It is a remarkable coincidence that the size and distance of the moon in our time just happens to cover the disk of the sun during an eclipse. If the moon were closer or farther away from us, the effect of solar eclipses would be quite different. Indeed, sometimes the moon is a little farther away in its elliptical orbit and the eclipse is called annular, such that the sun shows a brilliant ring during an eclipse, and the sky is not so dark. But most solar eclipses are dramatic – the disk of the moon moves gradually over the brilliant disk of the sun, as if a bite is being taken out of it. The temperature drops and the sky darkens to an eerie twilight. If you are in the right place, the moon’s shadow can be seen in the landscape, racing across the earth toward you, and quite suddenly there is darkness: all is night. Planets near the sun appear; the stars are out. A brilliant corona rings the blackened sun. If you are with other people, there will be a shared sense of stillness, even reverence for this mystery in the heavens. There is nothing quite like it. Something primitive is awakened – one feels a deep awe; everyone around you senses it. And this in an era where we understand how and why eclipses occur.
These were my thoughts on witnessing the total solar eclipse of August 21, 2017, off a rural road in eastern, Oregon. We drove from the town of Maupin in the early blue-sky morning to a place close to the centerline of totality, the center stripe, if you will, of the road the shadow would take as it whisked across our state at 3,500 kilometers per hour (about 2,200 mph). Even though I had seen a total eclipse before, this one was special because I had in tow my whole family including four eager young grandchildren. And it was interesting in another way, too – a historical way. To give you the full picture I will mention two “personal” eclipses that I did not see.
The first was on July 20, 1963, the year I graduated from high school. On that day the moon’s shadow raced from Alaska through northern Canada then down through Maine and out to sea. I desperately wanted to go to Maine to witness it (where a school friend was at summer camp there) but couldn’t. Its path – I would realize years later – would be strikingly parallel to the track of the more southerly August 21, 2017, eclipse through Oregon and America’s heartland that I saw with my family. Here is view of the track of the 1963 eclipse, cutting through Canada and Maine, side by side with track of the 2017 eclipse, shifted south [1]:
The second eclipse I missed was the one of March 7, 1970, whose path crossed Central America before speeding through northern Florida (and the town of Atlantic Beach where I lived as a boy) then skimming up the Eastern seaboard. I missed it because I was in Hawaii busy meeting the girl I’d marry later that year. I missed the eclipse, but hope to be redeemed on April 8, 2024, by the solar eclipse that begins in Mexico and arcs through the center of the United States, on a path again parallel to, but somewhat north of, the 1970 eclipse track.
I
What is the common link between these two pairs of near-parallel eclipses: between July 20, 1963, and August 21, 2017 (in Saros Series 145); and between March 7, 1970, and April 8, 2024 (in Saros Series 139)? You’ve probably already figured out that they are both 54 years and one month apart. And it turns out that that difference, called exeligmos by ancient writers, has rich historical significance going back two and a half millennia. And to understand exeligmos, we must step back and look at the more ancient roots of a shorter eclipse cycle called Saros.
Saros
If we go back in time – as far back as the first half of the seventh century bce – we discover the earliest records of eclipse observations from Mesopotamia. Letters and reports from Assyrian and Babylonian scholars to the Assyrian court make it clear that attempts even then were being made to predict eclipses before they were observed [2]. Accurate predictions gave order to their cosmos and allowed time for the priests to prepare the appropriate rituals to counter eclipse omens, or even to install a substitute king until the danger passed. Systematic records of astronomical phenomena began to be kept about a century earlier on clay tablets now known as the Astronomical Diaries. Scholars sorting through myriad cuneiform tablet fragments of Diaries learned to their surprise that the eclipse predictions were surprisingly accurate. But mathematical competence didn’t arise immediately; it was an evolving effort. By the last three centuries bce, however, Babylonian astronomers had created sophisticated mathematical algorithms for predicting portentous appearances of the moon and planets.
You may have heard of the Saros cycle, which according to Claudius Ptolemy the ancients called ‘periodic’ [3]; as we’ll see, it is the basis of the exeligmos cycle. Saros was discovered quite early in the progression of Babylonian computational achievement, probably around 500 bce. It is an eclipse recurrence cycle of about 18 years, or 6585⅓ days [4], so it is not a perfect integer cycle. The extra third of a day means that the earth has turned about eight hours or 120º west in longitude from its orientation at the previous eclipse. To illustrate, let’s look at the eclipse of March 29, 2006, the Saros eclipse immediately preceding the one due on April 8, 2024. It has the same parallel slant as the one we’ll expect in 2024, shown in the right-hand diagram above, but at a sharply different longitude.
Here, the moment of greatest eclipse on March 29, 2006, occurred over Africa at longitude 16.7º E. The moon’s shadow at greatest eclipse on April 8, 2024 will darken central Mexico at longitude 104.1º West, evidencing the westward shift in longitude over the course of the 18-year Saros cycle. The two eclipses will be 6,585 days plus 8 hours (6585.338 days) apart. That extra fraction will bring the consecutive eclipses in Saros Series 139 (encompassing both eclipses) from Africa to America [5].
That the Saros cycle is a third of a day more than an integer value implies that it may have been first discerned from analyses of lunar rather than solar eclipse records. This is because lunar eclipses can be seen from broad swaths of the Earth’s surface, and repetitions more easily noted, whereas solar eclipses have a far narrower path. For example, if you saw the solar eclipse in Babylon, you would not see it there 18 years later because of the third-of-a-day longitude shift. Just as, if you saw the March 2006 eclipse in Africa you would not necessarily intuit that it was a part of the cycle that would occur next in America. But with long records of solar and particularly lunar eclipses, you might start connecting the dots, just as the Babylonians did. From Claudius Ptolemy’s perspective, ancient records of lunar eclipses were indeed the more reliable measures of the moon’s motion, “[f]or these are the only observations which allow one to determine the lunar position precisely; all others, whether they are taken from passages [of the moon] near fixed stars, or from [sightings with] instruments, or from solar eclipses, can contain a considerable error due to parallax [6].”
Let’s examine some ancient lunar eclipse records of the type Ptolemy mentioned to confirm the Saros and show how it led to the exeligmos cycle [7]. The table below lists six lunar eclipses (in Saros series 45) dating from – 594 to – 503 [8]. The events in this series are particularly noteworthy total eclipses of the moon, for in each the moon passed remarkably close to the very center of the earth’s shadow, as indicated by the gamma column (γ) in the table. The closer γ is to zero, the more dead-on center the eclipse is [9]. The time is given in the TD (dynamical time) column of the moment of greatest eclipse. Julian date is the date in days according to the Julian Date system used by astronomers. The final column shows the separation in days from the previous eclipse. Note the tidy confirmation in the last column of the length of the Saros cycle given by this one sample of ancient eclipses.
The deviation from an integer number of 6585 days slightly exceeds a third of a day [10]. The mean of Δ days is 18 years and 11 days. You can imagine what sense of comforting regularity the discovery of Saros must have conveyed to the ancient Babylonians in an otherwise chaotic world controlled by arbitrary gods and demons.
From Saros to Exeligmos
Yet, despite the regularity of Saros, the nagging third-of-a-day excess hinted at a deeper truth. The extra fraction practically begs to be tripled. Astute reckoners, the Babylonians realized that three saroi would form an integer number of days: 19,756, which is 54 years and 32 days, the same number I alluded to above in comparing the two sets of modern eclipses. This to them must have seemed to be a divine coincidence, evidence of the great turning cosmic wheel that returns things back to where they started. (Or nearly so, because of the slight shift in latitude.) It is an astonishing match-up. This 3-saroi period came to be called exeligmos which means “turning of the wheel [11].” To see exeligmos for yourself on the chart, compare columns one, two, and three with columns four, five, and six respectively. They confirm the exeligmos cycle, 19,756 days apart: 54 years and about thirty-two days [12]. They show about 1/10th day (2.4 hours) error.
And Beyond
But alas, mathematicians, that restless bunch, are rarely satisfied with even small approximations if they can do better. And so Ptolemy, who had access to Babylonian data through Hipparchus, raised the bar even higher. Here, just after his reference to exeligmos, he cites evidence from ancient times, of a longer, even more accurate cycle:
“However, Hipparchus already proved, by calculations from observations made by the Chaldeans and in his time, that the above relationships [the exeligmos] were not accurate. For from the observations he set out he shows that the smallest constant interval defining an ecliptic period in which the number of months and the amount of [lunar] motion is always the same, is 126,007 days plus 1 equinoctial hour [13].”
That’s a cool 345 years, accurate to an hour. How true is that, really, and how did Hipparchus, and the ancient astronomers before him do that? To answer those questions we need to look at lunar motion and see it as the Babylonians did.
As Babylonian observations a mathematical competence increased, they did more than find patterns and record repetitions. In the Seleucid era they learned about certain repeating characteristics of lunar motion [14]. Most obvious from ancient times was that the moon has a fixed synodic month, which is sometimes called the “lunar month.” It is the time from new moon to new moon. The Babylonians also learned that the moon’s speed was not constant in its orbit; it exhibits motions that depart from regular circular motion – anomalies. Through careful observations they found the fastest point in its orbit – the perigee – and discovered that its perigee point regresses slowly in a direction opposite the moon’s counterclockwise motion. We moderns know that happens because the moon’s orbit is elliptical, and its apsis precesses: that is, the long axis of its elliptical orbit moves slowly clockwise. Because of this, the time it takes for the moon to go from perigee to perigee is quicker by about 2 days than the synodic month; it’s called the anomalistic month. The Babylonians also knew, crucially, that eclipses only occurred in the vicinity of the lunar nodes. The nodes also regress in the direction opposite the moon’s motion. The time for the moon to get from one node the same node is called the draconic month.
Many people get puzzled by this last one; I’ll try to clarify it.
Imagine the moon’s orbit is a circle on a piece of paper. In the middle of the paper you draw the earth. Then you fold the paper in half so that the fold goes through the picture of the earth. Flatten the paper out so you can still see the fold. That fold is where the plane of the moon’s orbit intersects the plane of the Earth’s orbit. (Recall from your geometry class that the intersection of two planes creates a line; here the line is the nodal line.) The moon’s orbit intersects it because it is at a slight 5° inclination – a tilt – with respect to the plane of the Earth’s orbit. (You have to imagine another paper slicing through the moon’s orbit at a tilt.) Put the paper on a table and some distance away place an object we’ll call the sun. Now slowly turn the circular piece of paper counterclockwise: this represents the slow rotation of the nodes. Eclipses only occur when the fold – the nodal line – is pointing toward the sun. If the fold is not pointing toward the sun, it means the moon is either above or below the Earth’s orbit and will not cast a shadow on the earth for a solar eclipse or be caught in the Earth’s shadow for a lunar eclipse. But when the fold in our paper has rotated to that sweet spot where the nodal line is pointing to the sun at the same time the moon also happens to be passing through the node during the lunar month, the moon will appear to cross the disk of the sun or the shadow of the earth, as the case may be, as it travels through space. The time of the eclipse will be heavily affected by the location of the perigee; hence, knowing where it is and how it lines up with the lunar month is also important. Here are the lengths of each of these types of months [15]:
Think of each of these periods as separate rings in a cylindrical lock. They are slightly different from one another, but with the requisite number of turns, different for each, they will line up perfectly and the key will fit: there will be an eclipse. But there is play in this lock. Aligning the synodic month with the draconic month is the most important. Yet when things are not perfect there will still be what scholars call “eclipse possibilities.” These occur primarily because there are two nodes, because partial eclipses can occur when nodes are not exactly lined up, and because eclipses can be seen in different parts of the world [16].
As you may have gleaned from our discussion above, perfect lineups of these three types of periods don’t occur often. Remarkably, for the Saros cycle the Babylonians found that 223 synodic months = 239 anomalistic months = 242 draconic months [17]. Tripling these for exeligmos means that 669 synodic months = 717 anomalistic months = 726 draconic months. This is the 19,756 days cycle (54.1 years) mentioned above.
The Babylonians seemed to have stumbled upon this shorter cycle too: 251 synodic months = 269 anomalistic months = 726 draconic months [18]. This is about 7,412 days (20.3 years), which times 17 yields 4,267 synodic months = 4,573 anomalistic months and the 126,007 days (345 year) period mentioned by Hipparchus. Ptolemy also mentions an astonishingly long period in which the synodic months line up with the nodes quite perfectly. Here, 5,458 synodic months = 5,923 draconic months for a total for each of 161,178 days, or 441.3 years! Scholars, examining and translating tablets and fiddling with number combinations have derived all sorts of other interesting combinations, some even longer [19]. It remains a fruitful field of study even after two-and-a-half millennia!
NOTES
[1] For an outstanding resource on solar eclipses with maps, see https://eclipse.gsfc.nasa.gov/SEpubs/5MCSE.html and the links provided there. [2] See generally the discussion in Steele, John M., Eclipse Prediction in Mesopotamia, Arch. Hist. Exact Sci. 54, p. 421-454 (Springer-Verlag: 2020) from which these historical notes have been drawn. [3] Claudius Ptolemy, Ptolemy’s Almagest, trans. and annotated by G.J. Toomer (Princeton: Princeton University Press, 1998) IV.2 (hereafter cited as Almagest). [4] Saros is defined as 223 times the duration of the synodic month (new moon to new moon) of 29.53059 days. It comes out to a mean value of 6585.3223 days, or 6585 days, 7 hours 43 minutes. The Babylonian value for saros in sexagesimal notation is 29;31,50,8,20 and "appears over and over again in ancient and medieval astronomy, first mentioned in our Greek sources a century before Ptolemy…" Otto Neugebauer, A History of Ancient Mathematical Astronomy, 3 vols. (Berlin: Springer, 1975), 310 (hereafter cited as HAMA). It is within a second of the modern value. This value is mentioned in Ptolemy's Almagest, IV.2.
[5] For those liking to work out the numbers, the moment of greatest eclipse on March 29, 2006, occurred at 10:12 TD (JD 2453823.9245); the upcoming eclipse of April 8, 2024, will be greatest at 18:18 TD (JD 2460409.261957). [6] Almagest, IV.1. In other words, because the moon is close to the earth, its apparent position against the background stars during a solar eclipse will appear to be different depending on where on the surface of the earth one observes it. [7] These eclipses were selected for their remarkable centeredness (low gamma) and do not purport to have been found, for example, on Babylonian tablets. Ptolemy's reports of Babylonian observations, however, go back at least this far, and even further, the first being the lunar eclipse seen in Babylon on -720 March 19/20. Almagest IV.6. [8] The data in columns 2, 3 and 5 of the table was drawn from https://eclipse.gsfc.nasa.gov/SEpubs/5MCLE.html . [9] Gamma is the “distance from the center of the shadow cone axis to the center of the Moon (in units of Earth's equatorial radii) at the instant of greatest eclipse.” See https://eclipse.gsfc.nasa.gov/SEcat5/SEcatkey.html . [10] The mean of the Δ Days in this sample is 6585.3658 days. Because of the elliptical orbit of the moon and other factors unique to the earth-moon revolutions around their barycenter, snapshots from different times will deviate from what one obtains on the assumption of uniform motion. [11] The name appears to have been coined by Geminus in his Introduction to the Phenomena. Claudius Ptolemy also refers to it in his Almagest IV.2: "In order to obtain a period with an integer number of days, they [the ‘even more ancient astronomers’] tripled the 6585⅓ days, obtaining 19756 days, which they called ‘Exeligmos’.” As we will note later, Ptolemy mentioned that Hipparchus proved that even the exeligmos the cycle is not perfect. [12] The mean of the sample is 19,756.1 days. Thus, exeligmos is not perfect because the Saros cycle is not perfect. [13] Almagest, IV.2. [14] The mechanics of the moon’s orbital motion are famously complicated. We touch upon the simplest aspects here. [15] All these periods are subject to slow variation over long time scales. These periods from NASA are as of 2000 ce. [16] The mathematics of eclipse possibilities in short intervals is beyond the scope of this piece, but there is a good deal of literature on the subject. See e.g., Steele, John M., Eclipse Prediction in Mesopotamia, Arch. Hist. Exact Sci. 54, p. 421-454 (Springer-Verlag: 2020) and the sources cited by him. [17] Almagest, IV.2; HAMA, 502 et seq. [18] Almagest, IV.2. [19] See e.g., de Jong, Teije, On the Origin of the Lunar and Solar Periods in Babylonian Lunar Theory. Studies on the Ancient Exact Sciences in Honor of Lis Brack-Bernsen. Berlin Studies of the Ancient World 44 (2017), 105 (J. Steele and M. Ossendrijvr, eds.)
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