How the Saros Interval got its name

Although, as I said earlier, the existence of the 18-year interval had been known in antiquity, it wasn't called a Saros until 1691, when the term was introduced by Edmond Halley (he of the comet). Halley had been regularly observing the Moon as part of his work on the problem of accurately calculating its orbit, partly as a check on the newly-emerging orbital theories of Isaac Newton but also because it was hoped that observing the position of the Moon in the sky might be a way of determining a ship's longitude while at sea - a vital requirement for accurate navigation. Halley, like other astronomers, knew of the 18-year periodicity but he was the first to recognise that it could form the basis of accurate predictions of eclipses, not just general indications.

Importantly though, he was a student of the history of astronomy (amongst many other topics) and therefore studied the works of earlier writers. In the course of these studies he came across a 10th Century Byzantine work known as the Suda, or the Lexicon of Suidas. This is part dictionary and part encyclopaedia with some 30,000 entries in (Greek) alphabetic order, drawing upon more ancient sources some of which have now been lost. The Lexicon contains an entry for a word spelled in Greek as Saroi ( Σαροι )¹, which can be translated as "A measure and a number among the Chaldeans. For 120 saros-cycles make 2222 years according to the Chaldeans' reckoning, if indeed the saros makes 222 lunar months, which are 18 years and 6 months"² - Chaldea was a country in Mesopotamia which was absorbed into Babylonia in the late 6th Century BC. Note that for the arithmetic to be correct all reckoning must be in lunar years, each of 12 lunar months (so 222 = [18 x 12] + 6, as stated) and that, even so, the figure 2222 is incorrect - it should be 2220 (120 x 222/12). Note also that the entry seems to be saying that if the saros is a certain duration then 120 of them would be 2222 years, rather than calculating the duration of a saros from the observation that 120 of them are 2222 years. This would perhaps imply that the compiler of the lexicon was aware that the Chaldeans knew of the 18-[solar]year interval and was remarking that 120 saroi totalled 2222 [lunar] years, rather than calculating the length of a saros interval from observational evidence. Finally, and most critically, although there is an association with lunar months (which is not unexpected, as ancient peoples tended to measure time by lunar months as their passage is straightforward to observe) no explicit mention is made of a connection of the word Saros with eclipses.

The earliest-known writer to mention the eclipse-cycle is the Roman naturalist Pliny the Elder, whose major work Naturalis Historiae, completed in 76 AD, purports to cover all ancient knowledge in encyclopedia-like form and is one of the largest single works to have survived from the Roman Empire. In Chapter 13 of Book II of this multi-volume treatise he addresses "The Recurrence of the Eclipses of the Sun and the Moon" and starts by saying that "It is certain, that all Eclipses in 222 Months have their Revolutions, and return to their former Points". It is likely that the Lexicon of Suidas drew inspiration from this work but convolved what Pliny said with other knowledge that not only did the Babylonians know about the 18-year period but also had a measure of time called the sar, often used to describe the ruling period of early dynasties of kings. The most usual definition of the length of a sar is 3600 years, despite the fact that this would mean that each king (who the records show ruled for typically 20 sars) had a regnal period of 72,000 years! However, no-one can be sure about the duration of a sar because there is uncertainty about how intervals of time were actually written.

The Babylonians had three terms for cycles of time - the Soss, the Ner, and the Sar. Conventionally, these are given the values 60 years, 600 years and 3600 years (based on the fact that the Babylonians used an alternate base-60 and base-10 counting system). This does result in the regnal length problem mentioned above though, which could indicate that either the conventional values may not be correct, or that the regnal lengths given in the Babylonian texts are merely to emphasise the length of the Babylonian civilisation or the mythical status of the early kings. It has even been suggested that these terms might just simply be the names of the cuneiform symbols used to represent the terms, or that the cuneiform representations of numbers have been mis-read, possibly due to there being no zero in the counting system which would make it difficult to distinguish between (in modern terms) 1, 10 & 100, or because a given symbol could represent a variety of values such as 1, 60 or 3600³. Whatever the answer, it is pretty clear that at no time in pre-history was there a term "Saros" equal to 18.5 lunar years. Furthermore, the word Saros did not even exist because the plural of Sar is Saroi - the exact word quoted in the Lexicon but then wrongly assumed to be the plural of the invented Greek word Saros.

That is not quite the end of this part of the story though as, interestingly, the total duration of the reigns of the first 10 Chaldean kings is given by the 3rd Century BC writer Berossus as 120 sars. Recognise that number? It's the first number used in the Lexicon entry. Furthermore, there is some evidence that, while they would not have known the astronomical reason for it, the Sumerians (a civilisation pre-dating even that of Chaldea) knew about some of the effects caused by the Precession of the Equinoxes and ascribed profound significance to them⁴. We now know that precession is caused by the fact that the spin axis of the Earth is not fixed in space: the direction of Celestial North actually describes a circle in the sky over a period of about 26,000 years. For observers on the Earth this slow rotation results in an apparent shift in the relative positions of the stars and the planets, with a consequent change in the apparent position of the Sun in the astrological Zodiac (as determined by noting the Zodiacal sign that rises above the horizon just before sunrise). Over a very long period the Sun seems to "move backwards" through the Zodiac. The length of time that the Sun spends in each of the 12 traditional signs is therefore about 26000/12 = 2167 solar years or 2236 lunar years of 354 days (12 lunar months). Does that number also sound almost familiar? Yes, it's nearly the second number mentioned in the Lexicon. The fact that it is "nearly" the same number isn't that important, as although ancient peoples would have known about the effects of precession it is highly unlikely they would have had an accurate knowledge of the duration of one complete "circle". The number in the Lexicon (2222 lunar years) would have been derived from observations of the movement of the Sun over a relatively limited period of time and so would have been prone to error. It is difficult to accurately compare the Lexicon value with the modern value as the precession period is not a fixed quantity because of the variation of the parameters from which it is calculated - especially the orbital period of the Moon. It is clearly pretty close though!

Is it possible then that the Lexicon compiler noticed that by arithmetically combining a mythically important duration (the regnal period of the first 10 kings - 120 sars) with the 18.5 lunar year eclipse-cycle which the Chaldeans were known to be aware of he magically came up with an astrologically important duration (the time the Sun spends in each Zodiac sign)? Could he then have put 2 and 2 together to make 5 and simply assumed this indicated that the eclipse-cycle of 222 lunar months mentioned by Pliny was the same as the period the Chaldeans seemed to be calling a Sar, and documented this appropriately in the Lexicon? We will never know, but the correspondence is tempting!

Which brings us to the third number mentioned in the Lexicon, 222, assumed to have been simply copied from Pliny's writing. The compiler of the Lexicon was clearly happy with this value, as it enabled all his arithmetic to work nicely, and he presumably was not himself an astronomer and so had no reason to query it. However, Herschel realised that the value was in fact wrong and should, by his calculation, actually be 223. He thus suggested that the early translation of Pliny from which the Lexicon compiler must have been working had a mis-reading here, with the word assumed to be "duobus" in the number "ducentis viginti duobus" (222) actually being "tribus", making the number 223. This was subsequently confirmed by experts on manuscripts of this period, cementing the value of the (mis-named) Saros as 223 lunar months or 6585.32 solar days (18.03 tropical solar years). That this error did indeed exist in early translations would seem to be born out by the fact that versions of Pliny's work with both variations of this number can be found on the Internet. A neatly printed translation by Bostock & Riley dated 1855 and held in the US National Library of Medicine⁵ has 223 while a copy in the library of the University of Califoria dated as being printed in 1847 but "on the basis of that by Dr. Philemon Holland. Ed. 1601" ⁶ i.e. before Herschel's time, has 222.

The basic story is thus complete, which is that the Chaldeans discovered the 223 lunar month period but did not call it anything in particular; Pliny then mentioned the period in his Natural History but his writing was either mis-read or mis-translated so that 223 became 222; Suidas then picked up the 222 value from Pliny and erroneously connected it with the word Sar or Saros, and finally Halley discovered the word in the Lexicon but altered its duration to 223 lunar months because he knew from his own work that this was the correct value. Phew!

The time (in solar years) the Sun actually spends in each of the 12 traditional Zodiacal signs as computed from the current (2020 AD) value of the precession period (25,769) is 2147.42 years. Comparing this with the value derived from the calculation in the Lexicon (120 x 18.03 = 2163.60 solar years) shows that the two differ by just 0.75%. This agreement could have been even closer in ancient times, as the precession period is decreasing over timescales of thousands of years as a consequence of the reducing effect of the Moon on the Earth as it slowly recedes because of tidal dissipation. The question thus arises as to whether this close apparent connection between the Saros interval and the "time in a Zodiac sign" period (and by extension the precession period) is merely a bizarre numerical coincidence or is telling us something about the underlying planetary dynamics. This was not at all clear to me though so I felt I had to take it further. I thus contacted the eminent Belgian "astro-calculator" Jean Meeus. He and I have had several discussions similar to this one so I was sure that if anyone knew the answer to my puzzle then he would. His reply was "Concerning your question about the Saros, I think it is just a coincidence. If it is not, I really have no idea how to explain it". So there you have it - if Jean Meeus thinks it's just a coincidence I'm quite happy to agree with him!

Based upon this opinion, I re-visited the topic of "mystically important time periods". The document mentioned in Ref.3. says that a duration of 25,920 years was a significant one, but this seems to be only on the basis of being generated by multiplying up other significant numbers, not as a value derived from observation (nor, of course, derived from the Saros period!). The document claims a derivation as 360 times 72, based on the assertion that 72 was the number of weeks of 5 days each in a Sumerian [solar] year, excluding the five feast days. However, other references state that Sumerian weeks were 7 days long, the same as those of most other civilisations, so this is probably unlikely.

It is true though that 360 was very significant, being the number of "calendar" days in the lunisolar year used by many ancient civilisations, which had 12 months of 30 days each - the extra 5 days were festival days. It also ties into the Chaldean counting system, which was based on both 6 and 10 (as opposed to just 10, as ours is today) which has left its legacy in our use of a circle of 360 degrees, an hour of 60 minutes, a minute of 60 seconds and, arguably, a day of 24 hours - four periods of 6 hours each, as still used as "watches" on ships.

It should perhaps be noted that the document does admit that 25,920 is only actually "special" as being twice 12,960, claimed to be the basis of ancient multiplication tables. This number is 360 x 36 of course, which does at least fit in with the "6 and 10" theory. It seems to me, therefore, that the writer of Ref.3 is straining a little too hard to find an association with a number which is close to the precession period, indicating that the coincidence with 1440 Saros periods is not only just that but also something that the Chaldeans did not recognise as important. The final thing to note in this context is that even if 25,920 was derived as 360 x 72, if we factorise the Saros period into 3 x 6 the multiplication suggested by the Lexicon can be written as (120 x 3) x (12 x 6) - which is exactly the same sum as 360 x 72. The arithmetical connection between the Saros and precession periods thus occurs purely by chance, simply because the Saros period can be conveniently factorised.

Moving on to the mathematics of the precession period, the equation which allows one to calculate the period (derived from a complex analysis of the gravitational effects of both Moon and Sun on the tilt of the Earth's axis) involves just the Moon's sidereal orbit period - the period relative to the stars. There is no mention of any of the periods used together to calculate the Saros - the new-moon to new-moon (or synodic) period; the node-crossing (or draconic) period, and the perigee-to-perigee (or anomalistic) period. The synodic period is calculated from the sidereal period and the orbital period of the Earth round the Sun of course, but although this second factor does appear in the calculation of the precession period it does not do so in a way which would generate the synodic period. The other two periods relate solely to eclipses, which would seem to have no obvious connection with precession and so it is perhaps not surprising that they do not appear in the defining equation.

Which is not to say that the interactions which cause the draconic and anomalistic periods to differ from the sidereal period don't have an effect on precession, because they do. As mentioned on the main page, the "pointing direction" of the nodal axis of the Moon's orbit moves round in a period of 18.61 years - this is another type of precession, in fact, which results in the draconic period being shorter than the sidereal period. Importantly for this discussion, the variation in the Moon's effect on the Earth which nodal precession causes does influence axial precession - but not the basic period. Rather, it is the major contributor to the phenomenon of nutation, which causes a very small "nodding" of the Earth's spin axis as it describes the circle of precession and a much larger variation in the instantaneous rate of precession (i.e. a slowing down and speeding up of the circling rate) but no change in the basic rate averaged over an entire circuit. In a similar way, the fact that the Moon's orbit is not perfectly round (which results in it having an anomalistic period) adds a further component to this nutation motion without affecting the basic period. So, given that neither of these orbital periods is involved in the calculation of the basic precession period, it is very hard to see why they should play any role in causing a mathematical or physical connection between it and the Saros period.

The above discussion presents several reasons why the close agreement between the Earth's axial precession period and 120 x 12 Saros periods is just a coincidence. There is no specific evidence that ancient peoples felt that the number generated by this sum was particularly important; the arithmetic which produces the coincidence is facilitated by the fact that the Saros period just happens to factorise conveniently; the equation for calculating the precession period does not involve any of the lunar orbital periods used to define the Saros period; such effects as are produced by the lunar orbit being neither fixed in space nor exactly round (which result in the draconic and anomalistic periods) cause only second-order corrections to precession, without changing the period itself, and - Jean Meeus says so!

I therefore feel that, to the extent that the matter can be settled by arguments of a general nature and without attempting any foray into complex mathematics, one must concede that the correspondence is indeed just a coincidence - if a very strange one!

References