Continuing our investigation of measurement used by other ancient cultures we next turned away from India and back to Europe. There was once a Castilian Spanish unit of measurement known as the ‘vara’, which also became popular in Spanish Central America. The vara is generally considered to be equal to 83.5905 centimetres,5 making it about 0.75 per cent larger than the Megalithic Yard.
There are Megalithic structures in Castile but Alexander Thom did not survey them, so we do not know whether the Megalithic Yard was employed there. Old Castile was originally a province of the kingdom of Leon with Burgos as its capital. Today it is central and northern Spain, traditionally divided into Old Castile and New Castile, and now split into Castile-La Mancha and Castile-Leon. It does seem possible that this region could have retained a unit from prehistory but the Spanish vara cannot be confidently considered more than a coincidence without further corroboration, but the strong possibility of a connection remains.
We looked next to the Far East and found that the oldest Japanese measure was known as the ‘shaku’, which is believed to have been imported from China more than 1,000 years ago. At 30.30 centimetres this unit is almost indistinguishable from a Minoan foot, which is just 0.6 millimetres larger. It follows therefore that 366 Megalithic Yards are almost the same as 1,000 Japanese shaku, with a fit accuracy of 99.8 per cent. Connection or coincidence? It could be either, so we decided not to develop this area of investigation unless something new occurred to strengthen the possibility of a connection. The only relatonships with the Megalithic builders of the British Isles and other cultures that could be described as conclusive are with the Minoans, the Sumerians and now the Harappan culture.
The next civilization we turned to is the most famous of them all – Ancient Egypt. The Ancient Egyptians have long fired the imagination because they left such stunning artefacts behind them – both in terms of scale and beauty. They began their development just on our side of the Great Wall of History and they appeared to blossom from nowhere. There are many pyramids in Egypt but the three splendid ones on the Giza Plateau are by far the most famous, along with the mysterious Sphinx, which sits nearby on the sands of the desert. The largest of the three pyramids is associated with King Khufu and has an estimated volume of some 2.6 million cubic metres. It is believed that 2.5 million blocks of stone, each weighing an average of 2.5 tonnes, were used to build it. Each side of this pyramid is approximately 230 metres long and its height is about 146 metres.
Unquestionably, these 4,300-year-old structures represent an almost superhuman feat of engineering and this obvious skill has caused many people to wonder if such buildings were more than grand burial mounds.
It is universally agreed that Egyptians were good practical astronomers. It has been suggested that some of the mysterious ‘shafts’ deliberately built into the sides of the Khufu pyramid were specifically angled to cosmological events. If this were the case, it undoubtedly had a religious meaning, because the Egyptians were obsessed with death and the afterlife.
Ancient and modern mathematics
It might be thought that people who could build on such a colossal scale would also be excellent mathematicians and this is true, though only up to a point. Most experts agree that Egyptian expertise with mathematics dealt mainly with the practical aspects of life and did not veer much, or often, into the area of theory (which would become so important to the Ancient Greeks). The Egyptians had a form of geometry, knew how to make right angles and appear to have followed broadly similar principles to their contemporaries, the Sumerians, though without some of the flair exhibited by the mathematicians of Mesopotamia.
It is worth repeating here the fundamental difference between modern mathematics and the mathematics of the ancients. The variety used in the British Isles, Mesopotamia, India and Egypt was what is now known as ‘algorithmic mathematics’ and that used today (invented by the Greeks) is called ‘dialectic mathematics’. The following definitions are provided by Emeritus Professors Philip J. Davis, Brown University, and Reuben Hersh, University of New Mexico.
Algorithmic mathematics as used by the ancient civilizations represents a tool for solving real-world problems. It is concerned not only with the existence of a mathematical object but also with the credentials of its existence. This approach allows mathematics to vary according to the urgency of the problem in hand.
Dialectic mathematics is a rigorously logical science where statements are either true or false and where objects with specified properties either do or do not exist. It is an intellectual game played to rules about which there is widespread consensus. Throughout the 20th century mathematics became increasingly dialectic and many amateur mathematicians have mistakenly assumed it is the best, or even the only form the subject can take.
NASA could never have put a man on the moon if the trajectories had not been computed with dialectic rigour combined with algorithmic pragmatism. In short, dialectic mathematics invites contemplation while algorithmic mathematics invites action and delivers results. We believe that it is fair to say that both approaches have their value and most great achievements required the use of both, though there is also some tension between them. Leading mathematicians Professors Davis and Hersh believe that there is sometimes a conflict in the minds of users:
‘There is a distinct paradigm shift that distinguishes the algorithmic from the dialectic, and people who have worked in one mode may very well feel that solutions within the second mode are not “fair” or not “allowed”. They experience paradigm shock.’6
The Ancient Egyptians created the pyramids using an algorithmic approach and it is also true to say that they were fantastic at logistics. They had to be, because gathering perhaps tens of thousands of people together in one place, for example to build a huge pyramid, involved planning on a grand scale. Not only had the craftsmen and work gangs to be organized, but also their raw materials had to be sourced and prepared, and a massive support team would have been needed to feed and water such huge numbers.
What the Egyptians seem to have been less good at was building a calendar that could be said to show a high degree of accuracy to the actual year. The reason for this was no lack of intelligence on the part of the Egyptian astronomer-priests, but rather that it was tied to necessity. It rarely rains in Egypt and the region is not particularly subjected to seasons in the accepted sense of the word. Egypt owed its prosperity to the annual flooding of the Nile, the great river that was the lifeblood of all the towns and cities comprising the civilization.
The River Nile rises many hundreds of kilometres beyond the boundaries of Egypt itself, in areas that do experience significant changes in rainfall. The Egyptians themselves were almost certainly unaware of this fact but they had noticed that the flooding of the Nile took place each year just after the helical rising (first visible, brief appearance on the eastern horizon before sunrise) of the star Sirius. The Nile flood brought with it extremely fertile silt, which ran across the fields alongside the river. When the flood had receded, crops were planted in the silt and were simply harvested when they had reached maturity. All crops would be harvested well ahead of the next flooding of the Nile, eventually creating a society that was really not at all concerned about tremendous accuracy in terms of the length of the year.
The best the Egyptians ever achieved in terms of a calendar, at least until the time of Alexander the Great, was to celebrate a 360-day year, with 5 extra days each year added as holidays. The true solar year is 365.2564 days in length, so every year the Egyptian calendar was out by over a quarter of a day. Nobody cared much, just as long as those dedicated to watching for the event saw the helical rising of Sirius and alerted everyone to the fact.
It is certain that Egyptian scholars were good at dealing with areas and volumes, in fact any aspect of mathematics that had a sound, practical reason, though the methods used developed early and did not move forward for well over 2,000 years. By inference, the Egyptians probably knew about a 360-degree circle, thoug
h they never seem to have understood its significance in the way the Sumerians did because they opted very early in their history for a 24-hour day, which essentially divorces the measurement of time and Earth geometry. There is no evidence we are aware of to suggest that the Egyptians knew of, or cared about, the Sumerian second or minute of time.
The ‘DNA’ of the Great Underlying Principle
What we wanted to know was whether any aspect of Egyptian measures carried any of the ‘DNA’ of the Great Underlying Principle we had identified among the Megalithic folk and the Sumerians. Looking at the available information it seemed as though neither Megalithic geometry nor its linear measurements were known to the Ancient Egyptians.
The basic unit of linear measurement in use for nearly the whole of Egyptian history was the ‘royal cubit’. Opinions vary very slightly on the length of this unit, some putting it at 52.372 centimetres and others at 52.35 centimetres, while Professor Livio Stecchini considered it to be about 52.4 centimetres. He concluded that the length of the sides of Khufu’s pyramid (also known as the Pyramid of Cheops) was intended to be 230,560 millimetres and after exhaustive research he went on to state:
‘It is agreed amongst serious scholars that the side was calculated as 440 Egyptian royal cubits. Borchardt drew the conclusion that the cubit had a length of 523.55 mm, but in my opinion, one must take into account the difficulty of proceeding in a perfectly straight line without telescopic instruments. Cole, as an experienced surveyor, calls attention to this factor. Since other dimensions, such as those of the King’s Chamber, indicate the use of a cubit very close to 524 mm, one can assume that the theoretical length of the side was 230,560 mm.
The length of 524 mm for the cubit of the Pyramid has been confirmed by the endless measurements that have been applied to every detail.’7
The range of opinions is within a fraction of a millimetre and we are happy to take Stecchini’s highly authoritative opinion and consider the Egyptian royal cubit as being 52.4 centimetres. It did not take us long to arrive at our first assumption that this cubit did not appear to have any direct connection with either the Sumerian or Megalithic systems.
We then turned to another Ancient Egyptian unit closely related to the royal cubit called the ‘remen’. The remen had a relationship with the royal cubit in that if a square had sides of one royal cubit the diagonal from opposite corners will be one remen. len
This Ancient Egyptian relationship between two major units of length uses the geometric principle supposedly invented 1,500 years later by Pythagoras, who is credited with observing that ‘the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides’.
A simple example of this principle is the classic 3,4,5 triangle. If the triangle has a 3-inch base, a 4-inch high side and a sloping side of 5 inches, and we create a square on each side, the result is three squares of 9, 16 and 25 square inches. Adding the first two together gives 25 square inches, which is equal to the third side. The attribution of this very ancient principle to Pythagoras appears to be another case of the Greeks unwittingly reinventing old knowledge.
It is now accepted that this geometric principle was also important to the Babylonians (and possibly therefore the Sumerians).
THE ORIGIN OF PYTHAGORAS’ THEORUM
The basis of this so-called Pythagorean relationship is actually the study of the square root of two. This is because the length of the hypotenuse (the corner-to-corner diagonal) in a square is the square root of the sum of the squares of the other two sides. The Sumerian/Babylonian solution was written down as 1, 24, 51, 10 in their base 60 notation, which would be written today as the decimal number 1.414212963.
If the royal cubit was indeed 52.4 centimetres, then the remen must have been a length equal to 74.1 centimetres. Once again, however, we could find no apparent connection to the Megalithic or Sumerian principles. But we continued to look at the patterns of the cubit/remen relationship. Like the Megalithic people and the Sumerians, the Ancient Egyptians considered halves and doubles to be as valid as the full dimension of most units. The principle of using the side of a square and its hypotenuse lead directly to a sequence of doubling or halving. This can be seen when a series of squares is developed using the hypotenuse.
If the smallest square has sides of a royal cubit, the second square formed on the diagonal will have sides of one remen, and a third square build on the next diagonal will have sides of a double royal cubit. The next obvious step seemed to be to introduce a circle because the Megalithic and Sumerian systems worked with circles.
We could draw an infinite series of squares and circles and they would produce an alternating series of cubit and remen dimensions, doubling as they move outwards, and halving as they go inwards.
Squares on the hypotenuse of a royal cubit.
A circle within a cubit.
The power of the circle.
All of this was very basic, although it was beautiful and intriguing. Here we had the power of the circle that could be said to define the two major units of Egyptian measurement. The next obvious question was, ‘What is the length of circles created by a royal cubit and remen series?’ The answer was very interesting.
Taking a square with sides of one quarter-remen (18.526 centimetres) we find that the circle that encloses it is very close indeed to a Megalithic Yard in circumference! At 82.31 centimetres it was 99.2 per cent of Thom’s Megalithic Yard found in the British Isles. The next square has sides of a half-cubit and the next one a half-remen; the circle that encompasses this square is two Megalithic Yards in circumference. There is a small discrepancy between the quarter-remen square and the Megalithic Yard circle around it, but we had to remember that a pendulum swing is inversely proportional to gravity, which reduces towards the equator and causes a pendulum with the same period for its swing to be shorter in length. This boils down to the fact that anyone following the rules for a Megalithic Yard would get a noticeably smaller result at the latitude of the pyramids than they would in Orkney, for instance. Alexander Thom’s Megalithic Yard was an average derived from all of his measurements taken from Megalithic sites from northern Scotland down to Brittany, and the vast majority were derived from northerly sites. Our conclusion is that the tiny variances in the available data are greater than the inaccuracy found in the principle of the royal cubit and the remen defined by a Megalithic pendulum.
A Megalithic Yard made in Egypt according to the Megalithic pendulum method would be 82.7 centimetres long. This shows that the pendulum method of reproduction must originally have been intended to work only in and around the British Isles. At this southern latitude the same process does not produce a correct geodetic unit. However, for the theoretical Egyptian Megalithic Yard to be the circle in the remen/cubit series the royal cubit would have to be 52.648 centimetres – less than half a per cent larger than Stecchini’s estimate.
When our manuscript was being checked for scientific accuracy by Peter Harwood, he had been very surprised and eventually impressed with our findings. Peter was doing a great job for us, pointing out some errors in our calculations and drawing our attention to issues we had missed. When he read this section on the possible use of the Megalithic Yard to define the royal cubit he suggested that we appeared to have inferred a significant discovery about the Khufu pyramid that we had missed in fact. He reminded us of John Taylor’s book The Great Pyramid, written in 1859, where it is observed that if one divides the height of the pyramid into twice the size of its base, the result is pi. While some people believed this demonstrated that the ratio we now call pi, must have been sacred to the Egyptians, others had a more prosaic explanation.
Critics of the ‘sacred pi’ theory pointed out that if a wheel had been made with a diameter that was a subdivision of the height, and it was used to roll out a certain number of revolutions along the sides, the height and sides would automatically have a pi relationship without the builders even realizing it.
Peter Harwo
od’s email went on to say:
‘If you have a wheel a foot in diameter, say, then construct a pyramid by making each side of the base square exactly one roll of the wheel long, and the height two diameters of the wheel, you have your pi ratio without actually knowing what pi is. But supposing instead of a foot you use a half cubit diameter wheel. You will end up with a copy of the great pyramid 1 cubit high, with the length of each base side 1 MY! Now that made my pulse race. I can’t believe you’d miss such a sexy result.’
Peter was quite right; we had missed a very significant point. The use of a Megalithic-Yard wheel would explain a very old mystery. We checked out the height of the pyramid and found that it is estimated to be 146.59 metres and the sides are estimated to be 230.56 metres. Because all of the estimates of the royal cubit vary a tiny amount we decided to standardize and make the assumption that the Megalithic Yard principle, used in Egypt, had been the starting point. So, taking a Megalithic Yard as 82.7 centimetres and a half royal cubit as 26.324 centimetres, we found the following for the Great Pyramid of Khufu at Giza:
height
= 279 royal cubits
side of base
= 279 Megalithic Yards
corner to corner
= 279 remens
All of the measuring units appeared in the same number when used in Khufu’s pyramid. We could only assume that some kind of ancient numerology had made the value ‘279’ deeply meaningful to the architects. Checking other pyramids we found they all appear to have been made to differing requirements, although the other two pyramids at Giza had perimeters that appear to have been measured in Egyptian Megalithic Yards:
Civilization One: The World is Not as You Thought it Was Page 14