Almagest – Page 11 – Following Kepler

January 23, 2021February 7, 2021

Almagest Book V: Lunar Distance Adjustments for Eccentre

So far in this chapter, we’ve reviewed how to calculate the lunar parallax for certain limits of the lunar position and looked at what’s necessary to estimate the effects for lunar positions away from those limits due to the epicycle. Now, we need to discuss the impact of the eccentre and how that we can estimate the effect on parallax due to it bringing the moon closer and further.

So let’s set up a generic diagram of our eccentric model, ignoring the epicycle and only concerning ourselves with the mean moon:

Continue reading “Almagest Book V: Lunar Distance Adjustments for Eccentre”

January 17, 2021February 7, 2021

Almagest Book V: Lunar Distance Adjustments for Epicycle

In the last post, we explored how to calculate parallax if the distance to an object is known and its distance from the zenith. This was done for the sun and the moon at four different distances. However, because the moon varies so widely in distance in Ptolemy’s model, we need a way to estimate between those positions and we’ll begin by looking at the effect the epicycle has on distance for various points throughout its cycle. To help us, we’ll start with a new diagram:

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January 16, 2021February 6, 2021

Almagest Book V: Parallaxes of the Sun and Moon

Now that Ptolemy has worked out his model for the sun and moon in earth radii, we can use this result to calculate parallax for any position in our model. To begin, we will calculate

the parallaxes with respect to the great circle drawn through the zenith and body.

This is essentially the reverse calculation of what we did in this post, so we can reuse the same diagram:

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January 1, 2021January 10, 2021

Almagest Book V: Distance to the Sun

Ptolemy starts the chapter reminding us that we found distance to the moon at its maximum distance was $64;10^p$. This was comprised of the distance from earth the mean distance at syzygy being $59^p$ and the radius of the epicycle being an addition $5;10$. Using that,

let us see the size of the sun’s distance which results. Continue reading “Almagest Book V: Distance to the Sun”

January 1, 2021January 1, 2021

2020 in Review

Welcome to the dumpster fire of a year in review!

Gulf Wars: CANCELLED! Lilies War: CANCELLED! Pennsic: CANCELLED!

Plague for everyone!

Well, I guess I didn’t need to spend any time working on new garb, fixing my armor, or many other things. So while this year has been dismal in many respects, it was a pretty good year for my project. Continue reading “2020 in Review”

December 29, 2020September 6, 2021

Almagest Book V: Angular Diameter of the Moon and Earth’s Shadow at Apogee During Syzygy

In order to determine the relationship between the true distance to the sun and moon, one of the key pieces of information we’ll need is their apparent angular diameters at syzygy. I’ve mentioned several time that both are around half a degree. For this to work, we’ll need to be more accurate than that, but measuring small angles like this is especially tricky. Ptolemy mentions a few ways astronomers prior to him tried to tackle the issue which include

measuring [the flow of] water¹ or by the time [the sun and moon] take to rise at the equinox.

However, he rejects these stating that they are not sufficiently accurate. Instead, he states he used a dioptra which is a surveying instrument. While he doesn’t get into the detail of its construction or use, he does give a summary of some of his key findings:

First, he states that the sun’s apparent diameter does not appear to change, but the moon’s apparent diameter does. According to Ptolemy, it has the same angular distance as the sun only when it is at its maximum distance² which he states disagrees with his predecessors who claimed that the moon’s diameter matched that of the sun only at mean distance. Ptolemy’s position is easily refuted as the existence of annular eclipses by necessitates a smaller angular diameter of the moon than the sun.

The second conclusion Ptolemy gives is that the angular diameters he has determined are “considerably smaller than those traditionally accepted.” Although he doesn’t give any of those previous values, Toomer notes that Hipparchus had a value of “a six hundred and fiftieth of its circle” which is about $0;33,14º$. Ptolemy also doesn’t give away his figure just yet either, but does indicate that it didn’t actually come from use of the dioptra he just mentioned, but “on certain lunar eclipses.”

So in this post, we’ll explore that method, but as a warning, while Toomer praises this section as “elegant and theoretically correct” this section has many problems. Continue reading “Almagest Book V: Angular Diameter of the Moon and Earth’s Shadow at Apogee During Syzygy”

December 25, 2020December 25, 2020

Almagest Book V: Scale of the Lunar Model

Now that we’ve worked out the distance to the moon at the time of the observation, we can put this information back into our lunar model diagram to work out the true scale. We’ll begin with a drawing of our lunar model at the time depicted:

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December 25, 2020

Almagest Book V: Calculation of Lunar Distance

Now that we’ve demonstrated a parallax of $1;07º$ for the moon, we can use that do determine a distance to the moon. As a forewarning, some of the math may seem suspect here, so I’ll do my best to explain it.

To being, let’s start off with a new drawing:

Continue reading “Almagest Book V: Calculation of Lunar Distance”

December 25, 2020December 25, 2020

Almagest Book V: Calculation of Lunar Parallax

So far in this book, we’ve refined our lunar model, shown how to use it to calculate the lunar position¹, 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. $1$²], $5 \frac{5}{6}$ 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 $50 \frac{11}{12}º$.

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December 24, 2020

Almagest Book V: Lunar Parallactic Observations

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:

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