50 Historical and Mathematical Background of the Wave Principle
The Fibonacci (pronounced fibehnah´chee) sequence of numbers was discovered (actually rediscovered) by Leonardo Fibonacci da Pisa, a thirteenth century mathematician. We will outline the historical background of this amazing man and then discuss more fully the sequence (technically it is a sequence and not a series) of numbers that bears his name. When Elliott wrote Nature’s Law, he explained that the Fibonacci sequence provides the mathematical basis of the Wave Principle. (For a further discussion of the mathematics behind the Wave Principle, see "Mathematical Basis of Wave Theory," by Walter E. White, in a forthcoming book from New Classics Library.)
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Leonardo Fibonacci da Pisa
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The Dark Ages were a period of almost total cultural eclipse in Europe. They lasted from the fall of Rome in 476 A.D. until around 1000 A.D. During this period, mathematics and philosophy waned in Europe but flowered in India and Arabia since the Dark Ages did not extend to the East. As Europe gradually began to emerge from its stagnant state, the Mediterranean Sea developed into a river of culture that directed the flow of commerce, mathematics and new ideas from India and Arabia.
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During the Middle Ages, Pisa became a strongly walled citystate and a flourishing commercial center whose waterfront reflected the Commercial Revolution of that day. Leather, furs, cotton, wool, iron, copper, tin and spices were traded within the walls of Pisa, with gold serving as an important currency. The port was filled with ships ranging up to four hundred tons and eighty feet in length. The Pisan economy supported leather and shipbuilding industries and an iron works. Pisan politics were well constructed even according to today’s standards. The Chief Magistrate of the Republic, for instance, was not paid for his services until after his term of office had expired, at which time his administration could be investigated to determine if he had earned his salary. In fact, our man Fibonacci was one of the examiners.
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Born between 1170 and 1180, Leonardo Fibonacci, the son of a prominent merchant and city official, probably lived in one of Pisa’s many towers. A tower served as a workshop, fortress and family residence and was constructed so that arrows could be shot from the narrow windows and boiling tar poured on strangers who approached with aggressive intent. During Fibonacci’s lifetime, the bell tower known as the Leaning Tower of Pisa was under construction. It was the last of the three great edifices to be built in Pisa, as the cathedral and the baptistery had been completed some years earlier.
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As a schoolboy, Leonardo became familiar with customs houses and commercial practices of the day, including the operation of the abacus, which was widely used in Europe as a calculator for business purposes. Although his native tongue was Italian, he learned several other languages, including French, Greek and even Latin, in which he was fluent.
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Soon after Leonardo’s father was appointed a customs official at Bogia in North Africa, he instructed Leonardo to join him in order to complete his education. Leonardo began making many business trips around the Mediterranean. After one of his trips to Egypt, he published his famous Liber Abaci (Book of Calculation) which introduced to Europe one of the greatest mathematical discoveries of all time, namely the decimal system, including the positioning of zero as the first digit in the notation of the number scale. This system, which included the familiar symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, became known as the HinduArabic system, which is now universally used.
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Under a true digital or placevalue system, the actual value represented by any symbol placed in a row along with other symbols depends not only on its basic numerical value but also on its position in the row, i.e., 58 has a different value from 85. Though thousands of years earlier the Babylonians and Mayas of Central America separately had developed digital or placevalue systems of numeration, their methods were awkward in other respects. For this reason, the Babylonian system, which was the first to use zero and place values, was never carried forward into the mathematical systems of Greece, or even Rome, whose numeration comprised the seven symbols I, V, X, L, C, D, and M, with nondigital values assigned to those symbols. Addition, subtraction, multiplication and division in a system using these nondigital symbols is not an easy task, especially when large numbers are involved. Paradoxically, to overcome this problem, the Romans used the very ancient digital device known as the abacus. Because this instrument is digitally based and contains the zero principle, it functioned as a necessary supplement to the Roman computational system. Throughout the ages, bookkeepers and merchants depended on it to assist them in the mechanics of their tasks. Fibonacci, after expressing the basic principle of the abacus in Liber Abaci, started to use his new system during his travels.
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Through his efforts, the new system, with its easy method of calculation, was eventually transmitted to Europe. Gradually Roman numerals were replaced by the Arabic numeral system. The introduction of the new system to Europe was the first important achievement in the field of mathematics since the fall of Rome over seven hundred years before. Fibonacci not only kept mathematics alive during the Middle Ages, but laid the foundation for great developments in the field of higher mathematics and the related fields of physics, astronomy and engineering.
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Although the world later almost lost sight of Fibonacci, he was unquestionably a man of his time. His fame was such that Frederick II, a scientist and scholar in his own right, sought him out by arranging a visit to Pisa. Frederick II was Emperor of the Holy Roman Empire, the King of Sicily and Jerusalem, scion of two of the noblest families in Europe and Sicily, and the most powerful prince of his day. His ideas were those of an absolute monarch, and he surrounded himself with all the pomp of a Roman emperor.
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The meeting between Fibonacci and Frederick II took place in 1225 A.D. and was an event of great importance to the town of Pisa. The Emperor rode at the head of a long procession of trumpeters, courtiers, knights, officials and a menagerie of animals. Some of the problems the Emperor placed before the famous mathematician are detailed in Liber Abaci. Fibonacci apparently solved the problems posed by the Emperor and forever more was welcome at the king’s court. When Fibonacci revised Liber Abaci in 1228 A.D., he dedicated the revised edition to Frederick II.
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It is almost an understatement to say that Leonardo Fibonacci was the greatest mathematician of the Middle Ages. In all, he wrote three major mathematical works: the Liber Abaci, published in 1202 and revised in 1228, Practica Geometriae, published in 1220, and Liber Quadratorum. The admiring citizens of Pisa documented in 1240 A.D. that he was "a discreet and learned man," and very recently Joseph Gies, a senior editor of the Encyclopedia Britannica, stated that future scholars will in time "give Leonard of Pisa his due as one of the world’s great intellectual pioneers." His works, after all these years, are only now being translated from Latin into English. For those interested, the book entitled Leonard of Pisa and the New Mathematics of the Middle Ages, by Joseph and Frances Gies, is an excellent treatise on the age of Fibonacci and his works.
Although he was the greatest mathematician of medieval times, Fibonacci’s only monuments are a statue across the Arno River from the Leaning Tower and two streets that bear his name, one in Pisa and the other in Florence. It seems strange that so few visitors to the 179foot marble Tower of Pisa have ever heard of Fibonacci or seen his statue. Fibonacci was a contemporary of Bonanna, the architect of the Tower, who started building in 1174 A.D. Both men made contributions to the world, but the one whose influence far exceeds the other’s is almost unknown.
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50 Historical and Mathematical Background of the Wave Principle

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