Tag: mathematics

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Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries

In my last post, I mentioned that entered a paper based on the rising sign calculations presented in this post into an A&S competition. This was a very interesting piece to do because it showed how well woven the roots are, as doing so made use of almost every section we’ve gone through previously. As such, it felt like a good capstone for book II. But it doesn’t end there.

Rather, Ptolemy decides to go on for several more chapters as this book is focused on the great circles on the celestial sphere. While we’ve covered the ecliptic and celestial equator pretty extensively, we have done less with the horizon and meridian which is where Ptolemy seeks to go for the last few chapters in this book. Specifically, we’ll be covering:

  • The angles between the ecliptic and meridian
  • The ecliptic and horizon
  • The ecliptic and an arc from horizon to the zenith (an altitude circle)

All followed by another summary chapter at various latitudes. As the title of this post may have indicated, we’ll be covering the first of these in this post1. Continue reading “Almagest Book II: Angle Between Ecliptic and Meridian – Symmetries”

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Almagest Book II: Applications of Rising-Time Tables

At this point we’ve spent some considerable time doing the work to develop our rising time tables. Now Ptolemy answers the question: What can we do with them?

Ptolemy provides several algorithms:

Length of a Day

Seasonal Hours (Alternative Method)

Seasonal Hours to Equinoctial Hours

Horoscopes

Upper Culmination (Alternative Method)

Rising Point

Continue reading “Almagest Book II: Applications of Rising-Time Tables”

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Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs

With the previous theorem about the ascensional differences complete, it’s time to move on to determine how to figure out the rising time of arcs of the ecliptic for 10º segments at various latitudes using what Ptolemy promises to be a shortcut in the math. In the modern sense it really doesn’t seem to be much of a shortcut, but that’s because with the assistance of calculator’s, the equations we were using previously seem much less daunting. If it had to be done by hand, I’m sure it would be far more tedious.

Instead, Ptolemy reduces the number of calculations by going through the proof regarding ascensional differences as well as making use of some previously calculated values to avoid having to do other calculations.

To get started, Ptolemy revises the previous drawing, making it a bit simpler by removing the ecliptic and renaming a few of the points, as well as changing a few definitions.

Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for 10º Arcs”

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Almagest Book II: Ascensional Difference

Not content to simply figure out how long it would take a zodiacal constellation to rise at latitudes other than the equator, Ptolemy sets out to further divide the ecliptic into 10º arcs and he’s promised an easier method than what we’ve done previously. But before we can get there, Ptolemy gives a brief proof which he’ll make use of later.

To start, we begin with the vernal equinox on the horizon:

Continue reading “Almagest Book II: Ascensional Difference”

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Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for Remaining Arcs

In the last post, we explored the rising time for one zodiacal sign which would comprise 30º of the ecliptic. Since the preliminary math is now out of the way, we can quickly do 60º of the ecliptic which constitutes Aries and Taurus. From there, we’ll be able to more quickly compute the remaining constellations as well. Continue reading “Almagest Book II: Calculation of Rising Times at Sphaera Obliqua for Remaining Arcs”

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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 together2, 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: 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).

Continue reading “Almagest Book II: Ratio of Gnomon Equinoctial and Solsticial Shadows”

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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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