geometry – Following Kepler

May 9, 2021May 10, 2021

Almagest Book V: Components of Parallax – Corrections

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.

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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:

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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 10, 2021

Almagest Book V: Size of the Sun, Moon, and Earth

Now that we’ve worked out a distance to the sun and moon, as well as the angular diameter, we can put these together to determine the sizes which is what Ptolemy seeks to do in this chapter. Fortunately, we don’t need a new diagram and can simply make reference to the one we used in the last post:

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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”

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:

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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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