Tracing the history of medieval astronomy
We’re now ready to make the correction for the third opposition in this second iteration. And the good news is that this is the last time we’ll need to calculate a correction.
Ptolemy still does a third iteration which relies on these corrections, but since he won’t be doing a fourth iteration, further corrections aren’t necessary. So let’s get to it. Continue reading “Almagest Book X: Second Iteration Correction for Equant – Third Opposition”
The final few chapters of Book VI are rather odd. Now that we’ve completed the discussion of eclipse prediction, Ptolemy wants to do an “examination of the inclination which are formed at eclipses.” However, he doesn’t appear to provide any motivation for doing so. Toomer and Neugebauer both indicate that the actual reason was likely weather prediction1, but the Almagest doesn’t contain any information on how this is to be used. Neugebauer indicates that,
[T]he technical term connected with this problem is “prosneusis”…developed from the original meaning of the verb νευειν (to nod, to incline the head, etc…). According to the terminology of hellenistic astrology, the planets or moon can, e.g., give their consent by “inclining” toward a certain position, i.e., by being found in a favorable configuration.
However, aside from these astrological purposes, these last few chapters are essentially left as a free-floating bit of material. Continue reading “Almagest Book VI: Angles of Inclination at Eclipses”
We have finally reached the final in this run of eclipse timing feasibility checks. In it Ptolemy wants to demonstrate that it is impossible to have two eclipses separated by one month
even if one assumes a combination of conditions which could not in fact all hold true at the same time, but which may be lumped together in a vain attempt to provide a possibility of the event in question happening.
In short, we’re going to assume an overly ambitious “best case” scenario which can’t actually happen because some of these best case conditions contradict one another. Continue reading “Almagest Book VI: Solar Eclipses Separated by One Month”
Having established that two solar eclipses separated by the five months from the same location are just barely possible, Ptolemy then works on whether it will be possible for the same to occur over a period of seven months concluding that it is possible, provided it happen in the “shortest $7$-month interval”2. Continue reading “Almagest Book VI: Solar Eclipses Separated by Seven Months”
Now that we’ve determined how much further from the nodes parallax can cause solar eclipses to occur due to the latitudinal parallax, we need to consider the longitudinal effect. As with the last post, Ptolemy is absolutely no help in this. He simply tosses out some values with no explanation or work stating
When [the latitudinal] parallax is $0;08º$ northwards3, [the moon] has a maximum longitudinal parallax of about $0;30º$ … and when its [latitudinal] parallax is $0;58º$ southwards4, it has a maximum longitudinal parallax of about $0;15º$…
Seeking some assistance, I again refer to Neugebauer and Pappus, but immediately run into an issue. Neugebauer minces no words and states
Ptolemy is wrong in stating that $p_\lambda = 0;30º$ and $p_\lambda = 0;15º$ are the greatest longitudinal components of the parallax for locations between Meroe and the Borysthenes. It is difficult to explain how he arrived at this result.
Well… this will be interesting to try to untangle then. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Longitudinal Parallax”
At the end of the last post, we noted that Ptolemy wasn’t quite satisfied with what we did previously because we used some rather faulty assumptions.
As Ptolemy states it:
For lunar parallaxes, we considered it sufficient to use the arcs and angles formed by the great circle through the poles of the horizon [i.e., an altitude circle] at the ecliptic, instead of those at the moon’s inclined circle. For we saw that the difference which would result at syzygies in which eclipses occur is imperceptible, and to set out the latter would have been complicated to demonstrate and laborious to calculate; for the distance of the moon from the node is not fixed for a given position of the moon on the ecliptic, but undergoes multiple changes in both the amount and relative position.
The key phrase here is the “at the ecliptic, instead of the moon’s inclined circle.” This got swept under the rug in that post because Ptolemy didn’t really explain why the algorithm he gave us should work. So to understand, let’s start by taking a harder look at what’s actually going on.
Continue reading “Almagest Book V: Components of Parallax – Corrections”
So far in this book, we’ve refined our lunar model, shown how to use it to calculate the lunar position5, discussed a new instrument suitable for determining lunar parallax, as well as an example of the sort of observation necessary to make the calculation.
Now, Ptolemy walks us through an example of how to calculate the lunar distance using an example entirely unrelated to the one we saw in the last post.
In the twentieth year of Hadrian, Athyr [III] $13$ in the Egyptian calendar [135 CE, Oct. $1$6], $5 \frac{5}{6}$ equinoctial hours after noon, just before sunset, we observed the moon when it was on the meridian. The apparent distance of its center from the zenith, according to the instrument, was $50 \frac{11}{12}º$. For the distance [measured] on the thin rod was $51 \frac{7}{12}$ of the $60$ subdivisions into which the radius of revolution had been divided, and a chord of that size subtends an arc of $50 \frac{11}{12}º$.
Continue reading “Almagest Book V: Calculation of Lunar Parallax”
In the last post we followed along as Ptolemy discussed the construction and use of his parallactic instrument, which he would use to measure the lunar parallax. To do so, Ptolemy waited for the moon to
be located on the meridian, and near the solstices on the ecliptic, since at such situations, the great circle through the poles of the horizon and the center of the moon very nearly coincides with the great circle through the poles of the ecliptic, along which the moon’s latitude is taken.
That’s pretty dense, so let’s break it down with some pictures, First, let’s draw exactly what Ptolemy has described above:
Continue reading “Almagest Book V: Lunar Parallactic Observations”
Syzygy is one of those words that has popped up very little in the Almagest so every time it does, I’m always thrown off a bit7. Especially when Ptolemy is going to spend an entire chapter discussing a topic that has scarcely even come up. But here we have Ptolemy spending the entirety of chapter $10$, to demonstrate that these modifications we’ve made to the lunar model have a negligible effect because he fears readers might think it does since
the centre of the epicycle does not always … stand exactly at the apogee at those times, but can be removed from the apogee by an arc [of the eccentre] of considerable size, because location precisely at the apogee occurs at the mean syzygies, whereas the determination of true conjunction and opposition requires taking the anomalies of both luminaries into account.
Continue reading “Almagest Book V: The Difference at Syzygies – Maximum Lunar 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 month8. 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”