Category: Almagest

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Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions

Now that we’ve revised our lunar model to include the position of the “mean apogee” from which we’ll measure motion around the epicycle, we need to discuss how we can use this to determine the true position of the moon.

As a general statement, we know how to do this: Take the position of the mean moon, determined by adding the motion since the beginning of the epoch, and add or subtract the equation of anomaly. The problem is that our revisions in this book mean the table for the lunar equation of anomaly we built in Book IV is no longer correct.

Instead, to determine the equation of anomaly, we’ll start with the motion around the epicycle1 and need to factor in the double elongation of the moon from the sun.

To see how to do so, let’s get started on a new diagram:

Continue reading “Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions”

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Almagest Book V: Second Determination of Direction of Epicycle

In the last post, we followed along as Ptolemy determined that the position of “apogee” used for calculating the motion around the epicycle is not the continuation of the line from the center of the ecliptic or center of the eccentre through the center of the epicycle. Rather, motion should be measured from the “mean apogee” which is defined from a third point opposite the center of the ecliptic from the center of the eccentre.

Ptolemy doesn’t give a rigorous proof for this and instead relies on proof by example. So in that last post, we went through one example, but in this post, we’ll do a second one

in order to show that we get the same result at the opposite sides of the eccentre and epicycle.

Continue reading “Almagest Book V: Second Determination of Direction of Epicycle”

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Almagest Book V: The Direction of the Moon’s Epicycle

When we built the first lunar model, it was done using observations only at opposition, which is to say, during eclipses which only happen during the full phase. In the last few chapters, we looked at quadrature, which is to say, during first and third quarter moon and derived a second anomaly. But what happens if we consider the moon when it’s somewhere between those phases?

Ptolemy gives the answer:

[W]e find that the moon has a peculiar characteristic associated with the direction in which the epicycle points.

So what does that mean? Continue reading “Almagest Book V: The Direction of the Moon’s Epicycle”

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Almagest Book V: Second Anomaly Eccentricity

So far, we’ve stated that the effect of the second anomaly is to magnify the first anomaly. In the last chapter, we worked out how much larger. Since this second model works by bringing the moon physically closer and further by offsetting the center of the lunar orbit with an eccentric and having that eccentre orbit the Earth, we can determine how far that center must be. In other words, the eccentricity of this second anomaly. Continue reading “Almagest Book V: Second Anomaly Eccentricity”

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Almagest Book V: Size of the Second Anomaly

So far, what we know about Ptolemy’s second anomaly is that it doesn’t have an effect at conjunction or opposition. Its at its maximum at quadrature, which is to say, a $\frac{1}{4}$ and $\frac{3}{4}$ of the way through each synodic month2. Its effect is to re-enforce whatever anomaly was present from the first anomaly. Ptolemy laid out a conceptual model in Chapter 2, but to determine the parameters of the model, we’ll need to first explore how much this second anomaly impacts things.

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Almagest Book V: Model for the Second Anomaly

In the last chapter, we introduced an instrument capable of determining the ecliptic latitude and longitude of an object so long as the position of the true sun or fixed star is known. Using this on the moon, Ptolemy found

that the distance of the moon from the sun was sometimes in agreement with that calculated from the above [Book IV] hypothesis, and sometimes in disagreement, the discrepancy being at some times small and at other times great.

How so? Ptolemy provides details. Continue reading “Almagest Book V: Model for the Second Anomaly”

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Almagest Book V: On the Use of an Astrolabe

Now that Ptolemy has described how to construct an astrolabe, he covers its usage. Specifically, Ptolemy begins when

both sun and moon could be observed above the earth at the same time.

As a brief reminder, what Ptolemy is really discussing here is what was necessary for him to make the observations that revealed the moon’s second anomaly. So what is given here is really in that context. As such, neither the sun nor moon is actually required for use of this instrument. In this case, Ptolemy is using the sun to align the instrument for use, and considering the moon to be the target, hence why the sun and moon are required. But, as we’ll see, a star can be used for alignment and there’s no reason one couldn’t be used as a target either.

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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,”3 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 translation4.

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Almagest Book IV: Hipparchus’ Two Values of Lunar Anomaly Second Triad

In the last post, we followed Ptolemy as he reviewed three eclipses Hipparchus used to determine the parameters for his model showing they were different from Ptolemy’s. In this post, we’ll repeat the procedure for the second set of three eclipses, again showing that Hipparchus’ calculations did not match those of Ptolemy.

Continue reading “Almagest Book IV: Hipparchus’ Two Values of Lunar Anomaly Second Triad”