Читать книгу Mysteries and Secrets of Numerology - Patricia Fanthorpe - Страница 9

The earliest artifacts relating to mathematics were various tally sticks, some dating back nearly 40,000 years. As with ancient cave paintings, which are believed to have been magical in the eyes of the early peoples who created them, the ancient mathematical tally sticks may well have had a magical numerological significance as well as a purely practical, numerical one.

The Lebombo bone is estimated to be 35,000 years old — or perhaps more. It was found in the Border Cave in the Lebombo Mountains of Swaziland in East Africa. Originally the fibula of a baboon, it has 29 distinct notches, which has led some archaeologists to believe that it was intended as a lunar calendar.

Lebombo bone

The Wolf bone was discovered in Moravia by Karl Absolon in 1937. Estimated at approximately 35,000 years old, it was found close to a Venus figurine. The bone has 55 marks carved into it. Its association with the Venus figurine suggests some kind of numerological or magical function as well as a straightforward counting or measuring function. Was this a situation in which mathematics and numerology overlapped?

The Ishango bone is rather younger, dating back some 20,000 years. It was discovered in 1960 by a Belgian explorer named de Braucourt in what was then known as the Belgian Congo, near the upper reaches of the Nile. Like the Lebombo bone, the Ishango bone was once the fibula of a baboon. At one end there is a piece of quartz, which suggests that the Ishango bone was used for marking or engraving things. It is thought that the clusters of marks cut into the bone are more complex than those on the Lebombo bone, which might indicate that the Ishango bone is something more mathematically complicated than a basic tally stick or calendar.

Mathematical historians are of the opinion that mathematical thinking started when our earliest ancestors began to form concepts of number, magnitude, and form. What precisely do we mean by number? Although there is still some controversy over whether to include “0,” what are described as natural numbers are the following: 0, 1, 2, 3, 4, 5 … and so on. Integers include negative numbers and can be illustrated as -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5…and so on — the positive integers being the same as the natural numbers 1, 2, 3, 4, 5.... Rational numbers are able to be written as “a/b” but neither “a” nor “b” can be “0.” Irrational numbers cannot be written as rational a/b expressions. They are numbers such as π (pi in the Greek alphabet), which represents the number 3.14159, with an infinite decimal trail. Pi is the ratio of a circle’s circumference to its diameter.

Ishango bone

The earliest mathematical thought was also concerned with magnitude — the size of an object compared to other objects of the same kind. To Palaeolithic hunter-gatherers, how many objects there were and how big they were was an important piece of survival data, as was form. This refers to the configuration of an object, its visual appearance, and, basically, its shape.

Recent studies of animal intelligence have reached very interesting conclusions about the basic levels of mathematical ability of this elementary type that some animal species seem to share with human beings. Numerous “counting” dogs and horses have featured as circus and vaudeville acts, and they certainly seem to show some basic number skills.

An impressive university study on animal mathematical ability was conducted by Dr. Naoko Irie in Tokyo. Elephants from the Ueno Zoo watched as apples were dropped into buckets, and the elephants were then offered their choice of the buckets. Human subjects were also involved in the experiment to compare their results with those of the elephants. The elephants scored 74 percent while the human beings scored only 67 percent. The experiment suggested that when more than a single apple was dropped, the elephants had to carry out the equivalent of running totals in their heads.

The history of mathematics indicates that as civilizations developed, the demand for mathematics increased. The old commercial civilizations, such as Sumer in the region of the Tigris and Euphrates, needed to make careful records of commercial transactions: jars of oil, measures of corn, units of cloth, slaves and animals bought and sold. The Sumerians developed writing, irrigation, agriculture, the wheel, the plough, and many other things. Their writing system, known as cuneiform, used wedge-shaped characters cut into clay tablets that were then baked. As a consequence, they have lasted thousands of years and archaeologists have studied them closely for centuries. In the Sumerian civilization there was the need to measure areas of land and to calculate taxes. Sumerians developed calendars and were keenly interested in observing and recording the stars and planets in their courses. They developed the use of symbols to represent quantities. A large cone stood for “60.” A clay sphere stood for “10,” and a small cone was a single unit. In addition to these developments, they used a simple abacus.

Just as the popular base-10 decimal system of numbering is almost certainly based on the fact that we have 10 fingers, so it is suggested that the Sumerian and later Babylonian sexegesimal system (base-60) is based on the 12 knuckles of 1 hand and the 5 fingers of the other, which create 60 when multiplied together. Five hands would be thought of as containing 60 knuckles.

This base-60 system had many advantages. For example, “60” is the smallest number into which all numbers from 1–6 will divide exactly. The number “60” is also divisible by 10, 12, 15, 20, and 30. The convenience of “60” can still be seen in the concept of having 60 seconds in a minute, and 60 minutes in an hour. The 360 degrees of a circle is based on 60 multiplied by 6.

The Babylonians also used an early version of the “0,” although they seem to have employed it more as a place marker than as a symbol representing nothing. Five thousand years ago the Sumerians and Babylonians were making complicated tables filled with square roots, squares, and cubes. They could deal with fractions, equations, and even algebra. They got as close to π as regarding it as 3 1/8, or 3.125, which isn’t far from our contemporary 3.14159….

They also had the square root of 2 (1.41421) correct to all 5 decimal places. The square root of 2 is very useful for calculating the diagonal of a square. The formula is:

side of square×√2=the diagonal of that square

As a maths tutor, co-author Lionel passes that useful shortcut to his students along with the square root of 3 multiplied by the side of a cube to calculate the diagonal of a cube. The formula is:

√3×side of cube=diagonal of cube

Other Babylonian tablets provide the squares of numbers up to 59 (59×59=3481): a major achievement for mathematicians without calculators or computers!

The rich leisure culture of Babylon had numerous games of chance, and the dice they designed for these provided further archaeological evidence of their mathematical knowledge. This would seem to suggest an area of early thought where mathematics and numerology share the territory. Gamblers enjoy using systems of “lucky” numbers to try to beat the odds. In the old Babylonian games of chance, players may well have played their luck with numbers that they hoped would prove to be influential in moving the odds in their favour. Outstanding mathematicians like Marcus du Sautoy have examined these theories and suggested among other things that picking consecutive numbers can increase a gambler’s chances of winning a lottery.

Their buildings were also geometrically interesting, and the Sumerians and Babylonians had no problems calculating the areas of rectangles, trapezoids, and triangles. Volumes of cuboids and cylinders were also well within their mathematical capabilities.

One of many interesting problems in the history of mathematics and numerology is the famous Plimpton 322 tablet. It came from Senkereh in southern Iraq, and Senkereh was originally the ancient city of Larsa. The tablet measures 5 inches by 3.5 inches, and was purchased from Edgar J. Banks, an archaeological dealer. In 1922, he sold the mysterious tablet to George Plimpton, a publisher, after whom it was named. Plimpton placed it in his collection of archaeological treasures and finally bequeathed them all to Columbia University.

Written some 4,000 years ago, the tablet contains what seem to be Pythagorean triangle measurements — written centuries before Pythagoras lived! The classical “3, 4, 5” Pythagorean triangle is right-

angled because 32+42=52, also expressed as 9+16=25. Any triangle

with 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 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 are equal.

Pythagoras lived from approximately 570–495 BC and was the leader of a group known as the Pythagoreans. He and his followers in Croton (what is now Crotone in southern Italy) lived like a monastic brotherhood and were all vegetarians. All their joint mathematical discoveries were attributed to Pythagoras, so it is impossible to tell how much of the work was his alone. Because of this cult aspect of his life, it may be more appropriate to think of him as a numerologist than as a scientific mathematician. The safest conclusion is that he was both. The theorem associated with him is that in any right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other 2 sides. The proof of this theory is a very interesting one. Draw a right-angled triangle with sides a, b, and c as shown below. The longest side, the hypotenuse opposite the right-angle, is side c. What has to be proven is that a²+b²=c². Now draw the square of side c as shown in the diagram, and draw the original 90-degree triangle, “a, b, c,” in the 4 corners as illustrated here.

The big square with the c² inside it has the area (a+b)², or we can say:

A (the area of the big square)=(a+b)(a+b).

The area of the tilted internal square is c².

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Diagram of Pythagoras' Theorem

Each of the right-angled triangles has the area 1/2ab. There are 4 of these, so their total area is: 4(1/2ab). We can re-write this as 2ab. The entire area of the tilted square and the 4 right-angled triangles is A=c²+2ab. This then becomes (a+b)(a+b)=c²+2ab. The term (a+b)(a+b) can be multiplied out to produce a²+2ab+b², further resulting in the equation a²+2ab+b²=c²+2ab. The next step is to subtract the 2ab from each side. This leaves a²+b²=c², which is the proof that we were looking for!

One of Pythagoras’s most able disciples was Parmenides, who applied his mathematical skills to cosmology and came up with the idea of a spherical Earth inside the spherical universe. One of his disciples, Zeno of Elea, devoted his mathematical and philosophical skills to creating paradoxes, and some of these shaped the development of mathematics for centuries after his death. One of Zeno’s best-known paradoxes is called Achilles and the Tortoise. Zeno argued that a fast runner like Achilles could never overtake the tortoise. If it started 16 metres ahead of Achilles, but could run at only half his speed, by the time he covered those 16 metres, the tortoise would have gone 8. By the time he had covered those 8 metres, the tortoise would have gone another 4. By the time he had covered those 4, the tortoise would have gone another 2 … and so on indefinitely.

Another important early Greek mathematician was Hippocrates of Chios (470–410 BC) who, like many others since his day, attempted to square the circle: that is, to construct a square with the same area as a given circle. It is impossible to do this with perfect accuracy because π (3.14159…) is not a rational number.

The curve, known as the “quadratrix,” or “trisectrix,” was the work of Hippias of Elias around about 430 BC. It can be used to trisect an angle, or to divide an angle into a given number of equal parts. He, too, was one of the early mathematicians involved in unsuccessful attempts to square the circle.

Eudoxus of Cnidas (410–350 BC) developed a system of geometric proofs, based on what became known as the exhaustion method. When he attempted to show that 2 areas, a and b, were equal, he would begin by trying to prove that a was greater than b.

When that proof failed (was exhausted) he would attempt to prove that b was greater than a. When that proof also failed (was also exhausted) Eudoxus argued that as neither area was bigger or smaller than the other, they must be equal.

Eudemus of Rhodes (350–290 BC) was not so much a great early mathematician in his own right as an historian of mathematics. His 3 very informative books in this genre were histories of arithmetic, geometry, and astronomy. He also wrote another volume dealing with angles.

Euclid of Alexandria (325–265 BC) has gone down in mathematical history as the greatest mathematical teacher of all time. His ideas are still quoted authoritatively today. In addition to all his well-known work on geometry, his theory of the infinite number of prime numbers has stood the test of time.

Aristarchus of Samos (310–230 BC) was a remarkably able mathematician and astronomer, who came up with a well-argued heliocentric theory of the universe. Nearly 2,000 years later, Copernicus delved into Aristarchus’s work and agreed with his conclusions.

Apollonius of Perga (262–190 BC) focussed his mathematical skills on cones and the curves that are derived from slicing them. His book on conics introduced the terms “parabola,” “ellipse,” and “hyperbola.”

Hipparchus of Rhodes (190–120 BC) worked mainly as the pioneer of trigonometry. Every angle has a sine, a cosine, and a tangent, and these can be used to find angles or sides in a 90-degree triangle. The 3 sides are referred to as the hypotenuse, the adjacent, and the opposite. The basic trigonometrical formulae are:

sine=opposite÷hypotenuse

cosine=adjacent÷hypotenuse

tangent=opposite÷adjacent

These formulae make it clear that whenever any 2 of the measurements are known, the third can readily be calculated.

Claudius Ptolemy (85–165 AD) enjoyed, as his name implies, a rich mixture of Greek and Roman culture and learning. His mathematical works, principally on astronomy, were honoured with the Arabian title Almagest, meaning “The Greatest.”

Diophantus (200–284 AD) did remarkable early work on number theory, and his book Arithmetica provided a great deal of inspiration for Pierre Fermat (1601–1665). Fermat’s Last Theorem states that if we call 3 positive integers a, b, and c, then the equation an+bn=cn will only be possible if n is not greater than 2. Proving it became a leading mathematical problem for centuries, and even made its way into the Guinness Book of Records!

A brilliant Persian mathematician named Al-Khowarizmi (780–840) was also a gifted scientist and astronomer. His additional interest in astrology made him something of a numerologist — like Pythagoras — as well as a scientific mathematician. The modern word algebra was transliterated from his book Hisab al-jabr w’al-muqabala, where it

was rendered as “al-jabr.”

Francesco Pellos (1450–1500) was the inventor of the decimal point — a tremendously useful part of contemporary mathematics. The gifted Scots theologian John Napier (1550–1617) indulged in mathematics more or less as a hobby when he wanted a break from theology. He was largely responsible for creating logarithms, which were perfected by Henry Briggs. At around the same period, Sir Isaac Newton (1642–1727) published his epoch-making Principia — a masterpiece of mathematics and science. There is some justification for those who regard him as the greatest scientist who has yet lived. His work on gravitation and the 3 laws of motion are unforgettable.

Another important milestone in the history of mathematics was William Jones’s (1675–1749). He used the Greek symbol “π” to show the result of dividing the circumference of a circle by its diameter. This feat was published in his book, New Introduction to Mathematics, in 1706.

Calculus was the particular brainchild of the Italian maths genius Maria Agnesi (1718–1799). Her famous textbook on it, Istituzioni Analitiche, was an authoritative teaching aid on calculus for many years.

David Hilbert (1862–1943) was one of the most outstanding mathematical leaders in the late nineteenth and early twentieth centuries. His great contributions were to invariant theory and the axiomatization of Euclid’s geometry. One of his other theories that was essential to functional analysis was named after him as the theory of Hilbert spaces. Tragically, he and a number of other brilliant academic mathematicians at the University of Göttingen were persecuted by the Nazis.

Benoit B. Mandelbrot (1924–2010) was a superb French-American mathematician whose name is associated with the mathematical idea of fractals. The word comes from the Latin fractus, which means broken. A fractal could be described as a rough or fragmented geometric shape that can be divided repeatedly into smaller and smaller parts. These smaller parts resemble — not always exactly — the original larger whole. This characteristic is described as self-similarity. Fractals play a large part in such varied sciences as soil chemistry, seismology, and medicine. After a lifetime dedicated to mathematics, Mandelbrot became the oldest professor at Yale. One of the great things about Mandelbrot’s work was the way in which his fractals extend throughout nature. His demonstration of them appearing in natural environments in so many ways prompts the thoughtful reader to wonder about the extent to which fractals are numerological as well as scientifically mathematical.

Andrew Wiles, working at Princeton University in the 1990s, finally succeeded in proving Fermat’s Last Theorem from the seventeenth century, stating that the equation an+bn=cn will only be possible if n is not greater than 2. The gap of 300 years between its formulation and its proof emphasize the sequential nature of mathematical history. Contemporary mathematicians so often depend upon, and build upon, the work of their founding fathers.