From 0 to Infinity in 26 Centuries (10 page)

Khayyám also wrote a book that tackled Euclid’s fifth postulate, which had long rankled a contingent of mathematicians. The fifth postulate Euclid wrote concerned
parallel lines, and it is therefore normally referred to as the
parallel postulate
.

Imagine two lines (PQ and RS) with a third (XY) crossing them. Inside PQ and RS we now have four angles, two on each side of the XY: a, b, c and d:

The parallel postulate suggests that if you add the pairs of angles on the same side of XY together (e.g. a+b and c+d) then PQ and RS will cross on the side of the line where
the sum of the angles is less than 180°. If the angles on each side add up to 180° then PQ and RS are parallel and therefore will never cross.

Mathematicians, however, have argued over the ages that this postulate is not quite as obvious as Euclid made out. Khayyám was the first to come up with a counter-example, arguing that
Euclid’s parallel postulate does not always work if the surface you are drawing on is curved. Thus Khayyám instigated the ideas of
elliptical
and
hyperbolic
geometry, a
direct challenge to the simple
Euclidean
geometry that had gone before. This kind of thinking would eventually help Albert Einstein to come up with his ideas of space-time and gravity.

The Middle Ages in Europe

Despite Europe’s plunge into the Dark Ages – so-called because it was thought that following the fall of the Roman Empire the continent had reverted back to a
barbaric state of tribal warfare and religious fundamentalism – there remained a coterie of individuals intent on pushing the boundaries of mathematics even during these difficult times.

B
EDE
(672–735)

The Venerable Bede is known more perhaps for his contribution as a historian than for the role he played in the development of mathematics. Bede was a monk living in
north-eastern England and his translation of a number of scholarly works into the English of the time helped to spread an enormous amount of knowledge.

Bede’s contribution to mathematics began when he attempted to develop a way to calculate accurately when Easter would fall. At the time it was thought to fall on the
first Sunday after the first full moon following the spring equinox. Missing Easter mass following the calculation of an incorrect date would have resulted in excommunication, and therefore
damnation, so Bede’s was no trivial task.

Dating in the Dark Ages

In order to calculate the date of Easter, it was necessary for Bede to rationalize the date of the spring equinox with the lunar calendar. This was a difficult task in itself
because the date of the equinox varied because the Julian calendar in use at the time was unreliable. Because the date of the equinox varied each year, and full moons come at alternate 29- or
30-day intervals, it meant that there was a 19-year cycle of possible dates for Easter. The procedure for calculating the date of Easter has been known as computus (meaning
‘computation’) ever since.

Once Bede had completed the computus, he decided to sort out dating the rest of history as well. Prior to Bede’s endeavours, historians had been dating things in reference to the lifetime
of the current emperor or king, for example: ‘the Vikings first attacked in the third year of Aethelred’s reign.’ This method, of course, relied on the reader knowing when
Aethelred was around in the first place. Bede decided that it would be far more sensible to date everything occurring either before or after the birth of Jesus Christ. Although not originally
Bede’s idea – that responsibility lay with Dionysius Exiguus, a south-eastern
European monk active during the sixth century – such was his influence that we
have been using
AD
(
Anno Domini
, Year of the Lord) and
BC
(Before Christ) ever since.

Finger Talk

Bede also wrote a book called
On Counting and Speaking With the Fingers
, which allowed the reader to use hand signals for numbers into the millions
– a super-sized version of the systems we saw Stone Age cultures using. Again, such was his influence, people were still referencing Bede’s book 1,000 years later.

A
LCUIN OF
Y
ORK
(730–804)

A gifted poet, scholar, teacher and mathematician, Alcuin of York began his academic life under the instruction of Archbishop Ecgbert of York, who in turn had been tutored by
Bede. Alcuin’s main mathematical work was a textbook for students titled
Propositiones ad acuendos juvenes
(
Problems to Sharpen the Young
). The book contains many word-based
logic puzzles, a few of which have become quite famous, including the following two river-crossing problems.

Heavy load

The first problem relates to a man trying to cross a river with a wolf, a goat and a cabbage. The man’s boat is very small and he can only fit one thing in the boat with
him at a time. However, if he leaves the goat and the wolf together, the wolf will eat the goat. If he leaves the goat and the cabbage together, the goat will eat the cabbage. How does he get them
all safely across the river?

Answer: this is a good medieval example of lateral thinking. Clearly, on his first run the man can only take the goat across the river. On his second trip he brings the wolf across but
takes
the goat back with him
; he then leaves the goat there and takes the cabbage across, and then makes a final trip for the well-travelled goat.

Family matters

In the second problem, a couple, who are of equal weight, have two children, each of whom weighs half the weight of one of the adults. All four people need to cross a river, but
their boat will only hold the weight of one adult. How can they cross in safety?

Answer: the children cross the river in the boat. One child stays on the far bank while the other child returns. Dad crosses to the far bank and the child returns with the boat to be with the
mum and the other child. The two children cross again and one remains on the far side with Dad. The other child returns to be with Mum. Mum crosses to the far side, and the child with the dad
returns to collect the other child to reunite the family.

An Education

In 781 Alcuin joined the court of Charlemagne, the King of the Franks, where his skills as a teacher were in great demand. While there Alcuin introduced
the
trivium
and
quadrivium
, which he had encountered during his time in York.

During medieval times only seven subjects were taught in schools and universities. The trivium (Latin for ‘three roads’) comprised logic, grammar and rhetoric.
Logic was seen as the way to organize one’s thinking, grammar the way to express these thoughts without confusion and rhetoric was the way to persuade others that your correctly expressed
thoughts were worth listening to.

After graduating in the trivium, worthy students could attempt the quadrivium (Latin for ‘four ways’): geometry, arithmetic, astronomy and music.

The trivium was the equivalent of an undergraduate course and the quadrivium a Master’s degree. Succeeding in these courses of study gave access to the Doctorates,
either of Philosophy or Theology.

G
ERBERT
D’A
URILLAC
, P
OPE
S
YLVESTER
II
(946–1003)

Born in France, D’Aurillac joined a monastery during his teenage years, from where he was sent to Spain for further education. Under significant Arabic influence, Spain
exposed D’Aurillac to the wonderful discoveries of the Islamic mathematicians. D’Aurillac carved a name for himself as an excellent teacher and was taken on as a royal tutor. His
political career soon followed, culminating in him becoming the first French Pope in the year 999.

In his elevated position, D’Aurillac introduced the Hindu-Arabic number system to Europe (see
here
), although it did not immediately become widely accepted. He was also responsible for
re-introducing the abacus, which had not been used since Roman times but which soon became commonplace.

L
EONARDO OF
P
ISA
(F
IBONACCI
) (
c.
1170–1250)

The son of an Italian trader, Fibonacci lived near Algiers in North Africa, where he gained his first taste for Arabic mathematics. He travelled widely around the Islamic world
to further his learning and published a seminal book on his findings called
Liber Abaci
(
Book of the Abacus
). Fibonacci’s approach to writing the book
showed a keen head for business – not only did he expound the advantages of the Hindu-Arabic number system, he also applied it directly to banking and accounting. Fibonacci’s book
became very popular among medieval European scholars and businessmen, and his success earned him the patronage of the Holy Roman Emperor. A triumph, Fibonacci was then able to continue his
mathematical work in the fields of geometry and trigonometry.

Fibonacci’s name is well known for the sequence of numbers named in his honour. The sequence derived from, of all things, a problem about rabbits that he posed in his
Liber
Abaci
.

At it like rabbits

Fibonacci numbers
were known to Hindu mathematicians long before Fibonacci encountered them, but, much like Blaise Pascal (see
here
), Fibonacci became eponymous with
the sequence after it appeared in his book. In his problem, Fibonacci considers the growth of a rabbit population in a field. Fibonacci conjectured rabbits could start mating after they’d
reached the age of one month, and could reproduce every month thereafter. Therefore, if you start with one pair of newborn rabbits (one male and one female) in a field, how many pairs will you have
in one year (if each female rabbit continues to breed one male and one female)?

You can see the pattern emerging in the right column: 1, 1, 2, 3, 5. You can calculate the next number in the sequence by adding the previous two numbers in the sequence
together. So, next month there will be 3 + 5 = 8 pairs in the field. If you continue with this pattern until there are 13 terms in the sequence (which takes you to the end of the twelfth month of
the year), you get the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Therefore there will be 233 pairs of rabbits at the end of a highly sexed, not to mention very
incestuous, year.

All Part of Nature

Although the rabbit example is biologically inaccurate, the Fibonacci numbers do crop up in all manner of natural settings:

The number of petals on some flowers form part of the Fibonacci number sequence.

Plant shoots often split in such a way that the number of stems follows a Fibonacci
pattern.

The scales on a pineapple make three spirals, each of which contains a Fibonacci number of
scales.

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