on the other hand, the footnotes in the translation make specific note that the information in this chapter is not referenced again.

Continue reading “Almagest Book II: Exposition of Special Characteristics, Parallel by Parallel”

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Almagest Book II: Ratio of Gnomon Equinoctial and Solsticial Shadows

First off, what’s a gnomon?

Apparently it’s the part of a sundial that casts shadows. Now you know.

To start this next chapter, Ptolemy dives straight into a new figure, but I want to take a moment to justify it first. To begin, let’s start with a simple diagram. Just a side view of the meridian, the horizon, and the north celestial pole. The zenith is also marked (A).

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Almagest Book II: For What Regions, When, and How Often the Sun Reaches the Zenith

The fourth chapter in book two is a very short one. In fact, it’s a single paragraph so I almost didn’t dedicate an entire post to it but ultimately decided to as it didn’t really fit with either the previous or next chapter.

In this, Ptolemy says,

it is a straighforward computation to determine for what regions, when, and how often the sun reaches the zenith. For it is immediately obvious that for those beneath a parallel which is farther away from the equator than the 23;51,20º (approximately), which represents the distance of the summer solstice, the sun never reaches the zenith at all, while for those beneath the parallel which is exactly that distance, it reaches the zenith once a year, precisely at the summer solstice.

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Almagest Book II: Symmetries of Arcs and Day/Night Lengths

For the past few Almagest posts, we’ve been working on the following diagram:

As a refresher, AEG is the celestial equator and BED is the horizon of an observer. Z is south. H is the point of the winter solstice as it crosses the horizon (or rises).

In the first post we determined $arc \; EH$. In the second post, we used that to determine $arc \; BZ$. In the third, we’ve determined $arc \; E \Theta$ and then came full circle and showed another way to get $arc \; EH$.

Now, Ptolemy wants to generalize. Continue reading “Almagest Book II: Symmetries of Arcs and Day/Night Lengths”

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Almagest Book II: Difference Between Length of Solstice Day vs Equinox From Latitude

The next demonstration Ptolemy does is actually a reverse of what we did in the past 2 posts. Now, given latitude Ptolemy asks what the difference in length between the longest (or shortest) day and the daylight on the equinox would be. Again, Ptolemy has actually already given us the answer for the case we’re considering of the Greek city of Rhodes which is at 36º N latitude for which the longest day (the summer solstice) is 14.5 hours. Since the length of the day on the equinox is 12 hours, the answer will be 2.5, but Ptolemy simply wants to demonstrate that this can be achieved mathematically.

Again, we’ll use Menelaus’ Theorem II.

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Almagest Book II: From Length of Longest Day Finding Elevation of the Pole

In our last post, we explored how to find the angular distance around the horizon from the ecliptic and celestial equator. In this chapter, we explore another value that can be derived from knowing the length of the longest day (i.e. on the summer solstice): the elevation (or altitude) of the celestial pole (which is also the latitude).

Once again, we’ll start with the same diagram we’ve been using for awhile now:

Continue reading “Almagest Book II: From Length of Longest Day Finding Elevation of the Pole”

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Almagest Book II: Arcs of the Horizon between Equator and Ecliptic

Let us take as a general basis for our examples the parallel circle to the equator through Rhodes, where the elevation of the pole is 36º, and the longest day $14 \frac{1}{2}$ equinoctial hours.

Immediately starting the second chapter, we’re given a lot to unpack. First off, Ptolemy chooses to work through this problem by means of an example, selecting Rhodes, a city in Greece as the exemplar. I’m assuming that the “elevation of the pole” is the latitude, as Rhodes’ latitude is 36.2º. But what of these equinoctial hours? Continue reading “Almagest Book II: Arcs of the Horizon between Equator and Ecliptic”