If I were to summarize the books of the Almagest so far, I’d say that Book I is a mathematical introduction to a key theorem1 and an introduction to the celestial sphere for the simplest case of phenomenon at sphaera recta. In Book II, much of that work is extended to sphaera obliqua, but in both cases, we’ve only dealt with more or less fixed points on the celestial sphere: The celestial equator, ecliptic, and points within the zodiacal constellations based on the immovable stars.
But the ultimate goal of the Almagest and my project isn’t to study the unchanging sky; it’s to understand the changing sky: The sun, moon, and planets. Ptolemy decides to start with the position of the sun is a prerequisite to understanding the phases of the moon, and planets are more complicated with their retrograde motions. And to kick off the investigation of the motion of the sun, Ptolemy first begins by carefully defining a “year” noting
when one examines the apparent returns [of the sun] to [the same] equinox of solstice, one finds that the length of the year exceeds 365 days by less than $\frac{1}{4}$-day, but when one examines its return to the fixed stars, it is greater [than 365 $\frac{1}{4}$-days].
In other words, Ptolemy is pointing out the difference between the tropical year (in relation to the solstices and equinoxes) and the sidereal year (in relation to the distant stars) due to the precession of the equinoxes. To explain this, Ptolemy gives one of his rare citations, noting that Hipparchus believed that the sphere of the fixed stars had a slow motion of its own.
So which of these systems to use? Ptolemy states
for the purposes of the present investigation, it is our judgement that the only reference point which we must consider when examining the length of the solar year is the return of the sun to itself, that is [the period in which it travels] the circle of the ecliptic defined by its own motion.
In other words, Ptolemy is going to go with the tropical year.
Next, Ptolemy returns to Hipparchus’ investigation of the year, pointing out that Hipparchus wasn’t entirely sure that a tropical year was consistent. Ptolemy’s big goal for this chapter will be to reassure readers that it is indeed consistent. He quotes Hipparchus as having said,
it is clear that the differences in the year-length are very small indeed. However, in the case of the solstices, I have to admit that both I and Archimedes may have committed errors of up to a quarter of a day in our observations and calculations [of the time].
This indicates that Hipparchus wasn’t entirely certain about whether or not the tropical year was or was not consistent because the possible amount of discrepancy was smaller than the error in measurements.
Ptolemy does not discuss it in much detail here, and only quotes Hipparchus as having used a “bronze ring situated in Square Stoa in Alexandria.” This instrument is undoubtedly an equatorial ring. This is an impressively simple device that works by placing a ring in the plane of the celestial equator. On the equinox, the sun would precisely trace the path along that plane. The result would be that the shadow of the side of the ring closer to the sun would cast a shadow covering the opposite side. Once past the equinox, the sun would either be above that plane (in the case of being past the vernal equinox) or below it (in the case of being past the autumnal equinox), allowing for some amount of light to illuminate the interior of the opposite side. Fancier models place a semisphere on the side opposite the sun, which could be marked with angular divisions to measure the angle of the sun from the equator. But while the instrument is simple, Ptolemy notes that there are potential problems with its use:
For suppose that the instrument, due to its positioning or graduation, is out of true by as little as $\frac{1}{3600}$ of the circle through the poles of the equator: then, to correct an error of that size in declination, the sun, [when it is] near the intersection [of the ecliptic] with the equator, has to move $\frac{1}{4}º$ in longitude on the ecliptic. Thus the discrepancy comes out to about $\frac{1}{4}$ of a day.
In other words, even a tiny misalignment can cause a rather significant observational error. And that’s assuming that the instrument is carefully set up and calibrated before each use. Ptolemy notes that some, such as a set in a palaestra near his home, were intended to be permanently fixed in the plane of the equator, but
when we observe with them, the distortion in their positioning is apparent, especially that of the larger and older of the two, to such an extend that sometimes the direction of illumination of the concave surface in them shifts from one side to the other, twice on the same equinoctial day.
Here, Ptolemy claims the ring or the hemisphere on the back end were sufficiently warped that the shadows appeared to wander2. So how to know?
Hipparchus’ method 3 was to compare the times of equinoxes and solstices which he considered there to be excellent observations and compare the duration between the Kallippic and Egyptian calendars to see if they were always the same. Since the Kallippic calendar is meant to more accurately track the solstices and equinoxes and the Egyptian calendar is simply a fixed number of days, the differences over similar intervals would not be consistent if the length of the year were not consistent.
Ptolemy has followed this same method. Below, are the dates that Ptolemy listed of Hipparchus’. I have listed the results below in tabular form as opposed to the paragraphs as Ptolemy has, following Pedersen’s lead in Survey of the Almagest as I find it easier to read.
| Observation | Date According To Ptolemy4 | Julian Date According to Manitus |
| Autumnal Equinox | Kallippos III 17 Mesore 30 @ Sunset |
161 BCE, Sept 27 |
| Autumnal Equinox | Kallippos III 20 Epag. 1 @ Morning |
158 BCE, Sept 27 |
| Autumnal Equinox | Kallippos III 21 Epag 1 @ Noon |
157 BCE, Sept 27 |
| Autumnal Equinox | Kallippos III 32 Epag. 3 @ Midnight |
146 BCE, Sept 26 |
| Vernal Equinox | Kallippos III 32 Mechir 27 @ Morning |
145 BCE, March 24 |
| Autumnal Equinox | Kallippos III 33 Epag. 4 @ Morning |
145 BCE, Sept 27 |
| Autumnal Equinox | Kallippos III 36 Epag. 4 @ Evening |
142 BCE, Sept 26 |
| Vernal Equinox | Kallippos III 43 Mechir 29 @ Midnight |
134 BCE, March 23 |
| Vernal Equinox | Kallippos III 50 Pham. 1 @ Sunset |
127 BCE, March 23 |
To Hipparchus’ observations, Ptolemy adds a few of his own.
| Observation | Date According To Ptolemy | Julian Date According to Manitus |
| Autumnal Equinox | Hadrian 17 Athyr 7 @ 2h after noon |
132 CE, Sept 25 |
| Autumnal Equinox | Antonius 3 Athyr 9 @ 1h after sunrise |
139 CE, Sept 26 |
| Vernal Equinox | Antonius 3 Pachon 7 @ 1h after noon |
140 CE, March 22 |
| Summer Solstice | Antonius 3 Mesore 11 @ 2h after midnight |
140 CE, June 25 |
From these to lists of dates, Ptolemy first selects two values of the autumnal equinox: The one from the first table in 146 BCE and the one from his table in 139 CE. If one uses the Kallippic calendar, this is a difference of 285 years which is only 365 days, excluding the additional $\frac{1}{4}$ day, which means to calculate the intercalary days, we would divide the number of years by 4 since an intercalary day was inserted every 4th year. Thus, we get that there would have been $71 \frac{1}{4}$ intercalary days in addition to the $285 \cdot 365$ days in the Kallippic calendar.
So how does that compare to the Egyptian calendar?
For that, we recall that there are 365 days in the Egyptian calendar with no intercalary days. Which means the difference between the periods won’t be a fraction of a day, but some number of days. It still has the $285 \cdot 365$ days, but now, the days to return to the equinox will be from the 3rd epagomenal day to Athyr 9. Since there are 5 epagomenal days in the Egyptian calendar, this means that, from the 3rd epagomenal day at midnight, there is 6 hours until the sunrise which defines the next day and then 1 day until the end of the 5 epagomenal days. From there, we have 2 months of 30 days each (Thoth and Phaopi), plus 9 days into Athyr gets us to sunrise on the day in question. That is $70 \frac{1}{4}$ days. But Ptolemy noted the equinox at 1 hour after sunrise, so there’s an additional hour or $\frac{1}{24}$ of a day, which Ptolemy rounds off to $\frac{1}{20}$ of a day.
So by the Kallippic calendar we have $71 \frac{1}{4}$ and, in the Egyptian calendar $70 \frac{1}{4} + \frac{1}{20}$ of a day. The difference between the two is $\frac{19}{20}$ of a day out of 285 years. Which is about 1 day different in 300 years.
Ptolemy repeats this calculation for another pair of observations, this time at the vernal equinox5. Again, he finds the discrepancy to be right about 1 day in 300 years.
To really drive the point home, Ptolemy digs up some really old records from Babylonian observers:
| Observation | Date According To Ptolemy | Julian Date According to Manitus |
| Summer Solstice | Apseudes Phamenoth 21 @ Morning |
431 BCE, June 27 |
| Summer Solstice | Kallippos I 50 End of year |
279 BCE |
Again, Ptolemy does the calculations , using the solstice from 431 BCE to his solstice observation in 140 CE, and finds that it comes out to about 2 days in 600 years which reduces to 1 day in 300.
From there, he states that he
finds the same result from a number of other observation of our own, and we see that Hipparchus agrees with it on more than one occasion.
Thus, every way Ptolemy slices it, this appears to be the rate at which the calendars diverge consistently, which demonstrates that the tropical year, the return of the Sun back to the same solstice or equinox, is an astronomical constant.
However this 1 in 300 in also important in another way, because this describes how many days the Kallippic calendar is off in that interval, gaining an extra day every 300 years. And since the Kallippic calendar assumes a year is exactly $365 \frac{1}{4}$ days, this means that the true period of a year is shorter than that by $\frac{1}{300}$ of a day. In other words a year is $365 + \frac{1}{4} – \frac{1}{300}$ which comes out to 365 days, 5 hours, 55 minutes, and 12 seconds6. Expressed in sexagesimal, that’s 365;14;48.
Next, Ptolemy divides the full 360º that the sun travels in a year by the number of days to determine the number of degrees along the ecliptic it travels per day determining it to be 0;59,8,17,13,12,31º per day7.
If we then divide that by 24 to get the hourly motion along the ecliptic, we find it to be 0;2,27,50,43,3,1º per hour.
If that is then multiplied by 30, that gives the number of degrees in the Egyptian month: 29;34,8,36,36,15,30º.
Lastly, multiplying the daily value by 365, we get the number of degrees per Egyptian year as 359;45,24,45,21,8,35º8.
His last step is somewhat difficult to follow, but Ptolemy next multiplies that value by 18
since this number will produce symmetry in the layout of the tables.
Here, Ptolemy is stating that the next chapter will contain a set of tables laying out the position of the sun over various intervals in the Egyptian calendar. We’ll say more on that when we get there9, but for now, let’s explore the calculation.
When we multiply the yearly value by 18, we get 6475;37,25,36,20,34,30º. But a full circleis only 360º, so we need to subtract out intervals of 360º until we’re back in a real range which gives us 355;37,25,36,20,34,30º which is the position the sun would be along the ecliptic after 18 years.
In the next chapter, I’ll present those tables and we’ll have a bit more to say about them.
- Menelaus’ theorem.
- Or perhaps not. Toomer cites several later authors who argue about this. This includes Manitius who claims that the ring was positioned correctly but it was atmospheric refraction near dawn that caused the issue. Or Britton whose PhD thesis examining the accuracy of observations in the Almagest, who concluded that even with a perfectly functioning instrument, this is normal.
- These calculations would have been in his books, On the Length of the Year, and/or On Intercalary Months and Days, from which Ptolemy only quotes the final result.
- For an explanation of these dates, see this post.
- Specifically he used the the one in -145 BE and the one in 140 CE.
- This value is high by about 6 minutes. The true value is 365 days, 5 hours, 48 minutes, and 45 seconds. Ptolemy being a bit high is unsurprising since Ptolemy did round the $\frac{1}{24}$ of a day to $\frac{1}{20}$. This can, of course be forgiven since the inherent uncertainty in the measurements was much larger, but Pedersen does note that this will have “many consequences for the various theories of planetary motion. In particular it tends to make the mean daily motions in longitude too small.”
- This level of precision is, of course, ridiculous and the translator even notes that Ptolemy hardly takes it seriously as he uses rounded values when performing his actual calculations later.
- Keep in mind this isn’t the full 360º because the Egyptian year is slightly short of an actual tropical year being 365 days instead of ~365.25.
- The one thing I will say now since it’s part of the above quote is that the “symmetry” that Ptolemy is referring to appears to be a pattern in the last division which will be apparent when we can see the table laid out in front of us.