WHY DOES MONDAY COME ROUND SO QUICKLY?
How the numbers of the moon combined to make our week
In a book about everyday maths, what better place to start than the mathematics of the day itself, and of Monday in particular?
The seven-day week is so embedded in our culture that it is easy to forget that the idea of a week was just a convenient human invention. Why can’t we really love somebody ‘eight days a week’ (to quote the song)? Or ten, for that matter? And why does the working week have to start on Monday?
In fact, like many things at the heart of our modern culture, the seven-day week came about thanks to a combination of superstition, coincidence, human error, a need for order – and some elementary mathematics. These things weren’t just responsible for the week having seven days, they also determined the order of the days in the Western calendar. The reason why the week runs Monday, Tuesday, Wednesday, and not, for example, Wednesday, Monday, Tuesday, is to do with the way that numbers combine together.
To see how the modern week evolved requires a rapid sweep through some of the story of the calendar, much of which, it should be pointed out, is still a source of debate among historians. The maths, however, is sound.
Where did the week come from? In the earliest tribes, there was no notion of a week mainly because it wasn’t needed. The critical periods of time were the days and the seasons. For man, just as for animals, the day determined the basic survival routines of finding food, eating and sleeping. The seasons influenced the longer-term routines of hunting, harvesting and protecting against the climate.
Any tribe that could predict and plan for the different seasons would clearly have had better chances of surviving and prospering. Even the crudest of calendars would have given a tribe some advantage over rivals without such a tool.
So apart from the rough-and-ready clues of temperature and rain, how could ancient man have worked out what time of year it was? There is evidence that the first calendar was based on a very convenient clock in the skies – the moon.
The moon calendar – and the number twelve
After the sun, the moon is by far the most dominant presence in the sky, and it goes through an obvious cycle from one night to the next. A full moon slowly diminishes to a crescent and then no moon at all, and then returns to a full moon.
Archaeologists have found various clues that suggest that as far back as 30,000 BC the moon’s cycle was being followed closely. Etchings found on bone resemble the moon in its various phases and there are scratches that tally closely with the days in each cycle.
There are good reasons why the moon would have been important to primitive people. The period of full moon to full moon coincides almost exactly with the period between ovulations for women. Whether any sort of family planning existed thousands of years ago we don’t know, but the moon cycle, from which we derive the word menstruation, would at least have been a handy guide to fertility. Moon rituals, which still exist today, may have begun as fertility rites at that time.
There is another reason why the moon would have had such an attraction as a timekeeper. The cycles of the moon are a natural way of dividing up the year. There are roughly twelve moons in a year, so that became the obvious dividing number (the word ‘month’ also comes from the word ‘moon’, of course).
Quiz: How big is the moon?
Which of the following circular objects when held at arm’s length is about the same size as the moon?
(a) a pea
(b) a penny
(c) a ping-pong ball
(d) an orange
The answer is (a), a pea – a petit pois, in fact. For something that so dominates the night sky, this is surprisingly small. Thanks to the workings of the human brain, we perceive it to be much larger.
The exact number of moons in a year is, we now know, 12.36, and 12 just happens to be the nearest whole number. However, it takes quite a bit of rounding to trim 12.36 down to 12, and this was to prove a source of endless headaches to calendar makers from the Egyptians to Julius Caesar and beyond, as they sought to make the months coincide with the years. If the moon’s orbit around the earth had been just a fraction faster, we would probably have ended up with a calendar of thirteen months, and as a result the number 13 might have become regarded as a lucky number because of its link with the year. But it was not to be.
If the fluke of there being twelve moons per year first established this number as a fundamental measure of time, then the discoveries of the early civilisations in Egypt and Greece set the number 12 in stone.
Twelve is a conveniently small number, and it has other useful properties. One of the most important is that the number can be divided into two equal parts, or three, or four or six – which makes it a practical quantity for measuring and sharing. Indeed the convenient divisibility of twelve helped it to survive in British culture in both the currency (twelve pennies in a shilling) and the main unit of length (twelve inches in a foot) until late in the twentieth century.
The number 12 has connections with a circle, too. One of the easiest ways of dividing up a circle is to use a pair of compasses.
Marking around the circumference, the circle can be divided into six portions
And once a sixth of a circle has been found, it is easy to divide it into halves:
This means that a circle can very easily be divided into twelve equal pieces – very handy for dividing the sky into twelve portions to be symbolised by the signs of the zodiac, and later on, for dividing the clock face into hours.
The moon, the planets and the number seven
Every night, the familiar moon arches across the sky against a backdrop of twinkling stars. From the earliest times, it would have been observed that these stars were themselves slowly rotating, and that their rotation, like the sun’s, took exactly a day. However, there were some exceptions. Some bright stars didn’t follow the rest but moved at different speeds, and occasionally even looped back on themselves.
The small group of celestial objects with their own unique cycles acquired a special status. They became known as the wanderers, for which the Greek word was planetes. Since these planets each had their own distinct movements, it was only natural to give them names. By the time of the Romans, they were known in today’s English as the Moon, the Sun, Jupiter, Saturn, Venus, Mars and Mercury.
All whole numbers have factors – that is, smaller whole numbers that divide exactly into them (except for 1, of course). The factors of 12 are 6, 4, 3, 2 and 1, which add up to 16. A number whose factors add up to more than the number itself is called ‘abundant’ and 12 is the smallest such number. Abundant numbers are in fact extremely common. Looking for something to get their teeth into, mathematicians therefore began to look not just at whether numbers are abundant or not, but how abundant they are. For example, 12’s abundancy is 16/12 or 1.33. It is beaten by 24 (36/24 = 1.5) and this abundancy ratio increases with every multiple of 12 up to 60, whose factors add up to 108. (108/60 scores a whopping 1.8). The number 60 is highly abundant because so many numbers divide into it, and this Is why it became a popular base for counting. There seems to be no upper limit to how abundant a number can get, If you go high enough.
Does abundancy matter? Apart from enriching the understanding of numbers, no it doesn’t. But ancient Greeks, particularly Pythagoras and his cronies, were always looking for evidence to support their view that numbers controlled the entire universe. Any obscure property of numbers took on a significance that might now be regarded as excessive.
It is almost certainly because of this belief that there were seven so-called planets that this number acquired a mystical status. But, as with the number twelve, it took another coincidence to seal the significance of the number seven in the calendar. It just so happens that the number seven is linked to the cycle of the moon. From full moon to no moon is about fourteen days – two lots of seven. Full moon back to full moon is just over 29 days – not far from four lots of seven, or 28. And 28 is also a number loaded with mathematical significance, as the following section shows.
The factors of 28 are 1,2, 4, 7 and 14 – and it so happens that these add up to 28. The Greeks spotted this coincidence, and labelled this number, and others with the same property, as ‘perfect numbers’. Rather an arbitrary definition of perfection, you might think.
Perfect numbers are rare. The Greeks discovered only four of them: 6, 28, 496 and 8128, and as far as we know they didn’t find any others. This is hardly surprising since the next smallest perfect number is 33, 550, 336. It is believed that all perfect numbers end in 6 or 8, but it isn’t known if there are infinitely many of them.
The belief that numbers with curious mathematical properties were in some way mystical would certainly have helped to establish those numbers as part of the culture. The perfect number 28 and the abundant number 12 were two of the main beneficiaries.
This second occurrence of the number seven in the skies is a fluke, of course - the umber of days in the moon cycle has nothing to do with there being seven ‘planets’ – but, given man – kind’s natural tendency to read meaning into any coincidence, it is hardly surprising that early civilisations believed that the number seven had a fundamental link with the heavens. For convenience and for ritual, the moon cycle was therefore divided into partitions of four lots of seven days.
‘In the beginning’, according to the Bible, God took seven days to create the earth. Is this an even more fundamental reason for the mystical importance of the number seven? Or did the people who wrote this story choose a number that already carried significance for them because it divided up the lunar month and represented the number of the planets? It doesn’t really matter. The fact is, this story firmly established the idea of a seven-day week in Jewish culture with the seventh day being named the rest day, or Sabbath. The Jews spread the idea across the Middle East, and the Romans adopted it, too, despite the fact that they already had a separate eight-day ‘market’ week of their own whose origin is obscure.
Florence’s symbol of the eight-day week
In medieval Florence, they believed in an eight-day week – of sorts. In front of Florence cathedral is an octagonal building, the Baptistry. The shape of the building has a significance. Seven of its eight sides represent the seven days of the earthly week. The eighth side of the octagon represents the eighth, eternal day, the day we spend in heaven (or the other place) after we die.
It also happens to be a lot easier to build an eight-sided building than a seven-sided building, so the eight-day symbolism got the architect out of an awkward design problem.
Linking the hours and the planets
We’ve seen how, quite independently, the numbers 12 and 7 became crucial to the measurement of time, 12 because it divided up the year and, later, the day, and 7 because it divided up the month. The modern week was now ready to evolve, and the names of the days were to emerge because of one further link between those two mystical numbers.
It had been established early in astronomical history that each of the planets took a different amount of time to complete a cycle and return to its starting position – one planet ‘year’. This led the planets to be given a hierarchy, with Saturn, the planet with the longest cycle, being the most senior. The full hierarchy follows at the top of the next page.
Now you might expect that, with seven planets and seven days, it would be a logical and simple step just to name the days in the order of the planets: Saturnday, Jupiterday, Marsday and so on. But, by a quirk of astrological practice that is still not fully understood, this isn’t what happened.
The Egyptians were the first to divide the daylight into twelve hours, and then, around 1000 BC, the Babylonians, based in the region around modern-day Iraq, divided the day and night into 24 hours. Instead of naming the days after the planets, they decided to name the hours after the planets. The first hour was allocated to the most senior planet, Saturn, the second hour to the second planet, Jupiter, the third to Mars, and so on, repeating the cycle of the seven planets throughout the 24 hours and continuing it into the following days. The result looks like this (read down the columns and continue the cycle from one column to the next):
Because 7 doesn’t divide exactly into 24, the planet at the top of each day changes. In fact, because 24 divided by 7 has a remainder of 3, the planet at the top of the list skips three planets down the planet hierarchy each day. On Day Two the Sun tops the list, on Day Three the Moon, and so on. After seven days each of the seven planets has been top of the list, and on Day Eight the cycle begins again.
The planet at the top of each day became known as its ‘dominant’ planet, and it became the custom to name the day after this dominant planet. So we have:
Sound familiar? To make it explicit, here is this planet week set against the modern English and French weeks:
The days highlighted in bold have retained their planet names. And remember that this order has been determined by the fact that 24 divided by 7 has a remainder of 3.
This seven-day planet week eventually reached the Romans, who spread it across their empire, and it became the norm throughout Europe, with one small amendment. When Christianity took hold in the Roman Empire in the fourth century AD, the Romans felt it important to symbolise the change by making their week distinguishable from that of other religions. Since the Jews’ most holy day was Saturday (the Sabbath), the Christians appointed another day – Sunday – as their day of rest. They scrapped the pagan Sun god and renamed it ‘The Day of the Lord’ (Dies Dominici). Countries closest to the centre of the Roman Empire were the likeliest to adopt this, and, in many European countries today, Dies Dominici survives in a slightly corrupted form (Domenica in Italy, Dimanche in France, Domingo in Spain). Apart from this change, these romance languages have stuck with the planets, and always in the Babylonian order. Which means, of course, that the first working day of the week became Moon-day, the day after the rest day.
The Britons, who were that much further away from Rome’s religious influence, preserved the Sun-day, but sacrificed the last four planet days to Anglo-Saxon gods, Tiw, Woden, Thor and Frig. Well, when you’ve just been invaded and had your village pillaged, you don’t have much choice.
The time of day, the weeks, months and year are a great reminder of how numbers and mathematics are at the very root of our culture. The seven-day week owes a lot – perhaps everything – to early civilisations counting what they believed were seven planets.
How different things would have been if Neptune, Pluto and Uranus had been close enough to be visible to the naked eye. People of the Stone Age would have counted ten planets – one for each finger – which would surely have put ten beyond any question at the top of the number tree and would no doubt have led us to the ten-day week and the three-week month. Think of the consequences. On the downside, it might have meant more work and less play. But at least we would have 30 per cent fewer Monday mornings.
