Tag: ecliptic

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Almagest Book V: On the Construction of an Astrolabe

Book IV was all about setting up a preliminary lunar model with a single anomaly which Ptolemy modeled using the epicyclic model. But throughout, Ptolemy kept referencing a second anomaly he discovered, without ever saying how. In his introduction to Book V, Ptolemy finally gives the answer:

We were led to awareness of and belief in this [second anomaly] by the observations of lunar positions recorded by Hipparchus, and also by our own observations, which were made by means of an instrument which we constructed for this purpose.

That instrument was, at the time, called an “astrolabe” which simply means “for taking the [position of] stars,”1 but today we would call it an armillary sphere. Ptolemy describes how one should be constructed which is what we’ll be exploring in this post. To help us, here’s the image of one labeled from Toomer’s translation2.

Continue reading “Almagest Book V: On the Construction of an Astrolabe”

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Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon

We’re almost finished with chapter 6. All that’s left is to determine the position of the mean moon during one of the eclipses which will tell us the equation of anomaly at that point. To do so, we’ll add a few more points to the image we ended the last post with:

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon”

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Data: Summer Solstice Observation 6/20/2020

This past weekend was the summer solstice. Since the sun achieves its highest declination on this day, which is related to the obliquity of the ecliptic, this is a fundamental parameter that I try to observe the altitude on the solstice when possible. I did this back in 2018 using the solar angle dial from Book I of the Almagest. This time, I brought out the quadrant as I did for the autumnal equinox in 2018 as well.

Continue reading “Data: Summer Solstice Observation 6/20/2020”

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Almagest Book III: On the Epoch of the Sun’s Mean Motion

We’ve come a long way in this book establishing a working model for solar motion. In fact, we’ve explored two models and derived a table that shows how far the sun would be away from its mean motion based on the mean position. However, at this point, everything has been done in terms of apogee and perigee.

In this chapter, we’ll be defining the “epoch“. What that means is that Ptolemy is going to pick a point in time, and define where the sun was on that date. Then, applying the tools we have developed in this book, we’ll be able to determine where the sun is at any other given date using the epoch as the starting point. For those that like things in a bit more mathy terms, it’s the location of the sun on the eccentre at time = 0, wherein Ptolemy will decide what that date is. To get there, we’ll first establish precise point on the eccentre at a known point in time, and then use the methods from this chapter to go backwards until we get to the chosen epoch date. Continue reading “Almagest Book III: On the Epoch of the Sun’s Mean Motion”

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Almagest Book III: Equation of Anomaly from Perigee using Eccentric Hypothesis

In the past couple Almagest posts, we’ve demonstrated that you can derive the equation of anomaly if you know the angle from apogee of either the mean or apparent motion using either the eccentric or epicyclic hypothesis3. Now, we’ll do the same using the angular distance from perigee again using 30º as our example. Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Eccentric Hypothesis”

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Almagest Book III: On the Anomaly of the Sun – Basic Parameters

Now that we’ve laid out how the two hypotheses work and explored how they sometimes function similarly, it’s time to use one of them for the sun. But which one? Or do we need both?

For the sun, Ptolemy states that the sun has

a single anomaly, of such a kind that the time taken from least speed to mean shall always be greater than the time from mean speed to greatest.

In other words, the sun fits both models, but only requires one. Ptolemy chooses the eccentric model due to its simplicity.

But now it’s time to take the hypothesis from a simple toy, in which we’ve just shown the basic properties, and to start attaching hard numbers to it to make it a predictive model. To that end, Ptolemy states,

Our first task is to find the ratio of the eccentricity of the sun’s circle, that is, the ratio of which the distance between the center of the eccentre and the center of the ecliptic (located at the observer) bears to the radius of the eccentre.

Continue reading “Almagest Book III: On the Anomaly of the Sun – Basic Parameters”

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Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses

So far in Book III, we’ve been looking at the mean sun4. However, the sun’s motion along the ecliptic is not even. In other words, its speed along the ecliptic is not constant. This poses a problem because, to Ptolemy:

the rearward displacements of the planets5 with respect to the heavens are, in every case, just like the motion of the universe in advance, by nature uniform and circular. That is to say, if we imagine the bodies or their circles being carried around by straight lines, in absolutely every case the straight line in question describes equal angles at the center of its revolution in equal times.

Here, Ptolemy states that he accepts uniform circular motion as true. So the question becomes how can something moving at a constant speed appear to have not constant speed? Continue reading “Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses”

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Almagest Book III: Table of the Mean Motion of the Sun

To help us determine where the sun is on a given date, Ptolemy sets out a table to allow us to more quickly look this up instead of needing to calculate how far around the ecliptic the sun has moved. This table is very similar to another set of tables Ptolemy would produce later: The Handy Tables. However, there are some notable differences between this table and the Handy Tables. Continue reading “Almagest Book III: Table of the Mean Motion of the Sun”

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Almagest Book II: Table of Zenith Distances and Ecliptic Angles

Finally we’re at the end of Book II. In this final chapter6, Ptolemy presents a table in which a few of the calculations we’ve done in the past few chapters are repeated for all twelve of the zodiacal constellations, at different times before they reach the meridian, for seven different latitudes.

Computing this table must have been a massive undertaking. There’s close to 1,800  computed values in this table. I can’t even imagine the drudgery of having to compute these values so many times. It’s so large, I can’t even begin to reproduce it in this blog. Instead, I’ve made it into a Google Spreadsheet which can be found here.

First, let’s explore the structure.

Continue reading “Almagest Book II: Table of Zenith Distances and Ecliptic Angles”