Chap. 3. Methods and measurement
Pages 22 - 26
11. [p. 22] For the general questions of the principles of the arrangement of a triangulation, and of the reduction of the observations, we must refer to the two appendices on these subjects. They are so purely technical, and uninteresting to any but a specialist, that they are therefore omitted from the general course of this account. We begin here with lineal measure, and then proceed to angular measure, including tbeodolite work in general.
For lineal dimensions, I always used the system of a pair of rods butting end to end, and laid down alternately, instead of making marks at each rod length. In testing measures, the value of the sum of two rods can also be obtained more accurately than the exact butt length of either of them alone. But for the more important points, the direct measurement of a space by a rod has been often abandoned for the more accurate method of referring all parts to horizontal and vertical planes of known position. This is a necessary refinement when precision is needed, and it specifies a form in every element of size, angle, and place. In the passages, where the use of horizontal planes was impracticable, a plane at a given angle was adopted, and the roof and floor were referred to that.
In the Great Pyramid, the King's Chamber was measured by hanging a plumb-line from the roof in each corner of the room; and measuring the offsets from the lines to the top and bottom of each course on each side of the corner. Then the distances of the plumb-lines apart were measured by the steel tape on the floor. The heights of the courses were read on a rod placed in each corner. For the levels, the 5-inch theodolite was placed just about the level of the first course; then at 24 points round the side a rod was rested on the floor, and the level and the first course read on the rod.
The coffer was measured by means of a frame of wood, slightly larger than the top, resting upon it; with threads stretched just beyond the edges of the wood, around the four sides. The threads gave true straight lines, whose distances and diagonals were measured. Then offsets were taken to the coffer [p. 23] sides from a plumb-line hung at intervals over the edge of the wood; its distance from the straight stretched thread, being added to the offsets, thus gave the distances of the coffer sides from true vertical planes of known relation to each other, at various points all over the sides. Similarly, the inside was measured by a frame, slightly smaller inside it than the coffer. The bottom was measured by raising the coffer 8 or 9 inches; the theodolite was placed to sight under it, and offsets were thus read off to the outside bottom from a level plane, also reading the height of the plane of sight on a vertical rod; then the theodolite was raised so as to sight over the top of the coffer, the height of its plane on the same fixed rod was read off to give its change of level, and then long offsets were taken to points on the inside bottom of the coffer. Thus the thickness of the bottom is determined by the differences of level of the theodolite, minus the two offsets. Besides this, a check on the sides was taken by a direct measurement of their thickness with the pair of calipers already described.
The antechamber was measured in the common way; but the granite leaf in it had a bar placed across the top of it, with a plumb-line at each end of the bar, i.e., N. and S. of the leaf. The distances of the lines apart were taken below the leaf, and offsets were taken all up the leaf on each side; this was done at each end and in the middle of the leaf.
In the Queen's Chamber two plumb-lines were hung from the ends of the roof-ridge, their distance apart observed, and offsets taken to the side walls and to the ends. Offsets were also taken to the niche, which was, beside this, gauged with rods between its surfaces all over. The heights of the courses were also measured in each corner. The angles of the air channels were read by the goniometer already described.
The subterranean chamber was measured in the common way, with rods along the sides, but the irregularity of the floor, and the encumbrance of stones left by Perring made it very difficult to measure.
12. Turning next to measurements made with the theodolite, these generally included some determination of angular as well as lineal quantities. The straightness of the sloping passages was uniformly observed by clamping a theodolite in azimuth, pointing along the passage, and having a scale held as an offset against the wall at marked intervals; thus variations in azimuth of the passage were read. On reaching the end, the assistant holding the scale stopped, the theodolite was clamped in altitude instead of in azimuth, and the assistant returned, holding the offset scale to the floor or roof; thus variations in slope of the passage were read. The whole length of the entrance passage, and the ascending passage and gallery in one length, were thus measured. For the air channels on the outer face, where the floor is unbroken, a slip of board carrying a perpendicular mirror was let down the channel by a string, in lengths of 10 feet at a time; and the dip to the reflection in the [p. 24] mirror was noted by a theodolite at the mouth. It is then a matter of mere reduction to obtain the variations from a straight mean axis.
The horizontal measurements outside the Pyramid were entirely performed by triangulation; and this included in a single system the bases of the three larger Pyramids, the pavement of the Great Pyramid, the trenches and basalt pavement on the E. side of the Great Pyramid, and the walls around the Second and Third Pyramids. The Great Pyramid was comprised in a single triangle. This triangulation by means of the 10 inch theodolite occupied some months in all; some angles being read 14 times, and the fixed stations being about 50 in number, besides about as many points fixed without permanent marks. The first-class points were fixed with an average probable error of .06 inch; and the least accurate points, such as those on the rough stone walls, were fixed within I or 2 inches. For fixing the points uncovered by excavation, a rod was placed across the top of the hole, and a plumb-line dropped from it to the point to be fixed. A theodolite was then placed near it, and was fixed in the triangulation by reference to known stations; tbe distance of the plumb-line from the theodolite, was then measured by the angle subtended by divisions on the horizontal rod which supported it.
For connecting together the inside and outside measurements of the Great Pyramid, a station of the external triangulation was fixed on the end of the entrance passage floor, thus fixing the position of the passage on the side of the Pyramid. From this station the azimuth down the passage was observed thus fixing the direction of the passage. And levelling was also carried up from the pavement and casing stones of the N. face to this station; thus fixing the level of the passage, and hence that of all the interior of the Pyramid. The positions of the passages of the Second and Third Pyramids, on their faces, were also fixed in the triangulation.
The base of the survey was thrice measured, with a probable error of ± .03 inch (or 1/260,000 of the whole) by the steel tape. To avoid the need of. a truly levelled base line, a series of blocks of stone was put between the terminals of the base, which are 659 feet apart; a stone was placed at each tape length (1,200 inches), and at each chain length (1,000 inches); and a sufficient number of stones were placed also between these, as to support the chain or tape in catenary curves throughout, with the usual 10 lbs. tension. The stones thus varied from 140 to 393 inches apart. Then, the distances and levels of the stones being known, the reduction to be applied to the tape as it lay on them to ascertain its horizontal length, were easily applied. No attempt was made to place a mark at exactly each tape length on the stones; but a scale of 1/50ths of an inch was fixed temporarily on each stone at which the tape lengths joined; then the two ends of the tape were read simultaneously on the scales several times over, slightly shifting the tape each time in order to [p. 25] equalize the friction of its support: thus the distances of the zeros of the scales placed all along the base were ascertained, and hence the total length of the base.
For the height of the Great Pyramid a line of levelling was run up the SW. corner, across the top, and down the N.E. corner, stepping 15 to 20 feet at each shift. Separate lines of level were twice run round the Pyramid, (including the basalt pavement, &c.), and the differences were under ¼ inch, both between them and from the levels of Mr. Inglis, excepting his S.E. socket. Thus a complete chain, from N.E. to S.E., to S.W., to top, across top, and to N.E. was made; and the difference was only ¼ inch on the return, the total run being 3000 feet distance, and 900 feet height. Besides this, an independent measurement by rods had been carried up each of the four corners of the Pyramid to the top; generally two, and sometimes three or four, steps were taken in one length, and levelled to the nearest, 1/10 inch, from the upright rod to the upper step, by a reversible horizontal rod with level attached. The intermediate courses in each length were also measured off This gives all the course heights, and is regulated at every 10 or 20 courses by the accurate levelling on the N.E. and S.W. The same point was always used on each step, both in the measuring and the levelling, so as to avoid errors of levelling and dressing in the steps; and each tenth course has a cross scored on the stone, at the point used in the levelling. The Third Pyramid was only measured by rods up the courses.
The angles of the ascending passages were not retaken, as Professor Smyth had already done that work fully; but the angle to the bottom of the entrance was observed by the 10-inch theodolite, placed on a shelf across the mouth of the passage. The levels of the horizontal passages were taken with the 5-inch theodolite, placed in the middle, and reading on both ends. The level from the entratice passage to the ascending passage was read off on a single vertical rod placed in Mamun's Hole; a theodolite being put first in the lower and then in the upper passage to read on it.
As a general principle, in observing down a passage with a theodolite, no dependence was placed on measuring the position of the theodolite, which was usually outside the passage in question; but in all cases a signal was fixed in the passage near the theodolite, as well as one at the farthest point to be observed, and the azimuths of both were noted; the distances being roughly known, the minute corrections to be applied to the azimuth of the further signal could be readily determined. The azimuth observations of Polaris always included a greatest elongation. For the dip of the entrance passage the l0-inch theodolite was clamped in altitude, at closely the true angle; an offset was taken to the roof at the bottom, and the theodolite was reversed and re-read as usual to get the dip, reading level at the same time. Offsets were then read to points all up the roof keeping the telescope clamped in its second position; thus [p. 26] it was not necessary to know the exact height of the plane of the roof above the theodolite. The azimuth of the entrance passage was determined down to Mamun's Hole, by connection with the triangulation, whose azimuth is otherwise known; and it was also determined down to the bottom by Polaris observations. The azimuths of the horizontal subterranean passages were read by the signal at the top, and on candles placed in the passages;1 the S. end of the S. passage being invisible from the theodolite, its candle was sighted on in line with its N. end candle, and the line measured off in the chamber. The azimuth of the ascending passages was measured by three theodolites used together; all of the telescopes were set to infinite focus, so as to see each other's cross wires plainly when a candle was held behind the telescope observed on. The 10-inch was put in the entrance passage, reading on a signal at top, and on the 5-inch; the latter was placed on the rubbish in Mamun's Hole, reading on the 10-inch and 4-inch; and this last was placed just above the granite plug blocks, reading on the 5-inch and on a signal at the top of the ascending passage. Thus a chain of angles was formed from signal to signal, quite free from any errors of centring the theodolites or station marks. For the angle of the Great Pyramid casing stones in situ, the 10-inch theodolite was placed on the steps above; the dip was read to points on the top of the casing stones, and on the pavement in front of them; and then offsets were measured from these points to the face of the stone. The Second and Third Pyramid casing was measured by goniometer and protractor.
Thus it will be seen that several fresh methods of observation have been introduced, in order to obtain greater accuracy and more information : in particular the methods of plumb-lines and optical theodolite-planes, with offsets from these, have yielded good results. A fresh feature in the discussion of observations is the introduction of "concentrated errors;" on the principle of showing all the divergences from regularity on their natural scale, while reducing the distances of the parts so that they may readily be compared together. This is the essential basis of the method of graphic reduction, described in the Appendix (shown in Traces of Observations, Pl. xvi; and it renders possible the use of graphic methods in work of any delicacy; it is also exemplified in the diagrams of the King's Chamber walls Pl. xiii, and of the relation of the casing and pavement Pl. x
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