THE HALF-SURFACE LINE Up to now I have followed mainly the presentation of Petrie, because I am interested in showing how he concluded positively that the floor of the King’s Chamber coincides with the half-surface line. I have added more precise figures in order to show that the coincidence is perfect. But the mathematical analysis of Petrie was incomplete, because he totally ignored the role of the factor f. As I already explained, when he went to measure the Great Pyramid, he was still under the spell of the theories of Taylor and Smyth, which emphasized the factor p and completely ignored the factor f, which happens to be the one mentioned by Herodotus. Because he ignored the factor f, Petrie missed some of the further peculiarities of the half-surface line. The half-surface line not only divides the pyramid into half-surfaces by means of the meridian section, but also in terms of the surface of the faces. The half-surface line divides the typical face in such a way that - The square of the apothem (slant height) above the half-area line is half of the square of the total apothem.
- The surface of the face above the half-area line is half the surface of the face.
According to my reconstruction the basic face of the pyramid has the following dimensions: The square of the apothem
is 355.5636605 In any quadrangular pyramid
there is a level which has remarkable mathematical characteristics. If
in the meridian section of a pyramid there is drawn a line parallel to
the base at a height such that this line forms a smaller triangle which
has half the surface of the original triangle, this line The length of the line is ½Ö2 of the base line. This means that the line is equal to half of the diagonal of the base. The horizontal square at the level of the line has a diagonal which is equal to the length of the base line. DE = ½Ö2 B C Ö2 DE = B C Area ADE = ½ area ABC (DE)² = ½(BC)² In less mathematical language this means: The horizontal surface above level DE equals one half the horizontal surface above level BC. The diagonal of a horizontal square erected at level DE is equal to the distance BC. Conversely, DE is equal to half the diagonal of a horizontal square erected at level BC. Since according to my interpretations in the Great Pyramid BC was calculated as 419.75 cubits, DE must be 310.77343 cubits. That the half-surface line might be important in the planning of the Great Pyramid might occur to any person who has a feeling for geometry. For this reason when, following the publication of the Howard–Vyse report about the dimensions of the Great Pyramid, there was a great upsurge of interest in the geometry of the Great Pyramid, the half-surface line was mentioned as important by more than one writer. I have not been able to identify who was the first to suggest that the floor of the King Chamber’s was at the level of the half-surface line, but the idea was current at the middle of the nineteenth century. Petrie concluded the report of his survey by stating that the hypothesis that the King’s Chamber is at the half-surface level, is proved valid, since all figures agree with it with a maximum discrepancy of 3 inches. He thought that this was the kind of error that one could have expected in the actual construction of the Pyramid. In my opinion the builders were much more accurate than Petrie assumed them to have been (within the millimeter) and the height of the King’s Chamber agrees within the millimeter with the figures I can derive by combining Petrie’s survey with that of Cole. Hence, the half-surface theory can be considered positively demonstrated. * * * since Petrie did not have the opportunity to clear the corner of the pyramid from debris. But Petrie was able to establish by measuring the inside of the Pyramid that the distance between the base of North face and the top line of the Great Step was 4534.5 ± 0.9 inches, and concluded that the face of the Great Step marked the East-West axis of the pyramid; but he did not believe that the builders had been accurate in setting the position of the Great Step, because from his faulty measurements of the sides of the pyramid he had arrived at a figure of 4534.1 inches for the distance of the East-West axis from the base of the North face. According to my figures, which are based on the Cole report, the distance is 219.75 cubits = 115.181,58 meters = 4534.72 British inches. The Great Step marked exactly the East-West axis. Petrie is most articulate in stating that the positive result of his survey has been that of proving the validity of the half-surface hypothesis for the level of the floor of King’s Chamber. I will quote in full the conclusion of his book (p. 220–221): “Having now… for the… the form and size… the Queen’s Chamber.… The angle of the slope from the entrance passages is 1 rise on 2 base.” Later, I shall discuss in detail the problem of the inclination of passages. What I need to point out here is that before Petrie conducted his survey, several investigators had argued that the inclination of the passages was according to arc tan 0.5 = 26° 33’ 54.” 1842. This means that the floor of the passages constitutes the hypothenuse of a triangle in which the height is one and the base is 2. What these investigators failed to see is how this angle is related to the slope of the pyramid and, hence, to the height of the King’s Chamber. Petrie failed to see it, because he was obsessed by the idea inherited from Taylor and Smyth, that the slope was determined by the factor . Petrie never discussed the factor , although it had been mentioned by Herodotus. If he had considered the factor , he might have seen how a slope determined by the relation 1:2 is related to . It is enough to consider that a right triangle with sides 1 and 2 has a hypothenuse 5 and that = (5–1)/2 and that 1+2 = 3 and a right triangle with sides and 1/ has a hypothenuse 3. Earlier in the chapter entitled “Theories compared with fact,” (pp. 186-187) Pliny wrote: “We will now note some connections which appear between the exact dimensions...” According to my calculation of the base and of the inclination of the North face and the West face, the half-area level is 1689.49 inches. Therefore, the empirical findings of Petrie are closer to my figure than the three inches that Petrie grants as possible leeway. All these properties of the line DE are related to the properties of 2. I have explained that 2 was practically important in agrarian units of surface where squares were halved and duplicated. If one takes the diagonal of a square, the square constructed on the diagonal has twice the surface and the square constructed on half the diagonal has half the surface. I have explained that the formation of the royal cubit of 7 hands was related to these problems of duplications of squares and that the rough first draft of the pyramid was related to these calculations. My calculations assume that in principle the pyramid was calculated by the factor. If we assume that the pyramid had a slope determined by the exact factor , the meridian triangle would be the following: I assume that was calculated as 610/377. Accordingly, the meridian triangle would be: . The height of the half-surface line is 81.876162063 cubits = 42.91286903 meters = 1689.486095 English inches. I have concluded that the Western face of the pyramid was intended to embody the factor and was given a slope calculated by the round figure 51° 51’ (according to the exact value of the slope should be 51° 51’ 14”). I have explained how in order to achieve the result that the Western slope reach the same height as the North slope, the North-South axis of the pyramid was displaced to the West, so that the Western face of the pyramid corresponded to the following meridian triangle: the height of the half surface line is 42.972,85572 meters = 1689.4856 inches. Petrie dedicates a great deal of his analysis to refuting Smyth’s theory that the passages of the Great Pyramid had a slope 26° 18’ 10.” He concludes that one might argue that the Great Gallery was built according to this angle, but certainly not the other passages. Then, Petrie continued by observing that Smyth’s contention about the angle of the passages, is a derivation of his equal area square theory which must be rejected. At the end of the discussion of the theories about the angle of the passages, Petrie concludes (p.191): There then remains only the old theory of 1 rise or 2 base, or an angle of 26° 35’ 54”; and this is for within the variations of the entrance passage angle and very close to the observed angle of 26° 31’ 23”; so close to it, that two or three inches on the length of 350 feet is the whole difference; so this theory may at least claim to be far more accurate than any other theory. What Petrie means is that from the time of the first surveys of the passages there has been formulated the theories that their slope is determined by triangles in which the sides relate as 1:2 [see drawing] This type of triangle assumed a tangent 0.5. and an angle 26° 33’ 54.” 1842. pyramid. Petrie does not deny that it may apply at least to the entire descending passage. I shall show how it can fit exactly this passage, and I will explain why the ascending passage was given a slope which slightly less (26° 16’ 40” according to Petrie). As I have already stated, the official academic world greeted Petrie’s report with enthusiasm, because they understood that it refuted Piazzi Smyth’s theories, which is not entirely true. Then, they drew the next absolutely unwarranted conclusion that Petrie’s figures sealed a tombstone on all the mathematical theories about the structure of the Great Pyramid. The fact is that the academy did not like Petrie’s method of approaching the Great Pyramid in terms of precise measurements; hence, they referred to it as for it was convenient to quote or misquote it for polemic purposes, but never bothered to read it more in a skimming fashion. Even though Petrie’s survey of the base lines of the Great Pyramid was considered epoch-making at the time and it is even quoted today, it was proved basically incorrect by the later Cole survey. This should not be considered a reflection on Petrie’s diligence and skill as a surveyor: the distortion resulted from the circumstance that Petrie as a private investigator, not supported by academic institutions, did not have the opportunity to lay bare from the rubble the four comers of the pyramid. However, he was able to make an important contribution to the empirical archeology of the pyramid: he uncovered a set of blocks that marked the original line of the base on the North side near the center, below the entrance door to the pyramid. Starting to this line he inferred from what he could see what might have been the original base line for the other three sides. But he assumed incorrectly that the pyramid was intended to have a base that was an exact square except for minor errors in … * * * But surprisingly the floor of the Grand Gallery does not end at the level of the horizontal passage: it ends below it forming what is called the Great Step. A number of observers before Petrie had wondered about what was the significance of this step that forms a sort of stumbling block on the way of ascent. The face of the step runs in direction East-West; its height is reported by Smyth as 36 pyramid inches which would be equal to 917.2594879 = 36.11264 One reaches the remarkable result that the level of the half-surface line is the same according to the meridian triangle of North face and according to the meridian triangle of the West face. Calculating with the maximum theoretical precision, the difference in level would be less than 1/100 of millimeter. The interesting fact is that the same mathematical relations applies as well to a face calculated by the factor ¹. According to my figures for a face calculated by the factor ¹, the face would be [see drawing] As I have explained his measurements of the inside of the pyramid are much more reliable than those of the outside. Hence, Petrie should have concluded that the Great Step gave him the best figure for the North-South exterior of the Pyramid. He tried to prove that possibly the Great Step is 0.3 inches= 7.63 mm South of East-West axis. But he qualified this statement by adding that the Great Step is at a position 0.3 ± 0.9 inches South of the West Axis. He looked for another line which could be identified as being set on the East-West axis. This he found to be the middle line of the roof of the Queen’s Chamber. But this roof is formed by two gigantic granite blocks meeting at an angle so as to form a sloping roof. Given the extraordinary size of the heavy blocks their meeting line could not be set with absolute precision; hence, the middle line of the roof is somewhat irregular in terms of the precision of the Great Pyramid. Petrie chose to measure the middle of the roof line * * * well preserved. There are some damages and dislocations due to earthquakes, but there is a possibility of taking these factors into account. For instance, Petrie in calculating the dimensions of the King’s Chamber took into account that one should deduct the gaps between the blocks caused by the earthquakes. By assuming that originally the blocks were in contact, he arrived at the conclusion that about 1/5000 (see numbers written on paper) * * * The Great Step would provide a more reliable datum since it embodied a line that the builders could easily control. There is an amazing agreement between my figure obtained by the combined calculation of the distance of the East-West axis and the North base line and Petrie’s figure for the distance between the top edge of the Great Step and the North base line. My figure 4534.716 English inches Petrie’s Figure 4534.5 ±0.9 Petrie did not rely on the Great Step in order to calculate the size of the Pyramid, because he had obtained the following figures for the sides of the pyramid East side 9067.7 inches (1/2 = 4533.85) West side 9068.6 inches (1/2 = 4534.3) He compromised between the two figures choosing 9068.2 as the average on the assumption that the base of the pyramid was intended to be a perfect square. But after the Cole report we know that the pyramid was not intended to be square. Further we know that Petrie’s findings are unreliable because he did not have the means to clear from rubble the corners of the pyramid. This figure for the West side which is 9068.6 English inches = 260,342.02 mm. is not too different from Cole’s figure which is 230,357 mm. = 9069.19 inches. My figure is … * * * But Petrie was able to uncover
the blocks making the base of the pyramid in the area near the entrance
door on the North side. This was a datum that was not available to Piazzi
Smyth when conducted his survey, as he himself remarks in the Fifth Edition
of The Smyth theory, which would be called the equal area theory was a modification of the previously presented half-area theory. The half-area theory happens to be valid, as I have indicated, and also happens to be simpler; it has been found that simpler scientific theories are usually more likely to be valid than more complex ones, although why it should be so is a highly disputed matter among philosophers of science. The half-area theory declares that if one takes the meridian triangle and marks a line parallel to the base which divides the triangle into two area of equal surface, this line marks the level of the floor of the King’s Chamber. [see drawing] The area ADE is half of the area ABC. 115.181.5843 4534.716 Petrie calculated that the middle point of the pyramid was indicated by the position of the Great Step at the top of the Gallery and the middle point of the roof of the Queen’s Chamber. He gave the following data for the distance from the base line of the North side. Queen’s Chamber angle line of roof 4533.8 ± 0.8 Top edge of Great Step 4534.5 ± 0.9 But the roof of the Queen’s Chamber is formed by two huge blocks of granite which of necessity are rather irregular. Hence, the meeting line of the two blocks would be an extremely reliable reference point. Petrie himself grants that he had to make a choice and chose a point about half way on the North side of the Chamber. * * * the Royal society.” I suspect that Petrie’s report on “The Inside of the Great Pyramid,” which constitutes Chapter VII of the first edition of his book, is incomprehensible except for those who have made themselves familiar with the language of surveyors’ reports. As Smyth observed in the fifth edition of his book, which was issued after the publication of Petrie’s report, Petrie had one great advantage over his predecessor: he had discovered the stones marking the original base line below the entrance to the Pyramid. Hence, Petrie had an absolutely reliable reference point: and one single solid reference point can make a world of difference. Petrie moved through the very innards of the pyramid counting from North to South, starting from the North base line. He came to that amazing and most peculiar piece of architecture which is the Grand Gallery, leading upwards to the King’s Chamber. The Grand Gallery ends at the top where there begins the horizontal passage * * * proves to be invalid, but in its general conception it is similar to the hypothesis that can be validated. Smyth conducted a survey of the inner dimensions of the pyramid, but unfortunately given the means available to him his survey was not sufficiently accurate to prove or disprove his theory. But the later survey conducted by Petrie, who trained himself for the task by choosing surveying as profession, proved that Smyth’s hypothesis had to be rejected. Unfortunately, the survey conducted by Smyth inside the pyramid was not sufficiently precise to succeed in either validating or invalidating Smyth’s hypothesis. The more accurate survey conducted by Petrie succeeded in proving that this hypothesis is not valid. * * * the actual construction of the four corners. As a result his best figure is that for the West side, since this side is actually at a right angle with the North side. Petrie reported that the West side is: 9068.6 inches = 230,342.03 mm. Cole reported a length of 230,357 mm. = 9069.19 inches According to my interpretation the length is * * * Petrie assumed that the East side had about the same length: 9067.7 inches = 230,319.16 mm. The Cole survey found an East side substantially longer, in terms of the extreme precision of the Great Pyramid: 230,391 mm. = 9070.53 inches. Petrie decided that the East-West axis was running at a distance from the North side of 4534.1 inches = 115,165.93 mm. He obtained this figure by averaging the part of the East side with that of the West side and dividing the result by 2. Actually the true figure according to my calculations is: 115,181.5843 mm. = 4534.716 inches Petrie’s own survey proves that my figure is the correct one. In his survey Petrie followed in the footsteps of Piazzi Smyth. Actually when Petrie began his survey his purpose was to collect data to prove or disprove Smyth’s theories. The later had dedicated most of his attention to the inside passages of the Great Pyramid for two reasons. The first, a practical one, namely that in spite of a technical difficulties one was more likely to obtain good data about inside passages to which access was possible than about the sides which were covered with rubble. The second is that Smyth, even before leaving for Egypt, had developed a theory about the rationale of the inside passages and their angles. Smyth had started from the correct scientific assumption that the passages must be related to the total dimensions of the Pyramid and that it should be possible to deduct the overall dimensions, lengths and angles, from the lengths and angles of the passage. The assumption is correct but Smyth’s theory about the mathematical rationale of the passages was correct only in a minor part. Petrie began to survey the inside of the Pyramid repeating Smyth’s operations and he actually began by using as reference points grooves that Smyth’s has cut on the walls of the Descending Corridor. It is only in the course of the operation of surveying the inside that Petrie found that some of Smyth’s assumptions did not apply and began to operate with more independence. Smyth was not satisfied with the half-area hypothesis, because a good scientific theory must be as comprehensive as possible. It is a basic principle of scientific method that a scientific hypothesis is superior in relation to the breadth of events that it can explain. Smyth felt that there should be a hypothesis that explains not only the level of the King’s Chamber but also possibly the position of the other rooms and most of all the arrangements of the passages which is most peculiar. It is indeed amazing that the King’s Chamber should have to be reached first by descending and then by climbing inside the pyramid. Hence, he rejected the half-area hypothesis, and constructed the equal area hypothesis. Petrie’s survey proved that Smyth’s equal area hypothesis * * * Petrie’s survey of the inside of the Great Pyramid proves to be amazingly well conducted in spite of the material difficulties under which he had to operate: for instance, the passages had not yet been cleaned of rubble at key points. Petrie’s survey of the inside of the Pyramid had a fate it did not deserve. The official academy was glad to infer from it that Smyth’s theories had been disproved and applauded, but they never bothered to read and study it. The only people who read and studied Petrie’s figures have been the supporters of Piazzi Smyth, who conisdered whether a new version of his theories could be derived from Petrie’s figures. The only people who surveyed again the inside of the Pyramid in order to check Petrie’s figures and if necessary, to improve upon them have been supporters of Smyth’s general views, such as Morton Edgar and David Davison. As a first step it is necessary to correct an error in Petrie’s calculation of the triangle of the first part of the Descending Passage. According to his report the triangle is the following: [see drawing] These figures contradict the Theorem of Pythagoras. From Petrie’s own statements one learns that his procedure was to measure on the ground the hypotenuse (which is the floor of the Passage) and then by the sine of the angle of inclination to obtain the vertical dimensions. The horizontal dimension apparently was obtained from the other two sides by the Theorem of Pythagoras. He states specifically that he calculated by sine 26° 31’ 23" ± 5" * * * at a point (middle of the West side) which gave him the figure 4535.8 inches, so that by averaging this figure with the figure obtained from the Great Step, he would arrive at the figure of 4534.1 which he had obtained by measuring the outside of the pyramid. But actually his figure about the roof line of the Queen’s Chamber does not mean much since he qualifies it a being 4533.8 ± 0.8 inches = 115.158, 31 meters 0.020,32 In my opinion the line of the Great Step which the builders could more easily have set with absolute precision, should be presumed to be exactly on the East-West axis. A new survey would be in order to establish whether one arrives at the figure which I have obtained from mathematical interpretation 219.75 cubits = 115.181,58 meters = 4354.72 English inches. It is not to be noticed how close my figure is to the one Petrie proposed for the Great step: 4354.5 inches. Petrie concluded that the Great Step marked the middle point of the pyramid in the sense of its being on the line of the East-West axis. He found it to be at a distance of the top line of the step to be at a distance from North base line of 4534.5 ± inches = 117.176 of 14 mm. According to my calculations the North-West line is at distance of 219.75 cubits = 115.181,58 meters = 4354.72 inches. Petrie did not trust his conclusion that the Great Step marked exactly the position of the East-West axis, because from his unreliable measurements of the sides he had obtained a figure of 4534.1 inches. The important fact is that the ratio that the height of King’s Chamber was determined by the half–surface line, was duplicated by several writers at the time Petrie went to survey the Great Pyramid. Petrie tested the correctness of this hypothesis and stated most emphatically in the conclusion of the report of his survey, that this hypothesis proves to be valid. This is the most positive statement in the report of Petrie, but in spite of his expressing himself in most clear terms, the statement got lost in the fury of controversies about the hypotheses of Piazzi Smyth. * * * the meridian section. But this line marks a half surface not only vertically, but also horizontally: the surface of the horizontal section of the pyramid at level of the line is half of the surface of the base of the pyramid. The position of the line in question is easy to identify because its length is equal to half of the diagonal of the base. DE = 1/2 Ã2 BC [see drawing] Surface ADE = 1/2 surface ABC Surface ADE = Surface DECB surface of horizontal section at level DE = 1/2 surface of base of pyramid must be rejected and that the half-area hypothesis appears to be validated. I shall show that by reexamining Petrie’s survey on the basis of the additional data of the Cole survey, the half-area hypothesis can be positively validated. Further, I will show that the half-area hypothesis can be used also to explain the arrangement of the passages. That such a line might be important in the planning of the Great Pyramid might occur to any person who has a feeling for geometry. For this reason this line was mentioned by several authors as people began to speculate about the geometry of the Great Pyramid around the middle of the nineteenth century. The notion of the importance of the half-surface line would occur so readily, that I have not been able to establish who was the first who suggested that the level of the floor of the King’s Chamber corresponded to this line. Any pyramid will have a half-surface line, (line equal to 1/2 Ã2 of the base), but level of this line (its distance from the base) will depend on the inclination of the faces. People who had a sense for geometry suspected that the half-surface line had importance on the construction of the pyramid. On the basis of the survey of Howard-Vyse and Perring they suggested that the floor of King’s Chamber is the level of this line. Petrie’s survey proved that they were correct.[see drawing] DE = 1/2 Ã2 BC Ã2 DE = BC Area ADE = Area 1/2 ABC (DE) Since according to my interpretation in the Great Pyramid BC was calculated as 439.5 cubits, DE must be 310.77343 cubits. It is a reasonable assumption that some relation should have existed between the inner dimensions and the outer dimensions of the Great Pyramid. If this relation can be established then we have a basis by which we can establish or at least verify the outer dimensions in spite of ravages to which the outside was subjected by those who used the pyramids of Gizah as quarries. That this is the crux of the problem in the investigation of the Great Pyramid was realized by Piazzi Smyth, who, in spite of what his detractors may say, had clear understanding of scientific methodology. Before leaving for Egypt he had already developed a hypothesis about the relations between the outer dimensions and the inner divisions. Anticipating what I will explain later I can state that this hypothesis must be rejected. In the preceding pages I have tried to reconstruct what was the height to the Great Pyramid and, hence, the slope of the North side and the West side. I have tried to pursue any piece of evidence that may be available, including the careful interpretation of ancient authors. But in spite of all my efforts, one could still argue that evidence, using a legal term: The real direct and irrefutable evidence would be an examination of the inclination of the faces. Petrie did the best he could under difficult conditions and concluded that the slope of the North face could be taken to have been 51° 50’ 40" ± 1’ 0.5.” This figure does not disagree with my figure of 51° 49’ 39" for the North face. 51° 49" Piazzi Smyth. But the leeway of about one minute of are left by Petrie’s survey, is too broad to be satisfactory in terms of what I consider the Egyptian precision. Now that after the Cole survey we can establish the exact lines of the base of the pyramid, * * * In any quadrangular pyramid
there is a level which has remarkable mathematical characteristics. If
in the meridian section of a pyramid there is drawn a line parallel to
the base such that it marks on the original triangle a smaller triangle
which is half in surface, this line halves the surface not only DE = 1/2 Ã2 BC Surface ADE = 1/2 surface ABC, Horizontal surface at level DE = 1/2 horizontal surface at level BC. Diagonal of horizontal square at level DE = BC DE = 1/2 diagonal of horizontal square at level BC. [see drawing] Before Smyth developed his hypothesis, another hypothesis had been formulated on the basis of the measurement of Howard-Vyse and Perring. This could be called the half-area hypothesis. If we could drew the meridian section of any rectangular pyramid there is a line parallel to the base which has peculiar mathematical properties. This is the line that divides the meridian section of the pyramid into two halves of equal surface: that is the surface above the line of half of the total surface of * * * a datum that was not available to Petrie, it would be in order to proceed to a new test of the inclination of the faces. But given the present condition of the faces of the pyramid, it is doubtful that we could improve substantially on Petrie’s figure, although any advance would be a help to the solution of the problems raised by the architecture of the Great Pyramid. If a unequivocal result cannot be obtained on the matter of the height of the pyramid from the inclination of the faces, there may be found other possible sources of information. If the outer surface of the pyramid cannot provide absolutely certain data, because the outer casing has disappeared there is the likehood that precise figures can be obtained from the study of the passages and rooms inside the pyramid. These passages and rooms are generally… * * * Because of the geometry of any quadrangular pyramid the equal area line has also these two other mathematical characteristics: a) The surface of the horizontal section of the pyramid at the level of the line, is half of the surface of the base. b) The diagonal of this horizontal section is equal to the length of the sides of the base. c) The length of equal area line is half of the length of the diagonal of the base. In other words, the geometry of the half-area line is connected with the problem of constructing squares which are double in surface of others. I have explained that this was basic problem of ancient land surveying, was at the root of the origin of the royal cubit as a unit of measure, and explains the first general draft of the Great Pyramid. We must remember that given a square with side one, the square with double surface has a side Ã2 = 1.414 and the square with half the surface has a side 1/2 Ã2 = 0.7. The square constructed on a diagonal of a square has twice the surface of the original square. |