## The Pyramids and Temples of Gizeh by Sir Flinders Petrie - Outside of Great Pyramid

The Pyramids and Temples of Gizeh by W. M. Flinders Petrie

Chap. 6. Outside of Great Pyramid Pages 37 - 54

20 p. 37 The materials available for a discussion of the original size of the base of the Great pyramid are :

(1) the casing in situ upon the pavement, in the middle of each face;
(2) the rock cut sockets at each corner;
(3) the levels of the pavement and sockets;
(4) the mean planes of the present core masonary.

Since the time of the first discovery of some of the sockets in 1801, it has always been supposed that they defined the original extent of the Pyramid, and various observers have measured from corner to corner of them, and thereby obtained a dimension which waswithout further inquiryput down as the length of the base of the Pyramid. But, in as much as the sockets are on different levels, it was assumed that the faces of the stones placed in them rose up vertically from the edge of the bottom, until they reached the pavement (what-ever level that might be) from which the sloping face started upwards. Hence it was concluded that the distances of the socket corners were equal to the lengths of the Pyramid sides upon the pavement.

Therefore, when reducing my observations, after the first winter, I found that the casing on the North side (the only site of it then known) lay about 30 inches inside the line joining the sockets, I searched again and again for any flaw in the calculations. But there were certain check measures, beside the regular checked triangulation, which agreed in the same story; another clue, however, explained it, as we shall see.

The form of the present rough core masonry of the Pyramid is capable of being very closely estimated. By looking across a face of the Pyramid, either up an edge, across the middle of the face, or even along near the base, the mean optical plane which would touch the most prominent points of all the stones, may be found with an average variation at different times of only 1.0 inch. I therefore carefully fixed, by nine observations at each corner of each face, where the mean plane of each face would fall on the socket floors; using a straight rod as a guide to the eye in estimating. On reducing these observations to give the mean form of the core planes at the pavement level, it came out thus :

p 38

 Case Plane Sides. Azimuths. Socket Sides. Azimuths. N. E. S. W. 9002.3 8999.4 9001.7 9002.5 4' 35" 5' 26" 5' 23" 5' 39" 9129.8 9130.8 9123.9 9119.2 3' 20" 5' 21" + 1' 15" 7' 33" Mean Mean Difference 9001.5 1.0 5' 16" 20" 9125.9 4.4 3' 45" 2' 42"

Here, then, was another apparently unaccountable fact, namely, that the core masonry was far more accurate in its form than the socket square. It is, in fact, four times as accurate in length, and eight times as accurate in angle. This forced me to the conclusion that the socket lines cannot show the finished base of the Pyramid.

The clue which explains all these difficulties isthat the socket corners vary from a true square in proportion to their depth below the pavement, the sockets nearer the centre being higher.

This means that the sockets were cut to receive the foot of the sloping face, which was continued right down to their floors, beneath the pavement. (See Pl. xi.) Hence the sockets only show the size of the Pyramid, where it was started from varying levels, which were all under the pavement; and its true base upon the pavement is therefore 20 or 30 inches inside the lines of the sockets.

This exactly explains the position of the casing found on the N. side, as it was found to be inside the line of the sockets. The test, then, of this explanation, was to find the casing on the other sides, fix its position, and see if it was likewise within the lines of the sockets. The shafts were accordingly sunk through the rubbish, two or three feet inside the socket lines; and the casing was found on each side, just in the expected alignment. Without this clue, the narrow shafts might easily have missed the casing altogether, by being sunk too far out from the Pyramid.

Now having found the casing foot on each side of the Pyramid, it is settled that the faces must have passed through these fixed points, and when the casing was duly projected down at its angle of slope to the socket floors, it was found to fall on an average 4 inches inside the edges of the socket corners. This is what might be expected, as the socket sides are neither straight nor square; so that this margin would be much less at a minimum than it is at their corners; and it would be natural to allow some free space, in which to adjust the stone.

21 p 39 Having, then, four lines passing through the middles of the sides, what is to define the junctions of those lines at the corners ? Or, in other words, what defines their azimuth? Was each side made equidistant (1) from its socket's sides ? or, (2) from the core side at each of its ends ? Or was a corner made equidistant (3) from the sides of its socket corner? or, (4) from the sides of its core corner? The core may be put out of the question ; for if the sides followed it exactly in any way, they would run outside of the sockets in some parts. Which, then, is most likely: that the sockets were placed with an equal amount of margin allowed on the two ends of one side, or with an equal margin allowed at both sides of one corner? The latter, certainly, is most likely; it would be too strange to allow, say, 6 inches margin on one side of a socket, and only 2 inches on its adjacent side. It seems, then, that we are shut up to the idea that the socket corners lie in the diagonals of the Pyramid casing.

But there is another test of this arrangement, which it ought to satisfy. Given four diagonals, as defined by the socket corners; and given four points near the middles of the sides of the Pyramid, as defined by the existing casing: if we start from one diagonal, say N.E.; draw a line through the E. casing to S.E. diagonal; from that through the S. casing, to the S.W. diagonal; and so on, round to the N.E. diagonal again ; there is no necessity that the line should on its return fall on the same point as that from which we started : it might as easily, apart from special design, fall by chance anywhere else. The chances are greatly against its exactly completing its circuit thus, unless it was so planned before by the diagonals of the socket corners being identical with those of the square of the casing.

On applying this test to the diagonals of the sockets, we find that the circuit unites, on being carried round through these points, to within 1 inch far closer, in fact, than the diagonals of the sockets and the line of the casing can be estimated.

This is, then, a conclusive test; and we only need to compute a square that shall pass through the points of the casing found on each side, and having also its corners lying on the diagonals of the sockets. This square, of the original base of the Great Pyramid casing on the platform, is of these dimensions :

 Length. Difference from Mean. Azimuth. Difference from Mean. N E S W 9069.4 9067.7 9069.5 9068.6 + .6 1.1 + .7 .2 3' 20" 3' 57" 3' 41" 3' 54" + 23" 14" + 2" 11" Mean 9068.8 .65 3' 43" 12"

p 40 Thus the finished base of the Pyramid had only two-thirds of the irregularity of the core masonry, the mean difference of which was 1.0 inch and 20" ; this is what would be expected from a final adjustment of the work, after the rougher part was finished.

But it must always be remembered that this very small mean error of .65 inch and I2" is that of the sockets, and not that of the casing stones; these latter we can hardly doubt would be adjusted more carefully than the cutting of the sockets with their free margin.

Also it must be remembered that this result includes the errors of survey. Now the probable errors of fixing the plumb-lines in the triangulation were about .2 on E. side, .2 on S. side, .1 on W. side, and the casing .1 on N. side; the probable errors of the triangulation of the points of reference is in general much less than this; we may then say .3 for the absolute places of the plumb-lines. The exact amount of this is not of so much consequence, because the errors of estimating the original points of construction are larger. They are, on the N., .04; on the E., .2 ; but another less satisfactory estimate differed 1.1 on the S., .2; on the W., .5, taking the mean of two points that differed 1.1 inches. Besides this, the estimation of the socket diagonals cannot be put under .5 by the bad definition of the edges and want of straightness and orientation of the sides. If we then allow that the probable errors from all sources of our knowledge, of each of the original sides of the Pyramid amount to .6, we shall not over-estimate them. Hence it is scarcely to be expected that our determinations of the sides should agree closer than .65 inch, as they do on an average.

So we must say that the mean errors of the base of the Great Pyramid were somewhat less than .6 inch, and 12" of angle.

22 In computing the above quantities, I have used my final determination of the socket levels below the pavement; these, with the first approximate results, and Inglis's figures, stand thus :

 Accurate in 1882 Approx in 1881 Inglis in 1865 N.E. S.E. S.W. N.W. 28.5 39.9 23.0 32.8 28.7 39.9 22.9 32.6 28.6 42.4 23.0 32.8

the level of the pavement being zero. The approximation was very roughly done, and it is strange that it should agree as well with the accurate determination as it does. From Inglis's measures I have subtracted 28.6, in order to reckon them from the pavement level; by the exact agreement of my two p 41 levellings at the S.E. (which was taken second in the series each time, and hence is checked by others), I conclude that Inglis is there in error by a couple of inches; and his other work, in measuring the steps, contains much larger errors than this.

The relations, then, of the core masonry, the base of the casing on the pavement, the edge of the casing in the sockets, and the socket edges, are shown in Pl. x., to a scale of 1/50. The position of the station marks is also entered. The inclinations of the various sides of sockets and casing are stated; and it is noticeable that the core masonry has a twist in the same direction on each side, showing that the orientation of the Pyramid was slightly altered between fixing the sockets and the core. The mean skew of the core to the base is 1' 33", and its mean azimuth 5' 16" to true North. The diagram also shows graphically how much deformed is the square of the socket lines ; and how the highest socket (S.W.) is nearest to the centre of the Pyramid ; and the lowest socket (S.E.) is furthest out from the centre of the socket diagonals, and also from the mean planes of the core.

23. For ascertaining the height of the Pyramid, we have accurate levels of the courses up the N.E. and S.W. corners; and also hand measurements up all four corners. The levels were all read to 1/100 inch, to avoid cumulative errors; but in stating them in Pl. viii., I have not entered more than tenths of an inch, having due regard to the irregularities of the surfaces.* The discrepancy of .2 inch in the chain of levels (carried from the N.E. to S.E., to S.W., on the ground, thence to the top, across top, and down to N.E. again), I have put all together at the junction of levelling at the 2nd course of the S.W., as I considered that the least certain point. It may very likely, however, be distributed throughout the whole chain, as it only amounts to I.8" on the whole run.

* Owing to mistaking (in a photograph) the rock bed of the pavement for the pavement itself, Prof. Smyth has entered all the levels in his works (both of his own measures and those of others) from a datum 20 inches below the true pavement level. This has led him to reckon the first course as two; hence all his course numbers must have one subtracted, and all his levels about 20 inches subtracted, to reduce them to a true start from the pavement surface.

These levels, though important for the heights of particular courses, have scarcely any bearing on the question of the total height of the original peak of the casing of the pyramid; because we have no certain knowledge of the thickness of the casing on the upper parts.

The zero of levels that I have adopted, is a considerable flat-dressed surface of rock at the N.E. corner, which is evidently intended to be at the level of the pavement; it has the advantage of being always accessible, and almost indestructible. From this the levels around the Pyramid stand thus :

 N.E. E. S.E. S. S.W W. W.N.W. N.W. N. 2nd Course 1st Course Levelled rock Pavement Socket + 107.7 + 58.6 0 28.5 E.N.E. .15 N.N.E. .6? + 105.5 39.9 5.5? +111.2 + 57.6 23.0 + 1.1? 1.2? +106.6 + 58.0 32.8 +107.4 + 58.9 +

p 42 The pavement levels, excepting that on the N. side below the entrance, are not of the same accuracy as the other quantities; they were taken without an assistant, merely for the purpose of showing that it really was the pavement on which the casing was found to rest on each side. The differences of the 1st course levels, probably show most truly the real errors of level of the base of the Pyramid.

24. To obtain the original height of the Pyramid, we must depend on the observations of its angle. For this there are several data, as follows; the method by which the passage and air channels determine it being explained in detail further on, when the internal parts are discussed :

 Weight Casing stones, in situ, N. side, by theodolite (To 3 points on top and 3 on base.) by goniometer and level (To 3 points on top and 3 on base.) by steel square and level (To 3 points on top and 3 on base.) 5 overthrown by goniometer (To 3 points on top and 3 on base.) 18 fragments, all sides, goniometer (All above 2 inches in shortest length) N. face, by entrance passage mouth N. face, by air channel mouth 51 46' 45" 51 49' 51 44' 11" 51 52' 51 53' 51 53' 20" 51 51' 30" 2' 7" 1' 23" 2' 4' 1' 20" 7 2 0 0 0 10 5
 N. face, weighted mean S. face by air channel mouth 51 50' 40" 51 57' 30" 1' 5" 20"

In assigning the weights to these different data, the reason that no weight is given to the angles of shifted casing stones is that there is no proof that the courses did not dip inwards somewhat ; on the contrary, I continually observed that the courses of the core had dips of as much as to 1 so that it is not at all certain that the courses of the casing were truly level to 5' or 10', and occasional specimens showed angles up to 54 . The angle by means of the large steel square was vitiated by the concretion on the faces of the stones being thicker below than above, .1 inch of difference making an error of 6'. The small goniometer was applied to the clear patches of the stone, selected in nine different parts. These three casing stones in situ have not as much weight assigned to them as they would otherwise have, owing to their irregularities. One of them is 0.9 in front of the other at the top, though flush at the base a difference of 4'. The datum from the air channel, though far more accurate than that by the passage mouth (being on a longer length), is not so certainly intentional, and is therefore not worth as much. (See sections 32 and 33 for p 43 details.) From all these considerations the above weighting was adopted. It is clear that the South face should not be included with the North, in taking the mean, as we have no guarantee that the Pyramid was equiangular, and vertical in its axis.

25. The staff which was set up by the Transit of Venus party in 1874 on the top of the Pyramid, was included in my triangulation; and its place is known within inch. From this staff, the distances to the mean planes of the core masonry of the Pyramid sides, were determined by sighting over their prominent edges, just as the positions of the mean planes were fixed at the lower corners of the faces. Hence we know the relation of the present top of the core masonry to the base of the Pyramid. The top is, rather strangely, not square, although it is so near to the original apex. This was verified carefully by an entire measurement as follows :

 Mean of four readings, 1881. Mean of three readings, 1882. Mean of all. Center of Pyramid base horizontal to the : N. side E. side S. side W. side 226.0 .5 214.4 .4 215.0 .6 216.4 .5 223.7 .2 213.8 .6 215.0 .4 218.7 .5 224.5 .7 214.1 .3 215.0 .4 217.6 1.0

Now, at the level of these measurements, 5407.9 at N.E., or 5409.2 at S.W., above the base, the edges of the casing (by the angles of the N. and S. side found above) will be 285.3 2.7 on the North, and 30I.6 on the South side, from the vertical axis of the centre. Thus there would remain for the casing thickness 60.8 3 on the N., and 86.6 on the S.; with 77.6 for the mean of E. and W. Or, if the angle on the S. side were the same as on the N., the casing thickness would be 69.2 on the S. This, therefore, seems to make it more likely that the South side had about the same angle as the North.

On the whole, we probably cannot do better than take 51 52' 2' as the nearest approximation to the mean angle of the Pyramid, allowing some weight to the South side.

The mean base being 9068.8 .5 inches, this yields a height of 5776.0 7.0 inches.

26. With regard to the casing, at the top it mustby the above dataaverage about 71 5 inches in thickness from the back to the top edge of each stone. Now the remaining casing stones on the N. base are of an unusual height, and therefore we may expect that their thickness on the top would be rather less, and on the bottom rather more, than the mean of all. Their top thickness averages 62 8 (the bottom being 108 8), and it thus agrees very fairly with 71 5 inches. At the corners, however, the casing was thinner, averaging but 33.7 (difference of core plane and casing on pavement); and this is explained by the faces of the core masonry being very distinctly hollowed.

p 44 This hollowing is a striking feature; and beside the general curve of the face, each side has a sort of groove specially down the middle of the face, showing that there must have been a sudden increase of the casing thickness down the mid-line. The whole of the hollowing was estimated at 37 on the N. face; and adding this to the casing thickness at the corners, we have 70.7, which just agrees with the result from the top (71 5), and the remaining stones (62 8). The object of such an extra thickness down the mid-line of each face might be to put a specially fine line of casing, carefully adjusted to the required angle on each side ; and then afterwards setting all the remainder by reference to that line and the base.

Several measures were taken of the thickness of the joints of the casing stones. The eastern joint of the northern casing stones is on the top .020, .002, .045 wide; and on the face .012, .022, .013, and .040 wide. The next joint is on the face .011 and .014 wide. Hence the mean thickness of the joints there is .020; and, therefore, the mean variation of the cutting of the stone from a straight line and from a true square, is but .01 on length of 75 inches up the face, an amount of accuracy equal to most modern opticians' straight-edges of such a length. These joints, with an area of some 35 square feet each, were not only worked as finely as this, but cemented throughout. Though the stones were brought as close as 1/500 inch, or, in fact, into contact, and the mean opening of the joint was but 1/50 inch, yet the builders managed to fill the joint with cement, despite the great area of it, and the weight of the stone to be movedsome 16 tons. To merely place such stones in exact contact at the sides would be careful work ; but to do so with cement in the joint seems almost impossible.

The casing is remarkably well levelled at the base ; the readings on the stones of the North side, and the pavement by them being thus :

 W.End. Middle. E.End. Pavement by casing. Core 40ft.E. of casing. Casing Front Casing Back Core Pavement + 58.83 + 58.84 ( .56) + 58.84 ( .30) + 58.90 + 58.85 ( .05) .01 .03 + .02 .00 + 58.87

The pavement levels in brackets are on decidedly worn parts, and hence below the normal level, as shown in the fourth column. The average variation of the casing from a level plane of + 58.85 is but .02; and the difference to the core level, at the farthest part accessible in that excavation, does not exceed this. The difference of pavement level out to the rock at the N.E. corner is but .17 on a distance of 4,200 inches, or 8" of angle.

27. The works around the Pyramid, that are connected with it, are :

(1) The limestone pavement surrounding it;
(2) the basalt pavement on the E. side and
(3) the rock trenches and cuttings on the E. side, and at the N.E. corner.

p 45 The limestone pavement was found on the N. side first by Howard Vyse, having a maximum remaining width of 402 inches; but the edge of this part is broken and irregular, and there is mortar on the rock beyond it, showing that it has extended further. On examination I found the edge of the rock-cut bed in which it was laid, and was able to trace it in many parts. At no part has the paving been found complete up to the edge of its bed or socket, and it is not certain, therefore, how closely it fitted into it ; perhaps there was a margin, as around the casing stones in the corner sockets. The distances of the edge of this rock-cut bed, from the edge of the finished casing on the pavement (square of 9068.8) were fixed by triangulation as follows :

 N.N.W. N.side N.N.E. E.N.E. S.S.W. W.S.W. 616.9 near the corner; corner itself not found, nor any W.N.W. side. 615.9 at 570 E. of probable N.W. corner of pavement. 618.7 at 670 E. 616.2 at 890 E. 564 to 568 very rough and irregular, opposite entrance. 529.0 at N.E. corner, N. side of it. 538.8 at N.E. corner, E. side of it. 533.9 at 586 from N.E. corner. No cutting found at S.E. corner. 536.5 at 846 from S.W. corner. 533.0 at 520 from S.W. corner. 534.6 at 206 from S.W. corner. 529.6 at S.W. corner, S. side of it. 536.0 at S.W. corner, W. side of it. 627.9 at 751 from S.W. corner.

From these measures it appears that there is no regularity in the width of the cutting; the distance from the casing varying 99 inches, and altering rapidly even on a single side. The fine paving may possibly have been regular, with a filling of rougher stone beyond it in parts; but if so, it cannot have exceeded 529 in width.

The levels of the various works around the Pyramid are as follow, taken from the pavement as zero :

 Flat rock-bed of pavement W. of N.W. socket Flat rock-bed of pavement beside N.W. socket Flat rock-bed of pavement N. of N.W. socket Flat rock-bed of pavement N.E. of N.W. socket Flat rock-bed of pavement before entrance Flat rock-bed of pavement inner end of E.N.E. trench Basalt pavement, E. side of it Basalt pavement, W. side, in excavation 23.7 21.6 17.0 15.9 27.1 26.9 + 2.0 + 2.0

p 46 The Pyramid pavement must then have varied from 17 to 27 inches in thickness; it was measured as 21 inches where found by Vyse.

28. The basalt pavement is a magnificent work, which covered more than a third of an acre. The blocks of basalt are all sawn and fitted together; they are laid upon a bed of limestone, which is of such a fine quality that the Arabs lately destroyed a large part of the work to extract the limestone for burning. I was assured that the limestone invariably occurs under every block, even though in only a thin layer. Only about a quarter of this pavement remains in situ, and none of it around the edges the position of it can therefore only be settled by the edge of the rock-cut bed of it. This bed was traced by excavating around its N., E., and S. sides; but on the inner side, next to the Pyramid, no edge could be found ; and considering how near it approached to the normal edge of the limestone pavement, and that it is within two inches of the same level as that, it seems most probable that it joined it, and hence the lack of any termination of its bed.

Referring, then, to the E. side of the Pyramid, and a central line at right angles to that (see Pl. ii.), the dimensions of the rock bed of the basalt paving are thus :

 NORTH TO SOUTH. From mid-line of Pyramid Total length S. corner of opening on E. side N. corner of opening on E. side 1046.0 to N.E. 1077.7 to S.E. 2123.7 E. side. 321.0 to mid. 693.3 to mid. 1061.9 to N.W. 1062.8 to S.W. 2124.7 W. side. 756.7 to S.E. 352.7 to N.E.

 EAST TO WEST. Width traced. E. side, from Pyramid base. S. corner of opening on E. side of the base. N. corner of opening on E. side of the base. 1006.6 + . 2153.0 N. end. 2148.0 S. end. 2169.0 2160.0

Next, referring this pavement to the trench lines :

 NORTH TO SOUTH. N. trench, inner end from basalt S. trench, inner end from basalt 318.1 327.9

 EAST TO WEST. N.E. corner to N. trench axis N. trench axis there, to Pyramid S.E. corner to S. trench axis S. trench axis there, to Pyramid S.E. corner to N. trench axis, continued N. trench axis there, to Pyramid 1073.2 1079.8 1022.6 1125.8 1075.0 1073.0

p 47 Hence the plan of the basalt pavement seems to be two adjacent squares of about 1,060 inches in the side; the N. trench axis being the boundary of them, and there being a similar distance between that and the Pyramid. The outer side of the paving was laid off tolerably parallel to the Pyramid base ; but the angles are bad, running 15 inches skew.*

* The broken blocks of basalt, which border a track down the hill side E. of them Pyramid, are almost certainly from this pavement; they are of exactly the same stone, and have many worked faces remaining like those of the pavement. Their placing is quite rude, and looks as if done by some barbarian destroyers.

29. Next, referring to the rock-hewn trenches alone, the dimensions of the three deep ones are as follow :

 NORTH TO SOUTH. N. trench, outer end, to central line N. trench, axial length N. trench, inner end, to central line S. trench, inner end, to central line S. trench, axial length S. trench, outer end, to central line E.N.E.trench, outer end of axis N. of central line E.N.E.trench, axis cuts N. trench axis N. of central line EAST TO WEST. N. trench axis, outer end to base N. trench axis, inner end to base S. trench axis, inner end to base S. trench axis, outer end to base S. trench axis, E. of N. trench axis, at centre E.N.E.trench, outer end of axis to base E.N.E.trench, axial length from N. trench axis E.N.E.trench, axial length from actual bed of basalt E.N.E.trench, axial length from straight edge The slighter trenches are three in number : NORTH TO SOUTH. N.N.E.trench axis cuts N. trench axis N. of central line Trench by N.E. socket, end of axis from N. side of casing Trench by N.E. socket, on the axis, from N. side of casing Trench by trial passages, ends of axis N. of central line EAST TO WEST. N.N.E.trench, axis cuts pavement, from N.E. corner Trench by N.E. socket, end of axis from E. side of casing Trench by N.E. socket, on the axis, from E. side of casing Trench by trial passages, ends of axis E. of Pyramid base 3510.2 2130.2 1380.0 1390.7 2093.7 3430.4 848.3 68.5 1085.5 1080.6 1125.5 1122.9 497 4213.2 3231.1 2112.6 2124.7 116.0 643.3 1630.8 1563.3 1274.4 647.2 203.2 434.1 3161.6 3167.6

p 48 The subterranean passages are in one group :

 NORTH TO SOUTH. Trial passages axis, N. of central line, at the station marks EAST TO WEST. Trial passages axis, E. of central line, at the station marks 2233.6 1220.8 3446.7 3441.2

Hence it seems that the axial length of the E.N.E. trench outside the basalt paving is intended to be the same as the axial length of the North and South trenches. The angles of the axes of these trenches are as follow :

 N. trench S. trench E.N.E. trench N.N.E. trench Trench by N.E. socket Trench by trial passages Trial passages To E. Face of Pyramid. + 7' 53" + 4' 09" + 76 02' 26" + 24 25' 34" + 13 09' 38" 1 11' + 18' 40" To true North. + 3' 56" + 12' + 75 58' 23" + 24 21' 37" +13 5' 41" 1 15' + 14' 43"

Thus the angles between the trenches are: S. trench to E.N.E. trench, 104 1' 43" (or 2 X 52 0' 52"); and E.N.E. to N.N.E. trench, 51 36' 52."

With regard to the details of these rock cuttings, the forms of the ends of the N. and S. trenches were plotted from accurate offsets (see P1. iiia.); and there is little of exact detail in the cutting to be stated. The axes at the ends were estimated by means of the plans here given, but on double this scale; and the rock is so roughly cut in most parts that nothing nearer than an inch need be considered. The position of the inner end of the N. trench is not very exactly fixed, an omission in measurement affecting it, mainly from N. to S. In this trench I excavated to 110 below the present surface of the sand, or about 220 below the rock surface, without finding any bottom. The S. trench is more regular than the N. trench; at the outer end its width is 205 to 206, and at the inner end 134.2: it has a curious ledge around the inner end at 25 below the top surface. At the outer end the rock is cut, clearly to receive stones, and some plaster remains there; also some stones remain fitted in the rock on the W. side of this trench. Built stones also occur in the N., E.N.E., and N.N.E. trenches. From the inner end of the S. trench, a narrow groove is cut in the rock, leading into the rock-cut bed of the basalt pavement; this groove was filled for a short way near the end of the trench by stone mortared in. It was evidently in process of being cut, as the hollows in the sides of it were the regular course of rock-cutting. The rock beside the trenches is dressed flat, particularly on the E. of the p 49 N. trench, and the W. of the S. trench, where the built stones occur. There is a short sort of trench, on the E. side of the S. trench (not in plan); it is about 25 wide, 70 long, and 50 deep, with a rounded bottom; the length E. and W.

The E.N.E. trench is very different to the others; it has a broad ledge at the outer end, and this ledge runs along the sides of the trench, dipping downwards until it reaches the bottom towards the inner end: the bottom sloping upwards to the surface at the inner end. There are stones let into this ledge, and mortared in place, and marks of many other stones with mortared beds, all intended apparently to make good the ledge as a smooth bed for some construction to lie upon. The bottom of this trench I traced all over, by excavations across and along it; looking from the outer end, there first came two ledgesthe lower one merely a remainder of uncut rock, with grooves left for quarrying itthen the bottom was found about 200 inches below ground level; from this it sloped down at about 20 for about 200 inches; then ran flat for 300 or 400; and then sloped up for 300 or 400; then rose vertically, for some way; and then, from about 120 below ground level, it went up a uniform slope to near the surface, where it was lost at the inner end under high heaps of chips. At the outer end the width near the top is 152.8, and at 25 down 148.2 ; the lower space between the sides of the ledge widens rapidly to the middle, from the end where it is 43.0 wide above and 35.0 below. Towards the inner end the rock is very well cut; it has a row of very rough holes, about 6 diam., in the dressed rock along the N. edge of the trench, near the inner end. This dressed side of the trench ends sharply, turning to N. at 1603.6 from outer end of the trench axis; the width here is 170.I, or 172.3 at a small step back in S. side, a little E. of this point. The trench had not been clear for a long time, as many rudely-buried common mummies were cut through in clearing it; they were lying only just beneath the sand and rubbish in the bottom.

The N.N.E. trench was traced by excavations along the whole length of 2,840 inches, up to where it is covered by the enclosure wall of the kiosk. It is fairly straight, varying from the mean axis 2.1, on an average of five points fixed along it. The depth varies from 14 to 20 inches below the general surface. It is 38, 40, 39.2, and 36 in width, from the outer end up to a point 740 along it from the basalt pavement; here it contracts roughly and irregularly, and reaches a narrow part 18.2 wide at 644 from the pavement. The sides are built about here, and deeply covered with broken stones. Hence it runs on, till, close to the edge of the basalt pavement, it branches in two, and narrows yet more; one line runs W., and another turning nearly due S., emerges on the pavement edge at 629.8 to 633.4 from the N.E. corner of the pavement, being there only 3.6 wide. From this remarkable forking, it p 50 is evident that the trench cannot have been made with any ideas of sighting along it, or of its marking out a direction or azimuth; and, starting as it does, from the basalt pavement (or from any building which stood there), and running with a steady fall to the nearest point of the cliff edge, it seems exactly as if intended for a drain; the more so as there is plainly a good deal of water-weanng at a point where it falls sharply, at its enlargement. The forking of the inner end is not cut in the rock, but in a large block of limestone.

The trench by the N.E. socket is just like the N.N.E. trench in its cutting and size; and it also narrows at the inner end, though only for about 20 inches length. It has a steady fall like the N.N.E. trench; falling from the S. end 5.5 at 50, 8.5 at 100, 14.3 at 190, 21.0 at 300, and 27.0 at 400 inches. The inner end is turned parallel to the Pyramid, the sides curving slightly to fit it.

The rock cuttings by it are evidently the half-finished remains of a general dressing down of the rock; the hollows are from 3 to 6 inches deep, and so very irregular that they do not need any description beside the plan (P1. ii.).

The trench beside the trial passages is slight, being but 6 deep at N. and 17 at S.; it is 29.0 wide at N., 26.5 in middle, and 27.9 at S. Its length is 289, with square ends. The sides are vertical at the N., narrowing 3.5 to bottom at S.; ends shortening 3.0 to bottom. The bottom dips slightly to the S., the levels from the N. running 0, 1.7, 2.2, 3.2, and 5.8.

30. The trial passages P1. iii b.are a wholly different class of works to the preceding, being a model of the Great Pyramid passages, shortened in length, but of full size in width and height. Their mean dimensions -and mean differences from those dimensions- as against the similar parts of the Great Pyramid, are:

 26 32' mean difference .24' 41.46 mean difference .09 47.37 mean difference .13 23.60 mean difference .08 81.2 mean difference .6 28.63 mean difference .54 Pyramid passage angle 26 27' mean diff. 4' Pyramid passage widths 41.53 mean diff. .07 Pyramid passage heights 47.24 mean diff. .05 Pyramid ramp heights 23.86 mean diff. .32 Pyramid gallery widths 82.42 mean diff. .44 Pyramid well widths 28.2 mean diff. .3

The details of the measurements of each part are all entered on the section (P1. iiib.). The vertical shaft here is only analogous in size, and not in position, to the well in the Pyramid gallery; and it is the only feature which is not an exact copy of the Great Pyramid passages, as far as we know them. The resemblance in all other respects is striking, even around the beginning of the Queen's Chamber passage, and at the contraction to hold the plug-blocks in the ascending passage of the Pyramid (see section 38). The upper part of the vertical shaft is filled with hardened stone chips; but on clearing the ground over it, I p 51 found the square mouth on the surface. The whole of these passages are very smoothly and truly cut, the mean differences in the dimensions being but little more than in the finely finished Pyramid masonry. The part similar to the gallery is the worst executed part ; and in no place are the corners worked quite sharp, generally being left with radius about .15. The N. end is cut in steps for fitting masonry on to it; and I was told that it was as recently as 1877 that the built part of it was broken away by Arabs, and it appeared to have been recently disturbed ; in Vyse's section, however, the roof is of the present length, so the removal must have been from the floor. By theodolite observations the plane of the passage is straight and vertical within 5' or less.

31. Having thus finished the statement of the outside of the Pyramid and the works surrounding it, the next subject is the connection of the outside and inside of the building. To determine the exact place of the passages and chambers in relation to the whole Pyramid, a station of the triangulation was fixed in a hollow just on the end of the entrance passage floor and this was thoroughly connected with three main stations. Levelling was also carried up from the casing and pavement below, to this station, and to the courses near it. Thus the inside, as far as Mamun's Hole, is completely connected with the outside; and in the ascending passages beyond that, there is only 2' of azimuth in doubt.

32. The original length of the entrance passage has not hitherto been known, except by a rough allowance for the lost casing. But after seeing the entrances of the Third Pyramid, the South Pyramid of Dahshur, and the Pyramid of Medum, all of which retain their casing, there seemed scarcely a question but that the rule was for the doorway of a Pyramid to occupy the height of exactly one or two courses on the outside. That the casing courses were on the same levels as the present core courses, is not to be doubted, as they are so in the other Pyramids which retain their casing, and at the foot of the Great pyramid *

* The awkward restoration of the casing that Prof, Smyth adopted (Life and Work, P1. iii., 3) was forced on him by his mistaken assumption of the pavement level 20 inches under the truth (L. and W. ii. 137); hence by Vyse's casing stone measures he made the casing break joint with the core, in defiance of Vyse's explicit drawing of its position; and was obliged to reduce the pavement to 5 or 10 inches, in place of the 21 inches recorded by Vyse. The drawing of "hacking stones," at the foot of P1. 1., vol. iii., L. and W., is equally at fault; the casing stones which remain in the middle of the side, ending directly against the core masonry; and the core at the corners only leaving 34 inches for the casing thickness, No backing stones exist behind the casing of the Third Pyramid or the cased Dahshur Pyramid.

The next step is to see if there is a course equal to the vertical height of the doorway; and, if so, where such a course occurs. Now the vertical height of the doorway on the sloping face of the Pyramid (or difference of level of its top and base) would be 37.95 if the passage mouth was the same height as the present end, or 37.78 if the passage was exactly the same as the very carefully wrought courses of the King's Chamber, with which it is p 52 clearly intended to be identical. On looking to the diagram of courses (P1. viii.) it is seen that at the I9th course is a sudden increase of thickness, none being so large for 11 courses before it and 14 after it. And this specially enlarged course is of exactly the required height of the doorway, its measures running thus :

 By levelling at entrance 37.67 by measuring courses 37.8 ; by N.E. 38.1, S.E. 37.6, NW. 37.5, SW. 39.1 mean 37.94 .17 doorway height 37.95 or 37.78

Here the agreement is so exact that it is far within the small uncertainties of the two dimensions. Hence, if the passage emerged at the 19th course it would exactly occupy its height (see P1. xi.). * Besides this, it will be observed that there are two unusually small courses next over this, being the smallest that occur till reaching the 77th course. The explanation of these is clear, if the doorway came out in the 19th course ; an unusually thick lintel course was needed, so two thinner courses were put in, that they might be united for obtaining extra thickness, as is done over the King's Chamber doorway. These two courses are also occasionally united in the core masonry.

The crucial test then is, supposing the passage prolonged outwards till it intersects this course, how will its end, and the face of the casing, stand to the casing stones at the foot of the Pyramid? The answer has been already given in the list of determinations of the casing angle. It requires an angle of slope of 51 53' 20" 1'; and this is so close to the angle shown by other remains that it conclusively clenches the result to which we are led by the exact equality of the abnormal course height with the doorway height.

The data for calculating the result are (1) levels of the 19th course by entrance 668.30 and 705.97; (2) floor of passage at station mark, level 611.2 (3) which is inside the edge of the base of the casing horizontally, 638.4; (4) entrance passage angle at mouth 26 29' 1' ; (5) entrance passage height 47.26.

33. By a similar method the air channels give a determination of the angle of the faces. It is true that the channels did not occupy a whole course like the entrance; but as they are uniformly cut out as an inverted trough in the under side of a block' which is laid on a broad bed, it is almost certain that they similarly continued to the outside, through the oneor perhaps twostones now stripped off; and also that their floors thus started at a course level (see P1. xi.). **

* It should be explained that this is called the 20th course by Prof. Smyth, owing to his error about the 1st course and pavement level. His measure of it is 38 inches, and the two French measures give it as 37 and 38 inches.

** In the section of the S. air channel mouth published by Prof. Smyth, certainly "the joints are not put in from any measure," nor is any other feature of it. The passage, its bed, and top, are all about half of their true size, and the form of it is unlike anything that has been there, at least since Vyse's time.

If this, then, were the case (as the N. channel cannot by its position p 53 have come out in any but the 103rd course on the face, and the S. channel in any but the 104th), they would show that the casing rose on the N. face at 51 51' 30", and on the S. face at 51 57' 30", as before stated. The various data are entered on the diagram of the channel mouths. The levels were fixed by measuring several courses above and below the present mouths, and thus connecting them to the course levelling at the corners of the Pyramid. With regard to the main part of these air channels, the details are given further on in the measures of the King's Chamber (section 56); and it is disappointing that they vary so much in azimuth and altitude, that they are useless for connecting the measures of the inside and outside of the Pyramid.

34. The sloping blocks over the entrance to the Pyramid, and the space below them, were examined (partly by means of a ladder), and measured ; but the details are not worth producing here, as the work of them is so rough. The large blocks are as follows, in general size :

 Length on top Length below Breadth E. upper. (185) 117 80.0 to 91 W. upper. (194) 121 88.3 E. lower. 151 + 84 + 82.6 W. lower 167.7 107.6 81.6
 Height of mid-line Lean of face Angle on top Angle on base Angle on butment 20' to 2 in 35 40' to 39 50' 38 45' to 50' in 49 50' to 50 10' (114) 2 20' in mean 40 39 30 50 40 20' to 30' out 38 45' to 50' 39 20' to 50' hidden 91 25' to 30' in 39 30' 39 30' to 55' 50 30'

The measures in brackets are deduced from the angles and other measures. These blocks are much like a slice of the side of a casing stone in their angle; but their breadth and length are about half as large again as any of the casing stones. Their mean angle from 12 measures is 50 28' 5'. The thickness of these blocks is only 33 inches, and there are no others exactly behind them, as I could see the horizontal joints of the stones running on behind them for some inches. On the faces of these blocks are many traces of the mortaring which joined to the sloping blocks next in front of them. These were placed some 70 inches lower at the top, and were not so deep vertically. By the fragment left on the E. side, the faces of these blocks were vertical. In front of these came the third pair, similar, but leaning some 7 or 8 inwards on the face, judging by a remaining fragment. Probably a fourth and fifth pair were also placed here (see P1. ix.); and the abutment of the fifth pair shows an angle of 70 or 73 in place of 50. The successive lowering of the tops, leaning the faces in, and flattening the angle of slope of the stones as they approach the outside, being apparently to prevent their coming too close to the casing. These sloping blocks were probably not all stripped away, as at present, until recently, as there is a graffito, dated 1476 (half destroyed by the mock-antique Prussian inscription) on the face of the remaining block where it is now p 54 inaccessible, but just above where the next pair of blocks were placed. The sloping blocks are of remarkably soft fine-grained limestone, about the best that I have seen, much like that of the roofing of the chamber in Pepi's Pyramid; and it is peculiar for weathering very quickly to the brown tint, proper to the fine Mokattam limestone, darkening completely in about twenty years, to judge by the modern-dated graffiti.