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Near the end of the (Western) Roman Empire, the breakdown in trade brought about a new way of rural life the self-sufficient manor. This was a herald of things to come. In fact, the manor was exactly what the Dark Ages called for. The wide and well-built Roman roads (not all of which led to Rome) were hardly traveled with the exception of a few pilgrims on their way to holy sites. There were no universities and there was little need for mathematics or, for that matter, any intellectual endeavor. The priests were generally the only ones who were literate and the only justification for any mathematical activities on their part was the calculation of the date of Easter, which unlike Christmas, depends on the lunar calendar.

The average peasant lived and died within fifty miles of his birthplace. Under the new order, feudalism, he was a slave to his lord and worked the latter s land in return for a tiny portion of it for sustenance. With several minor exceptions, there was little or no scholarship throughout Europe, though monks in the monasteries wrote and rewrote ancient manuscripts and studied the Bible and the works of Roman authors such as Boethius (480-524). Very few people could read, let alone write. The priests read Bible passages to them and the church provided the rituals, holidays, and all of the other components of the social structure of that age.

The monks sang monophonic, free-flowing chants with haunting melodies of Pope Gregory (540-604) the Gregorian chants, of course! year after year, decade after decade. Life was stagnant. Virtue consisted of self-denial, prayer, and toil. The idea of a Greco-Roman bathhouse where men of letters congregated and debated philosophy was as distant as the outermost galaxies of the universe. To be sure, there was a brief period of learning in the so-called Carolingian Renaissance during the rule of Charlemagne, but it doesn t amount to a hill of beans.

The political agenda of the rulers of Europe was consolidation of power, repelling the Muslim invaders, and conversion of the barbarians to Christianity. This conversion effort spanned the five hundred years of the Dark Ages. There were no significant advances in mathematics in Europe during this time.

What changed There are volumes written about this question. It was the common belief that Christ would return to earth in the year 1000. As you probably are aware, he didn t. It was therefore logical to assume that civilization would endure another thousand years, as humankind had not yet acquired the terrifying nuclear weapons of the twentieth century. What followed was a church-building frenzy. This required a bit of carpentry, stonemasonry, transportation of goods from distant communities, and the hiring of laborers and artists. Feudal lords discovered that it was more efficient to buy armies than to extract servitude from their vassals. It was better to rent land to the peasants in return for much needed cash. This was the death knell of feudalism. Gradually a money economy began to take shape. Many restless young men ran off to the cities to join merchant guilds or join the army talk about upward mobility! The first crusade of 1095 woke Europe up and made it aware of a larger world. A greater demand for silk and other oriental products made the merchants of the Italian city-states prosper and these middle-class communities acquired power and influence, in some cases even self-rule. The twelfth century saw the rise of several European universities, such as Oxford, Paris, and Bologna, with many more soon to follow.

It was in the twelfth century that translations of works by Aristotle, Euclid, and other great Greek writers started to appear. This glimpse into the past glories of the classical world excited the imaginations of the Europeans.

The High Middle Ages was a time of growth and awakening. Huge gothic cathedrals such as Notre Dame and Chartres were monuments to the scope and vigor of activity in that period. New ideas appeared in Europe such as gunpowder, spectacles, mechanical clocks, and the flying buttress that permitted gothic churches to have large stained glass windows by freeing the walls from having to support the roof.

New mathematical ideas and the new Hindu-Arabic numerals gave scholars a shot in the arm. Commercial arithmetic was suddenly in demand and by the thirteenth century, merchants in Italy established private schools for their future heirs. At approximately this time, the Hanseatic League, an alliance of German, Swedish, Danish and English towns negotiated by their guilds to fight piracy and promote commerce, established schools that required commercial mathematics.

While the twelfth century saw a heightened interest in mathematics and learning, much of it was scholastic in nature. It was the epoch from which we draw the sarcastic question, How many angels can sit on the head of a pin The mathematics was astonishingly simple, as were the other subjects. The texts were handwritten versions of ancient authors. There was, of course, no empirical science and the highest authority in any matter was the Bible. To be sure, a battle was brewing between faith and reason between revelation and observation between authority and free inquiry.

Enter Thomas Aquinas (1225-1272) and his great peace making compromise. Reason, he said, is valid in matters of this world, while revelation is valid in matters of the next one. The word of God is absolute and final on matters such as birth, death, prayer, salvation, duty, and so forth, while one need not consult the Bible to find out how to boil water, hunt boar, or to solve a quadratic equation. Freedom at last! Use reason to your heart s content, but don t dare interfere with Church doctrine.

This truce lasted for several hundred years but, as we shall see later, was doomed from the start. In the interim, the teaching of the great philosopher Aristotle, to cite an example, was banned by the Church and hence removed from the curriculum of the University of Paris. The mathematics of Euclid wasn t deemed to be a threat to dogma and was spared the censor s cut. Little did the defenders of the faith realize that their greatest challenge would come from that seemingly harmless subject. We will see how this occurred when we get to the sixteenth century.

This is as good a time as any to look at the mathematics of Fibonacci.1 He listed the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, , where the means keep on going until we tell you to stop. In this case, the sequence continues forever. The trick is to see the pattern so that you can continue the sequence. The pattern is simple. Each term is the sum of the two preceding terms. Thus, 13 appears where it does because it is the sum of the two preceding terms 5 and 8. The third term of the sequence, 2, is the sum of the two preceding terms. This pattern does not explain the first two terms. They are simply given to us by Fibonacci.

1Fibonacci s sequence of numbers occurs in many places including Pascal s triangle, the binomial formula, probability, the golden ratio, the golden rectangle, plants and nature, and on the piano keyboard, where one octave contains 2 black keys in one group, 3 black keys in another, 5 black keys all together, 8 white keys, and 13 keys in total.

Now each term can be denoted using subscripts that identify the order in which the terms appear. We denote the Fibonacci numbers by u1, u2, u3, , and the nth term is denoted un. The sequence may now be presented by stating the first and second terms and the recursive relation which says, in an equation, that each term is the sum of its two predecessors.

The last equation in the box says that the (n + 2)nd term is the sum of the nth term and the (n + 1)st term. When n = 1 for example, this says that the third term, u3, is the sum of the first term, u1, and the second term, u2, which is, of course, correct. Sequences play an important role in mathematics today, and it s interesting to see that the concept is quite old.

It should be obvious to you that the sequence terms grow large fairly rapidly, eventually breaking through any arbitrarily set ceiling. If a macho mathematician were to lay down the challenge, Will your terms ever exceed 1,000,000,000,000 we would respond, with confidence, Absolutely! After some calculating, we would, to the accompaniment of a drum-roll, present a Fibonacci number larger than a trillion. Another way of saying all this is that the Fibonacci numbers approach infinity.

On the other hand, consider the sequence of ratios , , formed by dividing each term by its predecessor. Instead of consistently getting larger, they alternate between growing and shrinking! In decimal form, the first few ratios are . . . . Since this alternation is consistent, successive numbers narrow the range in which future ratios can fluctuate. It turns out that the ratios approach a single target which we can readily calculate using a clever argument.

Let us call the target (or limit as mathematicians say) L. Furthermore, let us denote the nth ratio Rn. In other words (symbols ), . Observe that if we divide the last equation in the box (the recursive relation) by un+1, we get

which, after applying the new symbols for the ratios, becomes

Notice that the first ratio on the right side of the equation is the reciprocal of the ratio we have defined. It is the ratio of the nth term over its successor the (n + 1)st term. Now let n approach infinity, and replace both ratios by their limiting value L, and we get

Notice that the first ratio on the right side of the equation is the reciprocal of the ratio we have defined. It is the ratio of the nth term over its successor the (n + 1)st term. Now let n approach infinity, and replace both ratios by their limiting value L, and we get

which, after courageously multiplying both sides by L, becomes

L2 = 1 + L

Finally, we transpose everything to the left side, obtaining the quadratic equation

L2 – L – 1 = 0

Recall the quadratic formula, which we now invoke to solve this equation. The general form of a quadratic equation is ax2+bx+c = 0, in which a, b, and c are the given coefficients which distinguish one quadratic equation from another. In the equation, 5×2 + 2x – 1 = 0, for example, a = 5, b = 2, and c = – 1. The quadratic formula gives us solutions in terms of a, b, and c. It says that

In our quadratic equation, L2 – L – 1 = 0, we see that a = 1, b = -1, and c = -1. Choosing the plus sign in front of the square root symbol yields the solution , which is approximately 1.618. (Choosing the minus sign in front of the square root symbol would have given us an absurd, negative answer.)

We have obtained a celebrated number dear to all mathematicians and to many artists. It is called the golden ratio and was known to the Ancient Greeks as the most pleasing ratio of the length of a rectangular painting frame to its width. Notice that very few paintings are square, by the way, and even more rare is it to see a painting on a canvas which is twice as long as it is wide.

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