by Chris Waring
The Mesopotamians could write up to the number 59 in this way, using a sub-column for each of the tens and units digits. To write numbers larger than 59, they would write a new number alongside the tens and units (just as we can go as high as 9 in one column before we have to move on to the next one). Whereas our columns follow the pattern units, tens, hundreds, thousands, etc., according to our way of thinking the Mesopotamians’ columns went units, sixties, three-thousand six-hundreds, two-hundred and sixteen-thousands. A Mesopotamian would think of our number 437 as being 7 lots of 60 (7 × 60 = 420) plus 17, which they would write like this:
Zero interest
Although this seems like a pretty decent number system, not unlike our own, there were a couple of problematic areas. The first problem was that until c. 500 BC the Mesopotamians had no symbol for zero, which means they had no way of showing an empty column. For example, if I write down 205, the zero tells you that I mean two-hundred and five, and nothing else. The Mesopotamian number system was flawed because empty columns in the middle or at the end of a number were missing. For example:
This looks like 60 + 10 = 70. But there could be an empty column in the middle, in which case the number would be 3600 + 0 + 10 = 3610. Or there could be an empty column at the end, in which case we would have 216000 + 60 + 0 = 216060 – quite a large difference. Apparently, Mesopotamians tended to rely on the context in which the numbers were used in order to read them in the most reasonable way.
Early Arithmetic
Many people consider the abacus to have been the ancient world’s version of the electronic calculator. In fact, the counting frame with beads – which most people think of when they hear the word abacus – is a relatively modern piece of technology, first made popular in China after AD 1000.
The word ‘abacus’ is thought to come from the Hebrew word for ‘dust’, and the first abacuses were simply that – a board or level surface strewn with dust that could be used as a scratch pad for calculations. Eventually the dust was replaced by a board with tokens that could be placed in columns to allow for the addition of large numbers without having to be able to count higher than ten.
Later, the Romans used pebbles or, in Latin, calculi (from which we get the words calculus and calculation). In England, we called the tokens ‘counters’, which is why shops had a counter-top to put their counting board on.
Multiplication madness
The second problem arose when the Mesopotamians tried to multiply numbers together. Whichever way you multiply using our decimal system, you need to have memorized your times tables up to 9 x 9 (because 9 is the highest digit we have). However, according to the Mesopotamian system, you needed to know your times tables up to 59 x 59! We think they used a few key times tables, written on small tablets, to help, but even so their times-table tests at school must have been a nightmare.
Archaeologists have found many hundreds of clay tablets littered with Mesopotamian mathematics. It seems the Mesopotamians were able to use fractions, to work out the areas of rectangles and triangles, and to solve quite complicated equations. My favourite fact is that many tablets found appear to have been maths homework! But maybe you need to be a maths teacher to appreciate that ...
ANCIENT EGYPTIAN MATHEMATICS
The Ancient Egyptians were a talented lot. In addition to building the pyramids, many of which are still standing over 4,000 years later, they also turned their hands to committing the written word to papyrus, a paper-like material made from interwoven reeds. Papyrus was a much more forgiving material to write on than the clay tablets the Mesopotamians were using further north. As such, unlike the Mesopotamians, the Ancient Egyptians were not limited to using a single symbol. However, because papyrus rots, especially if it gets wet, it does mean that a vast majority of the writing of Ancient Egypt has been destroyed over time.
Systems in place
It also seems that the Egyptians were not limited to one writing system.
Egypt is famous for its hieroglyphics – pictograms they carved on to their monuments, and which remained a complete mystery until French soldiers unearthed the Rosetta Stone in 1799.
Hieroglyphics were the Ancient Egyptian equivalent of calligraphy – decorative writing for use only on wedding invitations and inscriptions. The Egyptians had another writing system called hieratic, which they used for everyday stuff – a much easier and faster way to write script that scribes would then use for their calculations.
Hieroglyphic numbers had symbols for 1, 10, 100, 1000, etc., which the Ancient Egyptians would combine to make the required number:
So if an Egyptian wanted to refer to Rameses’ 1,234 chariots on his latest obelisk, he would have used the following symbols:
Because they added up the symbols in order to generate a total, the Egyptians could write the symbols in any order and direction they pleased – a handy tool when they wanted to be decorative.
The hieratic number system was a little more complicated because it used different symbols for each unit, each ten, each hundred and each thousand. The symbols for 40 and 50, for example, bore no relation to each other. It seems that this system relied on the fact that the scribes would be familiar with the symbols and would be able to perform calculations either in their heads, or by converting to the hieroglyphic system for tricky sums.
The business of numbers
Three important sources of mathematical information were left behind from Ancient Egypt: the Rhind Papyrus, the Moscow Mathematical Papyrus and the Berlin Papyrus. All three documents contained mathematical problems in arithmetic and geometry, alongside, interestingly, the first written information about pregnancy tests.
From these three sources we have learned that the Egyptians used fractions. However, they only used fractions that had a numerator of 1 – that is, the number on the top of the fraction could only be 1. They would talk about more complicated fractions by adding these unit fractions together. So, for example, they would think of ¾ as ½ + ¼. Although slightly cumbersome, this method stood the test of time – unit fractions were still used by mathematicians in medieval times.
The pyramid builders obviously had a pretty good grasp of geometry; the papyri contain detail about how they set about making these ancient structures. The pyramids were made with stacks of stone blocks in layers, and the steepness of a pyramid depended on the size of the overlap between two layers – the larger the overlap, the steeper the pyramid. The Egyptians devised a series of methods to work out what size of overlap was needed for different gradients. It has also been suggested the Egyptians had some idea of Pythagoras’ theorem (see here), which would have enabled them to work out the third length of a right-angled triangle if they knew the length of the other two sides.
There are, of course, many more far-fetched theories regarding the Ancient Egyptians, including the super-high technology they appropriated from the legendary island of Atlantis (or from aliens, or time travellers...). I cannot say whether such things were true, but I do know the Egyptians were pretty clever fellows.
A Tall Order
The fact about the Ancient Egyptians that I always find most extraordinary is that the Great Pyramid, which was completed c. 2560 BC, was the tallest building in the world until the central towers of Lincoln Cathedral were raised in AD 1311 – that’s the best part of 4,000 years!
THE MAYANS
By the first millennium AD Mayan civilization had reached a level of cultural and mathematical development similar to the Mesopotamians and the Egyptians. They declined somewhat as time went by, but when the Spanish conquistadores arrived in the early 1500s the Mayans had managed to recapture their previous levels of sophistication.
Born in isolation
The Mayans left behind a raft of evidence that demonstrated how they conducted their mathematics, but unfortunately virtually all of it was destroyed when the Spanish invaders arrived and sought to convert the region’s heathens to Catholicism. The Dresden Codex is one of three surviving examples of Mayan wri
ting. Although it was badly damaged during the Second World War, the book still contains a great deal of insight into the Mayan development of mathematics. Many surviving monuments in modern-day Mexico and Guatemala contain numerical information, such as dates, inscribed upon them.
Unlike the cross-pollination that occurred between the Mesopotamian and Egyptian cultures, the Mayans developed in complete isolation. They also failed to fulfil the last two criteria of ‘civilization’: they did not possess the wheel, perhaps because there were no beasts of burden in the parts of Central America where they lived; they also did not seem to be able to smelt metal. However, despite technically still existing in the Stone Age, the Mayans were able to build great cities, some of which contained populations of over 50,000 people.
Number crunching
So, what of their mathematics? The Dresden Codex is concerned only with astrology and astronomy, so everything we know about the Mayans’ mathematics is shone through this lens.
The Mayans used a base-20 system, within which lay a base-5 system (much like the Mesopotamians’ base-60 and base-10 system).
Like modern mathematics, Mayan mathematics had a grasp of place value: the value attached to the position of each digit. Unlike modern mathematics, Mayan mathematics placed numbers in vertical stacks, with the highest place value positioned at the top. Because the Mayans counted in groups of twenties, each level in the stack was twenty times the value of the level below. From the bottom it went something like this: 1s, 20s, 400s, 8000s, etc. So our number 8577 is one 8000, one 400, eight 20s and seventeen 1s, which in Mayan looked like this:
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’atuns 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.
THE RISE OF THE PHILOSOPHERS
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
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p; 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.
THALES (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.
