From 0 to Infinity in 26 Centuries (8 page)

Hypatia (
c.
370–415
AD
)

The Alexandrian mathematician, philosopher and astronomer Hypatia was the daughter of Theon, a mathematician who
produced an edition of Euclid’s
Elements
. He educated his daughter in the same way as his sons, which exposed Hypatia to the rich philosophical heritage of her Greek ancestors.

Hypatia was a teacher specializing in the philosophies of Plato and Aristotle, and as part of this she developed her own ideas in mathematics, physics and astronomy. She edited her
father’s editions of Euclid’s and Diophantus’ works, using her teacher’s eye to help the reader understand the more difficult sections.

Hypatia is widely considered to have been the first woman to make contributions to mathematics and science, although few of her original works survive. She dressed in scholar’s robes
rather than in female dress, and chose to navigate the city unaccompanied, often driving her own chariot, which at the time was considered very unladylike. Hypatia also stood for what by then the
Christian Romans considered to be a pagan religion. Her lectures, which were open to all comers, regardless of race or religion, were targeted by Christians and led to riots. This discrimination
reached an inevitably bloody conclusion, and in March
AD
415 Hypatia was brutally attacked and murdered by a Christian mob.

T
HE
E
ND OF THE
R
OMANS

The Roman Empire began to disintegrate in
c.
AD
380. In the absence of the sizeable bureaucratic machine and enforced discipline the Romans had
instilled, Western Europe entered what is sometimes referred to as the Dark Ages: a period when little intellectual development occurred. Allegedly.

Eastern Mathematics

The history of mathematics was not confined to Europe. China, India and the countries of the Middle East each have a tradition rich in the subject, and the flow of mathematical
knowledge was, generally speaking, from East to West. As Europe found itself plunged into the Dark Ages, mathematical discoveries in the East ensured the subject continued to go from strength to
strength.

C
HINESE
M
ATHEMATICS

Chinese history is populated by dynasties – a succession of ruling families, each of whom prioritized eradicating all evidence of the previous incumbent. As such, many
important Chinese mathematical works and artefacts have been lost over time.

Much of what we know about Chinese mathematics is
attributed to a scholar and bureaucrat called Qin Jiushao (1202–61). He wrote a book called
Mathematical Treatise
in Nine Sections
that discusses practical mathematics in a variety of fields relevant to government officials. Jiushao’s book also contains a detailed history of Chinese mathematics, and
sheds light on the country’s mathematicians and their advances in the field.

It All Adds Up

Chinese numbers were based on a system of
counting rods
: short sticks that, when placed in certain arrangements, denoted various numbers in a
decimal system. Their written numerals were simply drawings of the arrangement of these sticks.

In
c.
AD
700 the Chinese borrowed the concept of zero from India (see
here
), which means they were one of the first cultures to have a
fully fledged decimal number system.

Predicting the future

The
I Ching
(
Book of Changes
) is a famous Chinese text that dates from, at the very least,
c.
1000
BC
, and quite possibly before
then. The text allows you to divine your future using
trigrams
and
hexagrams
, both of which have their origins in mathematics.

A trigram is a stack of three horizontal lines, which can be either
yang
(solid) or
yin
(broken). It is possible to make eight different trigrams using this system, and each
trigram has various attributed meanings, including the Chinese elements: earth, mountain, thunder, water, lake, wind, heaven and fire.

Two trigrams could be combined to make a hexagram, and there are 64 (8×8) possible hexagrams to be made from the eight trigrams – which could then be used to predict
your future. Soothsayers would need to be familiar with the interpretations of each trigram and hexagram in order to use them to build up your reading.

The German philosopher Gottfried Leibniz (see
here
) was intrigued by Chinese philosophy and noticed that the
trigrams and hexagrams of the
I Ching
can be written
as
binary numbers
– a system of numbers that has 2 rather than 10 as its base – if the
yang
is seen as 1 and the
yin
as 0.

Leaps and bounds

Zu Chongzhi (
AD
429–500) was a Chinese astronomer and mathematician whose discoveries lay far ahead of his time. Chongzhi calculated various
astronomical constants to extremely high degrees of precision; he also worked out independently a value for π using Archimedes’ method of exhaustion on a polygon with over 12,000 sides.
His answer gave a working value of 355/133, which is accurate to six decimal places (see
here
). Europe would not achieve this level of precision for over 1,000 years.

The
Nine Chapters on the Mathematical Art
is one of the oldest and most important Chinese mathematical works, compiled over the centuries up to
c.
AD
100.
It gives us a very good idea of the state of Chinese mathematics that existed at approximately the same time as Greek civilization. The chapters covered the following topics:

1.
Areas of fields

2.
Exchange rates and prices

3.
Proportions

4.
Division; square and cube roots; area of a circle and volume of a sphere

5.
Volumes of other solids

6.
Taxation

7.
Solving equations

8.
Simultaneous equations

9.
Pythagoras’ theorem

While the Chinese may have been more concerned than the Greeks with practical matters, we can see that their development in mathematics was on a par.

Court eunuch Jia Xian (
AD
1010–70) is credited with being the first individual to investigate what later became known in Europe as
Pascal’s triangle
(see
here
). Chinese mathematician Yang Hui (
AD
1238–98) published Xian’s findings in 1261, four hundred years before Pascal’s discoveries would be
revealed. Xian was also interested in what we call
magic squares
, which had long fascinated Chinese mathematicians. A magic square is a square of numbers in which all the rows, columns and
diagonals add up to a particular number. For example:

In the example above, each row, column and diagonal has a
sum of 15. This particular magic square is known as the
Lo Shu
square because of its
connection to a legend in which the River Lo floods and a magic turtle carries the magic square on its shell to aid the afflicted people.

The Chinese Abacus

At some point in
c.
AD
1000 the Chinese began to use the
suanpan
(Chinese abacus) in favour of the counting rods,
although the
suanpan
had been around for some time and may have influenced the abacus in the West.

This particular abacus has counters suspended on rods, which are layered on two decks. On the lower deck there are five beads per rod, each of which is worth one; on the
top deck there are two beads, each worth five. Each rod represents a decimal column (unit, tens, hundreds, etc.) and pushing the beads towards the separator in the middle signifies the number.
For example, 123,456 would look something like this:

The extra bead on each deck can be used for performing calculations.

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