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”
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 month1. 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.
Continue reading “Almagest Book V: Size of the Second Anomaly”
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”
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.
Continue reading “Almagest Book V: On the Use of an Astrolabe”
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,”2 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 translation3.
Continue reading “Almagest Book V: On the Construction of an Astrolabe”
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”
Almagest Book IV: Hipparchus’ Two Values of Lunar Anomaly First Triad
For the last chapter of Book IV, Ptolemy diverges from his own narrative to address why his predecessor, Hipparchus, evidently had derived two different values for the lunar anomaly which did not match the value Ptolemy derived, or each other. Continue reading “Almagest Book IV: Hipparchus’ Two Values of Lunar Anomaly First Triad”
Almagest Book IV: Table of Lunar Equation of Anomaly (First Anomaly)
Although I don’t see any particular reason we couldn’t have done this several chapters earlier, Ptolemy has elected to hold off producing his table of Lunar Anomaly until almost the very end of Book IV4.
At the end of the previous chapter, he did toss in a few expository notes on this table noting that this will be useful in the future for “calculations concerning conjunctions and oppositions”5. To do so, Ptolemy doesn’t show his math, but notes that it was done “geometrically, in the same way as we already did for the sun.”
However, to use that method requires knowing the ratio of the radius of the deferent to that of the epicycle. Ptolemy states he takes a ratio of $60:5 \frac{1}{4}$. Using that, he calculates his table in the same manner as he did for the sun, which is in $6º$ intervals near apogee and $3º$ intervals near perigee. However, because this epicycle rotates opposite the direction of the deferent, we should subtract if the angle is in the first column, and add if it is in the second6
That being laid out, the table is below: Continue reading “Almagest Book IV: Table of Lunar Equation of Anomaly (First Anomaly)”
Almagest Book IV: Determining the Lunar Epoch for Latitude
Now that we’ve squared away the most accurate motion in latitude Ptolemy can muster, it’s time to use it to reverse calculate the lunar latitude at the beginning of the epoch, but to do so, we’ll need to calculate the precise position in latitude at some point in time. To do so, Ptolemy again turns to eclipses, with all the same considerations as the last post, except we want ones at opposite nodes instead of near the same one. Continue reading “Almagest Book IV: Determining the Lunar Epoch for Latitude”
Almagest Book IV: Correcting the Lunar Mean Motion in Latitude
Previously, we looked at how Ptolemy made corrections to the anomalistic motion for his lunar model. In this post, we’ll be doing something similar for the mean motion of lunar latitude.
Ptolemy explained that the value we originally noted was in error,
because we too adopted Hipparchus’ assumptions that [the diameter of] the moon goes approximately 650 times into its own orbit, and $2 \frac{1}{2}$ times into [the diameter of] the Earth’s shadow, when it is at mean distance in the syzygies.
In short, Hipparchus’ figures were a good starting point but now we can do better by
using more elegant methods which do not require any of the previous assumptions for the solution of the problem.
Continue reading “Almagest Book IV: Correcting the Lunar Mean Motion in Latitude”