Now that we’ve walked through how it’s calculated, here is Ptolemy’s Table of Mean Syzygies. As with previous tables, I’ve made this into a Google Sheet. I’ve also broken it into tabs to make the formatting easier since the first section has two tables and gives actual positions whereas the ones for years and months gives increments.
Almagest Book VI: On Conjunctions and Oppositions of Sun and Moon
Finally we’re on Book VI. So far in the Almagest, we’ve had a few books which laid out some preliminary tables and concepts, a book on the sun, and two on the moon1. Now it’s time to put the sun and the moon together to start looking at some of the most dramatic astronomical phenomena: eclipses. To introduce this topic, Ptolemy begins with an uncharacteristically short chapter which is a single paragraph. Continue reading “Almagest Book VI: On Conjunctions and Oppositions of Sun and Moon”
Almagest Book V: Calculating Solar Parallax Along a Great Circle Through the Zenith
Having computed the lunar parallax,
the sun’s parallax for a similar situation [i.e., as measured along an altitude circle] is immediately determined, in a simple fashion (for solar eclipses) from the number in the second column corresponding to the size of the arc from the zenith [to the sun].
Well that sure sounds easy. Let’s look at a quick example. Continue reading “Almagest Book V: Calculating Solar Parallax Along a Great Circle Through the Zenith”
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:
Continue reading “Almagest Book V: Parallaxes of the Sun and Moon”
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:
Continue reading “Almagest Book V: Size of the Sun, Moon, and Earth”
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”
Data: Summer Solstice Observation 6/20/2020
This past weekend was the summer solstice. Since the sun achieves its highest declination on this day, which is related to the obliquity of the ecliptic, this is a fundamental parameter that I try to observe the altitude on the solstice when possible. I did this back in 2018 using the solar angle dial from Book I of the Almagest. This time, I brought out the quadrant as I did for the autumnal equinox in 2018 as well.
Continue reading “Data: Summer Solstice Observation 6/20/2020”
Almagest Book III: On the Calculation of the Solar Position
This chapter is easily the shortest one in Book III. It literally consists of a paragraph (which I’ll quote in its entirely but break into two for ease of reading) that gives a very quick description of how one calculates the position of the sun at any given time from the epoch derived in the last chapter.
So whenever we want to know the sun’s position for any required time, we take the time from epoch to the given moment (reckoned with respect to the local time at Alexandria), and enter with it into the table of mean motion. We add up the degrees [and their subdivisions] corresponding to the various arguments [18-year periods, years, months, etc.], add to this the elongation [from apogee at epoch], 265;15º, subtract the complete revolutions from the total, and count the result forward from Gemini 5;30º rearwards through [i.e. in the order of] the signs. The point we come to will be the mean position of the sun.
Next we enter the same number, that is the distance from apogee to the sun’s mean position, into the table of anomaly, and take the corresponding amount in the third column. If the argument falls in the first column, that is if it is less than 180º, we subtract the [equation] from the mean position; but if the argument falls in the second column, i.e. is greater than 180º, we add it to the mean position. Thus we obtain the true or apparent [position of] the sun. Continue reading “Almagest Book III: On the Calculation of the Solar Position”
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”
Almagest Book III: Hypotheses for Circular Motion – Similarities in Time Between Apogee and Mean Between Models
In the last post, we introduced two different models that could potentially explain the anomalous motion of the sun (or other objects). Specifically, the sun sometimes appears to move faster along the ecliptic than at other times1. The first model was the eccentric model in which the observer was placed off center. The second was the epicyclic in which the object would travel around the deferent on an epicycle.
Ptolemy stated that for the simple motion of the sun, either of these models would be sufficient. However, he wanted to demonstrate a key equivalence. Specifically that
for the eccentric hypothesis always, and for the epicyclic hypothesis when the motion at apogee is in advance, the time from least speed to mean is greater than the time from mean speed to greatest; for in both hypotheses the slower motion takes place at the apogee. But [for the epicyclic hypothesis] when the sense of revolution of the body is rearwards from the apogee on the epicycle, the reverse is true: the time from greatest speed to mean is greater than the time from mean to least, since in this case the greatest speed occurs at the apogee.