THE PROBLEM OF THE GREAT PYRAMID.

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FEW months ago I endeavoured to trace out, in these pages, the probable origin of the week, as a measure of time, by a method which has not hitherto, so far as I know, been followed in such cases. I followed chiefly a line of à priori reasoning, considering how herdsmen and tillers of the soil would be apt at a very early period to use the moon as a means of measuring time, and how in endeavouring so to use her they would almost of necessity be led to employ special methods of subdividing the period during which she passes through her various phases. But while each step of the reasoning was thus based on à priori considerations, its validity was tested by the evidence which has reached us respecting the various methods employed by different nations of antiquity for following the moon's motions. It appears to me that the conclusions to which this method of reasoning led were more satisfactory, because more trustworthy, than those which have been reached respecting the week by the mere study of various traditions which have reached us respecting the early use of this widespread time

measure.

I now propose to apply a somewhat similar method to a problem which has always been regarded as at once highly interesting and very difficult, the question of the purpose for which the pyramids of Egypt, and especially the pyramids of Ghizeh, were erected. But I do not here take the full problem under consideration. I have, indeed, elsewhere dealt with it in a general manner, and have been led to a theory respecting the pyramids which will be touched on towards the close of the present paper. Here, however, I intend to deal only with one special part of the problem, that part to which alone the method I propose to employ is applicable-the question of the astronomical purpose which the pyramids were intended to subserve. It will be understood,

therefore, why I have spoken of applying a somewhat similar method, and not a precisely similar method, to the problem of the pyramids. For whereas in dealing with the origin of the week, I could from the very beginning of the inquiry apply the à priori method, I cannot do so in the case of the pyramids. I do not know of any line of à priori reasoning by which it could be proved, or even rendered probable, that any race of men, of whatever proclivities or avocations, would naturally be led to construct buildings resembling the pyramids. If it could be, of course that line of reasoning would at the same time indicate what purposes such buildings were intended to subserve. Failing evidence of this kind, we must follow at first the à posteriori method; and this method, while it is clear enough as to the construction of pyramids, for there are the pyramids themselves to speak unmistakably on this point, is not altogether so clear as to any one of the purposes for which the pyramids were built.

Yet I think that if there is one purpose among possibly many which the builders of the pyramids had in their thoughts, which can be unmistakably inferred from the pyramids themselves, independently of all traditions, it is the purpose of constructing edifices which should enable men to observe the heavenly bodies in some way not otherwise obtainable. If the orienting of the faces of the pyramids had been effected in some such way as the orienting of most of our cathedrals and churches-i.e., in a manner quite sufficiently exact as tested by ordinary observation, but not capable of bearing astronomical tests,-it might reasonably enough be inferred that having to erect square buildings for any purpose whatever, men were likely enough to set them four-square to the cardinal points, and that, therefore, no stress whatever can be laid on this feature of the pyramids' construction. when we find that the orienting of the pyramids has been effected with extreme care, that in the case of the great pyramid, which is the typical edifice of this kind, the orienting bears well the closest astronomical scrutiny, we cannot doubt that this feature indicates astronomical purpose as surely as it indicates the use of astronomical methods.

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But while we thus start with what is to some degree an assumption, with what at any rate is not based on à priori considerations, yet manifestly we may expect to find evidence as we proceed which shall either strengthen our opinion on this point, or show it to be unsound. We are going to make this astronomical purpose the startingpoint for a series of à priori considerations, each to be tested by whatever direct evidence may be available; and it is practically certain that if we have thus started in an entirely wrong direction, we shall before long find out our mistake. At least we shall do so, if we start with the desire to find out as much of the truth as we can, and not with the determination to see only those facts which point in the direction along which we have set out, overlooking any which seem to point

in a different direction. We need not necessarily be in the wrong track because of such seeming indications. If we are on the right track, we shall see things more clearly as we proceed; and it may be that evidence which at first seems to accord ill with the idea that we are progressing towards the truth, may be found among the most satisfactory evidence obtainable. But we must in any case note such evidence, even at the time when it seems to suggest that we are on the wrong track. We may push on, nevertheless, to see how such evidence appears a little later. But we must by no means forget its existence. So only can we hope to reach the truth or a portion of the truth, instead of merely making out a good case for some particular theory.

We start, then, with the assumption that the great pyramid, called the Pyramid of Cheops, was built for this purpose, inter alia, to enable men to make certain astronomical observations with great accuracy; and what we propose to do is to inquire what would be done by men having this purpose in view, having, as the pyramid builders had, (1) a fine astronomical site, (2) the command of enormous wealth, (3) practically exhaustless stores of material, and (4) the means of compelling many thousands of men to labour for them.

Watching the celestial bodies hour by hour, day by day, and year by year, the observer recognizes certain regions of the heavens which require special attention, and certain noteworthy directions both with respect to the horizon and to elevation above the horizon.

For instance, the observer perceives that the stars, which are in many respects the most conveniently observable bodies, are carried round as if they were rigidly attached to a hollow sphere, carried around an axis passing through the station of the observer (as through a centre) and directed towards a certain point in the dome of the heavens. That point, then, is one whose direction must not only be ascertained, but must be in some way or other indicated. Whatever the nature of an astronomer's instruments or observatory, whether he have but a few simple contrivances in a structure of insignificant proportions, or the most perfect instruments in a noble edifice of most exquisite construction and of the utmost attainable stability, he must in every case have the position of the pole of the heavens clearly indicated in some way or other. Now, the pole of the heavens is a point lying due north, at a certain definite elevation above the horizon. Thus the first consideration to be attended to by the builder of any sort of astronomical observatory, is the determination of the direction of the true north (or the laying down of a true north-and-south line), while the second is the determination, and in some way or other the indication of the angle of elevation above the north point, at which the true pole of the heavens. may lie.

To get the true north-and-south line, however, the astronomer would be apt at first, perhaps, rather to make mid-day observations than to observe the stars at night. It would have been the observation of these

which first called his attention to the existence of a definite point round which all the stars seem to be carried in parallel circles; but he would very quickly notice that the sun and the moon, and also the five planets, are carried round the same polar axis, only differing from the stars in this: that, besides being thus carried round with the celestial sphere, they also move upon that sphere, though with a motion which is very slow compared with that which they derive from the seeming motion of the sphere itself. Now, among these bodies the sun and moon possess a distinct advantage over the stars. A body illuminated by either the sun or the moon throws a shadow, and thus if we place an upright pointed rod in sunlight or moonlight, and note where the shadow of the point lies, we know that a straight line from the point to the shadow of the point is directed exactly towards the sun or the moon, as the case may be. Leaving the moon aside as in other respects unsuitable, for she only shines with suitable lustre in one part of each month, we have in the sun's motions a means of getting the north-and-south line by thus noting the position of the shadow of a pointed upright. For being carried around an inclined axis directed northwards, the sun is, of course, brought to his greatest elevation on any given day when due south. So that if we note when the shadow of an upright is shortest on any day, we know that at that moment the sun is at his highest or due south; and the line joining the centre of the upright's base with the end of the shadow at that instant lies due north-and-south.

But though theoretically this method is sufficient, it is open, in practice, to a serious objection. The sun's elevation, when he is nearly at his highest, changes very slowly; so that it is difficult to determine the precise moment when the shadow is shortest. But the direction of the shadow is steadily changing all the time that we thus remain in doubt whether the sun's elevation has reached its maximum or not. We are apt, then, to make an error as to time, which will result in a noteworthy error as to the direction of the north-and-south line.

For this reason, it would be better for any one employing this shadow method to take two epochs on either side of solar noon, when the sun was at exactly the same elevation, or the shadow of exactly the same length, determining this by striking out a circle around the foot of the upright, and observing where the shadow's point crossed this circle before noon in drawing nearer to the base, and after noon in passing away from the base. These two intersections with the circle necessarily lie at equal distances from the north-and-south line, which can thus be more exactly determined than by the other method, simply because the end of the shadow crosses the circle traced on the ground at moments which can be more exactly determined than the moment when the shadow is shortest.

Now, we notice in this description of methods which unquestionably were followed by the very earliest astronomers, one circumstance which clearly points to a feature as absolutely essential in every astronomical

observing station. (I do not say "observatory," for I am speaking just now of observations so elementary that the word would be out of place.) The observer must have a perfectly flat floor on which to receive the shadow of the upright pointer. And not only must the floor be flat, but it must also be perfectly horizontal. At any rate, it must not slope down either towards the east or towards the west, for then the shadows on either side of the north-and-south line would be unequal. And though a slope towards north or south would not affect the equality of such shadows, and would therefore be admissible, yet it would clearly be altogether undesirable; since the avoidance of a slope towards east or west would be made much more difficult if the surface were tilted, however slightly, towards north or south. Apart from this, several other circumstances make it extremely desirable that the surface from which the astronomers make their observations should be perfectly horizontal. In particular, we shall see presently that the exact determination of elevations above the eastern and western horizons would be very necessary even in the earliest and simplest methods of observation, and for this purpose it would be essential that the observing surface should be as carefully levelled in a north-and-south as in an east-andwest direction.

We should expect to find, then, that when the particular stage of astronomical progress had been reached, at which me not only perceived the necessity of well-devised buildings for astronomical observation, but were able to devote time, labour, and expense to the construction of such buildings, the first point to which they would direct their attention would be the formation of a perfectly level surface, on which eventually they might lay down a north-and-south or true meridional line.

Now, of the extreme care with which this preliminary question of level was considered by the builders of the great pyramid, we have singularly clear and decisive evidence. For all around the base of the pyramid there was a pavement, and we find the builders not only so well acquainted with the position of the true horizontal plane at the level of this pavement, but so careful to follow it (even as respects this pavement, which, be it noticed, was only, in all probability, a subsidiary and quasiornamental feature of the building), that the pavement "was varied in thickness at the rate of about an inch in 100 feet to make it absolutely level, which the rock was not."*

But now with regard to the true north-and-south direction, although the shadow method, carried out on a truly level surface, would be satisfactory enough for a first rough approximation, or even for what any but astronomers would regard as extreme accuracy, it would be open to

* It seems to me not improbable that the level was determined by simply flooding (though to a very small depth only, of course) the entire area to be levelled-not only the pavement level, but higher levels as the pyramid was raised layer by layer. By completing the outside of each layer first, an enclosed space capable of receiving the water would be formed (the flooding being required once only for each layer), and when the level had been taken the water could be allowed to run off by the interior passages to the well which Piazzi Smyth considers to be symbolical of the bottomless pit.