We are so accustomed to measuring time in years, months, days, hours, minutes and seconds that it is easy to forget that this way of dividing up time and space was invented by someone, once. Why should an hour contain sixty minutes, and a minute sixty seconds? Why should a circle contain 360 degrees, with each degree again divided into sixty minutes and each minute into sixty seconds? Every time we look at a clock, read a geographical coordinate or measure an angle, we are still using the remnants of an extraordinarily ancient way of counting.

Suppose you discover that a small integer possesses remarkable internal mathematical properties and that its multiples and powers also seem to organise the motions of the heavens. The obvious conclusion in a traditional cosmology might be that the number reveals something about the order of the cosmos itself.

Its origins are obscure. Sexagesimal mathematics is particularly associated with ancient Mesopotamia, where a sophisticated base-60 positional system was used, and its descendants passed into later Greek astronomy and ultimately into our own measurement of angles and time. But was its use much older and more widespread than is commonly thought? And why sixty was chosen in the first place? How the various applications of sexagesimal reckoning developed? The ultimate origin of the system is uncertain.

One way into the problem is to forget, for a moment, the way we practise astronomy today. Ancient astronomy was above all an astronomy of cycles: seasons, years, months, etc. The Sun returns. The Moon waxes and wanes. Planets disappear into the glare of the Sun and reappear. Eclipses return in recognisable patterns. Stars rise again at particular seasons. Some of these cycles are short enough to observe within a month or a year; others take decades, centuries or, at least in theory, thousands of years. The difficulty is that these celestial clocks do not keep the same time. A solar year is about 365.2422 days, while a synodic lunar month is about 29.53059 days. The Moon's return against the stars takes about 27.32166 days, while its return to perigee takes about 27.5545. The planets introduce still more periods. None divides neatly into the others. A calendar that could keep track of several of them at once, would have to focus on moments at which the different clocks come close to meeting again.

Some of the resulting numbers are familiar. Nineteen solar years correspond very closely to 235 synodic lunar months and 254 sidereal lunar month: the relationship we call the Metonic cycle. The Saros lasts a little over eighteen years and brings together 223 synodic months and nearly 239 anomalistic months. The eight-year cycle, or octaeteris, brings the Sun and Moon into a useful approximation in a relatively short period of time, and has the additional attraction of being very close

to five synodic cycles of Venus. Ancient astronomy was full of such periods. They were ways of making several moving heavens calculable at once.

There were other cycles which are much less familiar today. One of the strangest is 600 years. What's strange about it is that it seems both foundational, the number 6 being a key part of ancient mathematics, and completely obscure, it having been forgotten for a long time. The number occurs in ancient traditions discussed by Josephus and later writers, and was sufficiently puzzling to attract the attention of the great astronomer Giovanni Domenico Cassini. Josephus had called 600 years a “great year”, and Cassini realised that it made remarkable astronomical sense, being very close to 7421 lunar revolutions. Curiously, it also seems to function as a period of 600 years of 360 days. In addition, there is also a cycle of 60 years that seems to have been complimentary to this longer cycle of 600 years. So what are these cycles all about?

Our Guides: Bailly and Le Gentil

Two figures from the eighteenth century can serve as invaluable guides to the strange world of ancient astrnomy, both French astronomers, both members of the Paris Royal Academy of Sciences: Jean-Sylvain Bailly (1736–1793) and Guillaume Le Gentil (1725–1792). They both have great stories, so before we get into the business of 600 years, here they are. You may notice they both died around the time of the French revolution, but only one was guillotined.

Le Gentil was one of the astronomers sent to the four corners of the earth, as part of a global effort to observe the 1761 transit of Venus. This was a key moment in science, and fom it is was hoped that the scale of the solar system could be determined. Le Gentil was posted to India. En route to Pondicherry, he met with one misfortune after another. Indeed, after a momentous journey, when the destination was finally reached, war having broken out between the French and the English, and the city having been recaptured by the British,, he was unable to set foot on land. After all that, he was unable to make his observations from a stable vantage point. Rather than abandon his mission, Le Gentil decided to remain in India for the eight year wait until the second transit of Venus, and devote himself to the study of Indian astronomy. Finally, on the night of the second transit of Venus, the night sky was cloudy, and he was unable to see a thing - such is life.

On his eventual return to France, having been presumed dead and stripped of his property, as all his correspondence during his absence having somehow got lost, Le Gentil published a detailed account of both his travels and his astronomical studies. This provides an invaluable account of astronomical traditions that had survived until his day in India. Curiously, many aspects of this science were no longer understood by the keepers of this knowledge who taught him, and so much of his work was abbout piecing various elements together, and working out the original system. His studies influenced Bailly greatly. 

Jean-Sylvain Bailly’s Histoire de l’astronomie ancienne remains one of the most ambitious attempts to reconstruct the inheritance of ancient astronomy. Bailly drew on Greek authors, Biblical chronologies, Arabic reports, Indian tables, Chinese records, as well as the testimony of missionaries, and travellers such as Le Gentil. His aim was not to identify the underlying numerical practices that made precise astronomy possible long before the rise of modern instruments. Like Le Gentil, he looked at allthe various elements available to him, and attempted to reconstruct the system they would have originally beloneged to.

The cycles listed in this section, from four-year corrections to nodal periods and long planetary rhythms, are precisely the kinds of cycles that appear embedded, scaled, and interrelated in the Giza scheme explored in this book. Any architecture that meaningfully encodes such cycles presupposes a mode of astronomy closer to that of Bailly’s ancients or Le Gentil’s Indian scholars than to modern observatories. This study proceeds on that basis.

The astronomical cycles used throughout this study are not speculative reconstructions, nor are they drawn from a single culture or text. They are historically attested, repeatedly observed, and widely transmitted across the ancient world. Solar years, lunar months, eclipse periods, planetary revolutions, intercalary cycles, and long-period “great years” appear independently in the astronomical traditions of the Near East, Greece, India, China, and later the Islamic world. What varies is not the phenomena themselves, but the numerical frameworks used to organise and reconcile them.

In Histoire de l’astronomie ancienne (1775), Bailly systematically reviewed ancient Greek, Egyptian, and Eastern sources, combining textual scholarship with the practical judgment of a working astronomer. In 1787, he followed this with a major study of Indian and Eastern astronomy, recognising its depth and antiquity at a time when such traditions were often dismissed. He treated ancient astronomy as a science, understood the immense sophistication, and time required to compute the known cycles, and he took seriously the idea that very ancient cultures possessed refined astronomical knowledge. Bailly notes that one Indian solar year differs from the modern value by only a few minutes. His correspondence with Voltaire shows him thinking openly about deep time, lost sciences, and the continuity of human observation.

Bailly was a brilliant scientist, who wrote about everything from Jupiter's satellites to Plato's Atlantis. He corresponded with Voltaire, received Benjamin Franklin in his home, and was a friend of Pierre-Simon Laplace. Elected as the inaugural preseident of the National Assembly in June 1789, it was he who led the famous Tennis Court proceedings, and was the first to take the oath. Among his many political triumphs was his role in passing a decree that declared Jews to be French citizens, and allieviating Paris's food shortage crisis. He became the first mayor of Paris under the Commune. But all political careers end in failure, and in such dangerous times, this all too often meant losing one's head. Deemed responsible for the massacre of the Champ de Mars in 1791, along with Lafayette, he was guillotined in 1793.

Bailly’s broader philosophical position is equally important. He insists that astronomy progresses by recognising its own limits: by knowing not absolute truth, but the boundaries within which truth must lie. Each generation refines the work of its predecessors, not by discarding them, but by tightening those bounds. The view that knowledge advances through approximation, correction, and continuity rather than rupture is especially relevant when considering monumental constructions intended to endure across centuries. Seen in this light, the numerical cycles examined in this study belong to a well-attested historical tradition. They reflect a mode of thinking that is comfortable with very long timescales, adept at switching between schematic and physical time, and capable of embedding celestial order into durable forms. Whether expressed in tables, calendars, or stone, the same astronomical grammar is at work.

How Old Were These Cycles?

Before looking more closely at the 60- and 600-year periods, it is worth asking how old they are. The difficulty is that the first surviving written reference to a cycle need not come anywhere near its invention. This is particularly true of astronomy. A period such as nineteen years can only be recognised by comparing observations made over many years; a period of sixty years exceeds much of a human lifetime; and a period of 600 years, if it really was derived from observation, necessarily implies the transmission of information between generations. By the time such a cycle appears in a text, it may already be very old.

Bailly was acutely aware of this problem. He found the same astronomical numbers turning up in traditions separated by enormous distances. The nineteen-year lunisolar cycle, for example, was known in different forms among Greeks, Chaldeans, Indians, Chinese and others. The division of the Moon's path into twenty-seven or twenty-eight parts likewise appears across Asia. Alongside these observational periods were schematic numerical systems: years of 360 days, circles of 360 degrees, divisions by 60, and larger periods built from the same numbers.

China provides a particularly interesting example. Bailly recounts traditions surrounding Huangdi, conventionally placed in the third millennium BC, according to which his minister Yu-chi identified the pole star and constructed a spherical machine representing the celestial orbs. Bailly also associates the remote Chinese past with Fuxi, whom he describes as accomplished in astronomy, knowing the motions of the celestial bodies and constructing tables of them. The remembered beginning of Chinese civilisation is already populated by astronomers, celestial models and calculations.:

More important for us is the Chinese sexagenary cycle, formed by combining the ten Heavenly Stems (tiāngān) with the twelve Earthly Branches (dìzhī). These are two repeating series of signs: one runs through ten positions and the other through twelve. They advance together, first stem with first branch, second with second, and so on, until the same combination recurs after sixty steps, sixty being the lowest common multiple of ten and twelve. The result is a sequence of sixty distinct stem-and-branch combinations. Its antiquity is not in doubt. The system is already found in the Shang oracle-bone inscriptions of the late second millennium BC, where it was used to name days; indeed, it is one of the earliest securely attested Chinese methods of recording time. The application of the same sexagenary sequence to a cycle of years is attested later, and became customary during the Han period.

Bailly regarded this Chinese cycle as part of the same sexagesimal family that he found elsewhere in ancient astronomy. Chinese tradition placed its beginnings in the age of Huangdi, the Yellow Emperor, conventionally assigned to the third millennium BC. That tradition cannot establish that somebody devised the system in precisely 2697 BC. What we can say securely is already remarkable: the ten- and twelve-part cycles were being combined into a cycle of sixty by the Shang period, more than three thousand years ago. And by the time later Chinese chronological traditions attempted to explain its beginnings, its origin had been pushed back still further, into the foundational past.

There is something particularly interesting about the arithmetic of this arrangement. The decimal and duodecimal sequences are not alternatives but are designed to operate together: because 60 is the lowest common multiple of 10 and 12, the two sequences return simultaneously to their starting positions after sixty steps. The sexagenary cycle is therefore a particularly clear example of tenfold and twelvefold structures being integrated within a single system. The twelve Earthly Branches also came to be used to divide the day into twelve periods, each equivalent to two of our modern hours, so that the same twelvefold sequence could serve as a framework for measuring daily time. Thus 12, 24 and 60 are already linked within the Chinese system, while 10 provides the second sequence required to generate the cycle of 60. This is worth bearing in mind: rather than decimal, duodecimal and sexagesimal reckoning representing competing systems, here we can actually see different numerical structures working together within a single method of organising time.

There are other ways of glimpsing the depth of a sophisticated astronomical tradition in various parts of the world. One of the most beautiful involves the Pleiades. The Pleiades are particularly useful to an ancient observer because they are bright, distinctive and easy to recognise, while their rising and setting shift slowly in relation to the seasons because of precession. They consequently became markers of the year across an extraordinary geographical range. Hesiod used their rising to announce harvest and their setting to announce ploughing. In India they became Kṛttikā, associated with the system of twenty-seven or twenty-eight lunar nakṣatras. In Mesopotamia they were MUL.MUL, “the Stars”, and appear in calendrical rules concerned with the regulation of the year. In China they formed the lunar mansion Mao. Other peoples used their heliacal appearance to mark seasons or the beginning of a year.

Bailly was fascinated by something more specific. Ancient traditions appeared to remember a time when the Pleiades occupied a significant position in relation to the equinox. Al-Bīrūnī, writing in the eleventh century, referred to books attributed to Hermes which placed the equinox among the Pleiades, and understood this as evidence of a very remote epoch. In the seventeenth century, a genius Jusuit priest, Denis Pétau, attempted a similar reconstruction from ancient descriptions of their rising and setting and arrived at dates in the third millennium BC. The precise dates depend on what phenomenon the original descriptions were referring to, but the larger point is more important: ancient astronomical traditions preserved configurations of the sky that later astronomers believed belonged to a much earlier age.

The Pleiades therefore give us a useful glimpse of what astronomical knowledge looked like before the surviving technical texts. A small cluster of stars could serve simultaneously as a seasonal marker, a lunar station and a regulator of the calendar. Its behaviour could be watched from one generation to the next, and because the sky itself changes slowly, traditions concerning it could preserve traces of much earlier observations.

This is important when we come to periods such as 60 and 600 years. These numbers belong to a very long process of watching, counting and comparing. It must have taken many many geenrations to discover the now familiar nineteen-year cycle, which is almost exactly 235 lunar months. First the phases of the Moon and the return of the seasons have to be recorded accurately; then observations made years apart have to be compared; then somebody has to recognise that the two cycles, although individually incommensurable, approach one another again after nineteen years. Longer periods require longer records, or the inheritance of records made by earlier observers.

A 600-year period raises the problem in an even more dramatic form. No individual could observe one from beginning to end. If the period was genuinely astronomical rather than simply generated mathematically, its discovery presupposes either records extending over many generations or sufficiently advanced mathematics to derive the longer period from shorter, already well-established cycles.

That is why it is misleading to speak of ancient astronomy as though we were watching a science in its infancy. By the time it becomes clearly visible to us, we find calendars being corrected by intercalation, the Moon's path divided into stations, long lunisolar cycles being used to reconcile incompatible celestial motions, and sexagesimal periods embedded in chronology. What survives looks less like a first attempt to understand the heavens than the result of a very long habit of observation.

This was essentially Bailly's point. When he compared the astronomical traditions of India, China and Chaldea, he was struck not simply by what they knew but by the strange unevenness of that knowledge. Highly accurate methods could survive alongside much cruder ones; calculations were sometimes performed without any apparent understanding of why they worked; old observations were preserved after their original purpose had been forgotten. To Bailly, these did not look like the beginnings of a science. They looked like its remnants, the débris of an older astronomical tradition whose original structure had been partly lost.

Whether Bailly reconstructed that lost world correctly is another question. But the problem he identified is still with us. The ancient sources give us cycles whose origins they often do not explain, spread across cultures which sometimes preserve strikingly similar ways of dividing and measuring celestial motion.

And amongst those numbers, one keeps returning: 60.

There are 60-year cycles, 600-year cycles and periods of 3 600 years. The circle contains 360 degrees. Time itself came to be divided sexagesimally. If these really are fragments of a common mathematical language, then perhaps the place to begin reconstructing it is with the most mysterious of these ancient periods: the Great Year of 600 years.

A Period of 600 Years

Of all the ancient periods Bailly discusses, one particularly fascinated him: this “Great Year” of 600 years. His starting point was an intriguing passage in Josephus. Writing in the first century AD, Josephus claimed that the patriarchs before the Flood had lived for such extraordinary lengths of time partly because this allowed them to perfect geometry and astronomy. Six hundred years were necessary, he explained, because it was only after the revolution of six centuries that the “Great Year” was completed. Bailly took the story seriously, not necessarily because of the fabulous lifespans, but because the number itself appeared to preserve a genuine astronomical period. He also noted that Josephus cited earlier authorities, including Manetho, Hecataeus and Berossus, and took this to mean that the tradition of a 600-year Great Year was considerably older than Josephus himself.

The astronomer Giovanni Domenico Cassini had already noticed something remarkable about it. If the Great Year was astronomical, he reasoned, it ought to bring different celestial revolutions back into agreement. And 600 years does exactly that for the Sun and Moon with surprising accuracy.

Cassini calculated that 7 421 lunar months of 29 days, 12 hours, 44 minutes and 3 seconds amounted to approximately 219,146½ days. The same interval could be expressed as 600 solar years of 365 days, 5 hours, 51 minutes and 36 seconds. Bailly was deeply impressed by this, remarking that the resulting year differed by only a few minutes from the value known in his own day.

Using modern mean values changes the figures slightly but leaves the underlying relationship intact. Six hundred tropical years contain 219 145.32 days; divided by the mean synodic month of 29.53059 days, this gives 7 420.9597 lunations. In other words, 600 solar years fall only about 1.19 days short of 7 421 complete lunar months.

The importance of the 600-year period goes even further than the correspondence between solar years and lunar phases. Using modern mean values, 600 tropical years fall within about 1.19 days of 7 421 synodic months, about 1.72 days of 8 021 sidereal months, and about 3.98 days of 7 953 anomalistic months. Three different lunar clocks, the Moon relative to the Sun, to the stars, and to its own apsidal cycle, therefore approach whole numbers within the same six-century period. The draconic month, measured relative to the lunar nodes, does not fit 600 years nearly so closely, but at ten times the period something extraordinary happens: 6 000 tropical years come within about 0.31 days of 80 532 draconic months. In that sense the 600 / 6 000-year framework begins to resemble, on a much larger scale, a combination of the principles embodied by the Metonic cycle and the Saros: not simply a return of lunar phase, but a means of bringing several different lunar motions into a common system of reckoning.

Perhaps the purpose of the long period was not to make one celestial clock return perfectly, but to provide a common grid on which several imperfectly commensurable clocks could be compared. Six may have become sacred because it worked extraordinarily well.

Six has exceptional mathematical properties even before astronomy enters the picture. It is the first perfect number:

1 + 2 + 3 = 6

and also

1 x 2 x 3 = 6

It sits at the meeting point of 2 and 3, the first even and first odd prime:

2 x 3 = 6

From it an extraordinarily productive numerical architecture can be generated:

6 x 10 = 60

6 x 60 = 360

6 x 100 = 600

60^2 = 3600

600 x 360 = 216 000 = 60^3

These beautifully structured numbers aren't merely mathematically convenient. Some of them also land unexpectedly close to real celestial recurrences. Nature gives you a messy collection of incommensurable periods — tropical year, synodic month, sidereal month, anomalistic month, draconic month — yet a number generated naturally from this six-based family, 600, comes remarkably close to reconciling several of them. What is the origin of a “sacred number”? It need not mean arbitrary numerological superstition. It could originally mean something closer to a number perceived to reveal the mathematical order underlying nature. That would make the subsequent symbolism the result of mathematical astronomy rather than its cause. And historically, of course, six really does acquire exceptional status in later mathematical and philosophical traditions. The Pythagorean tradition's interest in perfect numbers, and later writers' theological interpretations of 6 as perfect, show that this way of thinking certainly existed. Perhaps 6, or 600, is Plato's perfect number of time. What we cannot establish from the numerical evidence alone is how far back that interpretation goes.  If astronomers then discovered that periods constructed from this numerical family also brought several otherwise incommensurable celestial motions into remarkably close agreement, six may have seemed to possess something more than computational convenience. Its later symbolic or sacred status could conceivably preserve a recognition that this number appeared to belong to the architecture of the heavens themselves. Could it be the origin of the idea of dividing up circles, of time and space, by 6 and multipels of 6?

That is impressive enough. But what interested Bailly almost as much as the accuracy of the period was the question of how anyone could have discovered it.

He saw two possibilities. Either the period had emerged from observations continued over an extraordinarily long time, or it had been calculated by astronomers who already possessed sufficiently accurate values for the movements of the Sun and Moon. Bailly imagined generations of observers recording new and full moons, preserving their observations and comparing them with much older ones. After centuries, they might discover that the Moon returned not merely to approximately the same date, but almost to the same hour. Alternatively, once astronomy had become sufficiently developed, the 600-year period could have been found mathematically from shorter known periods.

Either possibility interested him, because neither makes 600 years a primitive discovery. A cycle of that length presupposes a tradition behind it.

But 600 did not stand alone. This is where the story becomes much more interesting.

Bailly repeatedly places it within a family of periods based upon 60. He found a 60-year period in Indian chronology and in the traditions of China and Babylon. He also found a period of 3 600 years in Indian astronomical calculations, and regarded it as six periods of 600 years. As he put it, the Indians did not explicitly know the ancient 600-year period, but “they make use of it without knowing it”: their 3 600-year period contained six of them.

The same sequence appeared in accounts of Chaldean astronomy. In the terminology Bailly inherited from the ancient sources, the sossos was a period of 60 years, ten sossoi formed a neros of 600 years, and six neroi formed a saros of 3,600 years. Bailly recognised the resemblance to the periods he had found in India.

This ancient use of the word saros might be a bit confusing, because it is not the Saros we mean today. We now use the name for the familiar eclipse cycle of 223 lunar months, or a little over eighteen years, thanks to Edmund Halley. Bailly and Le Gentil were discussing a much older use of the term for a period of 3 600 years. The confusion itself is revealing: ancient names, numbers and periods had survived through different authors long after their original meanings had become uncertain.

So these periods were all linked: 60 years, 600 years, 3 600 years. And this was precisely the pattern that caught Le Gentil's attention when he encountered related periods in Indian astronomy.

Le Gentil was not content simply to catalogue them. He wanted to know what they were for. At first the obvious answer was lunisolar reconciliation. Sixty years gives a rough relationship between solar years and lunar months; 600 years gives a much better one; 3 600 years extends the system still further. It's easy to imagine the larger periods developing as accumulated errors in shorter periods became apparent. But Le Gentil eventually became dissatisfied with that explanation. He realised that periods of 60, 600 and 3 600 years could not be understood simply as intervals after which new moons, full moons or eclipses returned on a fixed date. He therefore wrote:

The “other point of view” he proposed was the Moon's apogee, and therefore the anomalistic motion of the Moon.

What's the anomalistic month? The synodic month measures the return from one new Moon to the next, but it is not the Moon's only cycle. Because the lunar orbit is elliptical, the Moon also moves between perigee and apogee, completing an anomalistic revolution in about 27.55455 days. Its distance and apparent speed therefore change continually, something that matters enormously when attempting accurate calculations of its position and of eclipses.

Le Gentil had encountered an Indian lunar table based upon a period of 248 days, after which, he tells us, the Brahmins supposed the Moon to return to the same point in its cycle. The number seems rather arbitrary until it is compared with the anomalistic month. Nine anomalistic months amount to 247.99095 days, less than thirteen minutes short of 248 days.

For Le Gentil, relationships of this sort suggested that the great periods were connected with a much deeper knowledge of lunar theory. He eventually concluded:

That is quite a statement. Le Gentil was not presenting 600 years as a curious piece of ancient chronology. He thought it belonged to a system capable of tracking one of the subtler irregularities in the Moon's motion.

And this is where Bailly and Le Gentil become particularly useful guides. They were both confronted by the same problem. The periods were clearly old. They appeared in more than one ancient tradition. They seemed to belong together. Yet their original purpose was no longer obvious.

There was another aspect of the anomalistic month that began to seem important to me. It did not merely measure another kind of lunar return. It measured an irregularity. The Moon does not travel across the sky at a constant speed, because its distance from the Earth changes through its orbit. Le Gentil's 248-day table was therefore a way of modelling variation within a cycle.

The Sun presents an analogous problem. A schematic year can be divided perfectly into twelve sectors of thirty degrees, but the apparent Sun does not spend exactly the same amount of time traversing each sector. The mathematical circle is uniform; celestial motion is not. Perhaps this was precisely the point of the system: not to pretend that the heavens moved uniformly, but to provide a uniform grid against which their irregularities could be calculated.

Bailly approached the problem principally through the sexagesimal system. Sixty, he observed, has many divisors and is exceptionally convenient for calculation. He believed this explained not only the 60-year period but a much wider family of ancient practices: the division of the circle into 360 degrees, the sexagesimal subdivision of time and angle, periods of 60 days and 60 years, and larger periods constructed by continuing the same numerical progression. The fact that these practices appeared so widely convinced him that they had a common origin.

Le Gentil approached the same numbers from the other direction. He was looking at the actual machinery of astronomical calculation and asking what celestial movements these periods might encode. His interest in the Moon's apogee is particularly revealing because it moves us beyond the simple idea that ancient astronomers were merely trying to keep a lunar calendar aligned with the seasons.

Taken together, their investigations leave us with an intriguing possibility. Sixty may be the mathematical framework, while 600 is one of the places where that framework locks unusually well onto the actual sky. Six hundred is ten times 60, but it is also very close to 7 421 synodic months. The larger period of 3 600 years is 60 squared and six times 600. The numbers are mathematically related before we even ask what astronomical functions they perform.

This may also help explain why such periods could survive after their purpose had been forgotten. A useful numerical framework can be applied to more than one celestial problem. A 60-year period need not have been invented for one single phenomenon, nor must 600 years have served only as a lunisolar cycle. The same numerical architecture could accommodate the Sun and Moon, lunar anomaly, planetary movements and progressively larger cycles. Different cultures might therefore preserve different pieces of the same system.

That possibility becomes particularly interesting when we turn to the enormous periods of Indian chronology. The Yugas are built from numbers such as 432 000, 864 000, 1 296 000 and 1 728 000 years. At first sight these seem to belong to an entirely different world: cosmology rather than practical astronomy. But they are saturated with the same numerical vocabulary, multiples of 6, 12, 36, 60, 72 and 432. If the smaller periods of 60, 600 and 3,600 can be shown to have genuine astronomical functions, it becomes reasonable to ask whether the enormous numbers of the Yugas are really disconnected from them, or whether they preserve the expansion of the same system onto a much larger scale.

Before going that far, however, there is a more immediate question to answer. Why 600?

We know that the period is ancient. We know that it was remembered as a Great Year. So Cassini found an extraordinarily good lunisolar relationship within it. And Bailly placed it within a widespread sexagesimal family of 60, 600 and 3 600 years. Le Gentil suspected these same periods had another astronomical function connected with the anomalistic motion of the Moon. So rather than assuming that 600 was chosen for one reason, can we find other astronomical relationships converge upon it, and whether they begin to reveal the structure of the larger system to which Bailly and Le Gentil suspected it once belonged?

How Astronomical Cycles Work

Before trying to work out what the 600-year period was actually for, it is worth looking more closely at what an astronomical cycle does. The basic problem is very simple. The Sun and Moon do not keep time in convenient whole numbers. The synodic month, which is the interval from one new Moon to the next, is about 29.53059 days. The tropical year is about 365.2422 days. Twelve lunar months therefore give only about 354.37 days, while thirteen give about 383.90. Neither fits the solar year.

For anyone trying to construct a calendar from both Sun and Moon, this is an unavoidable problem. The two clocks are continually slipping past one another.

The solution is to stop looking for an exact relationship within a single year and ask when the two clocks will come close to meeting again. The best-known ancient example is the 19-year Metonic cycle. Nineteen tropical years contain almost exactly 235 synodic months:

19 × 365.2422 ≈ 6 939.60 days.

The difference is only about two hours. After nineteen years, therefore, the phases of the Moon return to almost the same dates in the solar year.

This is the essential principle behind the great astronomical cycles. Two movements which refuse to fit together over a short interval may come extraordinarily close to doing so over a longer one. The trick is to find the right pair of whole numbers.

But there is never just one clock in the sky. The Moon alone has several different periods, depending upon what exactly we choose to measure. Its phases repeat according to the synodic month. It returns to approximately the same place among the stars according to the sidereal month. It returns to the same node of its orbit according to the draconic month, important for eclipses. And it returns to perigee or apogee according to the anomalistic month, which affects its distance, apparent size and speed.

This is why different ancient cycles solve different problems.

The modern Saros, for example, is not simply a Sun-Moon calendar cycle. Its 223 synodic months are also very close to whole numbers of draconic and anomalistic months. The Moon therefore returns not only to almost the same phase, but close to the same node and the same part of its elliptical orbit. That is why similar eclipses recur after a Saros.

The Metonic cycle does something different. Nineteen solar years correspond closely to 235 synodic months, but also to approximately 254 sidereal months. So one cycle can express several relationships at once. This is an important point. An ancient period need not have had one—and only one—function. A particularly useful period may have survived precisely because several celestial clocks come unusually close to agreement within it.

The eight-year cycle gives an even more intuitive example. Eight tropical years are about 2,921.94 days. Ninety-nine synodic months are about 2,923.53 days: not an exact match, but close. At the same time, five synodic periods of Venus amount to roughly 2,919.6 days. After eight years, then, the Sun, Moon and Venus have all completed numbers of their own cycles that bring them back surprisingly close to their earlier relationships.

This helps explain why ancient astronomy accumulated periods of different lengths. There was no single perfect cycle because there was no single celestial clock. A period might be chosen because it reconciled Sun and Moon particularly well; another because it incorporated Venus; another because it brought the Moon back to its node or apogee; another because it fitted conveniently into a calendar. Longer periods could combine or correct shorter ones.

The Antikythera mechanism gives us a wonderful surviving example of this way of thinking. Its back dials incorporated the nineteen-year Metonic cycle and the 223-month Saros, together with longer periods such as the seventy-six-year Callippic cycle and the fifty-four-year Exeligmos. These were not simply separate facts about the heavens. They were connected mechanically through ratios of gears. One celestial movement could, quite literally, be translated into another through number.

This is perhaps the easiest way to understand ancient astronomical cycles. They are translations between clocks.

Sometimes the clock itself is artificial. The Egyptians, for example, used a civil year of exactly 365 days. Twenty-five such years contain 9,125 days, remarkably close to 309 synodic lunar months. This provided a manageable lunar cycle within the civil calendar. But the 365-day civil year itself slowly slipped against the actual solar year. If we compare it with a schematic year of 365¼ days, the difference accumulates at exactly one day every four years. After 1,461 civil years, the two counts coincide again:

1,461 × 365 = 1,460 × 365.25 = 533,265 days.

The famous Sothic cycle is therefore another example of the same principle on a much larger scale: allow two clocks to drift, calculate the rate of the drift, and find the point at which they meet again.

This also explains something that can otherwise look rather odd in ancient astronomy: the simultaneous use of several different kinds of “year”. A year might contain 365 days for civil purposes, 365¼ for one kind of long-term calculation, approximately 365.2422 days when measuring the tropical return of the seasons, or approximately 365.256 days when measuring the Earth's return relative to the stars. None of these has to mean that an astronomer was confused about the length of the year. They are different measures for different purposes.

And then there is 360.

This number requires particular attention, because it belongs to a slightly different category. A year of 360 days is obviously not an accurate measurement of the solar year. Ancient astronomers were perfectly capable of knowing this. The Egyptian calendar itself makes the point rather neatly: twelve months of thirty days produce 360, after which five additional days have to be added to complete the 365-day civil year.

So why retain 360 at all?

Because 360 is extraordinarily useful.

A circle contains 360 degrees. A schematic year of 360 units therefore allows time and angular movement to be placed on the same numerical grid: one complete revolution corresponds to 360 units, and an average movement of one unit per day corresponds to one degree around the circle. The correspondence does not have to describe the physical year exactly to be computationally useful.

This distinction is crucial. An observed astronomical period and the numerical grid used to calculate it are not necessarily the same thing.

The tropical year of 365.2422 days belongs to the sky. So does the synodic month of 29.53059 days. They are quantities we discover by observation. But 60 and 360 can function differently. They provide a framework within which awkward natural periods can be divided, compared and recombined.

That is why I don't think we should ask simply, “Why did ancient astronomers use a 360-day year when they knew the year was longer?” A more interesting question is: why was a 360-unit revolution so useful that the same numerical structure could be applied to time, angle and astronomical cycles?

Once we make that distinction, the 600-year period becomes considerably more interesting.

As a straightforward lunisolar period, it is already impressive. Six hundred tropical years contain almost exactly 7,421 synodic months:

600 × 365.2422 = 219,145.32 days,

while

7,421 × 29.53059 = 219,146.50839 days.

The difference is only about 1.19 days in six centuries.

But that calculation uses the observed tropical year of 365.2422 days. What happens if 600 is also examined within the computational framework of 360?

Then something quite different appears:

600 × 360 = 216,000.

And 216,000 is not an arbitrary large number. It is exactly:

60 × 60 × 60 = 60³.

That changes the nature of the question. Six hundred may be interesting because it produces an excellent reconciliation of the actual Sun and Moon. But within a 360-unit framework it also produces the perfect sexagesimal cube.

So perhaps we are looking at two things at once: a number belonging to the mathematical grid, and an astronomical period that fits the real sky remarkably well.

And that is where I want to look next.

Taking Apart the 600-Year Cycle

We have arrived at a rather curious position. Six hundred years seems to belong simultaneously to two different worlds.

In the real sky, 600 tropical years come remarkably close to 7 421 synodic months. But in the schematic system of 360 units, 600 years become 216 000 days, or exactly 60³. One relationship comes from the observed movements of the Sun and Moon; the other comes from the internal architecture of the sexagesimal system.

This made me wonder whether the usefulness of 600 years lay precisely in its ability to connect the two.

Le Gentil had already followed the problem some way in this direction. He knew that the 600-year period gave a good lunisolar relationship, but he was not satisfied that this explained why periods of 60, 600 and 3 600 years had been preserved. If their only purpose was to bring lunar phases back into agreement with the solar year, there were shorter and more practical ways of doing it. The nineteen-year cycle, after all, does this extremely well.

So he began looking elsewhere.

The clue came from another rather peculiar number in the Indian astronomical calculations he was studying: 248.

Le Gentil describes a table which followed the changing daily motion of the Moon through a period of 248 days, after which the calculation began again. At first sight, 248 looks nothing like the numbers we have encountered so far. It is not a particularly elegant sexagesimal number. It does not divide 360. It does not belong obviously to the family of 60, 600 and 3 600.

But it belongs extremely well to the Moon.

The anomalistic month, which is the interval in which the Moon returns to the same part of its elliptical orbit, from apogee to apogee, is approximately 27.55455 days. Nine anomalistic months give 9 × 27.55455 = 247.99095 days. That is only 0.00905 days short of 248: a difference of about thirteen minutes.

So 248 is a very different kind of number from 60 or 360. There is no obvious reason to choose it because it is mathematically convenient. Its usefulness appears to arise from the behaviour of the Moon itself.

And this is where Le Gentil's calculations become fascinating. He compared this 248-day lunar period with the apparently quite different periods of 60 and 600 years, using the schematic year of 360 days.

Sixty such years give 60 × 360 = 21 600 days.

Divide this by 248 and we obtain 87 complete periods, with 24 days left over:

21 600 = 87 × 248 + 24.

Extend the same system to 600 years and something more striking happens:

600 × 360 = 216 000 days.

This contains 870 complete periods of 248 days, with 240 days remaining:

216 000 = 870 × 248 + 240.

But 240 is only eight days short of another complete 248-day period. In other words:

871 × 248 = 216 008.

So 600 schematic years and 871 periods of 248 days differ by only eight days in 216 000.

This was enough for Le Gentil to conclude that the 600-year periods had an “evident relationship with the Moon's apogee”, which in his view showed that the ancients had made “a fairly profound study of the theory of this celestial body”.

There is an important distinction here. We are not dealing with 600 actual tropical years. We have deliberately changed systems. Six hundred tropical years amount to about 219 145 days and give the extraordinary correspondence with 7 421 synodic months. Le Gentil's calculation uses 600 × 360 = 216 000 ideal units. And that is precisely what makes it interesting.

The same number, 600, is operating on both sides of the divide. Applied to the observed tropical year, it gives an excellent approximation to a whole number of lunar phases. Applied to the schematic 360-unit revolution, it gives a very close approximation to a whole number of 248-day anomalistic periods. So perhaps the 360-year calculation should not be thought of as an inferior version of the 365.2422-year calculation at all. It is doing something different. This suggests a useful distinction. Some numbers belong to the sky; others belong to the grid.

The 29.53059-day synodic month belongs to the sky. So does the anomalistic month of approximately 27.55455 days. The 248-day period is essentially observational too: nine anomalistic months happen to fall astonishingly close to 248 days.

But 60 and 360 have another character. They are extraordinarily divisible numbers. They provide a structure within which awkward astronomical quantities can be expressed, divided and compared. And when 600 is combined with 360, the result is not merely another round number 600 × 360 = 216,000 = 60³.

We are not dealing with 600 actual tropical years. Six hundred tropical years amount to about 219 145 days and give the extraordinary correspondence with 7 421 synodic months. Le Gentil's calculation instead treats the year schematically as 360 days, giving 600 x 360 = 216 000, which can then be compared numerically with the 248-day anomalistic period.

Bailly himself provides an illuminating explanation of what such a 360-day year could mean. Discussing Indian astronomy, he insists that the Indians did not mistake 360 days for the actual length of the year. Their lunar year was divided computationally into 360 “fictitious days”, and a similar procedure was applied to the Sun: its annual revolution was represented by supposing the Sun to move through one degree per day, so that the 360 degrees of the circle became a hypothetical year of 360 days. Bailly stresses that this was a supposition convenient for calculation, from which Indian astronomers possessed methods of reduction back to the real celestial movements. The 360-day year, in other words, belonged to the mathematical model rather than being an erroneous observation of nature.

This distinction is important. If the actual tropical year of 365.2422 days is mapped continuously onto 360 equal divisions, each ideal division corresponds to approximately 1.0145617 ordinary days. But in Le Gentil's calculation the 360 divisions are being used as schematic day-counting units. It is at this computational level that 871 x 248 = 216 008, producing the remarkable difference of only eight units. We should therefore resist treating the 216 000 as though it were simply another expression for the 219,145.32 actual days in 600 tropical years. It belongs to the grid.

And this is precisely what makes 600 so interesting. In the observed heavens, 600 tropical years give an exceptionally close approximation to 7 421 synodic months. In the schematic system, 600 ideal years give 216 000 = 60 x 60 x 60 units and come within eight of 871 x 248. The same number, 600, therefore operates on both sides of the divide: once in observation and once in computation. Some numbers belong to the sky; others belong to the grid, and the astronomical system provides the means of translating between them.

Bailly himself was quite explicit that a long astronomical period could not simply be translated into a calendar by counting identical years indefinitely. The discrepancy had to be managed through intercalation. Discussing the 600-year period, he attempted to reconstruct precisely such a system:

And I would translate it:

Bailly thinks that the recurrence of the same computational procedures in China, India and Egypt points towards inherited knowledge, and proposes and intercalation system. An ordinary four-year intercalation would insert 150 days, but only 146 are needed, so he proposes suppressing four of them, one every150 years

What, then, is the 60-year period?

This also changes the way we can approach the shorter period of 60 years.

If 600 were simply ten repetitions of a 60-year lunisolar cycle, we might expect ten times the shorter lunar count to produce the longer one. It does not quite do so.

Sixty tropical years contain about 742.096 lunations. If we approximate that as 742 lunar months and repeat the block ten times, we obtain only 7 420 months. But the 600-year approximation requires 7 421. That extra month is revealing.

The longer period is not simply ten identical copies of the shorter one. It absorbs the accumulated error of the shorter approximation. Ten 60-year blocks produce a discrepancy large enough that an additional lunation enters the count: 10 × 742 + 1 = 7 421.

This is exactly the sort of thing we should expect in a system built from approximate astronomical cycles. A smaller block provides a convenient working unit; a larger block corrects the error that accumulates within it.

And the 360-unit calculation displays the same principle in another form. At 60 schematic years, the 248-day cycle leaves a remainder of 24 days. At 600 years that remainder has become 240 days—almost another complete 248-day period. Again, the remainder is accumulating towards a correction.

This seems more interesting to me than the idea of a single miraculous “perfect cycle”. The system does not eliminate error. It keeps track of it.

Twelve lunar months do not make a solar year, so an additional month is periodically inserted. A 365-day civil year does not equal the actual solar year, so the discrepancy accumulates until another correction becomes necessary. The same principle can operate at much larger scales.

This raises the possibility that 60 and 600 were not rival astronomical periods at all. Sixty may have been a convenient computational block, while 600 provided a higher-order correction. And 3 600 years, another period which both Bailly and Le Gentil encountered, would take the same architecture one stage further.

One period, several clocks

There is another reason not to insist that 60 years must have had a single astronomical explanation. It happens to sit near several quite different celestial rhythms.

Jupiter takes about 11.86 years to orbit the Sun, while Saturn takes about 29.45 years. In sixty years Jupiter therefore completes approximately five revolutions and Saturn approximately two. More strikingly, Jupiter and Saturn come into conjunction about every 19.86 years. Three conjunction intervals amount to roughly 3 × 19.86 ≈ 59.58 years.

So a period of approximately sixty years also marks a larger rhythm of the two slowest visible planets. It is not an exact return, the conjunction shifts around the zodiac. but it is another celestial pattern that fits naturally into the same numerical block.

We tend to ask of an ancient cycle: what does this number represent? But perhaps that is the wrong question. A period such as 60 or 600 may not represent one celestial motion. It may be a common framework within which several motions can be compared.

That would make sense of something otherwise rather puzzling about Le Gentil's investigation. He keeps finding different possible meanings for the same periods. Six hundred years is lunisolar. Then it seems connected with the lunar apogee. Sixty years belongs to chronology, but also sits close to planetary rhythms. The periods expand into 3,600 years and beyond.

Perhaps this multiplicity is not a problem to be solved. Perhaps it is what the system was designed to achieve.

The sky supplies the awkward numbers: 29.53059, 27.55455, 365.2422, and so on. The mathematical system supplies numbers such as 60, 360, 600 and 3,600. Ancient astronomy consists, in part, of finding the places where the two sets come close enough to one another to be useful. Some numbers belong to the grid; others belong to the sky. Ancient astronomy is the art of making them communicate. And if that is what is happening, the next step is to follow the grid itself.

Because the sequence does not end with 600. Le Gentil found 60, 600 and 3,600 embedded within a still larger numerical structure, including a period of 24 000 years. And once 24,000 is brought into the picture, the enormous numbers of the Indian Yugas: 216 000, 432 000, 864 000 and beyond, no longer look quite so remote from the comparatively modest astronomical cycles with which we began.

A Sexagasimal System

Le Gentil challenged Halley’s interpretation of what he had termed the Saros, the cycle of eclipses, which is just over 18 years long. Le Gentil correctly stated, in his Voyages dans les Mers de l'Inde Vol 1. that technically, the Saros was a period of 3600 years. Whatever the case, Le Gentil's challenge opens an interesting question however on the importance of the number 6 and it's multiples in astronomy. As a result of this, Bailly notes that a day is divided into 60 hours in all the known peoples of the ancient world.

The Indians regulate their chronology by periods of sixty years. This period, as well as the division of the day, appears to us, as we have said (1), based solely on the property of the sexagesimal number (2). The Indians do not know the antediluvian period of 600 years; but, as M. le Gentil remarks, they make use of it without knowing it; they use in their astronomical calculations a period of 3600 years, which is luni-solar, composed of fixed periods of 600 years, & only a little less exact, because the error there is fixed times greater. We believe this to be a more modern invention than the others; & the fruit of the remark that the average movement of the sun, after an interval of 3600 years, needed correction.

More recently, Robert Temple has written about the period of 60 years associated with the Dogon of Mali:

The Sigui among the Dogon is celebrated every sixty years... The Egyptians had such a period associated with Osiris [principle of renwal]... My own predilection, when considering the period of sixty years, is to think in terms of a synchronisation of the orbital periods of the two planets, Jupiter and Saturn, for they come together in nearly sixty years ... Stonehenge has sixty stones in its outer circle... (This) outer circle is the oriental cycle of Vrihaspati... It is therefore interesting that the dogon say that sixty is the count of the cosmic placenta.

John Anthony West, after quoting these words in Serpent in the Sky, writes:

The sixty year cycle also provides a link between the Egyptian Sothic year and the Great Year of the precession of the equinoxes, which bear a relationship to each other similar to the Egyptian civil year of 360 days to the tropical year of 365 days. A precessional 'month' of 2160 years divides into three 'decans' of 720 years each. Two 'great' civil years of 360 years (6 x 60) plus five epagomenal years per 'great' year 2(360 x 5) make up the precessional 'decan' of 730 years. Conceivably, it is this relationship that determined the Egyptian year of 360 + 5 days to begin with; a reckoning which, as far as I know, has not been satisfactorily accounted for otherwise. In any case, 72 Sothic hemidemicycles of 360 years each plus one great epagomenal year (72 x 5 years) make up a precessional year. To construct a pentagon within a circle, it must be divided into 5 72 degree angles, and thus on a grand scale the Sothic and precessional cycles again reflect the relationships between 5 and 6, and their multiples and powers. The sixty-year Dogon cycle and Egyptian Osirian cycle is therefore a 'day' of the precessional scheme, Sirius plays a role similar to that played by Jupiter within the solar system; her Egyptian title of 'Great Provider' perhaps furnishes a clue that further research could elaborate upon.

A Sexagesimal System

By this point, I was beginning to wonder whether I had been asking the wrong question. Perhaps there was no single astronomical explanation for the periods of 60 and 600 years. Perhaps their usefulness lay precisely in the fact that they could accommodate several different celestial movements at once.

This was essentially the direction in which Bailly had gone. He did not simply collect examples of sixty-year periods from around the ancient world. He wondered whether their widespread use pointed towards something much more fundamental: the sexagesimal system itself.

Sixty is an extraordinarily convenient number. It divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. Bailly refers to « la propriété connue du nombre sexagésimal, qui a beaucoup de diviseurs », the familiar property of the sexagesimal number, which has many divisors, and suggests that this may have been « la source d'une infinité d'usages & de périodes »: the source of an infinity of uses and periods.

He believed the same principle had been extended in several directions. A circle could be divided sexagesimally; so could a day. But the progression could work upwards as well as downwards. Sixty units could form a larger unit, and sixty of those something larger again. Bailly consequently brought together periods of 60 days and 60 years with those of 600 and 3,600 years. He also pointed to their occurrence in Indian, Chinese and Chaldean traditions as evidence that the system itself was extremely old.

This is where the 600-year period begins to look less isolated. It belongs to a progression:

60 → 600 → 3,600

but it also produces, when combined with the 360-unit revolution:

600 × 360 = 216,000 = 60³.

There is something rather elegant about that. The 600-year period that works so well against the real movements of the Sun and Moon also occupies a precise position inside an ideal sexagesimal structure.

The distinction between the two is important. The heavens supply awkward quantities such as 365.2422 days, 29.53059 days and 27.55455 days. Mathematics supplies 60 and 360. Ancient astronomy had to find ways of making the two communicate.

And perhaps this is why the system survived. We still do exactly the same thing.

Time and Space

We are so accustomed to our divisions of time that they seem natural: hours, minutes, seconds. But minutes and seconds also belong to another system. A degree is divided into sixty minutes, and a minute into sixty seconds.

This is not merely a coincidence of terminology. In astronomy, time and angle are intimately connected, because the heavens appear to move through angles as time passes.

One complete rotation of the Earth corresponds to both a period of time and a circle in space. Twenty-four hours correspond to 360 degrees. One hour of rotation corresponds to 15 degrees; four minutes of time correspond to one degree. We can describe the same rotation by saying how long it takes or how far around the circle it has travelled.

Seen in this light, the schematic year of 360 units becomes much more interesting. It need not represent a mistaken estimate of the number of days in a year. It can function as a coordinate system: a complete revolution in time mapped onto a complete revolution in space.

Bailly was particularly interested in the fact that Indian astronomical reckoning used sexagesimal divisions of the day. The same culture could use other divisions for civil purposes. For him, that distinction mattered. Sexagesimal reckoning belonged particularly to astronomical calculation: it was a mathematical language for describing movement.

And this brings us to one of the most intriguing numbers encountered by Le Gentil.

54 Seconds and 24,000 Years

Among the Indian astronomical values reported by Le Gentil was an annual motion of the stars of 54 arcseconds. He interpreted this as a value connected with the precession of the equinoxes.

Whatever its original interpretation, the arithmetic is beautifully simple.

There are 60 arcseconds in an arcminute and 60 arcminutes in a degree. A complete circle therefore contains:

360 × 60 × 60 = 1 296 000 arcseconds.

If the stellar sphere moves by 54 arcseconds each year, one complete revolution requires:

1 296 000 ÷ 54 = 24 000 years.

So the apparently enormous period of 24 000 years is simply a circle translated into time.

And the smaller periods we have already encountered now acquire angular values of their own. At 54 arcseconds per year:

60 years = 54 arcminutes = 0.9°

600 years = 9°

3 600 years = 54°

24 000 years = 360°

This was the point at which the system began to look quite different to me. Sixty, 600 and 3 600 were no longer simply periods of years. They could also represent stages in a rotation.

Le Gentil regarded the 24 000-year period as an ancient estimate of precession. That may indeed have been its purpose. The modern precessional period is closer to 26 000 years, so 24 000 is not an especially accurate value by comparison with some of the lunar periods we have been examining.

But there is another possibility. The equation works equally well in reverse.

If you begin with a Great Year of 24 000 years and divides the complete circle by it, the resulting motion is automatically:

360° ÷ 24,000 = 54″ per year.

So which came first? Was 24 000 derived from an observed stellar motion of 54″ per year? Or was 54″ the rate implied by an already established 24 000-year cycle?

The arithmetic cannot answer that question.

What it does show is that 24 000 converts time into angular motion using exactly the same sexagesimal subdivisions that we still use for angles today.

There is another reason why 24,000 interests me. It contains exactly forty periods of 600 years, or 400 periods of 60 years:

24,000 = 40 × 600 = 400 × 60.

It also contains exactly 3 000 eight-year periods. This interested me because I had encountered the eight-year cycle elsewhere, in the relationship between the Sun, Moon and Venus, and in my investigation of ancient calendar systems. Three eight-year periods produce 24 years; twenty-five such 24-year blocks produce 600 years; and forty 600-year periods produce 24 000:

8 × 3 = 24

24 × 25 = 600

600 × 40 = 24 000.

I would not claim that this proves the system was historically constructed in this order. But it does show how naturally the periods can be nested. An eight-year astronomical cycle can be carried into 24, 600 and finally 24 000 years without leaving a framework of simple integer multiplication. And then the Indian Yugas enter the picture.

The Yugas

The traditional Indian system of Yugas initially seems to belong to an entirely different scale of thought. Instead of decades or centuries, we suddenly encounter hundreds of thousands and millions of years.

The four Yugas are conventionally given as:

Kali Yuga: 432 000 years

Dvapara Yuga: 864 000 years

Treta Yuga: 1 296 000 years

Krita Yuga: 1 728 000 years

Together they form the Mahāyuga of 4 320 000 years.

The familiar relationship between them is 1 : 2 : 3 : 4. But once the 24 000-year period has been introduced, another pattern appears:

432 000 = 18 × 24 000

864 000 = 36 × 24 000

1 296 000 = 54 × 24 000

1 728 000 = 72 × 24 000.

So the four Yugas are simply:

18, 36, 54 and 72 periods of 24 000 years.

The sequence 18, 36, 54, 72 is itself merely 18 multiplied by 1, 2, 3 and 4. The traditional Yuga proportions and the 24 000-year period therefore fit perfectly together.

Their sum gives:

180 × 24 000 = 4 320 000.

This does not in itself tell us what the Yugas originally meant. But it makes it difficult to regard their enormous durations simply as extravagant numbers chosen for effect. They have an extremely orderly mathematical construction.

And there is a second route to exactly the same numerical family.

We have already seen that:

600 × 360 = 216,000 = 60³.

Double it:

2 × 600 × 360 = 432 000.

But we have just arrived at the same number through the 24 000-year period:

18 × 24 000 = 432 000.

So:

2 × 600 × 360 = 18 × 24 000 = 432 000.

The first Yuga value can therefore be reached either through the 600-year period and the 360-unit revolution, or through eighteen revolutions of 24 000 years.

The complete Mahāyuga has the same structure:

4 320 000 = 20 × 216 000 = 20 × 60³.

The numbers that had looked impossibly large were beginning to look instead like expansions of a much smaller mathematical system.

One relationship is particularly interesting. A complete circle expressed in arcseconds contains: 360 × 60 × 60 = 1 296 000 arcseconds.

But 1 296 000 is also the duration of the Treta Yuga.

And because the 24,000-year cycle implies 54 arcseconds per year:

54 × 24 000 = 1 296 000.

So the same number can be generated in three ways:

360 × 60 × 60 = 54 × 24 000 = 1 296 000.

One expression describes a circle divided sexagesimally. Another describes angular movement accumulated through time. The third appears as one of the great periods of Indian cosmology.

Perhaps these vast “years” should not automatically be read in the same way that we read an ordinary solar year. They may preserve numerical structures whose original astronomical functions became absorbed into chronology and cosmology.

That was, after all, close to Bailly's suspicion: ancient traditions might preserve the numbers long after the system that generated them had been forgotten.

A Brief Return to Giza

Several of the numbers that had emerged independently from the astronomy were already familiar from my work on the pyramids. The Giza relationships don't prove anything here, but they provide interesting parallels. The numbers 19, 223, 248 or 254 already have independent astronomical meanings. The first example involves the two best-known ancient lunar cycles. The Metonic cycle contains nineteen solar years; the Saros contains 223 synodic months. Multiply the two numbers:

19 × 223 = 4 237.

Double this:

2 × 4 237 = 8 474.

Petrie's measured side of the Second Pyramid is approximately 8 474.9 inches.

The difference is less than one inch.

There is even a geometrical curiosity here: φ³ is approximately 4.236068, extremely close to 4.237. I would not want to build an argument upon that coincidence, but it is worth noting in a system in which astronomical and geometrical ratios may have been deliberately brought together.

The larger pattern arose from what I call the Great Giza Rectangle, whose length from Petrie's survey is approximately 35 713.2 inches. I had found that astronomical ratios involving the Metonic and Saros numbers transform this length into dimensions remarkably close to those of the pyramids.

For example:

35 713.2 × 223 / 235 = 33 889.55 inches.

This is close to the perimeter of the Second Pyramid, approximately 33,899.6 inches from Petrie's side measurement of 8 474.9 inches.

Using the sidereal-month count of the Metonic cycle:

35 713.2 × 254 / 1,000 = 9 071.15 inches,

compared ith Petrie's Great Pyramid side of approximately 9 068.8 inches.

And:

35,713.2 × 29.53059 / 254 = 4 152.09 inches,

which falls in the range of the Third Pyramid side measurements.

Again, these are numerical correspondences, not by themselves evidence of historical intention. But the reason I find them worth pursuing is that the numbers being used are not selected from nowhere. 19, 223, 235 and 254 belong to actual ancient astronomical cycles.

Le Gentil's 248-day period now adds another.

Divide the Great Giza Rectangle length by 248:

35 713.2 ÷ 248 = 144.00484...

Or reverse it:

248 × 144 = 35,712 inches.

That is just 1.2 inches short of the measured length of 35 713.2 inches.

And 144 is itself 12².

This one particularly interests me because 248 is not an obvious number to go looking for in a pyramid. It became significant for quite another reason: Le Gentil found a 248-day period operating in Indian lunar calculations, and nine anomalistic months are 247.99095 days. Only after understanding that did the Giza relationship acquire any astronomical meaning for me.

It does not prove that the Great Giza Rectangle was designed as 144 periods of 248 inches. But it gives me another reason to keep asking whether the astronomical numbers and the dimensions are speaking the same mathematical language.

There may be another way of thinking about the two principal axes at Giza. The north–south line is meridional: extended conceptually beyond the surface of the Earth, it belongs to the polar framework and the celestial axis about which the heavens appear to turn. The east–west line, by contrast, belongs naturally to the daily movement of the Sun, from sunrise to sunset, and to the equatorial circle swept out by the daily rotation of the heavens. This suggests a distinction between two forms of astronomical time. East–west and equatorial movement belong naturally to the day; north–south and meridional movement to the year, since the annual solar cycle reveals itself observationally through the changing declination of the Sun, moving north and south between its solstitial limits. It is interesting in this context that the metrological pattern I have been exploring appears to make a similar distinction, associating the equatorial circumference with the inch and astronomical periods expressed in days, while the polar circumference seems to belong to another family of measures.   

Plato provides a striking image with which to think about this geometry. In the Timaeus, the axis and circles of the cosmos belong to its geometrical construction before the visible celestial bodies are introduced as the instruments by which time is numbered. Only afterwards does the Demiurge, as it were, switch on the lights: Sun, Moon and planets are placed into their circuits, and their movements make the different measures of time perceptible, and only then is the sun lit up. Something similar may be useful as a way of thinking about Giza. Before there are days, months or years to count, there is an axis and a geometrical framework: north and south, east and west, pole and equator, meridian and horizon. The celestial bodies then move through that framework. Daily rotation gives the east–west circle its measure; annual solar motion gives the north–south dimension its changing measure. If Giza was conceived astronomically, its cardinal geometry may therefore have been more than orientation towards particular celestial events. It could have provided the coordinate framework within which different kinds of celestial time were expressed.

From the Great Year to the Measure of the Earth

Astronomy naturally converts time into angle. But angle also converts directly into distance once a circle is given a physical size. A degree on the Earth's surface is both an angle at the centre of the Earth and a distance travelled around its circumference. Divide the terrestrial circle into 360 degrees, each degree into sixty minutes and each minute into sixty seconds, and the sexagesimal system becomes a system of geodesy.

A complete terrestrial circle contains 360 × 60 × 60 = 1 296 000 arcseconds.

We have just encountered that number in the Yuga system.

This made me wonder whether some of the ancient units of length I had been studying might belong to the same conversation.

The equatorial circumference of the Earth is approximately 24 901.461 miles. Expressed in inches, this is:

24, 01.461 × 63 360 = 1 577 756 568.96 inches.

Now compare this with the Mahāyuga number of 4 320 000.

Dividing the terrestrial circumference by 4 320 000 gives:

1 577 756 568.96 ÷ 4 320 000 = 365.221428...

That immediately looks familiar. It is extremely close to the number of days in a year.

The comparison can also be reversed. If I take the tropical year of 365.2422 days and multiply it by the Mahāyuga number:

4 320 000 × 365.2422 = 1 577 846,304 inches.

The actual equatorial circumference used above is:

1 577 756 568.96 inches.

The difference is about 89,735 inches, or approximately 1.42 miles, around the entire circumference of the Earth.

In percentage terms, the discrepancy is only about 0.0057%.

I find that difficult simply to ignore.

The relationship can be written another way because:

4 320 000 = 60 × 60 × 12 × 100.

So the constructed circumference is:

60 × 60 × 12 × 100 × 365.2422 inches.

Here again a number of days in an astronomical year, a sexagesimal structure and a terrestrial distance appear in the same equation.

It is important to distinguish what I am doing here from the earlier historical evidence. I do not have an ancient text saying that the inch was defined by dividing the Earth's circumference in this way. This is my own metrological hypothesis. But it arose from my work on Giza, where I had increasingly come to suspect that the inch, or something extremely close to it, was the fundamental unit behind the dimensions, as well as observations made by Robin Heath.

There is another intriguing terrestrial value. A figure I have used for the polar circumference is approximately 1 575 000 000 inches. Unlike the modern equatorial figure above, this is an idealised round value, but its structure is extraordinary:

1 575 000 000 = 360 × 4 375 000

and it sits within the same broad world of highly structured terrestrial circumference numbers I had been finding in ancient metrology.

The question for me is therefore no longer simply whether one ancient culture knew the circumference of the Earth to some particular degree of accuracy. It is whether units of length themselves could have been generated from astronomical and terrestrial divisions.

That would dissolve some of the distinction we instinctively make between time and space.

A day measures a rotation. A degree measures part of that rotation. An arcminute and arcsecond subdivide the degree. On the surface of a sphere, those angular divisions correspond to distances. If the unit of length is then defined in relation to one of those distances, the same mathematical system has travelled all the way from time, to angle, to terrestrial measure. Perhaps that is what we are looking at.

What Survived?

At the beginning of this investigation, the 600-year Great Year looked like an isolated curiosity. Josephus had preserved the number; Cassini had rediscovered its extraordinary lunisolar accuracy; Bailly wondered why such an excellent period had apparently been forgotten.

But 600 did not remain isolated for long.

It belonged with 60 and 3,600. Six hundred schematic revolutions of 360 units produced 216 000 = 60³. Le Gentil's peculiar 248-day lunar period came within eight days of an integer relationship with those 216 000 units. A stellar movement of 54 arcseconds per year produced a complete revolution in 24 000 years. That period contained forty 600-year cycles. The Yugas were exact multiples of 24 000, while their smallest value, 432 000, could also be produced simply by doubling 600 × 360. The complete Mahāyuga was twenty times 60³.

And throughout it all, the same sexagesimal grammar kept reappearing: 60, 360, 600, 3 600, 24 000, 216 000, 432 000, 1 296 000, 4320 000.

Some of these numbers arise from observations. Some are mathematical constructions. Some belong securely to particular historical astronomical traditions; others are connections I have made while trying to reconstruct the system. I do not think they should all be treated as equivalent evidence.

But neither do I think they should automatically be treated as unrelated.

Bailly's image of ancient astronomy as débris, orscattered fragments of something once more coherent, keept returning to me. He imagined a very ancient science whose pieces had survived separately among different peoples. I would push the possibility much further back in time than Bailly could reasonably have done in the eighteenth century. The antiquity of humanity, and therefore the possible depth of accumulated observation, is vastly greater than he knew.

Perhaps the sexagesimal system itself is one of the largest surviving fragments.

We still divide the circle into 360 degrees. We still divide degrees into sixty minutes and minutes into sixty seconds. We still divide time sexagesimally. These conventions have survived every transformation of astronomy, geography and measurement around them.

Why?

Convenience is certainly part of the answer. Sixty is a wonderfully divisible number. But that does not tell us when the system was created, why it was first applied simultaneously to time and angle, or how far its original architecture extended.

My suspicion is that the answer lies much further back than the surviving written evidence.

If the reconstruction attempted here is even partly correct, the great periods of ancient astronomy may not be an assortment of unrelated inventions made by Babylonians, Egyptians, Indians, Greeks and others at different moments. Some may be local developments; others certainly were refined over time. But beneath them there may survive an older computational language: a way of mapping the imperfect cycles of the heavens onto a stable numerical grid.

The sky does not run on sixty. The tropical year is not 360 days. The lunar month does not divide neatly into anything. The planets stubbornly refuse to return on schedule.

That may be precisely why the system was needed.

Some numbers belong to the grid; others belong to the sky. Ancient astronomy was the art of making them communicate.

And perhaps, every time we look at a clock or measure an angle in degrees, minutes and seconds, we are still using a small piece of that forgotten language.

Conclusion: The Debris of a Science?

There is one final characteristic of this numerical system that I have encountered repeatedly, both in astronomy and in my work on ancient metrology: the scale can change while the numerical structure remains the same.

Numbers acquire a zero, or lose one; a period is multiplied by ten, by one hundred, or by 360; yet the underlying relationships are preserved. We have already seen this happening between 60 and 600, and again as the much larger Indian periods are generated from smaller numerical structures. A period of 600 schematic years contains 216 000 units; doubled, this becomes 432 000, the duration assigned to the Kali Yuga. The Mahāyuga expands the same family to 4 320 000.

I do not think this means that the system was purely sexagesimal. In fact, my research into ancient metrology has repeatedly suggested the opposite. The sexagesimal system seems to have been intended to work alongside a decimal one. Sixty provides the extraordinary divisibility required for fractions, angles and astronomical cycles, while ten provides a simple means of changing scale.

This is easily overlooked because we tend to speak of “decimal”, “duodecimal” and “sexagesimal” systems as though a culture must choose between them. But there is no mathematical reason why it should. A practical system can use different bases for different operations, and combine them when useful. Indeed, some of the numbers encountered here seem almost to invite precisely that treatment.

Thus 60 becomes 600 simply by changing the decimal scale. The square of 60 gives 3,600. Six times 60 gives the 360 divisions of the circle. Six hundred times 360 gives 216 000, or 60³. Double that and we have 432 000; multiply it by ten and we have 4 320 000. The transformations move backwards and forwards between multiplication by 6, by 60 and by powers of 10.

This is very familiar from metrology. In the systems I have been studying, the same underlying numerical value can recur at different orders of magnitude: a unit may be multiplied by ten, one hundred or one thousand while retaining its relationship to the other units around it. What matters is not always the absolute size of the number, but its place within a family of ratios.

That possibility may also change how we approach the enormous numbers of ancient chronology. Perhaps adding zeros did not necessarily imply the discovery of an entirely new astronomical period. It could represent a change of scale within an existing computational structure. Multiplication by 360 could perform another kind of transformation, converting one category of “year” or period into another.

The Indian Yuga system provides a particularly striking example. Its traditional figures can be generated from smaller numbers by multiplication by 360: 1 200 becomes 432 000; 2 400 becomes 864 000; 3 600 becomes 1 296 000; and 4 800 becomes 1 728 000. The four smaller numbers themselves preserve the simple proportion 1:2:3:4. The apparent enormity of the resulting chronology may therefore conceal an extremely simple numerical architecture.

This makes me reluctant to assume that an ancient number such as 432 000 must originally have meant precisely “432 000 ordinary solar years” in the modern sense. It may have done. But before making that assumption, we should ask what happens when the number is divided, rescaled or converted through the other operations of the system.

And this is perhaps another way of recognising Bailly's débris. What survives may sometimes be the number after the operation has been performed, while the rule that generated it has disappeared.

Bailly was confronted by a problem that has never entirely gone away. When he looked at the earliest astronomical traditions known to him in Chaldea, India, China and elsewhere, he did not think they looked like the tentative beginnings of a science. They looked strangely uneven. Sophisticated methods existed alongside apparently elementary gaps in understanding; accurate numbers survived without clear explanations of how they had been obtained; observations had been preserved whose original purpose was no longer obvious.

This led him to one of the most striking statements in his work:

That distinction between débris and élémens is crucial. Bailly was suggesting that what appeared in the earliest surviving records might not represent astronomy being invented. It might represent fragments of something older being preserved, copied and sometimes used without the principles behind it being fully understood.

He went further:

Bailly believed that by around 3000 BC astronomy was already reappearing after an earlier period of loss. I cannot demonstrate that history, and I do not think we need to accept the particular civilisation he eventually imagined in order to take his question seriously. What interests me is the observation behind it. How much of the astronomy that first appears in writing was already inherited?

The 600-year period is a good example of the problem. It appears in the historical material as something already ancient and obscure. Cassini discovered that it produced an extraordinarily good lunisolar period: 600 tropical years correspond very closely to 7 421 synodic months. Bailly was astonished that such a useful period could have been known and then apparently forgotten.

Le Gentil approached the problem differently. Working directly with Indian astronomers and their tables, he found that 600 years could not simply be dismissed as an unnecessarily long alternative to the nineteen-year cycle. It belonged to a family of periods including 60, 600 and 3 600 years, and those periods seemed to intersect with other forms of lunar calculation. In particular, his investigation of the 248-day period led him towards the Moon's anomalistic motion and its apogee. He concluded that:

This seems to me more important than deciding whether every reconstruction proposed by Bailly or Le Gentil was correct. Both men had recognised the same underlying problem. The numbers appeared to belong to a system whose original logic was no longer entirely visible.

My own investigation has led me to wonder whether that system was broader still.

The 600-year period is an excellent lunisolar period when measured against the real tropical year. But 600 also belongs naturally to sexagesimal arithmetic. Combined with the schematic revolution of 360 units, it gives:

600 × 360 = 216,000 = 60³.

Le Gentil's 24 000-year period extends the structure in another direction. At an annual stellar movement of 54 arcseconds, 24 000 years correspond to a complete revolution of 360 degrees. It contains forty periods of 600 years and 400 periods of sixty years.

Then the enormous Indian Yuga numbers fall naturally onto the same grid:

432 000 = 18 × 24 000

864 000 = 36 × 24 000

1 296 000 = 54 × 24 000

1 728 000 = 72 × 24 000

and together:

4 320 000 = 180 × 24 000 = 20 × 60³.

Perhaps these are simply the products of several historical traditions which happened to favour highly divisible numbers. That possibility cannot be excluded. Numerical coherence by itself does not give us a date, a place or a people.

But neither does the first surviving written appearance of a number tell us when it was invented.

That seems especially important in astronomy. A sophisticated long-period cycle is inherently retrospective. It presupposes observations to compare, or a mathematical system capable of extrapolating far beyond an individual human lifetime. The first person to write down such a period need no more have invented it than the first surviving scribe to record a myth need have invented the story.

So how old is the sexagesimal astronomical system? I do not know.

I think there is a good case that its essential architecture was already old by the third millennium BC. By then we are not looking merely at people noticing that the Moon returns every month or that the seasons repeat. We encounter organised calendars, long astronomical periods, numerical schemes, angular divisions and methods for reconciling celestial motions that do not naturally fall into whole numbers. How much further back those ideas go is much harder to establish. My suspicion is that they go considerably further.

Perhaps 60 began simply because it is an exceptionally useful calculating number. Perhaps 360 emerged because six sixties make an extraordinarily divisible representation of a complete revolution. Perhaps generations of observers then discovered that real celestial periods could be mapped onto this grid with remarkable effectiveness. Or perhaps some of these relationships were discovered first and helped to determine the grid itself. We cannot yet know which direction the process took.

The 600-year period may provide a glimpse of that process because it seems to stand exactly at the boundary. It is mathematically elegant, yet astronomically effective. It belongs to the grid, but it also fits the sky.

The same may be true of 24 000. It is forty periods of 600 years, but it is also a complete 360-degree rotation at 54 arcseconds per year. The same may be true of the Yugas, whose immense durations resolve into orderly multiples of 24 000 and powers and multiples of sixty. And perhaps the persistence of the same numerical language in the measurement of angles, time and, in the hypothesis I have been exploring, terrestrial distance is not accidental either.

Perhaps what survived was not a sexagesimal system in isolation at all, but a numerical language in which decimal scaling and sexagesimal division worked together. Ten changed the scale; sixty divided and recombined it; 360 mapped it onto a complete revolution. The same operations could then be applied to days, years, angles, astronomical periods and, perhaps, units of distance.

If so, the recurrence of 60, 600, 3 600, 24 000, 216 000, 432 000 and 4 320 000 becomes more interesting than the recurrence of any one number. What matters is the grammar connecting them.

I cannot prove that all of these fragments once belonged to a single prehistoric system. Nor can I put a date on its creation. The evidence does not allow that.

But we can make the case.

We can ask whether a number has an independent astronomical meaning before looking for it elsewhere. We can distinguish quantities generated by observation from those generated by a mathematical grid. We can examine whether the same numbers repeatedly translate between lunar periods, solar years, planetary cycles and angular movement. We can look for the same structures in surviving systems of chronology and measurement. And, where the evidence ends, we can say so, and then ask what reconstruction best explains what remains.

In that sense, I think Bailly's word débris remains wonderfully apt.

A fragment does not tell us what the whole object looked like. But enough fragments can begin to show that there was a whole.

Bailly believed that the scattered astronomical knowledge of the ancient world pointed backwards towards common ancestors. Le Gentil, working through the calculations themselves, found periods whose functions seemed deeper and more complicated than their surviving explanations suggested. More than two centuries later, with a much longer human past available to us than either man imagined, perhaps their questions deserve to be asked again.

Not: Which civilisation invented astronomy?

But: how much astronomy had already been inherited before history began to write it down?

And perhaps the strangest possibility is that part of the answer has never disappeared at all. We still measure the turning Earth in hours, minutes and seconds. We still divide the circle into degrees, minutes and seconds. The notation survives on our clocks, maps and instruments, even though its beginnings are lost.

If Bailly was right that we possess the débris rather than the beginnings of an ancient science, then the sexagesimal system may be one of its largest surviving pieces.

Notes

1. Bailly: “The knowledge here attributed to the earliest men is in no way surprising when we consider that which emerges from the Great Year, or astronomical period of six hundred years, which Josephus attributes to the patriarchs and which was undoubtedly their work. An astronomical period, when it concerns a single celestial body, is the time it takes to travel around the circle it describes. When several celestial bodies are involved, the period of their combined motions is the time which elapses from the moment when they all depart from the same point, or from certain relative positions, until they return to the same point or to the same relative positions. It is clear that a period of this kind must contain an exact number of complete revolutions of each of these bodies. The Great Year of six hundred years must have been a period of this kind, for the ancients called any revolution, whether of one or several planets, a ‘year’. They called a ‘Great Year’ one which encompassed a longer interval. The celebrated Giovanni Domenico Cassini was the first, on considering the account of Josephus, to be struck by the accuracy of this period and by the conclusions that could be drawn from it concerning the length of the year in the time of the patriarchs. He found that 7,421 lunar revolutions of 29 days, 12 hours, 44 minutes and 3 seconds amounted to 219,146½ days, and that the same 219,146½ days gave 600 solar years of 365 days, 5 hours, 51 minutes and 36 seconds, a duration differing by no more than three minutes from that observed today.”

    "Ces connoiffances accordées ici aux premiers hommes n'ont rien d'étonnant , quand on confidere celles qui réfulcenc de la grande année, ou de la période aftronomique de fix cens ans, que Jofeph attribueaux patriarches , & qui eft indubitablement leur ouvrage (1). Une période aftronomique, quand il s'agit d'un aftre feul , eftle tems qu'il employe à parcourir le cercle qu'il décrit. Quand il s'agit de plusieurs aftres , la période de leurs mouvemens combinés eff le cems qui s'écoule depuis qu'ils font tous partis du même point, ou de certains afpeéts , jufqu'à ce qu'ils reviennent au même point , ou aux mêmes afpelts. On voit que cette efpece de période doit comprendre exaéte. ment un nombre de révolutions complettes de chacun de ces aftres, La grande année de fixcens ans doit être une période de ce genre. Car les anciens appelloient année une révolution quelconque , foit d'une ou de plufieurs planetes (2). Ils appelloient grande année celle qui embraffoit un plus long intervalle. Le célebre Dominique Caflini eft le premier qui, ayant fait attention au récit de Jofeph, fut frappé de la jufteffe de cette période, & des conclufions quon en pouvoir tirer fur la longueur de l'année au tems des patriarches, Il trouva que 7421 révolutions lunaires de 29) 12n 44 3" (3), faifoient 219146 jours & demi , & ce même nombre de 219146 & demi donnent 600 années folaires de 365! st 51° 36" ; durée qui ne differe pas de trois minutes de celle qu'on obferve aujourd'hui."

2. Bailly, Histoire de l'Astronomie Ancienne