angled because 32+42=52, also expressed as 9+16=25. Any trianglewith those ratios will also be right-angled, for example “6, 8, 10” produces 36+64=100. Whoever carved the Plimpton 322 tablet millennia ago seems to have been well aware of that.
Evidence for the development of mathematics in ancient Egypt was found in a tomb at Abydos, where ivory labels with numbers on them had been attached to grave goods. The famous Narmer Palette was discovered in 1897 by J.E. Quibell at Hierakonpolis, the capital of Predynastic Egypt.
The palette reveals the use of a 10-base number system and accounts for thousands of goats, oxen, and human prisoners. Inscriptions on a wall in Meidum, near one of the mastaba (a low-lying, flat-bench-shaped tomb), carry mathematical instructions for the angles of the walls of the mastaba. These inscriptions involve the cubit as the unit of measurement. This was a unit based on the size of parts of the body, from the elbow to the fingertips, approximately 18 inches.
The Rhind Mathematical Papyrus, which is well over 3,500 years old, was acquired by Alexander Henry Rhind in Luxor in 1858 and is now in the British Museum. It was copied from a much older papyrus by a scribe named Ahmes, who described it as being used for “Enquiring into things … the knowledge of all things … mysteries and secrets.” It would seem from this that Ahmes had a numerological view of the power of numbers. They were not merely of use to solve mathematical problems: they had magical powers as well.
The Rhind Papyrus contains a number of problems in both arithmetic and algebra, which has led some antiquarians to suggest that it was perhaps intended as a teaching document. One interesting example shows how the Egyptian mathematicians of that period took the numbers from 1–9 and divided them by 10. They worked out that 7÷10 could be expressed as 2÷3+1÷30. The papyrus continues with interesting practical problems such as dividing loaves of bread among 10 men. There are then algebraic examples of linear equations such as x+1÷3x+1÷4x=2 in modern notation. When the equation is solved, x=1 and 5÷19 or approximately 1.263157894. The Rhind Papyrus goes on to provide methods of finding the volumes of cylindrical and cuboid granaries. The Rhind Papyrus also contains formulas for division and multiplication and its contents infer that the Egyptians of that period knew about prime numbers and the Sieve of Eratosthenes, which is a technique for finding prime numbers.
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
