Tag: eccentre

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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: 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 IV: Uneven Ratios in Epicyclic and Eccentric Model

In the last post, we explored how the eccentric and epicyclic models could produce the same result even if you didn’t have the same period of anomaly and ecliptic longitude. This was done by allowing the center of the eccentre to rotate around the Earth. In this post, we’ll explore how they can still produce the same result even

if [the members of] the ratios are unequal, and the eccentre is not the same size as the deferent…provided the ratios are similar Continue reading “Almagest Book IV: Uneven Ratios in Epicyclic and Eccentric Model”

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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”