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      The Narmer Palette

      A prime number is a natural number that has only 2 factors: itself and 1. They are, of course, also its only divisors. Oddly enough, “1” is not a prime number because it has only 1 factor, which is itself. The Sieve of Eratosthenes, who was an early Greek mathematician — not an Egyptian — will find all the prime numbers up to and including the end number of the specified range. This end number, the highest number, is always referred to as “n.” To use the Sieve of Eratosthenes, begin by writing out all the numbers in your list from 2 up to and including “n.” Now introduce the term “p,” which stands for “prime.” It has the starting value of “2,” which is the lowest prime number. Beginning with p itself, go through the list and cross off all the numbers that are multiples of p. Those numbers will be 2p, 3p, 4p … and so on. Now look for the lowest number that has not yet been crossed off. This will be the prime immediately above 2, which is “3.” The next step is to give “p” the value of this number (“3”), and repeat the process as often as necessary with each succeeding prime. Finally, when you have used all the known primes (2, 3, 5, 7, 11, 13 ... ) any unmarked numbers remaining will be primes.

      As a very short, simple example, suppose we are trying to find whether 17 is a prime. Call “17” “n” and begin with “p” as “2”:

      2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17

      Delete all the multiples of “p” when p=2:

      3, 5, 7, 9, 11, 13, 15, 17

      Therefore, “p” now becomes the next prime, which is “3.”

      Delete all multiples of 3:

      5, 7, 11, 13, 17

      In turn, “p” now becomes “5,” then “7,” then “11,” then “13.”

      This leaves only “17” — our original target — so “17” is a prime.

      The Rhind Papyrus certainly creates a high degree of respect for the Egyptian mathematicians who created it some 4,000 years ago.

      Another very interesting piece of evidence for the mathematical developments in early Egypt is found in what is known as the Egyptian Mathematical Leather Roll. This dates from well over 3,500 years ago and comes from Thebes. It found its way to the British Museum in 1864, but was not unrolled and deciphered until 1927. The roll contains numerous fractions added together to form other fractions. Examples include 1/30+1/45+1/90=1/15 and 1/96+1/192=1/64 together with 1/50+1/30+1/150+1/400=1/16. The Roll makes it clear that fraction calculations of this type were highly significant for the Egyptian mathematicians of this period. They regarded certain fractions as “Eye of Horus” numbers.

      In the legendary battles between the evil god Seth and Isis and Osiris, who were Horus’s parents, Seth tried to blind Horus, who was later healed by the good and wise Thoth. One piece of his eye, however, was missing, and Thoth used magic to make up for the missing piece. The Horus eye numbers were 1/2, 1/4, 1/8, 1/16, 1/32 and 1/64. These add to 63/64, leaving the missing piece of Horus’s eye with a value of 1/64. The eye legend combined with the missing fraction brings mathematics and numerology close together in Egyptian thought. Each Pharaoh thought of himself as Horus during his earthly reign, but at death he was transformed into Osiris.

      On the other side of the world, in South America, the ancient Mayans were developing a mathematical system using the vigesimal number base-20 instead of 10. This suggests to historians that the Mayans used toes as well as fingers in their counting system. Their basic numbering system was clear and effective: dots were used up to “5,” which was a short horizontal line. Dots were then added above the line until the sum of “9” was reached, illustrated as a horizontal bar, for “5,” with 4 dots above it. “Ten” was expressed with 2 of the short horizontal lines, worth 5 each. “Eleven” was the 2 lines that stood for “10,” with a single dot above it. The system continued in this way as far as “19,” which was represented with 3 of the horizontal 5-lines, 1 above the other, and then there were 4 dots above them.

      The Mayan calendar is made up of 20 sets of 13-day cycles, leading to 260 total days. These 13-day periods are known as trecena cycles, and are comparable to months in our modern calendar. Religious ceremonials are based on this unique system, and it is also used to predict the future. Mayan calendar mathematics also involves the numerology of divination.

      Over the centuries, various fundamentalist sects and cults — as well as some of the major religions —have preoccupied their thinking with eschatology and trying to find dates for the end of the world. Mayan calendar numerology is a case-in-point, and one which causes more unnecessary alarm than many of the others. According to what is referred to in the Mayan system as “Long Count,” a period of 5,126 years will come to an end on December 21, 2012. On that day, the winter solstice sun will be more or less in conjunction with what approximates to the galactic equator. The Maya regarded this as some sort of mystical sacred tree. Some pessimistic end-of-the-world enthusiasts are convinced that the event will mean the end of civilization as we know it.

      A similar end-of-the world obsession happened in July 1999. Co-author Lionel was then making the television program called The Real Nostradamus, and a great many Nostradamus readers had convinced themselves that the old French soothsayer had forecast terrible disasters for July 1999. One very sad interviewee on the show was totally convinced that everything was ending. He had given up a good professional job and his London home so that he could go back to the village where he had been born in order to await the end there. That is the kind of serious damage that misguided eschatological obsessions can induce. The authors — with over 50 years’ experience of investigating the paranormal and the anomalous — are totally confident that nothing bad will happen to our Earth in December 2012. Mayan mathematics was exceptionally advanced for its time, but well-designed as it was, their Long Count Calendar certainly does not herald the end of all things.

      Thales of Miletus lived from 624–546 BC. He was a pre-Socratic philosopher and a master of mathematics and astronomy, who also had a keen interest in ethics and metaphysics. Experts regard him as one of the traditional 7 Sages of Greece. He set out to explain natural phenomena through causes and effects, as opposed to the mythological explanations so prevalent at the time. It was characteristic of popular Greek thought during this period to depend upon exegetical myths — those that attempted to explain the origins of everything. For example, the changing of seasons was thought to be a result of the Greek earth goddess, Demeter, searching for her missing daughter, Persephone. Numerology features here in the myth of Persephone and the 6 pomegranate seeds that she ate out of temptation, leading to her curse of having to spend 6 out of every 12 months with Hades in the Underworld, which corresponds to the number of winter months we experience each year. Another example is provided by Aeolus, or Aiolos, who was appointed by Zeus to take charge of the storm winds. These were released when the gods wanted to cause damage and disaster. Instead of relying on these myths to explain the natural phenomena happening around him, Thales looked instead for rational explanations that he did his best to examine open-mindedly and objectively. This gave him the title of the Father of Science, although supporters of Democritus felt that he deserved the title instead.

      One of Thales’s excellent ideas was to calculate the height of a tall building, such as a pyramid, by standing in a position where he could measure his own shadow and the shadow of the target building. When his shadow coincided exactly with his own height, he measured the pyramid’s shadow and argued that if his shadow gave his height, then the pyramid’s shadow would reveal its height. He was also responsible for a number of important theorems: that any diameter bisects a circle; that the angle from the diameter to the circumference in a semi-circle is a right-angle; that the base angles of an isosceles triangle are equal; that the opposite angles formed by 2 intersecting straight lines are equal; and that triangles are congruent if 1 side and 2 angles

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