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by H. Peter Aleff |

| Numerals and constants | ||||||||

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Many people have investigated mathematical questions for their own sake. In the case of the ancient Egyptians, the natural curiosity about numbers that occurs so often spontaneously in our species was encouraged by their annual need to re-measure the fields, and also by their massive construction projects that required prodigious amounts of computations. In addition, the religious importance that they attributed to certain numbers must have further reinforced their interest in these intangible but nonetheless very real and apparently eternal entities. Numbers, to them, lived and acted in the same unseen world beyond ours that contained the gods and spirits, but unlike these, the numbers followed knowable laws. Numbers could therefore help those who studied them to come closer to whatever powers affected their lives from that invisible realm. The Egyptologist Richard H. Wilkinson described in his book on “
People as vitally interested in that divine planning as the ancient Egyptians were had thus a strong reason to take a close look at the relationships between numbers. For instance, from the First Dynasty on they devoted a large part of their annals, preserved in fragments of a later copy on the so- called Palermo Stone, to recording the heights of the yearly Nile inundations in cubits, spans, palms, fingers, and even fractional finger- widths down to about the nearest quarter inch[2]. These flood heights determined the agricultural yields and thus the tax revenues to be collected during the biennial “countings of the wealth”. It seems therefore obvious that some of the tax administrators or their scribes would have looked at these ever varying but all- important and life- affecting numbers from all conceivable angles to find some predictable patterns in their sequence. Like stock market analysts who try to guess future prices from cycles of past ups and downs, the ancients must have tried to relate their lists of Nilometer readings to everything they could think of but the hemline height of their kilts which changed too slowly for such comparisons. Such a steady and often useful preoccupation with numbers would easily have led the ancient sages to look at other properties and relationships of those mysterious entities that govern the visible world from the invisible one and that are more real and more permanent than the reality we see. These proposed explorations of simple number relationships were well within their reach because they require few mathematical tools beyond the four basic operations. Actually, many such explorations of the number world have been undertaken from scratch, even by children. Mathematical genius has a habit of popping up spontaneously among people, as demonstrated by the many child prodigies and untutored but distinguished contributors in the known history of mathematics. For instance, Blaise Pascal (1623 to 1662) is said to have re- invented much of Euclidian geometry as a child, and at sixteen he came up with “what is still the most important theorem of projective geometry”[3]. The mathematical giant Carl Friedrich Gauss (1777 to 1855) baffled his parents and schoolmasters with his precocious comprehension of numbers and found at the age of ten the formula for summing up a series[4]. The mostly self- taught genius Ramanujan (1887 to 1920) came up with several thousand theorems new to the mathematics of his time[5], and the “mathematician These are just a few examples of many that happen to be recorded, and it seems a safe bet that there were many others in earlier times whose names and achievements have not been transmitted. The brain capacity of our species has not significantly changed over the last 30,000 years, a mere blip in evolutionary terms even for our latecomer kind. It would therefore be quite parochial to suppose that such mathematical talents could occur and develop only in our last few centuries, with just a few exceptions in ancient Greece that jump- started Western science and opened the flood gates for its unprecedented stream of geniuses. More likely than not, individuals with similar abilities would have appeared in many places and at many times, long before we began to keep track of them. Indeed, people developed mathematical concepts long before our species evolved even to its present name. The hominids who left their stone tools in the African
Closer to us, but still some 30,000 years before our time, the makers of the oldest known cave paintings in the Grotte Chauvet left us examples of parallel lines, various geometric signs, and a half- circle made from dots. They accurately projected the abstracted profile shapes of absent animals onto the walls, rotated these to face the desired direction, including a vertical feline that seems to walk down an imaginary Y- axis. They also stacked groups of rhinos, lionesses, and horses in proper perspective, with the nearest ones correctly overlapping those farther away[8]. All this required a good grasp of complex geometric concepts. Some of those cave painters’ contemporaries carved sequences of notches into tally bones that give us glimpses at how they seem to have pondered the riddles of division. For instance, a 30,000 year- old wolf bone found in On the ancient Egyptians’ own continent, but still more millennia before the nation- founding Narmer than we live after him, we have the 11,000 year old Ishango bone from the Congo region around Lake Edward. This famous bone is incised on its three edges with notches in groups of 11, 21, 19, 9; then 3, 6, gap, 4, 8, gap, 10, 5, 5, 7; and on the third edge 11, 13, 17, 19. The interpretation of these notches is controversial because many people find it hard to accept that “primitives” from that long ago could have pondered mathematical concepts as “advanced” as these series seem to imply. However, as Dominic Olivastro explains in his book “ It looks as if the carver had worked on the problem of divisibility, long before the advent of farming and grain measuring and then taxing would require later generations to hone this art on a daily basis. Compared with these time spans in the overall development of our species’ mathematical skills and interests, there is no People in Egypt and throughout much of the ancient Near East practiced long- distance trade and navigation and field geometry and monumental construction. All these require much computing and create environments favorable to the flourishing of mathematics. In such settings, you can expect the usual sprinkling of mathematical geniuses among the population to have both a chance and an incentive for developing their skills. Before the advent of electronic computers, mathematical progress was never tied to technical advances or fancy equipment. People from thousands of years ago could explore numbers and lines and circles with pebbles, with their fingers drawing in sand or their styluses on clay, and then with ink on papyrus, and they could do this as easily as many modern mathematicians have done with only pencil and paper. You can also expect an accumulation of the so gained insights in Egypt because the priests of Seshat, Thoth & Company preserved their mathematical learning with great care and passed it on, along with their other traditions, speculations, and discoveries. Moreover, they did so in a relatively stable society that valued received wisdom and that experienced much less upheaval than any other country in the region. The proposed transmission of knowledge about constants between the times of Narmer's mace designer, around 3,100 BCE, and those of Pythagoras or even Archimedes, over 2,500 years later, seems therefore more likely than the transmission of a book attributed to Euclid through a greatly more turbulent and frequently book- burning millennium until Byzantine times, about 888 CE, when the oldest surviving copy of his "Elements" was written[11]. In other words, the anachronism in finding, for instance, the golden ratio phi or a fairly accurate value for the circle ratio pi before the Greeks is only apparent and disappears when you examine the alleged priority of the Greeks. | |||||||||

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