Understanding Ptolemy's Almagest (Book III)

Written in the 2nd century CE, Claudius Ptolemy's Mathematicae Syntaxis, better known as the Almagest, became the most important text in mathematical astronomy for the next 1500 years. But despite the historical significance of this text, it's nearly impossible to find detailed information on how its models were derived and functioned. So much so that when I began studying historical astronomy, I decided it was best to sit down and work through it myself. So in this series, I'll share what I've learned about one of the most important astronomical texts of all time. I'll begin by addressing things conceptually so viewers that aren't interested in the detailed mathematical workings can get an appreciation for Ptolemy's work. With that out of the way, I'll then explore select mathematical topics, taking the understanding from conceptual to functional, allowing viewers to understand how the models were practically applied. In this video, I'll be exploring Book 3, in which Ptolemy introduces the eccentric and epicyclic models to explain the inconsistent motion of bodies, and then adopts the former to develop his model of the Sun. So join me as we learn how Ptolemy defined astronomy before the Book 3 is all about the Sun. In this book, we'll be building the solar model, and that's going to require knowing some fundamental things about the Sun, starting with the length of the year. This sounds straightforward, but it's far less so than we tend to think. Typically, we think of a year as 365 days, with a leap year thrown in every fourth year, which really means that the year is 365 and a quarter days, but we don't count the fractional days until they add up to a whole. This isn't quite right, but that level of understanding is good enough to get us started. Calendars are fundamentally astronomical in nature, and thus they have an observational basis. So what does this 365 and a quarter days actually represent? It's the amount of time that it takes for the Sun to return to the same point along the ecliptic. For example, if we start counting at the vernal equinox, it is 365 and a quarter days until the Sun returns there. This is known as a tropical year. It should be noted that the Sun isn't actually at the same point in the sky when it does that. This is because of the precession of the equinoxes. This has the effect of moving the equinox up just a little, so the Sun gets to the equinox before it has actually completed that 360-degree circle with respect to the background stars, but only just. The amount the precession knocks off of the full 360-degree circle that the Sun would otherwise be traversing is only a bit over 1 one-hundredth of a degree each year. If we ignored precession and asked how long it took the Sun to return to its actual same position amidst the background stars, this is then known as a sidereal year, and as you'd expect with it only differing by that minute amount, it is only slightly longer than the tropical year. Using some modern values to give you an idea of the difference, a tropical year is 365.2422 days, or 365 days, 5 hours, 38 minutes, and 46 seconds. A sidereal year is 365.2564 days, or 365 days, 6 hours, 9 minutes, and 10 seconds. But, since we reckon our calendar by the equinox, Ptolemy decides he's going to stick with the tropical year. He also refers back to Hipparchus from the 2nd century BCE, stating that Hipparchus wasn't sure whether the length of the tropical year was consistent. Ptolemy then checks the length of the year by using records from Hipparchus on the dates of the various equinoxes and solstices. He compares this to his own observations of the date of the If you divide the number of days by the number of times the sun returned to that equinox, then you get the length of the year. What Ptolemy found when doing this is that the 365.25 days wasn't quite right. The calendar seemed to have picked up roughly an extra day between Hipparchus' time and his. Since that was a period of about 300 years, he decided that we should subtract that amount, 1 300th of a day, from the 365.25 days that had been assumed. Thus, his value for the length of the year is 365.24666 days. This is closer to the correct value, but it's still a bit high. To double check his figure, Ptolemy does this for a few pairs of equinoxes and solstices and arrives at basically the same answer each time. And to really drive the point home, he digs up some even older records from Babylonian observers and repeats the calculation with an even longer baseline. Again, he arrives at the same answer. Thus, he concludes that the length of the tropical year is constant and is 365.24666 days, which is quite accurate, only off by about 22 minutes. The length of the year is important because of an Aristotelian assumption about motion that Ptolemy adopts for his model of the solar system. That is, that all objects in the celestial realm move in perfect circles at a consistent speed. This means that we need to know the speed of the sun as it goes around its circle, known as its mean or average speed. The length of the year is that speed. It moves 360 degrees about its circle in one year. Ignore what we just said regarding a tropical year not actually being a 360 degree circle since Ptolemy is going to account for that in a different way later on. But Ptolemy was well aware that the sun didn't actually move across the sky at a consistent speed. So, he'll need to explain how the apparent speed that we perceive in the sky is created from but we'll come to that a bit later. For now, we'll just work with the sun moving about its own circle, free from any other assumptions. From high school physics, you should know that the speed multiplied by time equals distance. So, knowing the speed and how long the sun has moved, we can easily determine how far the sun has moved about its circle in a given amount of time. Except that doing multiplication in sexagesimal is annoying and we'd like to avoid having to do that as much as possible. So, to help us, Ptolemy again does the math for us, laying out how much the sun would have moved in the various intervals of time in another massive set of tables. In these, he calculates how much the sun would have moved in 18-year intervals, then single years, Egyptian months, which are always exactly 30 days, single days, and finally in single hours. He also conveniently subtracts out the full revolutions about the ecliptic, meaning the values are never greater than 360 degrees. This means that if you want to know how far the sun has moved in, let's pick a length of time, 27 years, 2 months, 4 days, and 6 hours, you would look up the value for an 18-year period, a 9-year period, a 2-month or 60-day period, a 4-day period, and 6-hour period. Then, you add up the values instead of having to do multiplication because addition is easier than multiplication. But, as we mentioned before, the sun's motion across the sky isn't constant. The values just discussed are averages. In reality, the sun speeds up and slows down throughout the year, moving faster in the winter and slower in the summer. In Ptolemaic astronomy, this deviation from the mean motion is known as anomalistic motion or simply the anomaly. Ptolemy notes that the appearance of irregularity can be explained by two hypotheses, which are the most basic and simple. In chapter 3, Ptolemy explores these two models. The first is known as the eccentric hypothesis. In this version, the center of motion of the object is simply moved off of Earth. This creates points along its path which are closer to the Earth, where the object appears to move faster, and ones where it's further away, where it appears to move more slowly, despite always moving about its own center at a consistent speed. The second is known as the epicyclic hypothesis. In this one, the object is placed on a second, smaller circle affixed to the edge of the primary circle and is allowed to rotate independently of the first. On the portion of the small circle, in which it carries the object in the same direction as the primary circle, the object appears to move faster. When being carried the opposite direction, it appears to move more slowly. When tangent to the line of sight from the observer, it's the same as the primary circle. As a bit of terminology, for the eccentric model, Ptolemy refers to the off-center circle as an eccentric. In the epicyclic hypothesis, the main circle is called a deferent, and the smaller circle, an epicycle. In both hypotheses, we get positions where the object is at its closest point to Earth, known as perigee, and is further away, apogee. Next, Ptolemy tells us that the Sun's motion is pretty simple. It only exhibits a single anomaly, speeding up and slowing down in a very regular pattern. But, he warns us, other objects, referring to the planets and the moon, are more complicated. Still, for the Sun, we can reproduce this single anomaly equally well with either the epicyclic or eccentric hypothesis, and the results will be mathematically the same, regardless of which he chooses. He demonstrates this with a mathematical proof. In chapter 4, Ptolemy is ready to get down to the business of building his solar model. But, in order to do so, he has to commit to one of the two hypotheses just described. He states that it would seem more reasonable to associate the anomaly with the eccentric hypothesis, since that one is simpler and is performed by means of one motion instead of two. To set up this model, there's a few things Ptolemy will need to know, which is how far the center of the Sun's circle is offset, and in which direction. You might expect Ptolemy to need to know the radius of the Sun's circle, but he actually ignores this and just treats it as his equivalent of a unit circle. In modern math, we define a unit circle to have a radius of one unit. Ptolemy uses a circle with a radius of 60 parts for his unit circles, where parts are arbitrary units. But since he's working in a base 60 system, this is not really all that different. His key observation here is that the time between each successive solstice and the subsequent equinox are not equal. Ptolemy notes that the amount of time from the vernal equinox to the summer solstice he observed to be 94.5 days. From the summer solstice to the autumnal equinox was 92.5 days. From the autumnal equinox to the winter solstice was a bit over 88 days. And from the winter solstice to the next vernal equinox was just over 90 days. But how did he determine this? I skipped it previously, but back in chapter one, when discussing determining the length of the year, Ptolemy mentions a bronze ring situated in the square stoa in Alexandria. This is likely a reference to an equatorial ring or an armillary sphere. At their most simple, this is simply a ring placed such that it is aligned with the celestial equator. At the moment of the equinox, the shadow from the ring towards the sun will be cast directly on the ring on the back, with no light being visible at the top or bottom of the rearward ring. If the side on which the shadow falls is replaced by a hemisphere with a scale, the angle of the shadow above or below that midline can be measured and the day on which it reaches the extremes would be the solstices, thereby allowing these dates to be known and thus revealing the anomalistic motion of the sun. Ptolemy essentially asks where he can move the sun's circle so that the amount of time it would spend in each of these quadrants of the zodiac were what he observed. Doing this, Ptolemy determines that the sun's circle is offset by about 2.5 parts out of the 60 parts of the radius of the sun's circle, and it is moved towards the point of the zodiac 24.5 degrees before the summer solstice, which is another way of saying 65.5 degrees from the spring equinox at the beginning of Aries, or 5.5 degrees into the constellation of Gemini. Before closing out this chapter, Ptolemy asks the question of where the sun's apparent position deviates the most from the position it would have if the sun's circle were not off-center. He determines that this is a bit under 92.5 degrees away from apogee. He also shows that, even if we used an epicyclic hypothesis, you get the same answer for this question, again demonstrating that both hypotheses produce effectively the same results. At this point, the mathematical model itself is complete. Ptolemy's focus for the rest of the chapter will be making it easy to use. I'll explain more as I get there, but one of the things we'll need is a table that tells us how much of an adjustment we need to make to the solar position based on its mean motion to account for the anomaly. In chapter 5, Ptolemy explains how this can be calculated, again walking us through the calculations with a sample problem. Interestingly, he also does this for the epicyclic hypothesis, really hammering home that these two models give the same results. It's important to understand here that the anomaly is dependent on how far around the circle the sun is from apogee, and is symmetric about the line of apsides, which is the line that goes through apogee and perigee. From apogee to perigee, the sun's position appears to lag the position it would have if the sun's circle were not off-center. Then, from perigee to apogee, the sun appears to be ahead of where it otherwise would. In other words, from apogee to perigee, the anomaly will be subtracted from the mean position, and from perigee to apogee, it will be additive. Regardless, chapter 5 is simply Ptolemy showing us how to calculate the correction we'll need to apply to the sun's position based on how far away from apogee it is. And just as before, having done one sample problem, Ptolemy is kind enough to do the rest of the math behind the scenes for increments of 6 degrees when nearer to the apogee, and 3 degrees when nearer to the perigee. This is because at perigee, the anomaly is larger, which makes sense since the perigee is when the sun is closer to the earth, and thus the discrepancy is more apparent. As such, Ptolemy wants to provide better granularity. This is all collected in a table, which is the entirety of chapter 6. But there's one more thing we'll need to know before we're ready to use the solar model. This is known as the epoch position. This is a date that Ptolemy chooses which we'll use as the start date to calculate from. Ptolemy chooses noon on the first day of the reign of Navanassar, which was February 26, 747 BCE. Why noon? It's because in Ptolemy's time, days were reckoned to start at noon. This date is around 900 years before Ptolemy's time and seems like an odd choice. I can't really justify why Ptolemy chose it other than that it is a sufficiently important date that he presumes most people would have known it and thus known how long it had been since then. Ptolemy uses his solar model to calculate the position of the sun about its own circle for that date and time. To do this, Ptolemy starts with an observation of the sun in his own time, mathematically determining where the sun was on its own circle in relation to apogee, and then he uses the mean motion table to rewind time until he gets to the epoch date in question. He finds that on that date, the sun was three quarters degrees into Pisces, which was 265.25 degrees before apogee. This gives us everything we need to know to use the solar model in a relatively easy manner, which Ptolemy explains in chapter 8. The first thing we need to do is determine how long it has been since the epoch date for our given date in question. With that in mind, we look up that interval of time in the table from chapter 2 and add it to the starting position calculated in chapter 7. That gives us the sun's mean position at the moment in question. Then, we need to make an adjustment for the anomaly. To do so, we figure out what the sun's position is relative to apogee, look that up in table 6, and add or subtract as necessary. And that's it! But before closing out book 3, there's still one more topic that Ptolemy wants to discuss regarding the sun. Referring back to the discussion of the definition of a year from the beginning of this book, we noted that Ptolemy used a year as a tropical year. If we break this down into individual days, then we have a tropical day, which is based on the return of the sun to the meridian each day. But because the sun has moved west to east from our point of view, it actually takes an extra 4 minutes for the sun to reach the meridian again. However, as we've discussed, the amount that the sun moves along its path day to day isn't constant. It appears to move faster in the winter and more slowly in the summer. This means that in the winter, the Earth has to rotate even more to bring the sun back to the meridian and a little less in summer. So when we talk about a day being the return of the sun to the meridian, what we're actually talking about is the return of the sun's mean position with no anomaly. In other words, what would happen if the sun's circle were not off-center? This is known as a mean solar time. Without it, the length of the day would vary throughout the year. On top of that, recall that different segments of the ecliptic do not rise in equal amounts of time as we explored in Book 2 Chapter 7 and created a table in Book 2 Chapter 8 to account for. The combination of these two effects is such that if you were to attempt to determine the time using the actual sun as people did before clocks, you would come up with a different value, as much as 17 minutes different as compared to the mean solar time. This is known as apparent solar time. The relationship between the two is referred to today as the equation of time, and it will become important in the next book when Ptolemy begins developing his lunar model in which he will make use of observations in apparent solar time, but he'll need them to be in standardized mean solar time. And with that, Ptolemy concludes Book 3. At this point, readers would have, in theory, been able to calculate the position of the sun on the ecliptic for any given date for any hour of the day. The model, of course, was not perfect. The main problem was the slight discrepancy in length of the year. While the discrepancy is small, it does add up over time. The result was that within a few hundred years, the problems with Ptolemy's model were quite apparent, as the calendar had drifted by a few days. In the next book, we'll begin examining the lunar model, which is surprisingly dependent on the solar model that we just derived. This is because Ptolemy didn't have instruments with sufficient accuracy to be able to determine the position of objects in the sky very easily. To manage this, Ptolemy will exploit the geometry of the cosmos to his benefit, using eclipses, since he knows that eclipses can only occur when the moon is either directly opposite the earth from the sun for lunar eclipses, or in front of it for solar eclipses. But we'll save that for next time, on Astronomy Before the Telescope.