by Chris Waring
Archimedes hit upon an important idea with this method of exhaustion – the idea that if an approximation is performed accurately enough it becomes indistinguishable from the true answer. This idea has been used in many other areas of mathematics, perhaps most noticeably in the calculus of Newton (see here) and Leibniz (see here) almost 2000 years later.
Archimedes proved other important results, including that the area of a circle is π multiplied by the radius squared. He also proved that the volume of a sphere is 2/3 the volume of the cylinder that it is able to fit into. Archimedes was so pleased with his discovery he had a sculpture of the sphere and cylinder erected on his tomb.
Lasting legacy
Archimedes died at the hands of a Roman soldier while working at his desk. Legend suggests Archimedes was so absorbed in his work he failed to respond to the soldier’s orders that he come with him. Insulted, the soldier killed Archimedes, and presumably faced the wrath of his commanding officer, who would have regarded the slain intellectual as a highly valuable scientific asset.
With the death of Archimedes we come to the end of Ancient Greece, when its territories were consolidated into the emerging Roman Empire. The mathematical legacy of the Greeks is long lasting and most people today will have encountered the discoveries made by many of the mathematicians mentioned in this chapter. I think the Ancient Greeks’ greatest contribution was to invent mathematics as a rich and diverse subject, moving it beyond the basic necessity of numeracy and arithmetic, the functional tools of an economy. They created a subject that would become the language of science and which would eventually allow humanity to create scientific ideas from first principles, basing discoveries on a concept rather than from fudging equations and formulae to match observations. Without this mode of thinking Sir Isaac Newton would have been unable to conduct much of his pioneering work.
The Romans
The Greek mainland was conquered by the Romans in 146 BC, and the empire reached its zenith 200 years later, occupying a vast area that covered the entirety of the Mediterranean on all sides.
A PRACTICAL PEOPLE
Discipline was a central aspect of Roman life, which extended to its education system. The wealthier young Romans were taught basic arithmetic, most likely at home, but the main thrust of their education was to understand the workings of their own society. Oration was seen as the pinnacle of education, along with physical training for boys, who would go on to do military service, and home economics for girls, who were in charge of running their homes.
In terms of higher mathematics, it appears that very little was taught to the Romans when compared to their Greek predecessors. The Romans were a far more practical people, focusing their attentions on developments in engineering and medicine; practicality is not the best mindset for exploring mathematics for its own sake.
A spanner in the works
The Roman number system, inherited from the Greeks, didn’t help matters. Roman numerals rely on the position they sit within a string of letters, which makes it very difficult to use them in arithmetic.
The basic Roman numerals are:
I: one
V: five
X: ten
L: fifty
C: one hundred
D: five hundred
M: one thousand
The Romans wrote their numbers with the largest starting from the left. Therefore, in order to read a Roman numeral you have to add up the numbers from left to right. For example:
MMMDCLXVII would be 1000+1000+1000+500+100 +50+10+5+1+1 = 3667
However, the Romans devised a useful shortcut for using when the value of a number was close to the value of the next letter. The method involved putting a letter out of sequence, which indicated it should be subtracted from the next letter in sequence.
For example, in longhand the number 999 should be written DCCCCLXXXXVIIII, but with the shortcut it could be written as IM. However, there seemed to be no written rules, and the Romans, it seems, didn’t like having an I before an M or a C if they could avoid it. Therefore, 999 would more likely have be written as CMXCIX which gives (1000 - 100) + (100 - 10) + (10 - 1) = 900 + 90 + 9 = 999. Needless to say, having more than one way to write a number did not make life easy!
Alexandria
The Roman Empire subsumed the old Greek Empire and, as such, the Greek mathematical tradition continued. It focused in Alexandria, Egypt, a remarkable centre of learning that had been founded in 331 BC by the leader of the Greeks, Alexander the Great, as he conquered his way East across Europe and Asia.
HERO (10–70 AD)
An Alexandrian scientist and mathematician, Hero is most famous for detailing a primitive steam engine, and for perhaps being the first person to harness wind power on land with the aid of a windmill.
Hero also made two significant contributions to mathematics:
1. He came up with a formula for working out the area of a triangle that only requires the lengths of the sides of the triangle.
2. He devised a way of working out square roots: a number that when multiplied by itself gives a specific quantity.
Hero’s formula
There are many ways to work out the area of a triangle. Most of us were taught at school that:
area of triangle = ½ × base × height
For this formula you need to choose which side is the base and then work out the height of the triangle, which, if it’s non-right-angled, may not be one of the other two sides:
Hero’s formula removed the need both to choose a base and to measure the height, although perhaps at the expense of simplicity:
area of triangle with sides of length a, b and c = ¼ × √[(a2 + b2 + c2)2 - 2(a4 + b4 + c4)]
The root of the problem
Hero’s method for working out square roots involved using a formula to generate a new value; this new value would then be put back into the formula and the process would be repeated a number of times with the answer getting closer to the true value.
This technique is called iteration – another important development in mathematics. For example, if you wanted to work out the square root of 2, which, as we saw earlier, is an irrational number – one which cannot be written as a fraction and whose decimal goes on for ever without repeating (see here) – Hero’s method would work like this:
new value = ½ × (old value + R ÷ old value)
where R is the number you want to know the square root of. The first time you use the formula there is no ‘old value’, so you have to take a guess. The square root of 2 must be between 1 and 2, because 1 × 1 = 1 and 2 × 2 = 4 and 2 lies between 1 and 4. Let’s opt for the middle value, 1.5, and see what happens:
new value = ½ × (1.5 + 2 / 1.5) = 1.41666666...
You can now repeat this process using 1.41666 as your old value:
new value = ½ × (1.41666 + 2 / 1.41666) = 1.414215686
new value = ½ × (1.414215686 + 2 / 1.414215686) = 1.41423562
new value = ½ × (1.41423562 + 2 / 1.41423562) = 1.41423562
At this point you should notice that the old value and new value are the same, so our work here is done – and this is indeed the square root of 2, accurate to 8 decimal places.
If you wanted to work out the square root of another number you would start with a different R. It’s important to note that if you make R a negative number the formula does not work. For example, if you make R = -2 and have 1 as your first guess you get:
new value = ½ × (1 - 2/1) = -0.5
If you repeat as before you get:
1.75
0.3035714286
-3.142331933
-1.252930967
0.1716630854
This process continues for ever without ever settling on a value. Why? Because negative numbers cannot have a square root – a negative number multiplied by a negative number always gives a positive answer. Hence the formula is searching for something that does not exist!
Hero did, however, postulate that it could be possible for a negative num
ber to have a square root, if you use a bit of imagination (see here).
DIOPHANTUS (c. 200–c. 284 AD)
A resident of Alexandria from c. 250 AD, Diophantus is sometimes referred to as the ‘Father of Algebra’ because of his contribution to solving equations. While today thoughts of algebra conjure up a process of replacing numbers with letters, Diophantus did not adhere to this principle. Before true symbolic algebra was invented, mathematicians were forced to write out equations longhand.
These days it’s very easy to write a simple algebraic equations, such as: 3a + 4a2. However, Diophantus’ method would have been far more laborious, involving something along the lines of: ‘three multiplied by the unknown number added to four times the unknown number multiplied by itself.’ This made solving equations a tricky process, both in terms of writing and reading them.
An imaginary triangle
Diophantus was interested in Pythagoras’ theorem. He noticed something strange when he tried to work out the sides of a right-angled triangle with a perimeter of 12 and an area of 7. It produced an equation that could not be solved, indicating a triangle with those specific dimensions cannot exist. However, Diophantus remarked that if negative numbers could have square roots he would be able to solve the equation – which would mean the triangle would then exist. Much later, these numbers were called imaginary numbers (see box here), because in order to get round the problem you have to imagine that there is a number, represented by the symbol ‘i’, that is the square root of -1.
Triple the fun
His interest in Pythagoras’ theorem also sparked another mathematical mystery that would take hundreds of years to solve. Diophantus was interested in Pythagorean triples, which are solutions to the theorem that are whole numbers. For example:
32 + 42 = 52
52 + 122 = 132
82 + 152 = 172
In his great work Arithmetica, Diophantus included instructions on how to find such numbers. In 1637 French mathematician Pierre de Fermat wrote in the margin of his copy of Arithmetica that it was not possible to find the Pythagorean triples where the numbers were raised to any power other than 2. He finished with a tantalizing comment that was to tease mathematicians for years to come: ‘I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.’
These innocuous words started a 350-year challenge to solve what became known as Fermat’s last theorem.
Unravel the Riddle
Although we know very little about Diophantus’ life, a charming riddle, sometimes known as ‘Diophantus’ Epitaph’, associated with him provides a brief overview of his days on this earth. The riddle was first noticed in a puzzle book by the Greek philosopher Metrodorus some time in the sixth century AD.
‘Here lies Diophantus,’ the wonder behold.
Through art algebraic, the stone tells how old:
‘God gave him his boyhood one-sixth of his life,
One twelfth more as youth while whiskers grew rife;
And then yet one-seventh ere marriage begun;
In five years there came a bouncing new son.
Alas, the dear child of master and sage
After attaining half the measure of his father’s life
chill fate took him. After consoling his fate by the
science of numbers for four years, he ended his life.’
Can you work out how old Diophantus was when he died?
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.
THE END OF THE ROMANS
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.
CHINESE MATHEMATICS
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 g
ave 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.
