Read From 0 to Infinity in 26 Centuries Online
Authors: Chris Waring
From 0 to Infinity in 26 Centuries (11 page)
A S
YMBOLIC
S
HIFT
Today, even the least mathematically minded schoolchild understands the four symbols we use for arithmetic: + - × ÷, and the sign we use to show our answers, =.
Before the invention of these shorthand ways of writing, the words were written out in full, which made following a mathematical treatise even more cumbersome than it is now.
Geoffrey Chaucer (1343–1400)
One of the great poets of the Middle Ages, Geoffrey Chaucer is not normally associated with the disciplines of science and mathematics. Chaucer, however,
also led a sideline in astronomy and alchemy, the latter of which sought to discover the philosopher’s stone, the means by which base metals could be turned into gold.
As part of these activities Chaucer became an expert at using a device called an astrolabe, a circular disc that allows you to find certain celestial bodies in the night
sky for any given latitude. Having detected his son Lewis’ interest in science from an early age, Chaucer wrote
A Treatise on the Astrolabe
in his honour. Understandably, perhaps,
it is a rather dry text, despite Chaucer’s attempts to enliven the subject by writing the book in verse.
The words
plus
and
minus
are, respectively, Latin for ‘more’ and ‘less’. In medieval times the letters ‘p’ and ‘m’ were used to
denote these two actions, until German mathematician Johannes Widmann first used the + and - symbols in his 1489 work
Nimble and Neat Calculation in All Trades.
Next came the equals sign: =, first used in Welsh physician and mathematician Robert Recorde’s catchily named book
The Whetstone of Witte
(1557) – one of the first books on
algebra to be published in Britain. In
The Whetstone of Witte
, Recorde states his intention to use symbols ‘to avoide the tediouse repetition of these words’.
The multiplication symbol, ×, came later on in Englishman William Oughtred’s book
The Key to Mathematics
, which was published in 1631. John Wallis, chief
cryptographer for Parliament, first used the ouroboros symbol,
∞
, to mean infinity, in his 1665 book
De sectionibus conicis
, in which he considers cones and planes intersecting to
form curves (which today is referred to as
conic sections
).
The division sign, ÷, is technically called an
obelus
, and it was first used in Swiss mathematician Johann Rahn’s book
Teutsche Algebra
in 1659. Ever concise, Rahn was
also the first to use ‘
.·.
’ to mean ‘therefore’.
The Dark Ages, it seems, weren’t quite so barren after all. The slow diffusion of mathematical knowledge from the East allowed Europeans to catch up gradually with their Islamic
counterparts. And what happened next allowed the Europeans to take the lead...
The Renaissance Onwards
The Renaissance began in Italy as early as the twelfth century and witnessed great advances in all fields of intellectual endeavour. It sparked a new-found interest in the
culture of the classical civilizations, which made an appreciation for – not to mention an investment in – science, culture and philosophy, the done thing among the wealthy patrons of
fourteenth-century Europe.
The initial epicentre of the Renaissance was Florence, Italy, where a wealthy merchant family, the Medicis, became sponsors of art and culture. One artist who benefited from their philanthropy
was Leonardo da Vinci.
L
EONARDO DA
V
INCI
(1452–1519)
Legendary for his talent in almost every field of intellectual pursuit, da Vinci was equally adept in the arts and the sciences. His superior ability and imagination enabled him
both to paint the
Mona Lisa
and to invent flying machines, among other extraordinary feats.
Perfectly proportioned
Da Vinci was also a keen anatomist, perhaps out of a desire to bring an element of realism to his art. He was very interested in the relative proportions of the human body, and
his famous drawing of the Vitruvian Man demonstrates his understanding.
The name of the picture harks back to a Roman architect called Vitruvius. He believed the proportions of the human body are naturally pleasing to the eye,
which led him to design his own buildings along similar proportions. Vitruvius considered the navel to be the natural centre of a man’s body. He believed a square and a circle drawn over an
image of a man with his legs and arms outstretched would represent the natural proportions of the body. Many artists tried to draw human figures that adhered to Vitruvius’ proportions, but
all looked somehow misshapen. Da Vinci, however, discovered the correct drawing could be made if the centre of the circle and the centre of the square are positioned differently.
A Helping Hand
Although da Vinci himself was not a trained mathematician, he did spend time receiving instruction from Luca Pacioli, a highly regarded maths teacher and
accountant. Da Vinci created many drawings of solids for one of Pacioli’s books, and his technical expertise with perspective helped to make the diagrams clear.
N
ICOLAUS
C
OPERNICUS
(1473–1543)
A Polish astronomer, Nicolaus Copernicus was one of the first to propose the heliocentric model of the universe: the earth is not the centre of the universe; it does, in fact,
orbit the sun. While this was not a mathematical discovery in itself, the way in which Copernicus devised his theory had significant implications for mathematics and science.
The march of science
According to the Bible, the earth was the centre of the universe, which was a perfectly reasonable, if slightly self-important, assumption to have made. After all, both the sun
and the moon appear to orbit the earth every day; indeed, all other objects in the night sky appear to do the same.
However, an exploration into the world of astronomy soon revealed problems with this assumption. For example, there are times when the planets appear to move in reverse, which could not be
explained if the earth is stationary.
Scientists at the time worked empirically, which means they made observations of phenomena and then came up with an explanation to fit what had been observed. But Copernicus did something that
was considered very backward by scientists at the time – he first came up with a theory about how the solar system might work and then tested it against observations, using mathematics as his
main tool.
While Copernicus’s heliocentric model did not cause much of a stir at the time, his way of working theoretically was one of the first examples of a new way of
conducting modern scientific methods.
J
OHN
N
APIER
(1550–1617)
A Scottish nobleman, John Napier was responsible for inventing a new kind of abacus called
Napier’s bones
. He also discovered
logarithms
.
Napier’s table
Logarithms are very important in many fields of mathematics and Napier’s book
Description of the Wonderful Rule of Logarithms
was quickly adopted by those who had
to conduct such calculations on a regular basis. It took Napier an astonishing twenty years to perform the calculations required for the tables of logarithms in the book – that’s some
dedication.
The logarithm of a number is the number we have to raise ten by in order to generate that number. For example:
The logarithm of 100 is 2 because 100 = 10
2
The logarithm of 1000 is 3 because 1000 = 10
3
The shorthand for writing this would be log (1000) = 3
We can find logarithms for numbers that are not whole powers of ten too:
log (25) = 1.39794 because 25 = 10
1.39794
We can also find the logarithm for numbers using any number, not just 10, as a base:
log
5
(25) = 2 because 25 = 5
2
Natural logarithms
(see
here
) are logarithms with a base of
e
, which is a very special number in mathematics that allows many difficult calculus problems to be
solved.
Logarithms today are computed using a calculator or computer, but originally they were worked out either by hand or using Napier’s tables. Before desktop calculators became commonplace
people would use logarithms and a
slide rule
(see
here
)to perform difficult calculations.
Napier’s bones
Napier also devised a faster and more convenient way of performing multiplication, based on a lattice method that Fibonacci had learned from the Arabs.
Napier’s bones was a useful tool that consisted of a set of sticks engraved with numbers, and each stick had a times table written on it, from 2 times the number up to 9 times the
number:
If you wanted to multiply 567 by 3, for example, you would collect together the three sticks that match the large number and then highlight the third row:
You then add the diagonal rows shown on the bones in order to calculate the answer.
Good Point
Napier was also one of the first people to use the decimal point. Although the Hindu-Arabic numeral system was in common use across Europe by this point,
a standard way of writing fractions was yet to be formalized. Because Napier needed a concise way to write them for his log tables, he adopted the Hindu-Arabic decimal fractions we use
today.
W
ILLIAM
O
UGHTRED
(1574–1660)
An English mathematician and teacher, William Oughtred continued Napier’s work on logarithms. He is credited with inventing the slide rule, a calculating device that
allowed the user to multiply large numbers together using a ruler marked with
logarithmic scales
(see
here
), which meant the answer to the multiplication could simply be read off the
ruler. Slide rules were used by scientists, engineers and mathematicians up until electronic calculators became commonplace in the 1970s.