H-Stonehenge.htm

The traveler who drives west across Salisbury Plain from Amesbury can hardly fail to notice Stonehenge close on his right hand. If he elects to stop and inspect this ancient monument, no doubt it will be the massive trilithons near the centre that will occupy his main attention (see Figure 2.1). It was so with me when I first visited Stonehenge about twenty years ago. I knew at the time that the stones of this central part are not local to Salisbury Plain, but from faraway Wales, from the Prescelly Mountains. I knew of the complex journey by which they were thought by archaeologists to have been transported, in part by sea, in part by river, and in part hauled bodily over land. It seemed obvious that, whoever had constructed this structure had been impelled by much the same motives as the builders of medieval cathedrals.

FIGURE 2.1. Trilithons near the centre of Stonehenge. These structures are tooled and belong to Stonehenge III. Probable date 1700‑2000 B.C. (Courtesy: Controller of Her Britannic Majesty's Stationery Office. British Crown Copyright.)

At that first visit I paid no attention to the remarkable ring of 56 holes which surrounds the inner structure of Stonehenge (see Figure 2.2). These holes were discovered by John Aubrey in the seventeenth century, and arc now generally known as the Aubrey holes. Apparently the first builders to arrive at Stonehenge dug these holes in the chalky ground, and having done so immediately filled them in again. If I had known this, I would certainly have been intrigued by why anyone would do anything so curious, but probably I would have been satisfied to have been told that archaeologists had discovered that fires had been lit in the holes, had also discovered crematorial remains in some of them, and hence had formed the opinion that they were connected somehow with the ceremonies and rites of an ancient people.

I did know of the connection between Stonehenge and midsummer day, but I was quite vague about it and did not even realise that the avenue which joins the centre to a distant stone, the Heelstone, lies closely in the direction of sunrise on midsummer day. Unless one knows about it, the Heelstone seems much less impressive than the central stones. It was not transported specially from Wales. It is just a natural stone obtained locally (see Figure 2.3). So are the places which mark a rectangle inside the circle of the Aubrey holes. The short sides of the rectangle point more or less in the same direction as the avenue from the centre to the Heelstone (see Figure 2.4). That is to say, they point more or less in the direction of midsummer sunrise (see Figure 2.5).

Quite certainly I would not have been surprised oil the occasion of my first visit to have been told that Stonehenge is a composite structure. The inner part, the part which so naturally is impressive on a first visit, is described by archaeologists as Stonehenge III. Its construction came later than that of the outer structure, different parts of Stonehenge III dating from about 2000 B.C. to about 1500 B.C.

FIGURE 2.2. Aerial view of Stonehenge. The Heelstone is at bottom centre. Those Aubrey holes which have been uncovered are clearly shown at middle left Office. British Crown Copyright.) (Courtesy: Controller of Her Britannic Majesty's Stationery

The first builders at Stonehenge were not concerned with transporting stones from Wales or with erecting trilithons. They dug a ditch and bank, probably about eight feet high, in the form of a circle rather more than 300 feet in diameter. The circle was broken toward the northeast, at the avenue to the Heelstone. The first builders were also responsible for the rectangle, for the 56 Aubrey holes, for several other stones‑again rough stones obtained locally ‑ within the circular bank, and also for a curious set of four postholes, which are outside the circular bank. For the moment I would like to leave open the date at which this apparently simple first structure, referred to by archaeologists as Stonehenge I, was built.

FIGURE 2.3. The Heelstone. Compare the natural form of this boulder with the elaborate structures shown in Figure 2.1. The Heelstone as seen from the centre of Stonehenge lies close to the direction of the midsummer Sun. It belongs to Stonehenge I, with probable date 2600‑2700 B.C. (Courtesy: Controller of Her Britannic Majesty's Stationery Office. British Crown Copyright.)

Between the times of construction of Stonehenge I and Stoneliengre III came, somewhat naturally, Stonehenge II. Stonehenge II is nothing but a set of stones and positions outside the main structure of Stonehenge III, but inside the bank of Stonehenge I. The interesting thing about Stonehenge II is that a program of erecting stones in circular patterns was begun, but not completed. The construction program seems to have been stopped in midflight, as if the builders had suddenly discovered a mistake.

FIGURE 2‑4‑ Plan Of Stonehenge. The post holes referred to as A1, A2, A3, A4 in the text are to be found immediately west of the Heelstone. (Courtesy: Controller of Her Britannic Majesty's Stationery Office. British Crown Coyright.)

Returning once again to my first visit to Stonehenge, undoubtedly the most astonishing thing I could have been told at the time was that Stonehenge is an ancient astronomical observatory of very great intellectual depth. The association of the direction of the avenue and of the Heelstone with midsummer sunrise had of course suggested some crude astronomical connection to investigators even in the eighteenth century. The emphasis now is on a wholly unexpected depth of perception by the stoneage builders of this ancient structure. Yet one rather obvious feature should have given me cause to wonder. There is a niche at a height of about five feet in an upright of one of the trilithons. This niche has been deliberately tooled. Why? Looking outwards from the centre, I found, on a much later visit to Stonehenge, that my right shoulder fitted snugly into this niche, so snugly that it was hard to believe this was not its purpose. But then I found myself looking through a stone arch towards the distant countryside, not inwards as one might expect if the centre were a special place where a priest was supposed to conduct some ceremony. A little detective work at this point might have set me wondering still more. Would people 4,000 years ago have been as tall as I am? It turns out from skeletal remains that they would have been the best part of a foot shorter. So at first sight the idea seems to collapse. But not on further investigation. Weathering has removed just about a foot of Salisbury Plain over the past 4,000 years; so today I stand a foot lower in relation to the trilithon than did its builders. This circumstance makes one wonder still more.

But I remained in ignorance of the wonders of Stonehenge until at least fifteen years after my first visit there. In October 1963 a remarkable letter was published in Nature by Gerald Hawkins. I noticed the letter at the time, but did not stop to give it more than a casual reading, because at that particular time several of my astro nomical colleagues and I were at work on problems connected with the newly discovered quasars. However, within a week or two of the appearance of Hawkins' letter, I had an urgent call from an archaeologist friend, Dr. Glyn Daniel. He sought to enlist my aid in checking Hawkins' calculations, which he explained would be of great importance if they were correct. So I returned to the publication in Nature, reading it now with more care. Unfortunately I accepted at its face value Hawkins' claim that a digital computer had been necessary for his analysis. Not having such a computer available to me at that time, I thought that a great deal of laborious calculation would be needed to make the checks that Daniel was asking for. And not wishing to make the calculations myself, I offered the problem to a graduate student. Fortunately or unfortunately, I don't quite know which, the student was un-attracted, and the matter soon lapsed.

It revived some two and a half years later, again with an appeal from Glyn Daniel. Apparently fierce controversy was just breaking loose between Hawkins and respectable archaeological opinion. Daniel assured me that Hawkins was seriously astray on certain of his archaeological assertions. Could he be just as much wrong in his astronomy? As it happened, I took a hill‑walking holiday in Scotland shortly after receiving this letter. It is relevant to my story that the Scottish mountains are bad‑weather mountains. It is advisable therefore to equip oneself with plenty of reading material for occupying wet days on such a holiday. I took with me a recently published book, Stonehenge Decoded by Hawkins, an elementary textbook on spherical astronomy, logarithmic tables, and plenty of writing paper.

FIGURE 2‑5. The midsummer Sun over the Heelstone, seen from within the structure of Stonehenge III. (Courtesy: Controller of Her Britannic Majesty's Stationery Office. British Crown Copyright.)

As it happened, the weather consisted of a few fine brilliant days interspersed by many wet ones. I began to read Stonehenge Decoded. From its style, the sort of style I had used myself in my youth, it was easy to see why archaeologists were annoyed, especially if there were mistakes in some of the archaeological speculations. But the main thesis, that Stonehenge was an astronomical observatory, seemed to me to have a genuine ring of truth. Even on a first reading it seemed that Hawkins had established a more convincing case than had earlier proponents of this theory‑always assuming that the calculations were correct, which was just what iny archaeological friends wanted to know. So at last there was nothing for it but to get down to the job of working out what I thought at first would be a vast amount of arithmetic. Luckily it soon appeard otherwise. I found the need for a digital computer to be an illusion. It took a couple of days to set up the mathematical form of the problem, and then no more than a few hours to work through the arithmetic. Within quite trivial margins, I confirmed all of Hawkins' results.

Now I must explain what these results were. The Sun rises at different points on the horizon on different days in the year. In the northern hemisphere, the Sun rises most to the north on midsummer day and most to the south on midwinter day (see Figure 2.6). At Stonehenge the swing between midwinter and midsummer is about 80 deg. What I had to do was to work out these midwinter and midsummer directions of rising for the latitude of Stonehenge and for a date of, say 2,000 B.C. The date makes a little difference, but not very much. The corresponding directions of sunset follow immediately, because they are symmetrically to the west of the north‑south alignment.

The situation for the Moon is more complicated. As can be seen in Figure 2.7, the Moon swings in the same way in any particular month. But the angle of swing changes slowly from one month to the next. At its least the angle of swing at Stonehenge is about 60 deg, and at its greatest the swing is about 100 deg. It was also necessary to calculate the greatest and least angles of swing. Once again the directions of moonset are symmetrically to the west of the north‑south direction.

Why does the Moon behave in this peculiar way? Why indeed does the Sun swing from north to south, and back again, in the course of the year? The Sun swings because the axis of rotation of the Earth is tilted at an angle of about 66 deg (2,000 B.C.) with respect to the plane of the Earth's orbit around the Sun. The Moon swings in each month for a similar reason, because the axis of rotation of the Earth is tilted with respect to the plane of the Moon's orbit around the Earth. The solar tilt is more or less constant, but the lunar tilt is not. It varies from about 61 deg to 71 deg, and it is this variation which produces the change in moonrise from one month to another. The situation is shown in Figures 2.8 and 2.9. In Figure 2.8, the Sun is considered to go round the Earth; that is to say, we describe the situation as it appears to us on the sky. The Earth itself then stays fixed. In particulari the axis of rotation of the Earth stays fixed with respect to the directions of distant stars. The Sun goes round its orbit in a year, whereas the Moon goes round in a month. For convenience, the two orbits are shown on the same scale in the two figures. (The fact that the Moon's orbit is actually much smaller than the Sun's orbit does not affect this particular problem.) All the time, as the Sun and Moon go around their orbits, the Earth spins on its axis. It is of course this spin with respect to the Sun which produces night and day. And it is the varying orientation of the Earth's axis with respect to the positions of the Sun and Moon which produces the variations in the directions of sunrise and moonrise. The situation in Figure 2.8 corresponds to midwinter in the northern terrestrial hemisphere. That is Figure 2.10 corresponds to midsummer.

The Sun's orbit can be considered to stay fixed, but the Moon's orbit does not stay fixed. This is best understood by combining Figures 2.8 and 2.9, as in Figure 2.11. The two orbits are inclined to each other at an angle of 5 deg 9 min. When they are drawn on the same scale, they intersect at the points N, N', known as the nodes of the Moon's orbit. Let this angle of 5 deg 9 min between the two orbits stay fixed, and imagine the points N, N', to move in a clockwise sense around the Sun's orbit. This will cause the Moon's orbit to slew around. It is this slewing motion which varies the tilt of the Earth's axis of rotation with respect to the plane of the Moon's orbit, and which causes the change in the monthly swing of moonrise and moonset. The time taken for the points N, N', to make a complete circuit of the Sun's orbit is 18.61 years, a number to be remembered in connection with Stonehenge.

What Hawkins found was that the extreme yearly directions of sunrise and sunset shown in Figure 2.6 are reproduced to a rather good approximation by directions at Stonehenge. Directions associated with the minimum and maximum monthly swings of the Moon are also present at Stonehenge. For Stonehenge I, he found the associations shown in Figure 2.12. The rectangle appearing in Figure 2.12 is just the rectangle built into Stonehenge I. The objects D, F, G, H, are stones or positions which Hawkins believed to be also associated with Stonehenge I. To interpret Figure 2.12 in relation to the above discussion, one should note that the terms ‘winter' moon and 'summer' moon do not refer to the seasons of the year, but to the most northerly and most southerly risings (A) and settings (A) to occur in the lunar month.

To go back now to my story, what I had to do was to check that the lines drawn in Figure 2.12 had been calculated correctly from an astronomical point of view. I left it to my friends to check that the lines also represented a correct joining of the various points in the structure of Stonehenge I. The controversy which was beginning to break loose was this: If one considers everything at Stonehenge, there is a tremendous number of stones and positions. If one proceeds to join every possible pair of positions, then among so many joins some are almost bound merely by chance to fall near the calculated astronomical directions.

This was the point of view my friends were seeking to maintain, but I soon found I couldn't agree with them‑although it was perhaps my natural instinct to do so! Figure 2.12 uses only positions belonging to Stonehenge I, whereas all the inner complexity belongs to Stonehenge II and Stonehenge III. If one wishes to test the hypothesis that Stonehenge I was an astronomical observatory, then surely the issue should be confined to only those positions which belong to Stonehenge I. There was no reason to confuse the supposed motives of the first builders with those of later builders. Even at this early stage, I had a half‑formed idea that the later parts of Stonehenge, the inner structure with the massive arches and trilithons, might be different. With only Stonehenge I positions used, it seemed to me that the argument of Hawkins could be maintained, even though some doubt was attached to the positions G, H, of Figure 2.12. These might, it seemed, have been natural, not man‑made. It was now hard or even impossible to decide which, because of the destructive archaeological methods used in excavating the site some fifty years ago. Yet even with G and H removed, the fact that both the short sides and the long sides of the Stonehenge I rectangle are astronomically significant seemed to me remarkable.

I should add that the rectangular alignments of Figure 2.12 had already been noted by C. A. Newham some six months before Hawkins' letter in Nature. Newham's findings were reported in the issue of the Yorshire Post for March 16, 1963. Although this is the daily newspaper of my native county, the article had entirely escaped my attention. Newham made a further point which (three years later) I found very impressive. The short sides of the Stonehenge I rectangle point north in the direction of midsummer sunrise and also south in the direction of midwinter sunset (because of the symmetry of rising and setting with respect to the north‑south direction). The long sides point towards the most southerly rising of the Moon in its 18.61 year cycle and also towards the most northerly setting of the Moon (again because of symmetry with respect to the north‑south direction). This property is only true for a rectangle if it is constructed at the latitude of Stonehenge. If Stonehenge had been built only a few tens of miles to the north or to the south, this dual property would have been lost. I found it impossible to dismiss this property as a mere coincidence. At last, then, I began seriously to examine the consequences of accepting the view favoured so strongly by Hawkins and by Newham, and which had also been strongly favoured by Lockyer as long ago as 1901, that Stonehenge (Stonehenge I, at any rate) was an astronomical observatory.

By taking the idea seriously, I mean taking seriously the idea that stoneage man knew a great deal about astronomy. It was necessary to reject from the outset that Stonehenge was used primarily to determine the seasons of the year. While it is true that such knowledge would have been of great practical value, and while it is true that, in a society not possessing accurate clocks, the seasons must be determined by observing the Sun (or by counting days, which presupposes a knowledge of the number of days in the year, and hence implies observation of the Sun), the necessary observations do not require anything as elaborate as Stonehenge. Once it is seriously accepted that Stonehenge is an astronomical construct, the feature which immediately strikes the attention is the large diameter, about 300 feet, more than twice the diameter of the dome of the largest modern optical telescope, on Mt. Palomar. This large diameter makes sense only if one is concerned to measure small angles. Using stones for sighting lines and using the naked eye, one could judge angles to well within 0.5 deg, perhaps even to within 0.25 deg. Yet the Sun moves just under 1 deg in its orbit in a single day. From a purely calendrical point of view, there would seem little advantage in working to such extreme accuracy. Practical considerations, such as the planting of crops or the migration of flocks, would hardly be affected by inaccuracies of as much as a week in the placing of the seasons. It would be sufficient to construct quite small circles of stones, such as do in fact exist in large numbers in many places in the British Isles. From the outset therefore, I rejected the calendrical theory advocated by Lockyer, as indeed archaeologists had done over the preceding fifty years.

The fact that the long sides of the Stonehenge I rectangle point towards the extreme southerly rising of the Moon implies that the builders knew of the 18.61 year cycle described above. Their astronomical knowledge was not just of a casual night‑by‑night kind. They had established correlations in the behavior of the Moon over long periods of time. This fact shows clearly that we must be on our guard against setting the intellectual sophistication of stoneage man at too low a level. From the outset therefore, I decided to accept the idea that the builders of Stonehenge I were thoroughly rational, that they knew a good deal about astronomy, and that when they behaved in an apparently mysterious way (the digging and immediate filling in of the white Aubrey chalk holes for example), they knew exactly what they were doing.

In June 1964 Hawkins had published a second letter in Nature, in which he followed up certain suggestions of R. S. Newall which led him to the view that the primary purpose of Stonehenge might have been the determination of eclipses of the Sun and Moon. This seemed a promising line of investigation, because the determination of eclipses requires a far higher standard of accuracy than does the simple calendrical theory of Lockyer. Hawkins also noticed that, since 56/3 = 18.67, if a stone were moved by three of the 56 Aubrey holes each year, it would make a complete circuit of the system of Aubrey holes in 18.67 years, close to the 18.61 year cycle of the nodes of the lunar orbit, shown in Figure 2.11. He therefore proposed that the Aubrey holes were a counting device, and this idea is fully discussed in Stonehenge Decoded.

At this point I ran into another issue of controversy between Hawkins and archaeological opinion. Professor Atkinson of Cardiff had objected that it was hardly necessary to go to the trouble of making 56 holes around a circle with a diameter as large as 320 feet merely to make a counting device. I found myself agreeing with this objection. Nor had I much confidence in the way the counting device was supposed to predict the occurrence of eclipses. I also had two further difficulties which I found overriding. Hawkins arrived at his counting system and at its calibration with the aid of tables of known past eclipses. Although such tables, extending back over long intervals of time, several centuries, might have been available to the, Stonehenge people, the method was based on noticing numerical coincidences. This method would have suited the mentality of Babylonian scientists in a later millennium, but it seemed to me at variance with the more directly observational kind of astronomy which was evidently implied by the structure of Stonehenge I. But this was only a subjective feeling I had about the problem. A more cogent objection was that the method suggested by Hawkins could only predict a small fraction of all eclipses. Would the builders really go to all this trouble if the best they could achieve would be to anticipate one eclipse in many? Wouldn't they be unbearably worried about the rest? It seemed better, then, to consider the eclipse problem in a more general way, and to see if Stonehenge I could be used (by us!) to predict essentially all eclipses. Let me add that, although at this stage I departed from Hawkins, his idea of connecting the 56 Aubrey holes with the 18.61 year cycle of the lunar nodes seemed to me inherently sound, and I determined to retain it in some form, if this should prove possible.

Eclipses occur when the Sun, Earth, and Moon are in line. If the Earth lies between the Sun and Moon, the Moon is eclipsed. If the Moon lies between the Sun and the Earth, the Sun appears eclipsed to us on the Earth. Because we are dealing with bodies of finite size, it is not necessary for their centers to be strictly in line; otherwise eclipses would be exceedingly rare. In fact, there can be as many as five solar eclipses in a year and as many as three lunar eclipses. Because the Earth is considerably bigger than the Moon, lunar eclipses are often seen to be total, whereas solar eclipses are rarely seen to be total.

The lunar nodes play a critical role in deciding whether eclipses occur or not, for unless the Moon is close to either N or N' in Figure 2.11, there is no possibility of an eclipse taking place, simply because the plane of the Moon's orbit is inclined to that of the Sun's orbit. However, the Moon passes through each of its nodes once a month, and on these two occasions an eclipse may occur, depending on where the Sun happens to be. Suppose in Figure 2.11 that the Moon is at N. If the Sun is within about 15 deg (forward or backward) of N, there is a solar eclipse. If the Sun happens, on the other hand, to be within about 10 deg of N' at this time, there is a lunar eclipse. When the Moon is at N' the situation is reversed‑there is a solar eclipse if the Sun is within about 15 deg of N, and a lunar eclipse if the Sun is within about 10 deg of N.

Let us suppose we know these simple rules. To decide whether eclipse conditions will occur or not at any time, say, over the next year, we have to know: (1) where the Sun will be in its orbit at all times, (2) where the Moon will be, and (3) where N, N' are with respect to the Sun's orbit. Let us discuss (1), (2), and (3) in sequence.

If we could determine where the Sun is at a particular moment, say, at midsummer, we could extrapolate the Sun's motion ahead, provided we know there are 365 1/4 days in the year. Since there are 360 deg around a circle, the Sun moves on the average by a little less than 1 deg per day. If we had a circle marked out accurately in degrees, we would simply move some pointer along the circle by rather less than 1 deg per day, starting the pointer at some agreed 'first point' on midsummer day. In fact, we would move the pointer by 360/365 1/4 degrees per day. This would be rather an awkward fraction, but by graduating our circle with sufficient accuracy we could manage even an awkward fraction.

To obtain accuracy in graduating our circle, we would naturally make it in metal, but this stoneage man could not do. If we were forbidden to use metal, our only recourse would be to make the diameter of the circle very large. But if we were to attempt to make a very large circle, say, from wood, how should we handle it? How should we prevent distortions occurring in its shape? Since we have no wish to move the circle, once it was suitably positioned, the obvious solution to the problem is simply to mark the circle out on a piece of flat ground.

Next we have to decide how to graduate the circle. We ourselves would be very likely to use the 360 units of division described above. But there is no reason why stoneage man should have used 360 units. Indeed, if stoneage man were a sophisticated astronomer, there is every reason why he would not have used the number 360. This number comes from bad astronomy. The interval betwedn successive times of full Moon is about 29 1/2 days. In a year of 365 ¼days, there are about 365 1/4/29 ½ such periods, which is rather more 12. If we make the mistake of supposing that the year is exactly 12 such periods, and if we make the further mistake of supposing that each period is exactly 30 days, then the year comes out at 360 days, and there would be exactly 1 deg per day of motion of the Sun along our circle. But I will suppose that stoneage man had meticulously observed the Sun and Moon, not just over a few years, or even over a few centuries, but over many millennia. I will suppose that he knew perfectly well that there are 365 1/4 days in the year. Then he would not divide his circle into 360 equal parts.

It would, in any case, be cumbersome to attempt so many subdivisions of the circle. Fine graduation is only convenient when one works in metal. So let us choose a smaller number of divisions, say, 56. Now try the following rule. Move a marker, a stone on the ground, by two divisions every thirteen days. It will take 56/2 X 13 days to move the marker around the whole circle, i.e., 364 days. At the end of a complete circuit, the time taken differs from a year by only 11 days, so that motion along the circle according to our rule can be considered to represent the motion of the Sun in its orbit to within an error of not much more than 1 deg. This is acceptable, because when we refer back to the rules which determine the occurrence of eclipses, we are concerned with the positioning of the Sun to something like 10 deg. And provided we reset our marker also with suitable accuracy every midsummer day, there will be no cumulative error piling up from year to year. Better still, if we reset our marker not just once a year but twice a year, on midwinter clay as well as on midsummer day, the maximum error will be halved, to about 0.5 deg.

Next let us consider the logic of resetting the marker. Every midsummer day, we wish to start our inarker from an assigned first point. It does not inatter which place on the circle we choose as first point, but once we have chosen a definite point, it does matter very critically that we reset our marker precisely every midsunnuer day. If we were five days wrong in our determination of midsummer day, our marker would be 5 deg wrong in its position. All hope of making reliable eclipse predictions would then be gone. The critical question now emerges. Do the sighting lines of Stonehenge I permit midsummer day to be accurately determined? If not, what is the represents an error in the determination of midsummer day of likely error?

At first sight it seems easy to determine midsummer day. Simply determine the day on which the Sun rises most to the north. But the angle which the direction of sunrise makes with the southerly direction depends on a trigonometrical cosine function of the form

where t is the time and T is the length of the year. This function has a maximum value of unity whenever t is equal to an integral multiple of T. If we were given t, and our problem were to determine cos (2pt/ T), there would be no difficulty, but the situation at Stonehenge is the other way round. Observation of sunrise gives cos (2pt/ T), and our problem is to find t. Suppose we are working with t near zero. The actual maximum of unity for cos(2pt/T) then occurs at t = 0, but the value of cos(2pt/T) differs very little from unity even though t is quite appreciably different from zero. To sufficient accuracy, cos(2pt/T) can be written as

Now observations with stones and by naked eye cannot be expected to yield an accuracy of better than, say, 0.3 deg. The swing in the direction of sunrise is about 80 deg at Stonehenge; so our accuracy in determining the maximum swing is of the order of one part in 250. That is to say, the best we can hope for in determining t, by a direct attempt to determine when the Sun rises most to the north, is represented by a value of t/T such that cos(2pt/T) differs from unity by about one part in 250. This gives

and t/T is easily seen to be about 0.014. Since T = 365 1/4 days, this represents an error in the determination of midsummer day of about five days, which means that the positioning of our Sun marker could be in error by as much as 5 deg, bad because the error is no longer small enough compared with the angles of order 10 deg which determine the occurrence of eclipses. Since we would expect further errors to arise in estimating the positions of the Moon and of the nodes N, N', it is clear that a direct attempt to measure the extreme values of angles‑such as the most northerly rising of the Sun ‑cannot lead to a satisfactory method of eclipse prediction.

An idea is needed to get the calibration of the Sun marker down to an error of no more than 1 deg. Suppose a sighting line is set up to determine not the most northerly rising of the Sun, but a point, say, 1.5 deg to the south of the most northerly rising. The situation is illustrated in Figure 2.13. The Sun will cross the stone going north some two weeks before midsummer day, and it will again cross the stone going south an equal time after midsummer day. So long as we can determine accurately the days on which the Sun crosses the stone, we can correctly set midsummer day as halfway between the stone‑crossing days. This provides the idea. Next it is necessary to examine more carefully the concept of 'crossing the stone.' Intuitively, we mean that one day the Sun rises to the south of the stone and the next day to the north of it, and vice versa. But what precise procedure would we adopt to decide such an issue?

Because the Sun has an apparent diameter of 32', the meaning of the Sun 'rising' is itself ambiguous. We could define sunrise as the first flash of light from the solar disk as the upper limb rises above the horizon. We could define sunrise as the moment when the whole solar disk first stands on the horizon. Or we could define sunrise as the moment when the solar centre first appears. This third possibility is more a theoretical concept than a useful practical definition. Unless there is a low‑lying absorbing mist, it is not possible to look directly at the full Sun, and hence it is not possible with the unaided eye to judge its position with adequate accuracy. The best defined moment during sunrise is undoubtedly at the first flash of light from the upper limb. The observer can watch for this moment without experiencing excessive glare. Yet it is not easy to judge the precise relationship of a bright point of light to the dark silhouette of a stone, unless we use the following more subtle procedure.

Suppose we arrange for our stone to project a little above t horizon. Then there will be a small arc over which the stone obscures the first flash of the Sun. As the Sun swings across the stone it will encounter this blank arc. The stone would prevent the first flash from being seen by an observer on one or possibly two days the Sun swings to the north, and then again on one or two days the Sun swings back to the south. By observing both these blank periods and by taking the middle day between them we could arrive at midsummer day with complete precision. How would we know that a day was blank? By having a colleague sited elsewhere, i.e. not at the centre of Stonehenge. He signals the moment of first flash. If we have not observed it our self, the day is blank.

Such a scheme would serve without further difficulty to set the Sun marker on our 56‑hole circle at its 'first point,' if we could sure that every morning was fine. But two or three days of weather at a critical 'blank‑arc period' would ruin the determination. For just one particular sighting line, nothing could be done about such an eventuality. But we could set ‑np several different sighting lines in slightly different directions, with blank arcs disposed differently with respect to midsummer day. We could, for example, set one sighting line 0.5 deg back from the most northerly rising, set another 1.0 deg back, another 1.5 deg, another 2 deg. This would greatly increase our chance of calibrating the sun marker.

At Stonehenge there is in fact just such a multiplicity of slightly different sighting lines, each a little on the south side of the direction of most northerly rising of the Sun. When I first checked the directions of the sighting lines given by Hawkins in his first letter to Nature (Figure 2.12), 1 was puzzled by why there should be so many of them and why they were all slightly different, and by why they all differed from what seemed the correct theoretical direction by amounts that were too large to be attributed to constructional errors. Constructional errors might have been about 1/4 or 1/3 deg or but not 1 deg to 2 deg, such as I actually found. But then I was looking for the extreme values. That is to say, I expected the sunrise sighting lines to coincide precisely with the most northerly rising of the Sun. It took some time to realize that this in fact would be a very poor procedure, indeed, that offsets of a degree or two were exactly what was needed. And it took still longer to understand why there should be so many slightly different sighting lines.

There is no profit in attempting to determine the position of the Moon in the same way, because the swing in the direction of moonrise does not repeat itself from month to month. All we need do to calibrate the Moon, however, is to notice that when the Moon is ‘new' it lies in the direction of the Sun, when it is 'full' it lies in the opposite direction. So we place a Moon marker on our circle, arranging for it to be suitably disposed with respect to the Sun marker at the observed times of new moon and full moon. The Moon goes round its orbit in 27.3 days (not the same as the time between successive times of full moon, because the Sun is moving). Remembering that we have 56 equal divisions on our circle, we see that the Moon averages a little more than two divisions per day. Because we can reset the Moon marker essentially every half‑circuit, it will be sufficient to arrange that the Moon marker is moved by just two divisions per day.

Finally, we need to know the whereabouts with respect to the Sun of the nodes of the Moon's orbit. If one understands astronomy, it is not difficult to arrive at the following rule:

In the month in which the Moon rises most to the north in its 18.61 ‑year cycle, the node N of Figure 2.11 follows by 90 deg the 'first point' on our circle ‑ i.e., the point where we place the Sun marker on midsummer day. The node N' is opposite to N and therefore precedes the 'first point' by 90 deg. The sense in which 'following' and 'preceding' is to be understood is the sense of the motion of the Sun marker.

But could the men (or women) of Stonehenge I really have understood anything so erudite as this? Such a question cannot be decided by preconceptions. One must look at the evidence. If we were attempting to apply the above rule, we would be ill‑advised to attempt to determine the most northerly rising of the Moon by erecting a sighting precisely in the direction of most northerly rising. This would run us into the same accuracy difficulty that we have already encountered in the determination of midsummer day. The problem is in some respects less difficult than it was for the Sun, because the Moon does not produce the overwhelming glare of the Sun, and also because the issue is not so urgent, in the sense that the nodal points N, N', move eighteen times more slowly than the Sun. If we were even a month wrong in the determination of the most northerly rising of the Moon, the consequent error in the positioning of N would be about 1.5 deg. Such an error would be unfortunate but not devastating. Because we are not pressed so tightly in time, we are not so much at the mercy of the weather.

Once again we would proceed by setting up a multiplicity of sighting lines slightly offset to the south from the direction of most northerly rising of the Moon. When I first encountered this question, I found (to my surprise) that an ordered set of sighting lines of this kind did in fact exist at Stonehenge I. They are formed by lines from the center to certain postholes known. as A1, A 2, A3, A4, placed a regular intervals near the Heelstone. These postholes are seen in the

plan shown in Figure 2.4. 1 was so moved by this discovery that in 1966 1 also wrote a letter to Nature. It will be useful here simply to reproduce this letter.

STONEHENGE‑AN ECLIPSE PREDICTOR

Professor Fred Hoyle

University of Cambridge

The suggestion. that Stonehenge may have been constructed with a serious astronomical purpose has recently received support from Hawkins, who has shown [Hawkins, G. S., Nature, 200, 306 (1963). ] that many alignments of astronomical significance exist between different positions in the structure. Some workers have questioned whether, in an arrangement possessing so many positions, these alignments can be taken to be statistically significant. I have recently reworked all the alignments found by Hawkins. My opinion is that the arrangement is not random. As Hawkins points out, some positions are especially relevant in relation to the geometrical regularities of Stonehenge, and it is these particular positions which show the main alignments. Furthermore, I find these alignments are just the ones that could have served far‑reaching astronomical purposes, as I shall show in this article. Thirdly, on more detailed investigation, the apparently small errors, of the order of ± 1 deg, in the alignments turn out not to be errors at all.

In a second article Hawkins [Hawkins, G. S., Nature, 202, 1258 (1964)] goes on to investigate earlier proposals that Stonehenge may have operated as an eclipse predictor. The period of regression of the lunar nodes, 18.61 years, is of especial importance in the analysis of eclipses. Hawkins notes that a marker stone moved around the circle of 56 Aubrey holes at a rate of three holes per year completes a revolution of the circle in 18.67 years. This is close enough to 18.61 years to suggest a connexion between the period of regression of the nodes and the number of Aubrey holes. In this also I agree with Hawkins. I differ from him, however, in the manner in which he supposes the eclipse predictor to have worked. Explicitly, the following objections to his suggestions seem relevant:

(1) The assumption that the Aubrey holes served merely to count cycles of 56 years seems to me to be weak. There is no need to set out 56 holes at regular intervals on the circumference of a circle of such a great radius in order to count cycles of 56.

(2) It is difficult to see how it would have been possible to calibrate the counting system proposed by Hawkins. He himself used tables of known eclipses in order to find it. The builders of Stonehenge were not equipped with such post hoc tables.

(3) The predictor gives only a small fraction of all eclipses. It is difficult to see what merit would have accrued to the builders from successful predictions at intervals as far apart as ten years. What of all the eclipses the system failed to predict?

My suggestion is that the Aubrey circle represents the ecliptic. The situation shown in Figure 2.14 corresponds to a moment when the Moon is full. The first point of Aries g has been arbitrarily placed at hole 14. S is the position of the Sun, the angle (0) is the solar longitude, M is the projection of the Moon on to the ecliptic, N is the ascending node of the lunar orbit, N' the descending node, and the centre C is the position of the observer. As time passes, the points S, M, N, and N move in the senses shown in Figure 2.14. S makes one circuit a year, M moves more quickly, with one circuit in a lunar month. One rotation of the line of lunar nodes N N' is accomplished in 18.61 years. In Figure 2.14, S and M are at the opposite ends of a diameter because the diagram represents the state of affairs at full Moon.

If the Moon is at N, there is a solar eclipse if the Sun is within roughly 15 deg of N, and a lunar eclipse if the Sun is within ± 10 deg of N'. Similarly, if the Moon is at N', there will be a solar eclipse if the Sun is within ± 15 deg of coincidence with the Moon, and a lunar eclipse if it is within roughly ± 10 deg of the opposite end of the line of lunar nodes. Evidently if we represent S, M, N, and N' by markers, and if we know how to move the markers so as to represent the actual motions of the Sun and Moon with adequate accuracy, we can predict almost every eclipse, although roughly half of them will not be visible from the position of the observer. This is a great improvement on the widely scattered eclipses predictable by Hawkins's system. Eclipses can occur as many as seven times in a single year, although this would be an exceptional year.

The prescriptions for moving the markers are as follows: (1) Move S anticlockwise two holes every 13 days. (2) Move M anticlockwise two holes each day. (3) Move N and N' clockwise three holes each year.

We can reasonably assume that the builders of Stonehenge knew the approximate number of days in the year, the number of days in the month, and the period of regression of the nodes. The latter follows by observing the azimuth at which the Moon rises above the horizon. If in each lunar month we measure the least value of the azimuth (taken cast of north), we find that the 'least monthly values' change slowly, because the angle f = NCg changes. The behavior of the 'least monthly values' is shown in Figure 2.15 for the range – 60 deg < f :5 60 deg. (The azimuthal values in Figure 2.15 were worked out without including a refraction or a parallax correction. These small effects are irrelevant to the present discussion.) The least monthly values oscillate with the period of f, 18.61 years. By observing many azimuthal cycles the period of f can be determined with high accuracy. At Stonehenge sighting alignments exist that would have suited such observations. With the periods of S, M, and N known with reasonable accuracy, the prescriptions follow immediately as approximate working rules.

Suppose an initially correct configuration for M, N, and S is known. The prescriptions enable us to predict ahead what the positions of M, N, and S are going to be, and thus to foresee coming events‑but only for a while, because inaccuracies in our prescriptions will cause the markers to differ more and more from the true positions of the real Moon, Sun, and ascending node. The lunar marker will be the first to deviate seriously the prescription gives an orbital period of 28 days instead of 27.32 days. But we can make a correcting adjustment to the M marker twice every month, simply by aligning M opposite S at the time of full moon, and by placing it coincident with S at new moon. The prescription for S gives an orbital period of 564 days, which is near enough to the actual period because it is possible to correct the position of S four times every year, by suitable observations made with the midsummer, midwinter, and equinoctial sighting lines that are set up with such remarkable accuracy at Stonehenge.

Stonehenge is also constructed to determine the moment when f = 0, that is, when N should be set at g. The line C to A1 of Figure 2.14 is the azimuthal direction for the minimum point of Figure 2.15. By placing N at g when the Moon rises farthest to the north, the N marker can b calibrated once every 18.61 years. The prescription implies only a small error over one revolution of N. If N started correctly, it would be out of its true position by only 1 deg or so at the end of the first cycle. The tolerance for eclipse prediction is about 5 deg, so that if we were to adjust N ever cycle, the predictor would continue to work indefinitely without appreciable inaccuracy. The same method also serves to place N at the be ginning.

But now we encounter an apparent difficulty. The minimum of Figure 2.15 is very shallow and cannot really be determined in the way I have just described. Angular errors cannot have been less than ±0.25 deg, an even this error, occurring at the minimum of Figure 2.15, is sufficient t produce an error of as much as ± 15 deg in f.

The correct procedure is to determine the moment of the minimum by averaging the two sides of the symmetrical curve, by taking a mean between points 2, for example. The inaccuracy is then reduced to not more than a degree or two‑well within the permitted tolerance.

What is needed is to set up sighting directions a little to the east of the most northerly direction. The plan of Stonehenge shows a line of postholes, A1, 2, 3, and 4, placed regularly and with apparent purpose in exactly the appropriate places.

The same point applies to solsticial measurements of the Sun. In summer the sighting line should be slightly increased in azimuth, in winter it should be slightly decreased.

Hawkins' gives two tables in which he includes columns headed – ‘Error Alt.'. These altitude errors were calculated on the assumption that the, builders of Stonehenge intended to sight exactly the azimuthal extremes. The test of the present ideas is whether the calculated 'errors' have the appropriate sign ‑ on the argument given here 'errors' should be present and they should have the same sign as the declination. In ten out of twelve values which Hawkins gives in his Table I this is so. The direction from C to the Heelstone is one of the two outstanding cases. Here the ‘error' is zero, suggesting that this special direction was kept exactly at the direction of midsummer sunrise, perhaps for aesthetic or ritualistic reasons. The other discrepant case is 91 > 94. Here my own calculation gives only a very small discrepancy, suggesting that this direction was also kept at the appropriate azimuthal extreme.

Negative values of the altitude error correspond to cases where it would be necessary to observe below the horizontal plane, if the objects in question were sighted at their extreme azimuths. This is impossible at Stonehenge because the land slopes gently upward in all directions. Such sighting lines could not have been used at the extremes, a circumstance which also supports this point of view.

It is of interest to look for other ways of calibrating the N marker. A method, which at first sight looks promising, can be found using a special situation in which full moon happens to occur exactly at an equinox. There is evidence that this method was tried at Stonehenge, but the necessary sighting lines are clearly peripheral to the main structure. Further investigation shows the method to be unworkable, however, because unavoidable errors in judging the exact moment of full moon produce large errors in the positioning of N. The method is essentially unworkable because the inclination of the lunar orbit is small. Even so, the method may well have caused a furore in its day, as the emphasis it gives to a full moon at the equinox could have been responsible for the dating of Easter.

An eclipse calibrator can be worked accurately almost by complete numerology, if the observer is aware of a curious near‑commensurability. Because S and N move in opposite directions, the Sun moves through N more frequently than once a year, in 346.6 days. Nineteen such revolutions is equal to 6,585.8 days, whereas 223 lunations is equal to 6,585.3 days. Thus after 223 lunations the N marker must bear almost exactly the same relation to S that it did before. If the correct relation of N to S is known at any one moment, N can be reset every 223 lunations; that is, every 18 years 11 days. The near‑commensurability is so good that this system would give satisfactory predictions for more than 500 years. It requires, of course, S to be set in the same way as before. The advantage is that in the case of N it obviates any need for the observational work described above. But without observations the correct initial situation cannot be determined unless the problem is inverted. By using observed eclipses the calibrator could be set up by trial and error. This is probably the method of the Saros used in the Near East. There is no evidence that it was used at Stonehenge. The whole structure of Stonehenge seems to have been dedicated to meticulous observation. The method of Stonehengc would have worked equally well even if the Saros had not existed.

Several interesting cultural points present themselves. Suppose this system was invented by a society with cultural beliefs associated with the Sun and Moon. If the Sun and Moon are given godlike qualities, what shall we say of N? Observation shows that whenever M and S are closely associated with N, eclipses occur. Our gods are temporarily eliminated. Evidently, then, N must be a still more powerful god. But N is unseen. Could this be the origin of the concept of an invisible, all‑powerful god, the God of Isaiah? Could it have been the discovery of the significance of N that destroyed sun‑worship as a religion? Could M, N, and S be the origin of the doctrine of the Trinity, the 'three‑in‑one, the one‑in‑three'? It would indeed be ironic if it turned out that the roots of much of our present day culture were determined by the lunar node.

The style of this letter is naturally a bit technical, but the situation concerning the postholes Al, A2, A3, A4 emerges (in Figure 2.14). The positioning of these postholes establishes for me that the builders of Stonehenge I did in fact understand the astronomical rule set out above. It is admittedly astonishing that they should have done so, but our astonishment in this respect could really be a measure of our ignorance rather than a reflection on the intellectual capacity of stoneage man. If this be true, and the evidence seems quite overriding, then we must dismiss the scientific achievements of Babylon as comparatively trivial. It is not until we come to Hipparchus and Ptolemy that anything of comparable stature can be found in the ancient world, and not until we move forward to Copernicus in the modern world. To paraphrase Brahms in his reference to Beethoven, we hear the tramp of the giant behind us.

All this refers to Stonehenge I. What of Stonehenge II and Stonehenge III? The odd and worrying thing is that we have been able to complete the story without referring at all to the later developments at Stonehenge. We might have hoped to find a culmination of intellectual quality in these later constructions, but in vain. Why, if one could succeed triumphantly in predicting eclipses‑no easy matter‑should one trouble to haul a great mass of stones all the way from the Prescelly Mountains of Wales? I could find no rationale for such an extraordinary procedure. Then the inner structures of Stonehenge are too small in their diameter to permit angles to be judged with sufficient accuracy. In short, I could find no sensible astronomical interpretation of the later activities at Stonehenge. It is true that Hawkins also found astronomical sighting lines in the inner structure, but these seemed less well‑chosen than the sighting lines of Stonehenge I. Nor could I find any evidence of the subtle method of fixing calibration positions in the later inner structure. And in this absence of subtlety there seemed to me a clue to what may have happened in the time interval between Stonehenge I and Stonehenge II.

Suppose the subtle ideas of Stonehenge I became lost, but that a memory of successful eclipse predictions lingered on. Suppose a later age attempted to recover the eclipse predictions, suppose indeed that incomplete prescriptions survived from earlier times. Suppose later attempts at the critical calibrations were made in terms of a direct assault on the determination of the most northerly swings of the Sun and Moon. Such attempts, however fiercely prosecuted, were certain to fail. Occasional partial successes might have been achieved, but these would inevitably have been followed by further disappointments. In such a situation, one might expect desperate measures to be taken. Strange ideas would be tried. It would be remembered that in an early age stones had somehow played a crucial role, and this might well have led to a mystical quality being associated with them. It may well have led to stones with a presumed special quality being hauled all the way from Wales.

So I was led to postulate an intellectual, although not necessarily a technological, decline between the times of Stonehenge I and Stonehenge III. The trouble with this idea was that carbon dating gave perhaps 2000 B.C. for Stonehenge I and perhaps 1750 B.C. for the main part of Stonehenge III. I found it hard to believe in a serious and essentially catastrophic intellectual decline taking place in only 250 years. I thought in terms of bronze age invaders from Europe breaking up an indigenous culture. There was an element of plausibility in this idea, but it scarcely seemed adequate.

Then what of the forerunners of Stonehenge I? The first builders at Stonehenge seemed to know exactly what they were doing. Their knowledge must have been derived from many centuries of earlier observations. What of the structures which they built in these earlier observations? It seemed attractive to suppose that the many small stone circles scattered over England represented such earlier structures. Yet when I came to examine the disposition of these circles, I could find no association with the ideas of Stonehenge I. So with reluctance I concluded that the stone circles were not forerunners of Stonehenge I, and because I found them primitive in concept, I was forced to ascribe to them a later date than Stonehenge I, to fit the idea of an intellectual decline following Stonehenge I.

Several months later I had a long chat with Professor Atkinson on the problems of Stonehenge. To my delight (and relief) he informed me that the stone circles are indeed younger than Stonehenge I. Then I had a great stroke of luck. I chanced to meet Hans Suess while waiting for a plane at London Airport. He told me that by using wood from a bristlecone pine some 7,000 years old, obtained by C. W. Ferguson of the University of Arizona, he had made an absolute calibration of the carbon‑dating method. To my questions, he told me that many ancient archaeological dates were now much changed, especially dates around 2000 B.C. Rushing to catch his plane, he pushed a sheaf of papers into my hand. They contained the new calibration. It had the effect of greatly stretching out the interval (on the old system) between 1500 B.C. and 2200 B.C. Thus 1500 B.C. became about 1700 B.C. on the new system, while 2200 B.C. on the old system became pushed back to about 2600 B.C. So on the new system the gap between Stonehenge I and the later parts of Stonehenge III widened to almost 1,000 years. This was amply sufficient to sustain the idea of a major intellectual change taking place between the stoneage and the bronzeage. Metal smelting was perhaps the greatest of all technological innovations. It would be strange indeed if it failed to produce great social changes, which need not have implied progress in astronomy.

The Stonehenge I system of eclipse prediction contains a trap. Given a calibration of the lunar nodes, the system would continue to work for almost a century without a further recalibration of the nodes being necessary. Since this is considerably longer than a human generation, the key idea connected with the postholes A1, A2, A3, A4 could be lost without the system appearing to go wrong for a very long time. This makes it easy to see how, in a community bemused by the new concepts of metal smelting, the old ideas of astronomy might come to be forgotten.

The issue of the forerunners to Stonehenge is as yet unresolved. I suspect earlier structures were built in wood and have not survived. Shakespeare wrote about everything under the sun except about himself. At Stonehenge we have a profusion of ideas, but little about the people themselves. I believe I understand the practical method by which they laid out their geometrical structure‑and this too is an interesting story‑but I have no idea how they made their astronomical calculations. I would expect them to have used devices similar to the abacus, but how they conceived of numbers and how they carried through the basic processes of arithmetic remains unknown.

Even so, it is interesting to speculate. Perhaps you may think I have been doing nothing else but speculate. So why start now? Then let me say this: It is not a speculation to assert that we ourselves could use Stonehenge I to make eclipse predictions. We could certainly do so without making any substantive changes in ‑the layout. While this does not prove that stoneage man did in fact use Stonehenge I for making eclipse predictions, the measure of coincidence otherwise implied would be quite fantastic. How does one prove any incident belonging to the past? Historians argue by documentary evidence. But how if their documents are false? A plethora of documents belonging to the present day are false, many of them made so deliberately. It is not possible to argue that Stonehenge I was falsified deliberately, to maintain a facade of astronomical subtlety by a people ignorant of astronomy. It will probably be hard for the historian to accept the idea of a geometrical arrangement of stones and holes providing evidence much stronger than a document, but I believe this to be true.

Now back to the speculation. To maintain our eclipse predictor we made a rule for following the motion of the Sun:

Move a marker stone by two Aubrey holes every 13 days.

How does one count two holes every 13 days? Not by memory, because sooner or later a mistake will be made. Not by making a note on paper‑at any rate, not for stoneage man. A simple procedure would be to mark out a circle on the ground. Make 13 divisions of the circle, numbering them in order 1, 2, . . . , 13. Start on a day when the sun stone is advanced one Aubrey hole by placing a counter on section I of our new circle. Move the counter one sector each day. When it reaches sector 7, move the sun marker by another Aubrey hole. When the counter comes back to sector I move the sun marker again, and so on.

This system can be refined by giving each of our 13 sectors a sunrise position and a sunset position, and by moving our counter twice a day, once in the morning, once in the evening. If the sun marker has been advanced by an Aubrey hole with the counter on the dawn position of sector 1, we again move the sun marker when the counter reaches the evening position of sector 7, and so on. This procedure has the effect of moving the sun marker by one Aubrey hole every 6 1/2 days, which is more precise than two holes every 13 days.

It is interesting to find that we have divided the days into cycles of 13, and that the number 7 has appeared. So we have the unlucky number 13 and also the mystic number 7.

The system I have described has the disadvantage that we must mark sectors 1 and 7 explicitly. We have to know which sector we are to count from. We could for example divide the circle in such a way that sector I was a bit longer than the others, and sector 7 a bit shorter. This would be foolproof but it would be inelegant.

Now I wish to consider another way to count one every 6 1/2 days. Divide a circle into 7 parts, this time all equal. Let each division again have two positions, a dawn position and a sunset position. Instead of a single counter use two, a dark counter and a light counter, and make the following rules.

1 The light counter must always be in a dawn position, the dark counter always in an evening position.

2 Counters always move in the same sense along the circle, for example, in a clockwise sense.

3 One counter cannot jump over another.

4 Start with the counters adjacent to each other, one in the dawn position, one in the evening position, with the dawn count ahead if the sun stone has been moved at dawn, but with the evening counter ahead if the sun stone has been moved in the evening.

5 So long as rule 3 can be obeyed, move whichever counter start ahead by one sector each day.

6 Continue until the counters come adjacent again, as they wi do after 6 days. Change then to the other counter at the suc ceeding dawn or sunset (according to its colour), and mov the sun marker by one Aubrey hole.

7 Proceed as before.

This method is illustrated in Figure 2.16,

Of these two methods, the first is simpler to understand, but the second is much the more elegant. It uses no special starting point on the circle. It requires no uneven division of the circle. It uses all sectors equally.

I will end now by making three apparently grotesque speculations. The notion of 13 being unlucky is a corruption arising from the method of 13 divisions, which was thought by stoneage man to be inferior. Because the method of 7 is harder, those who failed to understand it might perhaps have been considered 'simple,' and hence unlucky.

The arrangement in the second method of days into cycles of 7 was the origin of the week.

The game of checkers dates from the stoneage. The rules required by the method of 7 lead naturally to the concepts of counters, light and dark according to dawn and dusk, to counters not jumping (or jumping) over each other, and to the concept of counters always moving in a particular sense.

I suspect the mathematical methods of stoneage man are exemplified by this method of 7. I also suspect that much of our present day culture ‑ especially those things we grew up with and regard as so natural that we never bother to question them ‑ we owe to our stoneage ancestors.