Chapter IV (3)

The lower part of a pyramid would fulfil the conditions required for the stability of such a structure, and a square or oblong form would be suitable for the base of such a pyramid. We must not overlook the fact that a complete pyramid would be utterly unsuitable for an astronomical edifice. Even a pyramid built up of layers of stone and continued so far upwards that the uppermost layer consisted of a single massive stone, would be quite useless as an observatory. The notion which has been entertained by some fanciful persons, that one purpose which the great pyramid was intended to subserve, was to provide a raised small platform high above the general level of the soil, in order that astronomers might climb night after night to that platform, and thence make their observations on the stars, is altogether untenable. Probably no fancy respecting the pyramids has done more to discredit the astronomical theory of these structures than has this ridiculous notion; because even those who are not astronomers and therefore little familiar with the requirements of a building intended for astronomical observation, perceive at once the futility of any such arrangement, and the enormous, one may almost say the infinite disproportion between the cost at which the raised small platform would have been obtained, and the small advantage which astronomers would derive from climbing up to it instead of observing from the ground level. Yet we have seen this notion not only gravely advanced by persons who are to some degree acquainted with astronomical requirements, but elaborately illustrated. Thus, in Flammariou's "History of the Heavens," there is a picture representing six astronomers in eastern garb, perched in uncomfortable attitudes on the uppermost steps of a pyramid, whence they are staring hard at a comet, naturally without the slightest opportunity of determining its true position in the sky, since they have no direction lines of any sort for their guidance. Apart from this, their attention is very properly directed in great part to the necessity of preserving their equilibrium. In only one point in fact does this picture accord with à priori probabilities--namely, in the great muscular development of these ancient observers. They are perfectly herculean, and well they might be, if night after night they had to observe the celestial bodies from a place so hard to reach, and where attitudes so awkward must be maintained during the long hours of the night.

It is perfectly clear, and is in fact one of the chief difficulties of the astronomical theory of the pyramids, that it would only be when these buildings were as yet incomplete that they could subserve any useful astronomical purposes; nevertheless we must not on this account suffer ourselves at this early stage of our inquiry to be diverted from the astronomical theory by what must be admitted to be a very strong argument against it. We have seen that there is such decisive and even demonstrative evidence in favour of the theory that the pyramids were not oriented in a general, still less in a merely casual, manner, and this is, in reality, such clear evidence of their astronomical significance, that we must pass further on upon the line of reasoning which we have adopted--prepared to turn back indeed if absolutely convincing evidence should be found against the theory of the astronomical _purpose_ of the pyramids, but anticipating rather that, on a close inquiry, a means of obviating this particular objection may before long be found.

Let us suppose, then, that astronomers have determined to erect a massive edifice, on a square or oblong base properly oriented, constructing within this edifice such tubular openings as would be most useful for the purpose of indicating the true directions of certain celestial objects at particular times and seasons.

Before commencing so costly a structure they would be careful to select the best possible position for it, not only as respects the nature of the ground, but also as respects latitude. For it must be remembered that, from certain parts of the earth, the various points and circles which the astronomer recognizes in the heavens occupy special positions and fulfil special relations.

So far as conditions of the soil, surrounding country, and so forth are concerned, few positions could surpass that selected for the great pyramid and its companions. The pyramids of Ghizeh are situated on a platform of rock, about 150 feet above the level of the desert. The largest of them, the Pyramid of Cheops, stands on an elevation free all around, insomuch that less sand has gathered round it than would otherwise have been the case. How admirably suited these pyramids are for observing stations is shown by the way in which they are themselves seen from a distance. It has been remarked by every one who has seen the pyramids that the sense of sight is deceived in the attempt to appreciate their distance and magnitude. "Though removed several leagues from the spectator, they appear to be close at hand; and it is not until he has travelled some miles in a direct line towards them, that he becomes sensible of their vast bulk and also of the pure atmosphere through which they are viewed."

With regard to their astronomical position, it seems clear that the builders intended to place the great pyramid precisely in latitude 30°, or, in other words, in that latitude where the true pole of the heavens is one-third of the way from the horizon to the point overhead (the zenith), and where the noon sun at true spring or autumn (when the sun rises almost exactly in the east, and sets almost exactly in the west) is two-thirds of the way from the horizon to the point overhead. In an observatory set exactly in this position, some of the calculations or geometrical constructions, as the case may be, involved in astronomical problems, are considerably simplified. The first problem in Euclid, for example, by which a triangle of three equal sides is made, affords the means of drawing the proper angle at which the mid-day sun in spring or autumn is raised above the horizon, and at which the pole of the heavens is removed from the point overhead. Relations depending on this angle are also more readily calculated, for the very same reason, in fact, that the angle itself is more readily drawn. And though the builders of the great pyramid must have been advanced far beyond the stage at which any difficulty in dealing directly with other angles would be involved, yet they would perceive the great advantage of having one among the angles entering into their problems thus conveniently chosen. In our time, when by the use of logarithmic and other tables, all calculations are greatly simplified, and when also astronomers have learned to recognize that no possible choice of latitude would simplify their labours (unless an observatory could be set up at the North Pole itself, which would be in other respects inconvenient), matters of this sort are no longer worth considering, but to the mathematicians who planned the great pyramid they would have possessed extreme importance.

To set the centre of the pyramid's future base in latitude 30°, two methods could be used, both already to some degree considered--the shadow method, and the Pole-star method. If at noon, at the season when the sun rose due east and set due west, an upright A C were found to throw a shadow C D, so proportioned to A C that A C D would be one-half of an equal-sided triangle, then, theoretically, the point where this upright was placed would be in latitude 30°. As a matter of fact it would not be, because the air, by bending the sun's rays, throws the sun apparently somewhat above his true position. Apart from this, at the time of true spring or autumn, the sun does not seem to rise due east, or set due west, for he is raised above the horizon by atmospheric refraction, before he has really reached it in the morning, and he remains raised above it after he has really passed below--understanding the word "really" to relate to his actual geometrical direction. Thus, at true spring and autumn, the sun rises slightly to the north of east, and sets slightly to the north of west. The atmospheric refraction is indeed so marked, as respects these parts of the sun's apparent course, that it must have been quickly recognized. Probably, however, it would be regarded as a peculiarity only affecting the sun when close to the horizon, and would be (correctly) associated with his apparent change of shape when so situated. Astronomers would be prevented in this way from using the sun's horizontal position at any season to guide them with respect to the cardinal points, but they would still consider the sun, when raised high above the horizon, as a suitable astronomical index (so to speak), and would have no idea that even at a height of sixty degrees above the horizon, or seen as in direction D A, Fig. 1, he is seen appreciably above his true position.

Adopting this method--the shadow method--to fix the latitude of the pyramid's base, they would conceive the sun was sixty degrees above the horizon at noon, at true spring or autumn, when in reality he was somewhat below that elevation. Or, in other words, they would conceive they were in latitude 30° north, when in reality they were farther north (the mid-day sun at any season sinking lower and lower as we travel farther and farther north). The actual amount by which, supposing their observations exact, they would thus set this station north of its proper position, would depend on the refractive qualities of the air in Egypt. But although there is some slight difference in this respect between Egypt and Greenwich, it is but small; and we can determine from the Greenwich refraction tables, within a very slight limit of error, the amount by which the architects of the great pyramid would have set the centre or the base north of latitude 30°, if they had trusted solely to the shadow method. The distance would have been as nearly as possible 1125 yards, or say three furlongs.

Now, if they followed the other method, observing the stars around the pole, in order to determine the elevation of the true pole of the heavens, they would be in a similar way exposed to error arising from the effects of atmospheric refraction. They would proceed probably somewhat in this wise:--Using any kind of direction lines, they would take the altitude of their Polar star (1) when passing immediately under the pole, and (2) when passing immediately above the pole. The mean of the altitudes thus obtained would be the altitude of the true pole of the heavens. Now, atmospheric refraction affects the stars in the same way that it affects the sun, and the nearer a star is to the horizon, the more it is raised by atmospheric refraction. The Pole-star in both its positions--that is when passing below the pole, and when passing above that point--is raised by refraction, rather more when below than when above; but the estimated position of the pole itself, raised by about the mean of these two effects, is in effect raised almost exactly as much as it would be if it were itself directly observed (that is, if a star occupied the pole itself, instead of merely circling close round the pole). We may then simplify matters by leaving out of consideration at present all questions of the actual Pole-star in the time of the pyramid builders, and simply considering how far they would have set the pyramid's base in error, if they had determined their latitude by observing a star occupying the position of the true pole of the heavens.

They would have endeavoured to determine where the pole appears to be raised exactly thirty degrees above the horizon. But the effect of refraction being to raise every celestial object above its true position, they would have supposed the pole to be raised thirty degrees, when in reality it was less raised than this. In other words, they would have supposed they were in latitude 30°, when, in reality, they were in some lower latitude, for the pole of the heavens rises higher and higher above the horizon as we pass to higher and higher latitudes. Thus they would set their station somewhat to the south of latitude 30°, instead of to the north, as when they were supposed to have used the shadow method. Here again we can find how far they would set it south of that latitude. Using the Greenwich refraction table (which is the same as Bessel's), we find that they would have made a much greater error than when using the other method, simply because they would be observing a body at an elevation of about thirty degrees only, whereas in taking the sun's mid-day altitude in spring or autumn, they would be observing a body at twice as great an elevation. The error would be, in fact, in this case, about 1 mile 1512 yards.

It seems not at all unlikely that astronomers, so skilful and ingenious as the builders of the pyramid manifestly were, would have employed both methods. In that case they would certainly have obtained widely discrepant results, rough as their means and methods must unquestionably have been, compared with modern instruments and methods. The exact determination from the shadow plan would have set them 1125 yards to the north of the true latitude; while the exact determination from the Pole-star method would have set them 1 mile 1512 yards south of the true latitude. Whether they would thus have been led to detect the effect of atmospheric refraction on celestial bodies high above the horizon may be open to question. But certainly they would have recognized the action of some cause or other, rendering one or other method, or both methods, unsatisfactory If so, and we can scarcely doubt that this would actually happen (for certainly they would recognize the theoretical justice of both methods, and we can hardly imagine that having two available methods, they would limit their operations to one method only), they would scarcely see any better way of proceeding than to take a position intermediate between the two which they had thus obtained. Such a position would lie almost exactly 1072 yards south of true latitude 30° north.

Whether the architects of the pyramid of Cheops really proceeded in this way or not, it is certain that they obtained a result corresponding so well with this that if we assume they really did intend to set the base of the pyramid in latitude 30°, we find it difficult to persuade ourselves that they did not follow some such course as I have just indicated--the coincidence is so close considering the nature of the observations involved. According to Professor Piazzi Smyth, whose observational labours in relation to the great pyramid are worthy of all praise, the centre of the base of this pyramid lies about 1 mile 568 yards south of the thirtieth parallel of latitude. This is 944 yards north of the position they would have deduced from the Pole-star method; 1 mile 1693 yards south of the position they would have deduced from the shadow method; and 1256 yards south of the mean position between the two last-named. The position of the base seems to prove beyond all possibility of question that the shadow method was not the method on which sole or chief reliance was placed, though this method must have been known to the builders of the pyramid. It does not, however, prove that the star method was the only method followed. A distance of 944 yards is so small in a matter of this sort that we might fairly enough assume that the position of the base was determined by the Pole-star method. If, however, we supposed the builders of the pyramid to have been exceedingly skilful in applying the methods available to them, we might not unreasonably conclude from the position of the pyramid's base that they used both the shadow method and the Pole-star method, but that, recognizing the superiority of the latter, they gave greater weight to the result of employing this method. Supposing, for instance, they applied the Pole-star method three times as often as the shadow method, and took the mean of all the results thus obtained, then the deduced position would lie three times as far from the northern position obtained by the shadow method as from the southern position obtained by the Pole-star method. In this case their result, if correctly deduced, would have been only about 156 yards north of the actual present position of the centre of the base.

It is impossible, however, to place the least reliance on any calculation like that made in the last few lines. By _à posteriori_ reasoning such as this one can prove almost anything about the pyramids. For observe, though presented as _à priori_ reasoning, it is in reality not so, being based on the observed fact, that the true position lies more than three times as far from the northerly limit as from the southern one. Now, if in any other way, not open to exception, we knew that the builders of the pyramid used both the sun method and the star method, with perfect observational accuracy, but without knowledge of the laws of atmospheric refraction, we could infer from the observed position the precise relative weights they attached to the two methods. But it is altogether unsafe, or, to speak plainly, it is in the logical sense a perfectly vicious manner of reasoning, to ascertain first such relative weights on an assumption of this kind, and having so found them, to assert that the relation thus detected is a probable one in itself, and that since, when assumed, it accounts precisely for the observed position of the pyramid, therefore the pyramid was posited in that way and no other. It has been by unsound reasoning of this kind that nine-tenths of the absurdities have been established on which Taylor and Professor Smyth and their followers have established what may be called the pyramid religion.

All we can fairly assume as probable from the evidence, in so far as that evidence bears on the results of _à priori_ considerations, is that the builders of the great pyramid preferred the Pole-star method to the shadow method, as a means of determining the true position of latitude 30° north. They seem to have applied this method with great skill considering the means at their disposal, if we suppose that they took no account whatever of the influence of refraction. If they took refraction into account at all they considerably underrated its influence.

Piazzi Smyth's idea that they knew the _precise_ position of the thirtieth parallel of latitude, and also the _precise_ position of the parallel, where, owing to refraction, the Pole-star would appear to be thirty degrees above the horizon, and deliberately set the base of the pyramid between these limits (not exactly or nearly exactly half-way, but somewhere between them), cannot be entertained for a moment by any one not prepared to regard the whole history of the construction of the pyramid as supernatural. My argument, let me note in passing, is not intended for persons who take this particular view of the pyramid, a view on which reasoning could not very well be brought to bear.

If the star method had been used to determine the position of the parallel of 30° north latitude, we may be certain it would be used also to orient the building. Probably indeed the very structures (temporary, of course) by which the final observations for the latitude had been made, would remain available also for the orientation. These structures would consist of uprights so placed that the line of sight along their extremities (or along a tube perhaps borne aloft by them in a slanting position) the Pole-star could be seen when immediately below or immediately above the pole. Altogether the more convenient direction of the two would be that towards the Pole-star when below the pole. The extremities of these uprights, or the axis of the upraised tube, would lie in a north-and-south line considerably inclined to the horizon, because the pole itself being thirty degrees above the horizon, the Pole-star, whatever star this might be, would be high above the horizon even when exactly under the pole. No star so far from the pole as to pass close to the horizon would be of use even for the work of orientation, while for the work of obtaining the latitude it would be absolutely essential that a star close to the pole should be used.

A line along the feet of the uprights would run north-and-south. But the very object for which the great astronomical edifice was being raised, was that the north-and-south line amongst others should be indicated by more perfect methods.

Now at this stage of proceedings, what could be more perfect as a method of obtaining the true bearing of the pole than to dig a tubular hole into the solid rock, along which tube the Pole-star at its lower culmination should be visible? Perfect stability would be thus insured for this fundamental direction line. It would be easy to obtain the direction with great accuracy, even though at first starting the borings were not quite correctly made. And the further the boring was continued downwards towards the south the greater the accuracy of the direction line thus obtained. Of course there could be no question whatever in such underground boring, of the advantage of taking the lower passage of the Pole-star, not the upper. For a line directly from the star at its upper passage would slant downwards at an angle of more than thirty degrees from the horizon, while a line directly from the star at its lower passage would slant downwards at an angle of less than thirty degrees; and the smaller this angle the less would be the length, and the less the depth of the boring required for any given horizontal range.

Besides perfect stability, a boring through the solid rock would present another most important advantage over any other method of orienting the base of the pyramid. In the case of an inclined direction line above the level of the horizontal base, there would be the difficulty of determining the precise position of points under the raised line; for manifest difficulties would arise in letting fall plumb-lines from various points along the optical axis of a raised tubing. But nothing could be simpler than the plan by which the horizontal line corresponding to the underground tube could be determined. All that would be necessary would be to allow the tube to terminate in a tolerably large open space; and from a point in the base vertically above this, to let fall a plumb-line through a fine vertical boring into this open space. It would thus be found how far the point from which the plumb-line was let fall lay, either to the east or to the west of the optical axis of the underground tunnel, and therefore how far to the east or to the west of the centre of the open mouth of this tunnel. Thus the true direction of a north-and-south line from the end of the tube to the middle of the base would be ascertained. This would be the meridian line of the pyramid's base, or rather the meridian line corresponding to the position of the underground passage directed towards the Pole-star when immediately under the pole.

A line at right angles to the meridian line thus obtained would lie due east and west, and the true position of the east-and-west line would probably be better indicated in this way than by direct observation of the sun or stars. If direct observation were made at all, it would be made not on the sun in the horizon near the time of spring and autumn, for the sun's position is then largely affected by refraction. The sun might be observed for this purpose during the summer months, at moments when calculation showed that he should be due east or west, or crossing what is technically the _prime vertical_. Possibly the so-called azimuth trenches on the east side of the great pyramid may have been in some way associated with observations of this sort, as the middle trench is directed considerably to the north of the east point, and not far from the direction in which the sun would rise when about thirty degrees (a favourite angle with the pyramid architects) past the vernal equinox. But I lay no stress on this point. The meridian line obtained from the underground passage would have given the builders so ready a means of determining accurately the east and west lines for the north and south edges of the pyramid's base, that any other observations for this purpose can hardly have been more than subsidiary.

It is, of course, well known that there is precisely such an underground tunnelling as the considerations I have indicated seem to suggest as a desirable feature in a proposed astronomical edifice on a very noble scale. In all the pyramids of Ghizeh, indeed, there is such a tunnelling as we might expect on almost any theory of the relation of the smaller pyramids to the great one. But the slant tunnel under the great pyramid is constructed with far greater skill and care than have been bestowed on the tunnels under the other pyramids. Its length underground amounts to more than 350 feet, so that, viewed from the bottom, the mouth, about four feet across from top to bottom on the square, would give a sky range of rather less than one-third of a degree, or about one-fourth more than the moon's apparent diameter. But, of course, there was nothing to prevent the observers who used this tube from greatly narrowing these limits by using diaphragms, one covering up all the mouth of the tube, except a small opening near the centre, and another correspondingly occupying the lower part of the tube from which the observation was made.

It seems satisfactorily made out that the object of the slant tunnel, which runs 350 feet through the rock on which the pyramid is built, was to observe the Pole-star of the period at its lower culmination, to obtain thence the true direction of the north point. The slow motion of a star very near the pole would cause any error in time, as when this observation was made, to be of very little importance, though we can understand that even such observations as these would remind the builders of the pyramid of the absolute necessity of good time-measurements and time-observations in astronomical research.

Finding this point clearly made out, we can fairly use the observed direction of the inclined passage to determine what was the position of the Pole-star at the time when the foundations of the great pyramid were laid, and even what that Pole-star may have been. On this point there has never been much doubt, though considerable doubt exists as to the exact epoch when the star occupied the position in question. According to the observations made by Professor Smyth, the entrance passage has a slope of about 26° 27', which would have corresponded, when refraction is taken into account, to the elevation of the star observed through the passage, at an angle of about 26° 29' above the horizon. The true latitude of the pyramid being 29° 58' 51", corresponding to an elevation of the true pole of the heavens, by about 30° 1/2' above the horizon, it follows that if Professor Smyth obtained the true angle for the entrance passage, the Pole-star must have been about 3° 31-1/2' from the pole. Smyth himself considers that we ought to infer the angle for the entrance passage from that of other internal passages, presently to be mentioned, which he thinks were manifestly intended to be at the same angle of inclination, though directed southwards instead of northwards. Assuming this to be the case, though for my own part I cannot see why we should do so (most certainly we have no _à priori_ reason for so doing), we should have 26° 18' as about the required angle of inclination, whence we should get about 3° 42' for the distance of the Pole-star of the pyramid's time from the true pole of the heavens. The difference may seem of very slight importance, and I note that Professor Smyth passes it over as if it really were unimportant; but in reality it corresponds to somewhat large time-differences. He quotes Sir J. Herschel's correct statement, that about the year 2170 B.C. the star Alpha Draconis, when passing below the pole, was elevated at an angle of about 26° 18' above the horizon, or was about 3° 42' from the pole of the heavens (I have before me, as I write, Sir J. Herschel's original statement, which is not put precisely in this way); and he mentions also that somewhere about 3440 B.C. the same star was situated at about the same distance from the pole. But he omits to notice that since, during the long interval of 1270 years, Alpha Draconis had been first gradually approaching the pole until it was at its nearest, when it was only about 3-1/2' from that point, and then as gradually receding from the pole until again 3° 42' from it, it follows that the difference of nine or ten minutes in the estimated inclination of the entrance passage corresponds to a very considerable interval in time, certainly to not less than fifty years. (Exact calculation would be easy, but it would be time wasted where the data are inexact.)

Having their base properly oriented, and being about to erect the building itself, the architects would certainly not have closed the mouth of the slant tunnel pointing northwards, but would have carried the passage onwards through the basement layers of the edifice, until these had reached the height corresponding to the place where the prolongation of the passage would meet the slanting north face of the building. I incline to think that at this place they would not be content to allow the north face to remain in steps, but would fit in casing stones (not necessarily those which would eventually form the slant surface of the pyramid, but more probably slanted so as to be perpendicular to the axis of the ascending passage.) They would probably cut a square aperture through such slant stones corresponding to the size of the passage elsewhere, so as to make the four surfaces of the passage perfectly plane from its greatest depth below the base of the pyramid to its aperture, close to the surface to be formed eventually by the casing stones of the pyramid itself.

Now, in this part of his work, the astronomical architect could scarcely fail to take into account the circumstance that the inclined passage, however convenient as bearing upon a bright star near the pole when that star was due north, was, nevertheless, not coincident in direction with the true polar axis of the celestial sphere. I cannot but think he would in some way mark the position of their true polar axis. And the natural way of marking it would be to indicate where the passage of his Pole-star _above_ the pole ceased to be visible through the slant tube. In other words he would mark where a line from the middle of the lowest face of the inclined passage to the middle of the upper edge of the mouth was inclined by twice the angle 3° 42' to the axis of the passage. To an eye placed on the optical axis of the passage, at this distance from the mouth the middle of the upper edge of the mouth would (_quam proximé_) show the place of the true pole of the heavens. It certainly is a singular coincidence that at the part of the tube where this condition would be fulfilled, there is a peculiarity in the construction of the entrance passage, which has been indeed otherwise explained, but I shall leave the reader to determine whether the other explanation is altogether a likely one. The feature is described by Smyth as "a most singular portion of the passage--viz., a place where two adjacent wall-joints, similar, too, on either side of the passage, were vertical or nearly so; while every other wall-joint, both above and below, was _rectangular_ to the length of the passage, and, therefore, largely _inclined_ to the vertical." Now I take the mean of Smyth's determinations of the transverse height of the entrance passage as 47.23 inches (the extreme values are 47.14 and 47.32), and I find that, from a point on the floor of the entrance passage, this transverse height would subtend an angle of 7° 24' (the range of Alpha Draconis in altitude when on the meridian) at a distance 363.65 inches from the transverse mouth of the passage. Taking this distance from Smyth's scale in Plate xvii. of his work on the pyramid ("Our Inheritance in the Great Pyramid"), I find that, if measured along the base of the entrance passage from the lowest edge of the vertical stone, it falls exactly upon the spot where he has marked in the probable outline of the uncased pyramid, while, if measured from the upper edge of the same stone, it falls just about as far within the outline of the cased pyramid as we should expect the outer edge of a sloped end stone to the tunnel to have lain.

It may be said that from the floor of the entrance passage no star could have been seen, because no eye could be placed there. But the builders of the pyramid cannot reasonably be supposed to have been ignorant of the simple properties of plane mirrors, and by simply placing a thin piece of polished metal upon the floor at this spot, and noting where they could see the star and the upper edge of the tunnel's mouth in contact by reflection in this mirror, they could determine precisely where the star could be seen touching that edge, by an eye placed (were that possible) precisely in the plane of the floor.

I have said there is another explanation of this peculiarity in the entrance passage, but I should rather have said there is another explanation of a line marked on the stone next below the vertical one. I should imagine this line, which is nothing more than a mark such "as might be ruled with a blunt steel instrument, but by a master hand for power, evenness, straightness, and still more for rectangularity to the passage axis," was a mere sign to show where the upright stone was to come. But Professor Smyth, who gives no explanation of the upright stone itself, except that it seems, from its upright position, to have had "something representative of setting up, or preparation for the erecting of a building," believes that the mark is as many inches from the mouth of the tunnel as there were years between the dispersal of man and the building of the pyramid; that thence downwards to the place where an ascending passage begins, marks in like manner the number of years which were to follow before the Exodus; thence along the ascending passage to the beginning of the great gallery the number of years from the Exodus to the coming of Christ; and thence along the floor of the grand gallery to its end, the interval between the first coming of Christ and the second coming or the end of the world, which it appears is to take place in the year 1881. It is true not one of these intervals accords with the dates given by those who are considered the best authorities in Biblical matters,--but so much the worse for the dates.

To return to the pyramid.

We have considered how, probably, the architect would plan the prolongation of the entrance passage to its place of opening out on the northern face. But as the pyramid rose layer by layer above its basement, there must be ascending passages of some sort towards the south, the most important part of the sky in astronomical research.

The astronomers who planned the pyramid would specially require four things. First, they must have the ascending passage in the absolutely true meridian plan; secondly, they would require to have in view, along a passage as narrow as the entrance tunnel, some conspicuous star, if possible a star so bright as to be visible by day (along such a tunnel) as well as by night; thirdly, they must have the means of observing the sun at solar noon on every day in the year; and fourthly, they must also have the entire range of the zodiac or planetary highway brought into view along their chief meridional opening.

The first of these points is at once the most important and the most difficult. It is so important, indeed, that we may hope for significant evidence from the consideration of the methods which would suggest themselves as available.

Consider:--The square base has been duly oriented. Therefore, if each square layer is placed properly, the continually diminishing square platform will remain always oriented. But if any error is made in this work the exactness of the orientation will gradually be lost. And this part of the work cannot be tested by astronomical observations as exact as those by which the base was laid, unless the vertical boring by which the middle of the base, or a point near it, was brought into connection with the entrance passage, is continued upwards through the successive layers of the pyramidal structure. As the rock rises to a considerable height within the interior of the pyramid,[44] probably to quite the height of the opening of the entrance passage on the northern slope, it would only be found necessary to carry up this vertical boring on the building itself after this level had been reached. But in any case this would be but an unsatisfactory way of obtaining the meridian plane when once the boring had reached a higher level than the opening of the entrance passage; for only horizontal lines from the boring to the inclined tunnelling would be of use for exact work, and no such lines could be drawn when once the level of the upper end of the entrance passage had been passed by the builders.

A plan would be available, however (not yet noticed, so far as I know, by any who have studied the astronomical relations of the great pyramid), which would have enabled the builders perfectly to overcome this difficulty.

Suppose the line of sight down the entrance passage were continued upwards along an ascending passage, after reflection at a perfectly horizontal surface--the surface of still water--then by the simplest of all optical laws, that of the reflection of light, the descending and ascending lines of sight on either side of the place of reflection, would lie in the same vertical plane, that, namely, of the entrance passage, or of the meridian. Moreover, the farther upwards an ascending passage was carried, along which the reflected visual rays could pass, the more perfect would be the adjustment of this meridional plane.

To apply this method, it would be necessary to temporarily plug up the entrance passage where it passed into the solid rock, to make the stone-work above it very perfect and close fitting, so that whenever occasion arose for making one of the observations we are considering, water might be poured into the entrance passage, and remain long enough standing at the corner (so to speak) where this passage and the suggested ascending passage could meet, for Alpha Draconis to be observed down the ascending passage. Fig. 2 shows what is meant. Here D C is the descending passage, C A the ascending passage, C the corner where the water would be placed when Alpha Draconis was about to pass below the pole. The observer would look down A C, and would see Alpha Draconis by rays which had passed down D C, and had been reflected by the water at C. Supposing the building to have been erected, as Lepsius and other Egyptologists consider, at the rate of one layer in each year, then only one observation of the kind described need be made per annum. Indeed, fewer would serve, since three or four layers of stone might be added without any fresh occasion arising to test the direction of the passage C A.

It is hardly necessary to remind those who have given any attention to the subject of the pyramid that there is precisely such an ascending passage as C A, and that as yet no explanation of the identity of its angle of ascent with the angle of descent of the passage D C has ever been given. Most pyramidalists content themselves by assuming, as Sir E. Beckett puts it, "that the same angle would probably be used for both sets of passages, _as there was no reason for varying it_," which is not exactly an explanation of the relation. Mr. Wacherbarth has suggested that the passages were so adjusted for the purpose of managing a system of balance cars united by ropes from one passage to another; but this explanation is open, as Beckett points out, to the fatal objection that the passages meet at their lowest point, not at their highest, so that it would be rather a puzzle "to work out the mechanical idea." The reflection explanation is not only open to no such objections, but involves precisely such an application of optical laws as we should expect from men so ingenious as the pyramid builders certainly were. In saying this, let me explain, I am not commending myself for ingenuity in thinking of the method, simply because such methods are quite common and familiar in the astronomy of modern times.

While I find this explanation, which occurred to me even while this paper was in writing, so satisfactory that I feel almost tempted to say, like Sir G. Airy of his explanation of the Deluge as an overflow of the Nile, that "I cannot entertain the slightest doubt" of its validity, I feel that there ought to be some evidence in the descending passage itself of the use of this method. We might not find any traces of the plugs used to stop up, once a year or so, the rock part of the descending passage. For they would be only temporary arrangements. But we should expect to find the floor of the descending passage constructed with special care, and very closely fitted, where the water was to be received.

Inquiring whether this is so, I find not only that it is, but that another hitherto unexplained feature of the great pyramid finds its explanation in this way,--the now celebrated "secret sign." Let us read Professor Smyth's account of this peculiar feature:--

"When measuring the cross-joints in the floor of the
entrance-passage, in 1865, I went on chronicling their angles, each
one proving to be very nearly at right angles to the axis, until
suddenly one came which was _diagonal_; another, and that was
diagonal too; but, after that, the rectangular position was
resumed. Further, the stone material carrying these diagonal joints
was harder and better than elsewhere in the floor, so as to have
saved that part from the monstrous excavations elsewhere
perpetrated by some moderns. Why, then, did the builders change the
rectangular joint angle at that point, and execute such unusual
angles as they chose in place of it, in a better material of stone
than elsewhere; and yet with so little desire to call general
attention to it, that they made the joints fine and close to that
degree that they escaped the attention of all men until 1865 A.D.
The answer came from the diagonal joints themselves, on discovering
that the stone between them was opposite to the butt end of the
portcullis of the first ascending passage, or to the hole whence
the prismatic stone of concealment through 3000 years had dropped
out almost before Al Mamoun's eyes. Here, therefore, was a secret
sign in the pavement of the entrance-passage, appreciable only to a
careful eye and a measurement by angle, but made in such hard
material that it was evidently intended to last to the end of human
time with the great pyramid, and _has_ done so thus far."

Whether Professor Smyth is right in considering that this specially-prepared position of the floor was intended not for any practical purpose, but to escape the notice of the careless, while yet, when the right men "at last, duly instructed, entered the passage," this mysterious floor-sign should show them where a ceiling-stone was movable, on perceiving which they "would have laid bare the beginning of the whole train of those sub-aërial features of construction which are the great pyramid's most distinctive glory, and exist in no other pyramid in Egypt or the world," I leave the reader to judge. I would remark, only, that, if so, the builders of the pyramid were not remarkably good prophets, seeing that the event befell otherwise, the ceiling-stone dropping out a thousand years or so before the floor-sign was noticed; wherefore we need not feel altogether alarmed at their own prediction (according to Professor Smyth), that the end of the world is to come in 1881, even as Mother Shipton also is reported to have prophesied. For my own part, I am quite content with my own interpretation of the secret sign; as showing where the floor of the descending passage was purposely prepared for the reception of water, on the still surface of which the Pole-star of the day might be mirrored for one looking down the ascending passage.

Albeit, I cannot but think that this ascending passage must also have been so directed as to show some bright star when due south. For if the passage had only given the meridian plane, but without permitting the astronomer to observe the southing of any fixed star, it would have subserved only one-half its purposes as a meridional instrument. It is to be remembered that, supposing the ascending passage to have its position determined in the way I have described, there would be nothing to prevent its being also made to show any fixed star nearly at the same elevation. For it could readily be enlarged in a vertical direction, the floor remaining unaltered. Since it is not enlarged until the great gallery is reached (at a distance of nearly 127 feet from the place where the ascent begins), it follows, or is at least rendered highly probable, that some bright star was in view through that ascending passage.

Now, taking the date 2170 B.C., which Professor Smyth assigns to the beginning of the great pyramid, or even taking any date (as we fairly may), within a century or so on either side of that date, we find no bright star which would have been visible when due south, through the ascending passage. I have calculated the position of that circle among the stars along which lay all the points passing 26° 18' above the horizon when due south, in the latitude of Ghizeh, 2170 years before the Christian era; and it does not pass near a single conspicuous star.[45] There is only one fourth magnitude star which it actually approaches--namely, Epsilon Ceti; and one fifth magnitude star, Beta of the Southern Crown.

When we remember that Egyptologists almost without exception assert that the date of the builders of the great pyramid _must_ have been more than a thousand years earlier than 2170 B.C., and that Bunsen has assigned to Menes the date 3620 B.C., while the date 3300 B.C. has been assigned to Cheops or Suphis on apparently good authority, we are led to inquire whether the other epoch when Alpha Draconis was at about the right distance from the pole of the heavens may not have been the true era of the commencement of the great pyramid. Now, the year 3300 B.C., though a little late, would accord fairly well with the time when Alpha Draconis was at the proper distance 3-2/3° from the pole of the heavens. If the inclination of the entrance-passage is 26° 27', as Professor Smyth made it, the exact date for this would be 3390 B.C.; if 26° 40', as others made it before his measurements, the date would be about 3320 B.C., which would suit well with the date 3300 B.C., since a century either way would only carry the star about a third of a degree towards or from the pole.

Now, when we inquire whether in the year 3300 B.C. any bright star would have been visible, at southing, through the ascending passage, we find that a very bright star indeed, an orb otherwise remarkable as the nearest of all the stars, the brilliant Alpha Centauri, shone as it crossed the meridian right down that ascending tube. It is so bright that, viewed through that tube, it must have been visible to the naked eye, even when southing in full daylight.

But thirdly, we must consider how the builders of the pyramid would arrange for the observation of the sun at noon on every clear day in the year.

They would carry up the floor of the ascending passage in an unchanged direction, as it already pointed south of the lowest place of the noon sun at mid-winter. They would have to turn the tunnel into a lofty gallery, to increase the vertical range of view on the meridian. It seems reasonable to infer that they would prefer so to arrange matters that the upper end of the gallery would be near the middle of the platform which would form the top of the pyramidal structure from the time when it was completed for observational purposes. The height of the gallery would be so adjusted to its length, that the mid-winter's sun would not shine further than the lower end of the gallery (that is, to the upper end of the smaller ascending passage). In fact, as the moon and planets would have to be observed when due south, through this meridional gallery, and as they range further from the equator both north and south than the sun does, it would be necessary that the gallery should extend lower down than the sun's mid-winter noon rays would shine.

As it would be a part of the observer's work to note exactly how far down the gallery the shadow of its upper southern edge reached, as well as the moment when the sun's light passed from the western to the eastern wall of the gallery, and other details of the kind; besides, of course, taking time-observations of the moment when the sun's edge seemed to reach the edge of the gallery's southern opening; and as such observations could not be properly made by men standing on the smooth slanting floor of the gallery, it would be desirable to have cross-benches capable of being set at different heights along the sloping gallery. In some observations, indeed, as where the transits of several stars southing within short intervals of time had to be observed, it would be necessary to set some observers at one part of the gallery, others at another part, and perhaps even to have several sets of observers along the gallery. And this suggests yet another consideration. It might be thought desirable, if great importance was attached (as the whole building shows that great importance must have been attached) to the exactness of the observations, to have several observations of each transit of a star across the mouth of the gallery. In this case, it would be well to have the breadth of the gallery different at different heights, though its walls must of necessity be upright throughout--that is, the walls must be upright from the height where one breadth commences, to the height where the next breadth commences. With a gallery built in this fashion, it would be possible to take several observations of the same transit, somewhat in the same way that the modern observer watches the transit of a star across each of five, seven, or nine parallel spider threads, in order to obtain a more correct time for the passage of the star across the middle thread, than if he noted this passage alone.