From 0 to Infinity in 26 Centuries (4 page)

The Mayans had a symbol for an empty level in the stack, e.g. they possessed a concept of zero, which avoided the confusion faced by the Mesopotamians. So the number 419 (one
400, zero 20s and nineteen 1s) looked like this:

The Mayan calendar

As the information contained in
The Dresden Codex
would attest, the Mayans’ way of life was governed by astrology. Ritual human sacrifice was an integral part of
Mayan culture and was thought to aid the continuation of the Mayan people’s cosmology.

Tasked with working out which rites were necessary to appease
the gods, high-ranking priests were responsible for interpreting the positions of the sun, the moon and Venus.
They developed a system of several different calendars, which the Mayans used in parallel. They possessed a 365-days-a-year civil calendar called the
Haab
, which comprised 18 months of 20
days each, plus 5 ‘nameless’ unlucky days, called
Uayeb
, to make up the full total.

The Mayans’ main calendar was called the
Tzolk’in
, which worked on a 260-day cycle. The Mayans devised a 13-day week and believed that 20 gods were each associated with a day
of the year, and so 13 × 20 = 260 days in a cycle. The
Tzolk’in
was their everyday calendar, which they used to keep the date.

The 260-day cycle and the 365-day year would start together every 18,980 days, after 73 260-day cycles or 52 ‘vague’ years, as they were called. Fifty-two years was considered to be
a good, long life in those days, so in order to record anything longer than this the Mayans used yet another calendar – the
Long Count
. This calendar was used for recording dates of
important events, such as kings dying or volcanoes erupting; these were the dates they chiselled on to temples and statues using their stone tools. Considering their 360-day year (ignoring the
Uayeb
, the 5 nameless days – they did not want to bring bad luck to their monuments!) the Mayans, as base-20 people, deduced that 20 of those years made something called a
k’atun
, and 20
k’atun
s comprised a
b’ak’tun
. A
b’ak’tun
was approximately 395 years. The Mayans needed a starting point for their
dates – much as we use the birth of Christ for ours – which they decreed was 3114
BC
. All important dates were measured forward from that point.

The End of the World

It just so happens that the current
b’ak’tun
finishes in December 2012. Some people believe the Mayans, in their infinite wisdom,
predicted the world would end on this date. However, these people don’t realize the Mayans did in fact have a few more dates up their sleeves (if they had sleeves, that is) and that their
calendar could be extended up to 367 million years. So I wouldn’t worry about the world ending just yet.

The Mesopotamian, Egyptian and Mayan cultures had many things in common. Mathematically speaking, their work with numbers was functional, a means to an end – whether that
end was taxation, working out when the next eclipse was due or how to build a pyramid. Maths was certainly never performed for its own sake. The Mesopotamians and the Egyptians did amass a large
body of knowledge, which our next civilization – the Ancient Greeks – built upon.

The Gregorian Calendar

Since ancient times the idea that a year comprises 365 ¼ days has been well accepted. This system worked well for the Roman Empire after Julius
Caesar instigated the Julian calendar in 45
BC
. They even had a leap day every 4 years too.

But things started to get a little off kilter when it was observed that fixed points in the year, such as equinoxes and solstices, did not occur on the same day each year
as time progressed. The reason for this is that a year is actually eleven minutes shy of that quarter day – not a great deal of difference, but over hundreds of years it built up to
become quite an error.

By the sixteenth century the error had totalled ten days, which Pope Gregory XIII wouldn’t stand for. The majority of the Catholic countries in Europe changed to the
new Gregorian calendar, which got things back on track. Britain, as ever, mistrusted this newfangled European enterprise and stuck with the old calendar until 1752, by which point we had to
jump from 2 September to 14 September. The Russians kept the old calendar until the communist October Revolution in 1918, which, according to the new calendar, actually happened in
November.

The Ancient Greeks

Now it’s time to move on to the Classical period, when the great empires of the Greeks and, later, the Romans dominated vast swathes of the known world. Their respective
legacies were huge and their ways of doing pretty much anything were adopted and used for many hundreds of years after their demise.

Because we have only recently been able to translate and understand the cultures of the Mesopotamians and the Ancient Egyptians, it was for a long time believed the Greeks had been responsible
for the ancient world’s greatest discoveries and inventions.

T
HE
R
ISE OF THE
P
HILOSOPHERS

The philosophies of Socrates (
c.
470–399
BC
), Plato (427–347
BC
) and Aristotle (384–322
BC
), whose influence as mathematicians is explored later in this chapter (see
here
), were so significant that their modes of thought, minus their pagan beliefs, were later used by
Christian theologists to expound their doctrine. So it is no wonder that these three philosophers of Ancient Greece were held in awe for so long in Europe – their ideas, cobbled together with
stories from the Bible, were held to be the literal truth, and to disagree with them publicly was unwise.

However, despite their influence, I think they developed some quite strange ideas.

Illogical logic

The first philosophers (which means ‘lovers of wisdom’ in Greek) were often generalists because at that time they did not possess the specializations of science and
the humanities that we do today. Some of these philosophers used logic in its purest sense against clear evidence to the contrary. Zeno of Elea (
c.
490–430
BC
)
developed a series of paradoxes to help explain that motion was impossible. He argued, logically, that the great war hero Achilles could never catch up with a tortoise because, having started the
race 100 metres ahead, the tortoise would always be making slow progress as Achilles tried in vain
to catch up. Zeno also suggested that an arrow fired from a bow was
stationary because it could not be in two places at once. During its flight the arrow is constantly occupying a whole bit of space, and is therefore, in that instant, motionless. This
reductio
ad absurdum
method was given credence because it ‘proved’ that we should not trust the evidence of our senses, which were imperfect, whereas reasoning and logic were considered to
be flawless. Hmmm.

Greek mathematics

Because the Greeks were so interested in pure logic, they had a keen interest in maths for its own sake.

The Ancient Greeks split mathematics into two camps:
arithmetic
and
logistics
. Arithmetic, what we today call
pure mathematics
– the study of abstract ideas rather
than simple sums – was the sole preserve of intellectuals, the equivalent of today’s post-graduates. However, logistics, performing calculations, was an inferior trade that was better
left to numerate slaves.

The Greeks used two number systems. The first, in use from
c.
500
BC
, was the forerunner to the Roman system (see
here
), only it used Greek letters rather than
Latin: I for 1, ∏ for 5, Δ for 10, and so on.

The second system, which replaced the first by
c.
100 BC, was still based on the letters of the alphabet. The first ten letters, alpha (α) to iota (ι) represented numbers 1 to
10. After this point the letters went up in tens, so the eleventh letter, kappa (κ), stood for 20, and so on until rho (ρ), which stood for 100. The remainder
of the
alphabet then went up in hundreds. So the number 758 looked like ψνη. This number system still didn’t allow for calculations to be performed with the numbers themselves, so we
believe that sums would still have been carried out using counters. Despite the limitations of these numbers, they were used in Europe for over 1,000 years.

The Greek Alphabet

The Greeks, with the body of knowledge they acquired from their forebears in Egypt, Mesopotamia and elsewhere, wrote down and formalized many mathematical concepts that have been in use ever
since, and which we will now explore.

T
HALES
(
c.
624–
c.
545
BC
)

One of the first Greek philosophers, Thales (pronounced Thay-leez) hailed from present-day Turkey. Often considered to have been the first true scientist, at some point around
600
BC
Thales began to try to explain what he saw around him in terms of natural phenomena, rather than through the actions of deities.

When it came to mathematics, Thales was, like many Ancient Greeks, interested in geometry. He understood the principle of
similar triangles
and used it to predict the height of the
pyramids.

The two triangles on the preceding page are similar triangles because their angles are the same, and therefore their sides must be in proportion to each
other. Thales was able to determine the height of a pyramid by measuring the length of its shadow. He waited until his own shadow was the same length as his height to measure the pyramid’s
shadow in order to determine how tall it was.

P
YTHAGORAS
(
c.
570–
c.
490
BC
)

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