Europe Smells the Coffee 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 ...

1. Europe Smells the Coffee
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.

2. 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

3. 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.

4. 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

5. 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

6. 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, 5x2 + 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.