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THE SUNDAY TELEGRAPH
October 20 2002
Robert Matthews
"For years Nobel prizes have been won by sci-entists for achieve-ments almost as obscure as tlie sclentists themselves. Not tliis year, though: all the win-ners were A-list scientific celebrities, the only puzzle about their success being why they hadn't won years ago.
I was amazed to find that I evenrecognised the name of one of the winners of the Eco-nomics prize: Damel Kahrie-man of Princeton University, who in the 1970s - along with the late Amos Tversky of Stan-ford University - carried out a series of pioneering studies revealing how humans behave in: the face of uncertainty.
The upshot of these studies was that people tend to resort to rules of thumb -"heuris-tics" - which can mislead them when they try to make decisions based on limited or uncertain information.
Demonstrations of the fool-ishness of Joe Public would hardly seem the stuff of Nobel prizes. What made Kahneman and Tversky's work so influ-ential was their use of simple yet telling experiments and anecdotes to reveal the ubiq-uity of these flawed heuristics.
In their most famous paper, "Judgement under uncer-tainty: Heuristics and Biases", published in Science in 1974, Kahneman and Tversky illus-trated one such rule of thumb with a story about the Israeli Air Force. Officers in charge of training had discovered what they believed to be a brilliant insight into pilot mo-tivation. If they reprimanded pilots after a bad landing, next time they did better. In con-trast, praising pilots who landed their planes well had the opposite effect, virtually guaranteeing they would do worse next time.
The results appeared to stand orthodox motivational psychology on its head: while criticism produced greater effort, praise only led to com- placency. The implications for training were clear: the stick worked better than the carrot.
Yet as Kahneman and Tversky pointed out, the offi-cers had fallen into the . trap of believing that every landing was representative of each pilot's real ability. Yet each nat-urally had its peculiarities - sometimes better than the pilot's average, sometimes worse. An awful landing would, thus typically be followed by a better one through "regression to the mean", with the pilot's true ability only becoming clear in the longer term. As Kahne- man and Tversky pointed out, by failing to recognise this and concluding that criticism was better than praise, the office ran the risk of undermining pilot confidence.
A second rule of thumb. uncovered by Kahneman and Tversky centred on "avail-ability", where people judge the chances of an event according to whatever they can most readily bring to mind. A classic example is to ask people which are most' common: words beginning " with "r" , or words which have "r" as the third letter?
Most people bet the former - and lose, because it's not true. We think there are more words beginning with "r" just because they are easier to recall than words with "r" as the third letter. Such "avail-ability effects" also explain why many of those fatuous "All Time Greatest" polls that appear regularly in news-papers and magazines have some currently fashionable, name near the top.
The final rule of thumb described by Kahneman and Tversky in their paper was the so-called anchoring effect. In one of their experiments,they gave a group of students five seconds to estimate the value of 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8; meanwhile, they asked another group to do the same for 8 x 7 x 6 x 5 x 4 x 3 x 2 x l. The median result for the first group was 512, while the lat-ter came up with a figure over four times larger - because their guesstimate had been "anchored" to the higher numbers early in the sequence (but still nowhere near the correct number of 40,320).
So far, so trivial - until one thinks about how often we are presented with numerical lists which are similarly "loaded". For example, how .would you rate your satisfaction with your spouse, on a scale from 1 to lOO? Or how about 1 to 200? Research suggests that those asked to give a figure within the lower range will typically give a lower rating - because of the "anchoring" effect of the smaller end-figure (you have been warned).
In recent years, some psy-chologists have claimed that Kahneman and Tversky were too keen to show the failings" of human reasoning, and overlooked the fact that" some of these rules of thumb often work reasonably well , Even so, there can be few'.Nobel prizewinners whose research has implications that can give each of us pause for thought several times every day."
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8
8 x 7 x 6 x 5 x 4 x 3 x 2 x l
"nowhere near the correct number of
40,320"
FINGERPRINTS OF THE GODS
Graham Hancock 1995
Page 460 /1
"43,200
is not a random number"
Page 397
Part VII
Very large numbers also appeared in the Pyramid Texts, in the symbolic 'boat of millions of years', for example, in which the Sun God was said to navigate the dark and airless wastes of interstellar / Page 398 / space.44 Thoth, the god of wisdom ('he who reckons in heaven, the counter of the stars, the measurer of the earth') was specifically empowered to grant a life of millions of years to the deceased pharaoh 45 Osiris, 'king of eternity, lord of everlasting', was described as traversing millions of years in his life.,46 And figures like 'tens of millions of years' (as well as the more mind-boggling 'one million of millions of years')47 occurred often enough to suggest that some elements at least of Ancient Egyptian culture must have evolved for the convenience of scientifically minded people with more than passing insight into the immensity of time.
Such a people would, of course, have required an excellent calendar - one that would have facilitated complex and accurate calculations. It was therefore not surprising to learn that the Ancient Egyptians, like the Maya, had possessed such a calendar and that their understanding of its workings seemed to have declined, rather than improved, as the ages went by. 48 It was tempting to see this as the gradual erosion of a corpus of knowledge inherited an extremely long time ago, an impression supported by the Ancient Egyptians themselves, who made no secret of their belief that their calendar was a legacy which they had received 'from the gods'.
We consider the possible identity of these gods in more detail in the following chapters. Whoever they were, they must have spent a great deal of their time observing the stars, and they had accumulated a fund of advanced and specialized knowledge concerning the star Sirius in particular. Further evidence for this came in the form of the most useful calendrical gift which the gods supposedly gave to the Egyptians: the Sothic (or Sirian) cycle.49
The Sothic cycle was based on what is referred to in technical jargon as 'the periodic return of the heliacal rising of Sirius', which is the first appearance of this star after a seasonal absence, rising at dawn just ahead of the sun in the eastern portion of the sky.50 In the case of Sirius the interval between one such rising and the next amounts to exactly 365.25 days - a mathematically harmonious figure, uncompli-cated by further decimal points, which is just twelve minutes longer than the duration of the solar year 51
The curious thing about Sirius is that out of an estimated 2000 stars in the heavens visible to the naked eye it is the only one to rise / Page 399 / heliacally at this precise and nicely rounded interval of 365 and a quarter days - a unique product of its 'proper motion' (the speed of its own movement through space) combined with the effects of precession of the equinoxes. 52 Moreover, it is known that the day of the heliacal rising of Sirius - New Year's Day in the Ancient Egyptian calendar - was traditionally calculated at Heliopolis, where the Pyramid Texts were compiled, and announced ahead of time to all the other major temples up and down the Nile.53
I remembered that Sirius was referred to directly in the Pyramid Texts by 'her name of the New Year'.54 Together with other relevant utterances (e.g., 66955), this confirmed that the Sothic calendar was at least as old as the Texts themselves, 56 and their origins stretched back into the mists of distant antiquity. The great enigma, therefore, is this: in such an early period, who could have possessed the necessary knowhow to observe and take note of the coincidence of the period of 365.25 days with the heliacal rising of Sirius - a coincidence described by the French mathematician R.A. Schwaller de Lubicz as 'an entirely exceptional celestial phenomenon'?57
We cannot but admire the greatness of a science capable of discovering such a coincidence. The double star of Sirius was chosen because it was the only star that moves the needed distance and in the right direction against the background of the other stars. This fact, known four thousand years before our time and forgotten until our day, obviously demands an extraordinary and prolonged observation of the sky.58
It was such a legacy - built out of long centuries of precise observational astronomy and scientific record-keeping - that Egypt seems to have benefited from at the beginning of the historical period and that was expressed in the Pyramid Texts.
In this, too, there lies a mystery. . . '
Graham Hancock 1995
Page 454
Chapter 48
Follow these instructions carefully:
Draw two parallel straight lines vertically down a sheet of paper, about seven inches long and a bit under three inches apart. Draw a third line, also vertical, also parallel and of equal length, exactly mid- way between the first two. Write the letter'S' - for 'South' -at the top end of your diagram (the end farthest away from you), and the letter 'N' for 'North' at the bottom end. Add the letters 'E' for 'East' and 'W' for 'West' in their appropriate positions at either side of the diagram, to your left for East and to your right for West.
What you are looking at are the outlines of a geometrical map of Egypt incorporating a perspective very different from our own (where 'North' is always equated with 'Up'). This map where 'Up' is 'South' seems to have been worked out an enormously long time ago by cartographers with a scientific understanding of the shape and size of our planet.
To complete the map you should now mark a dot on the central of the three parallel lines about an inch to the south of ('up' from) the northern end of the diagram. Then draw two more lines diagonally down from this point, respectively to the north-east and north-west, until they reach the northern ends of the two outermost parallel lines. Finally link those parallel lines directly with horizontal lines running east to west at the northern and southern ends of the diagram.
The shape produced is a meridional rectangle {oriented north- / Page 456 / south). This rectangle is seven inches long by just under three inches wide and has a triangle demarcated at its northern (lower) end. The triangle represents the Nile Delta and the dot at the apex of the triangle represents the apex of the Delta - a point on the ground at 30° 06' north and 31° 14' east, very close to the location of the Great Pyramid.
Whatever else it may be, it has long been understood by mathemati-cians and geographers that the Great Pyramid serves the function of a geodetic marker (geodetics being the branch of science concerned with determining the exact position of geographical points and the shape and size of the earthl). This realization first dawned in the late eighteenth century when the armies of revolutionary France, led by Napoleon Bonaparte, invaded Egypt. Bonaparte, who had cultivated a deep interest in the enigmas of the pyramids, brought with him a large number of scholars, 175 in all, including several 'greybeards' gathered from various universities who were reputed to have acquired 'a profound knowledge of Egyptian antiquities', and, more usefully, a group of mathematicians, cartographers and surveyors:
One of the tasks the savants were set, after the conquest was completed, was to draw up detailed maps of Egypt . In the process of doing this they discovered that the Great Pyramid was perfectly aligned to true north - and of course to the south, east and west as well, as we saw in Part VI. This meant that the mysterious structure made an excellent reference and triangulation point, and a decision was therefore taken to use the meridian passing through its apex as the base-line for all other measurements and orientations. The team then proceeded to produce the first accurate maps of Egypt drawn up in the modem age. When they had finished, they were intrigued to note that the Great Pyramid's meridian sliced the Nile Delta region into two equal halves. They also found that if the diagonals running from the pyramid's apex to its north-eastern and north-western corners were extended (forming lines on the map running north-east and north- west until they reached the Mediterranean), the triangle thus formed would neatly encapsulate the entire Delta area 3
Page 457
Let us now return to our map, which also incorporates a triangle representing the Delta. Its other main components are the three parallel meridians. The eastern meridian is at longitude 32° 38' east- the official eastern border of Ancient Egypt from the beginning of dynastic times. The western meridian is at longitude 29° 50' east, the official western border of ancient Egypt. The central meridian is at longitude 31° 14' east, exactly midway between the other two (1° 24' away from each).4
What we now have is a representation of a strip on the surface of planet earth that is exactly 2° 48' wide. How long is this strip? Ancient Egypt's 'official' northern and southern borders (which bore no more relationship to settlement patterns than the official eastern and western boundaries) are marked by the horizontal lines at the top and bottom of the map and are located respectively at 31° 06' north and 24° 06' north.5 The northern border, 31 ° 06' north, joins the two outer ends of the estuary of the Nile. The southern border, 24° 06'N, marks the precise latitude of the island of Elephantine at Aswan (Seyne) where an important astronomical and solar observatory was located throughout known Egyptian history.6 It seems, that this archaic land, sacred since time began - the creation and habitation of the gods - was originally conceived of as a geometric construct exactly seven terrestrial degrees in length.
Within this construct, the Great Pyramid appears to have been carefully sited as a goedetic marker for the apex of the Delta. The latter, which we have indicated on our map, is located at 30° 06'N 31 14'E -a point in the middle of the Nile at the northern edge of modern Cairo. Meanwhile the pyramid stands at latitude of 30° N (corrected for atmospheric refraction) and at longitude 31 ° 09'E, an error of just a few minutes of terrestrial arc to the south and west. This 'error', however, does not appear to have resulted from sloppiness or inaccuracy on the part of the pyramid builders. On the contrary, a close look at the topography of the area suggests that the explanation should be sought in the need to find a site suitable for all the astronomical observations, that had to be taken for accurate setting- out, and with a sufficiently stable geological structure on which to park, for ever, a six-million-ton monument almost 500 feet high with a footprint of over thirteen acres.
Page 458
The Giza plateau fills the bill on all counts: close to the apex of the Delta, elevated above the Valley of the Nile, and equipped with an excellent foundation of solid limestone bedrock.
We were driving north from Luxor to Giza in the back of Mohamed Walilli's Peugeot 504 - a journey of just over 4 degrees of longitude, i.e., from 25° 42'N, to the 30th parallel. Between Asiut and EI Minya, a corridor of conflict in recent months between Islamic extremists and Egyptian government forces, we were provided with an armed escort of soldiers, one of whom wore plain clothes and sat in the passenger seat beside Mohamed fondling an automatic pistol. The others, about a dozen men armed with AK47 assault rifles, were distributed equally between two pick-up trucks which sandwiched us front and rear.
'Dangerous people live here,' Mohamed had confided out of the corner of his mouth when we had been stopped at a road-block in Asiut and ordered to wait for our escort. Now, although obviously rattled at being obliged to match the high speed of the escorting vehicles, he seemed to relish the kudos of being part of an impressive convoy, lights flashing and sirens wailing, weaving in and out of the slower traffic on the main highway from upper to lower Egypt.
I looked out of the car window for a while at the unchanging spectacle of the Nile, at its fertile green banks and the red haze of the deserts a few miles away to east and west. This was Egypt, the real organic Egypt of today and yesterday, which overlapped (but spread out far beyond) the strange 'official' Egypt of the map described, a rectangular fiction exactly seven terrestial degrees in length.
In the nineteenth century the renowned Egyptologist Ludwig Borchardt expressed what is still the conventional wisdom of his colleagues when he remarked, 'One must absolutely exclude the possibility that the ancients may have measured by degrees.'7 This was a judgement that seemed increasingly unlikely to be tenable. Whoever they may have been, it was obvious that the original planners and architects of the Giza necropolis had belonged to a civilization which knew the earth to be a sphere, knew its dimensions / Page 459 / almost as well as we do ourselves, and had divided it into 360 degrees, just as we do today.
The proof of this lay in the creation of a symbolic official 'country' exactly seven terrestrial degrees in length, and in the admirably geodetic location and orientation to the cardinal points of the Great Pyramid. Equally persuasive was the fact, already touched on in Chapter Twenty-three, that the perimeter of the pyramid's base stood in the relationship 2pi to its height and that the e.ntire monument seemed to have been designed to serve as a map-projection - on a scale of 1 :43,200 - of the northern hemisphere of our planet:
The Great Pyramid was a projection on four triangular surfaces. The apex represented the pole and the perimeter represented the equator. This is the reason why the perimeter is in relation 2pi to the height.8
The Pyramid/Earth ratio
We have demonstrated the use of pi in the Pyramid9 and need not go into this matter again; besides, the existence of the pi relationship, though interpreted as accidental by orthodox scholars, is not contested by them.10 But are we seriously supposed to accept that the monument could also be a representation of the northern hemisphere of the earth projected on flat surfaces at a scale of 1 :43,200? Let us remind ourselves of the figures.
According to the best modem estimates, based on satellite observations, the equatorial circumference of the earth is 24,90245 miles and its polar radius is 3949.921 miles.11 The perimeter of the Great Pyramid's base is 3023.16 feet and its height is 481.3949 feet.12 The scaling-down, as it turns out, is not absolutely exact, but it is very near. Moreover, when we remember the bulge at the earth's equator (our planet being an oblate spheroid rather than a perfect sphere), the results achieved by the pyramid builders seem even closer to 1 :43,200.
How close?
If we take the earth's equatorial circumference, 24,902.45 miles, and scale it down (divide it) by 43,200 we get a result of 0.5764 of a mile. There are 5280 feet in a mile. The next step, therefore, is to multiply 0.5764 by 5280, which produces a figure of 3043.39 feet. The / Page 460 / earth's equatorial circumference scaled down 43,200 times is there-fore 3,043.39 feet. By comparison, as we have seen, the perimeter of the Great Pyramid's base is 3,023.16 feet. This represents an 'error' of only 20 feet - or about three-quarters of 1 per cent. Given the razor- sharp accuracy of the pyramid builders, however (who normally worked to even finer tolerances), the error is less likely to have resulted from mistakes in the construction of the giant monument than in an underestimation of our planet's true circumference by just 163 miles, probably caused in part by failure to take account of the equatorial bulge.
Let us now consider the earth's polar radius of 3,949.921 miles. If we scale it down 43,200 times we get 0.0914 of a mile: 482.59 feet. The earth's polar radius scaled down 43,200 times is therefore 482.59 feet. By comparison the Great Pyramid's height is 481.3949 feet - just a foot less than the ideal figure, an error of barely one-fifth of one per cent.
As near as makes no difference, therefore, the perimeter of the Great Pyramid's base is indeed 1:43,200 of the equatorial circumfer-ence of the earth. And as near as makes no difference, the height of the Great Pyramid above that base is indeed 1 :43,200 of the polar radius of the earth. In other words, during all the centuries of darkness experienced by Western civilization when knowledge of our planet's dimensions was lost to us, all we ever needed to do to rediscover at knowledge was to measure the height and base perimeter of the Great Pyramid and multiply by 43,200!
How likely is this to be an 'accident'?
The commonsense answer is 'not very likely at all,' since it should be obvious to any reasonable person that what we are looking at could only be the result of a deliberate and carefully calculated planning decision. Commonsense, however, has never been a faculty held in high esteem by Egyptologists, and it is therefore necessary to ask whether there is anything else in the data which might confirm that the ratio of 1 :43,200 is a purposeful expression of intelligence and knowledge, rather than some numerical fluke.
The ratio itself seems to provide that confirmation, for the simple / Page 461 / reason that 43,200 is not a random number (like, say, 45,000 or 47,000, or 50,500, or 38,800). On the contrary it is one of a series of numbers, and multiples of those numbers, which relate to the phenomenon of precession of the equinoxes, and which have become embedded in archaic myths all around the world. As the reader can confirm by glancing back at Part V the basic numerals of the Pyramid / Earth ratio crop up again and again in those myths, sometimes directly as 43,200 sometimes as 432, as 4,320, as 432,000, as 4,320,000, and so on.
What we appear to be confronted by are two remarkable propositions, back-to-back, as though designed to reinforce one another. It is surely remarkable enough that the Great Pyramid should be able to function as an accurate scale-model of the northern hemisphere of planet earth. But it is even more remarkable that the scale involved should incorporate numbers relating precisely to one of the key planetary mechanisms of the earth. This is the fixed and apparently eternal precession of its axis of rotation around the pole of the ecliptic, a phenomenon which causes the vernal point to migrate around the band of the zodiac at the rate of one degree every 72 years, and 30 degrees (one complete zodiacal constellation) every 2,160 years. Precession through two zodiacal constellations, or 60 degrees along the ecliptic, takes 4320 years.13
The constant repetition of these precessional numbers in ancient myths could, perhaps, be a coincidence. Viewed in isolation, the appearance of the precessional number 43,200 in the pyramid/earth ratio might also be a coincidence (although the odds against this must be astronomical). But when we find precessional numbers in both these very different media - the ancient myths and the ancient monument - it really does strain credulity to suppose that coincidence is all that is involved here. Moreover, just as the Teutonic myth of Valhalla's walls leads us to the precessional number 432,000 by inviting us to calculate the warriors who 'go to war with the Wolf' (500 plus 40 multiplied by 800, as saw in Chapter Thirty-three), so the Great Pyramid leads us to the precessional number 43,200 by demonstrating through the pi relationship that it might be a scale model part of the earth and then by inviting us to calculate that scale.
"Moreover, just as the Teutonic myth of Valhalla's walls leads us to the precessional number
432,000
by inviting us to calculate the warriors who 'go to war with the Wolf'
(500 plus 40 multiplied by 800,.."
5 + 4 = 9 x 8 = 72
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Graham Hancock 1995
Page 457
"It seems, that this archaic land, sacred since time began - the creation and habitation of the gods - was originally conceived of as a geometric construct exactly seven terrestrial degrees in length".
"exactly seven terrestrial degrees in length."
"exactly seven"
"seven"
Page 458
"exactly seven terrestial degrees in length".
"exactly seven"
"seven"
Page 459
"exactly seven terrestial degrees in length".
"exactly seven"
"seven"
CATCHING THE LIGHT
Arthur Zajonc
Door of the Rainbow
Page 176
"Light rays entering a rainbow (above) and leaving (be-low) Notice how they crowd around ray seven
Here is where the rainbow will appear. "
"Notice how they crowd around ray seven"
RA
Y
"seven"
RAINBOW
RA - IN - BOW
Seven letters in Rainbow
RA Y SEVEN Y SEVEN RA
MARY Y RAM MARY MARY Y RAM
Y MARY Y
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