Mark Ronan
The following is an excerpt from a talk given by Professor Mark Ronan, University College,London at a research seminar. An historian of mathematics, Professor Ronan's contribution to the History of Banking research programme concentrated on the invention of numerals- that essential precondition for the calculation of assets and rates of interest. The full text of this talk will be published as a chapter in the Erasmus Forum'sHistory of Banking(2021)
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The origins of counting may have vanished in the mists of time, but the words we use today for numbers show that well-developed counting systems existed before the advent of civilisation. For example the Indo-European languages, now covering most of north India, Europe, and parts of western Asia derive their names for the numbers one to ten from a common source, and many other language families show similar evidence. Yet the words for numbers in one language family can be entirely unrelated to those in another language family, suggesting counting developed independently in different parts of the world.
Numbers were important for trade, where for example a certain weight of silver might be exchanged for suitable measures of cloth, but the origin of counting is earlier than this. It would probably have had quite a different purpose — recording the passing of days. The phases of the moon for instance were of huge importance, and some form of counting would be needed to decide the night of the full moon, which in ancient cultures would often be the occasion for a feast or celebration of some sort. The eagle bone below clearly shows the use of counting, and looks likely to have represented some form of lunar calendar.
At one time people would have understood the phases of the moon very well, and would have had no difficulty answering the following question: if you see a thin crescent moon in the sky, how do you know whether it is a first or last crescent?
Having largely lost our connection with the moon's phases, most people find difficulty answering this. Friends and acquaintances usually admit they do not know, or suggest it depends which side of the moon is lit up — but without being able to say for sure which side that should be. In fact it depends on whether you are in the southern or northern hemisphere, but that is not really the issue. The answer to my original question is simple: in the evening it's a first crescent, in the morning a last crescent. The first of my acquaintances to get it spot on happened to be Jewish, which probably helped because the Jewish calendar is lunar, and the same goes for the Islamic calendar. Each new month begins with the first crescent of the moon, which always occurs in the evening. This is why Jewish and Islamic days start at sundown.
By contrast a crescent at sunrise is always a waning moon, a fact brought vividly home to me at a conference on the south coast of Turkey. Staying in a hotel overlooking the sea, I rose at dawn to swim in the outside pool and noticed a crescent moon across the sea before sunrise. Each day it became thinner and closer to the horizon before disappearing from view as the sun rose on the same horizon. Then, one day, it was no longer there, and as I waited for it to rise the sun came up instead — there was no moon. For the first time in my life I had just seen what we call the new moon.
Ancient people before civilisation understood this very well: a waxing moon is always in the sky at sunset, before setting later in the night; and a waning moon rises some time during the night and remains visible at least until sunrise. Between the two is the full moon, which rises on the eastern horizon at sunset, remains in the sky all night, and sets on the western horizon as the sun rises in the east. The time from first crescent to full moon is about fourteen days, so if the first crescent appears on the first of the month, the full moon occurs on the fifteenth. For cultures using a lunar calendar the evening of the full moon remains important — for instance the Lantern Festival in China, and Passover in Judaism — and since such celebrations require agreement about which night to prepare for, it was important to be able to count down to the night of the full moon.
A vestige of such countdowns appears in the Roman calendar, which according to tradition was originally lunar. There was a countdown to the Ides in the middle of the month, which in a lunar calendar would be the date of the full moon, and in fact the whole calendar used countdowns: one to the Ides, one to the first day of the next month, called the Kalends (hence our term calendar); and one to a day called the Nones, eight days before the Ides. Although the Nones was eight days before the Ides, the Romans treated it as the ninth day before (hence the name), because when counting they always included the first and last days of the count. There was no concept of zero, and each countdown ended with a 1; thus the 'day before the Ides' (pridie idus) was preceded by the 'third day before the Ides' (ante diem tertium idus). Today countdowns end with zero, so for example we say, "Three, two, one, go", rather than "four, three, two, go".
Countdowns to the Kalends, Nones and Ides are now of no importance, but historians tell us that Julius Caesar was assassinated on the Ides of March, and in Act II of Shakespeare's play Brutus asks, "Is not tomorrow, boy, the Ides of March?" "I know not sir". "Look in the calendar and bring me word". Here is a picture of the Roman calendar as it would have been in 44 BC, the year of Caesar's death, following his great calendar reform two years earlier.
Each month starts with the Kalends (K). The Nones (NON) and Ides (EIDVS) are also marked, along with information about special days. The letters A to H in the first column of each month indicate that day's position in an eight-day cycle. This was the nunindal cycle, named after the number 9 because the last day of each cycle was the first day of the next one.
We call this 'inclusive counting', and the Greeks used it too. The New Testament of the Bible, written in Greek, states that after the crucifixion, Christ "rose on the third day", meaning two days later. Since the early Christian fathers reckoned the crucifixion to have taken place on a Friday, the third day was a Sunday, hence the choice of that day as the holy day of the Christian week.
When the Christian Church began to number the years of the Julian calendar they started with the year of Christ's birth, naming it year 1 AD (anno domini, year of the Lord). The preceding years were numbered using a countdown, ending with the year 1 BC (before Christ). There was no year zero.
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Counting preceded writing, and putting numbers in writing required some sort of shorthand — you would not want to make fifty marks to represent the number fifty, so consider how an ancient civilisation might write down numbers. Base ten is a natural method for oral counting because we have ten fingers (or eight fingers and two thumbs), so you make a mark for 1, two marks for 2, etc., and then have a special symbol for ten. Two of these make twenty, three make thirty, etc., and then you have another special symbol for one hundred, another for one thousand, and even ten thousand if the counting requires it. Combining as many of these as necessary you get any whole number, which is exactly what the Ancient Egyptians did. Here is an example of addition with the numbers written in Egyptian hieroglyphs.
The use of separate symbols for one, ten, hundred, thousand, etc. was perfectly natural, and the Romans supplemented this with special symbols for five, fifty, five-hundred, and so on. They also used an extra shorthand to write numbers such as four and nine: four was often written as IV, meaning one less than five, though it could also be written as IIII, as it is on many clock faces — London's famous Big Ben being a notable exception. Similarly nine could be written as VIIII or as IX, meaning one less than ten.
At one time the Greeks also had a system with special symbols for five, fifty and so on, though the exact symbols varied from one group of Greek islands to another. But in Classical times they adopted the alphabet from the Ancient Near East, along with the idea of using its letters for numbers. Alpha, beta, gamma, etc were used for 1, 2, 3, until they got to ten, which was iota[1]. For example ιδ meant 14 (10+4). After ten, the alphabet was used for twenty, thirty, forty, up to a hundred, then two hundred, three hundred, etc. Reaching 999 therefore required twenty-seven letters — nine for the units, nine for the tens (from ten to ninety), and nine for the hundreds (from one hundred to nine hundred). These twenty-seven letters comprised the usual twenty-four in the Greek alphabet, from alpha to omega, along with three obsolete ones.
In order to write the thousands from 1000 to 9000 they used the letters representing 1 to 9 with iota (ten) as a subscript or superscript. This gave all numbers from 1 to 9,999. For ten thousand they used the capital letter M, standing for the Greek word myriad. Numbers above ten thousand were written in two parts: the first representing a multiple of 10,000 (often written as M preceded by a number from 1 to 9,999), the second representing a number from 1 to 9,999. This gave any number up to eight digits, and mathematicians such as Apollonius and Archimedes devised methods for writing even larger numbers.
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Medieval Europe used Roman numerals, but in 1202 Leonardo of Pisa published his hugely influential Liber abaci (Book of Computation), introducing to Europe the place-value system we still use today. Leonardo's father, a wealthy merchant, appointed Consul for a community of merchants in the North African port of Bugia (now Béjaïa in Algeria), sent his talented son to study calculation with an Arab master. Leonardo later travelled to Egypt, Syria, Greece, Sicily and Provence where he studied various methods of computation. His book contained numerous computational exercises, all done using the system he had learned in the Arabic world, which he referred to as the modus indorum (method of the Indians). With just ten symbols 0, 1, 2, …, 9 you could write any whole number, the position of each digit determining whether it referred to units, tens, hundreds, thousands, etc. This is called the 'place-value' system, and although the symbols are written differently in Arabic, the principle is the same, and a single page of the book will give the idea of how Leonardo wrote them.
The numbers listed vertically in the red box on the right of the page are: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 337, the beginning of the well-known Fibonacci sequence, which the book introduces as the answer to a question about the breeding of rabbits. Each number is the sum of the preceding two: 3 is 1+2, 5 is 2+3, 8 is 3+5, 13 is 5+8, and so on. This sequence is ubiquitous in nature, from the arrangements of petals on a flower to the patterns on the surface of a pineapple, and the term Fibonacci is another name for Leonardo himself — filius bonacci (son of Bonacci).
The new system was a huge advance on what had gone before, and as the philosopher Alfred North Whitehead wrote in his 1911 book An Introduction to Mathematics:
By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties.
His term 'Arabic notation' meant what we now call Hindu-Arabic numerals, in other words the place-value system.
If the Arabs acquired their place-value system from India, did the Indians invent it, and if so, when? A lack of preserved texts makes this a tricky question to answer: stone inscriptions, and inscribed metal deed plates, exhibiting the system do not exist earlier than the second half of the first millennium AD, but in his essay on Indian Mathematics[2], Kim Plofker finds strong evidence from non-mathematical texts that the system existed by the third century AD. He even suggests it might go back to the first century AD, where a simile about "counting pits" used by merchants — one for units, one for tens, etc. — appears in writings by the Buddhist philosopher Vasumistra.
Buddhism started in India during the mid-first millennium BC, later passing to China, and communication between the two naturally prompts the question of whether the place-value system found its way to China, or possibly from China to India. Various forms of Chinese numerals appear on oracle bones from the fourteenth century BC, and later on coins from the sixth century BC, all operating to base 10 and showing whether a symbol represented units, tens, hundreds, etc.
A place-value system, however, arose later from the use of counting rods, laid out on a counting board to perform calculations. Texts from the Han dynasty (202 BC to AD 221) imply that these rods were already in use during the second century BC, and in later periods we find pictograms of them to represent different numerals. By the third century AD these had stabilised to the following forms.
Within each multi-digit number the vertical forms in the top row alternated with the horizontal forms in the bottom row, making it easy to distinguish between one digit and the next. For example five-thousand six-hundred eighty-two would be written
5 6 8 2
As another example, 506
would be written with no space between the digits because the first symbol cannot mean 50 since the lines are drawn vertically; it must mean 500, and this is effectively a place-value system with base 100. Eventually the circular symbol for zero came into use, appearing in print in the Shu Shu Chiu Chang (Mathematical Treatise in Nine Sections) in 1247 AD.
An intriguing feature of Chinese rod numerals was a distinction between positive and negative numbers. Negative ones were written in red, positive ones in black, a method that seems to have emerged from black and red counting rods, possibly as early as the second century BC[3]. The distinction between negative and positive numbers also appears in seventh century India in the work of Brahmagupta, and it was understood that the product of two negative numbers is positive.
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The Chinese use of counting rods on counting boards may have been the origin for their use of place-value, and gives a possible explanation for the place-value system in the Mayan civilisation of Central America. Their digits were written using bars and dots (rods and pebbles perhaps), though the system worked to base-twenty rather than base-ten, presumably because they used fingers and toes for counting. Mayan numbers therefore required nineteen digits instead of nine, along with a symbol or space for zero. The use of a specific symbol for zero goes back to the early Classic period (c. 250–900 AD), and Anna Blume[4] refers to explicit zeros on a stela dating to the mid-fourth century AD. Symbols for zero became standardised in the post-classic period, with the pictograph of a shell for zero, as in the table below.
When the Spanish conquistadors took control of Central America, their priests and bishops deliberately destroyed written documents, so our knowledge is limited and we now briefly return to Egypt, China and the Greco-Roman world to see how they wrote fractions, before going on to the great civilisation of Mesopotamia.
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Egypt used whole numbers and reciprocals of whole numbers, in other words 1/2, 1/3, 1/4, and so on. Apart from the fraction 2/3, the numerator always had to be 1. Here is how they wrote them:
Repeating a given reciprocal was not permitted, so 3 divided by 4 could not be written as three versions of ¼, but was represented as ½ + ¼. Dividing five loaves between four people was easy: each person got 1 loaf and ¼ of a loaf, but imagine dividing four loaves between five people. Mathematically speaking everyone gets 4/5 of a loaf, but as a practical matter you cannot cut five equal chunks from four loaves, and the Egyptian method gives an alternative way of writing the answer.
Take three of the loaves and divide each one in half, yielding six halves. Give one to each of the five people and then divide the sixth half into five equal pieces (yielding one-tenth of a loaf for each person). One loaf remains; divide it equally between the five people. Then everyone gets exactly the same: one-half, plus one-tenth, plus one-fifth of a loaf — in other words 1/2 + 1/5 + 1/10. This was the Egyptian answer to 4 divided by 5, clever in its way, but ill suited to more sophisticated mathematics.
Although the Egyptians did not use ratios of whole numbers, such as 4/5, the Chinese, the Indians, and the Greeks certainly did. These rational fractions ('rational' in the sense of ratio, not reasonableness) can also be used for approximations, such as 22/7 for the number π, but there is a simpler way. First make one estimate, then refine it with a better estimate just as we do with decimal fractions, working to the nearest tenth, then the nearest hundredth, thousandth, and so on. For technical purposes the Greeks did this, but in base-60 rather than base-10. In other words they worked to the nearest sixtieth, sixtieth-squared, sixtieth-cubed, and so on — rather than tenths, hundredths, thousandths, etc. — hence our use of hours, minutes and seconds. It is called the sexagesimal system rather than the decimal system (sexagesimal referring to sixtieths) and originated with the Babylonians, whose detailed astronomical data were used by early Greek astronomers, notably Hipparchos of Rhodes in the second century BC. His results were used and expanded in the second century AD by Ptolemy of Alexandria in the famous Syntaxis (Compendium), often referred to by its Arabic appellation as Ptolemy's Almagest.
To give one example from this work, the square root of 4,500 was approximated to 67 4' 55''[5], meaning 67 + 4/60 + 55/602. Details of how to calculate this square root are given by Theon[6] of Alexandria in the fourth century AD, and the result is good to three places of decimals. That may seem impressive, but the Babylonians themselves could work out square roots to even greater accuracy, two thousand years earlier, though their work was almost entirely lost, and we now turn to how it was rediscovered.
[1] this is now the ninth letter but they were using an older form that included a letter called digamma, between epsilon and zeta, as 6 [2] Kim Plofker, Mathematics in India, chapter in The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, ed. Victor J. Katz, Princeton Univ. Press 2009 [3] C. A. Ronan and J. Needham, The Shorter Science and Civilisation in China, Vol 2, p. 39 [4] A. Blume, Maya Concepts of Zero, Proceedings of the American Philosophical Society, Vol. 155, no. 1 (2011), pp. 51-88 (see p.58). [5] In Greek notation ξζ δ ́ νε´´, where ξ means 60, ζ means 7, δ means 4, ν means 50 and ε means 5. [6] Theon's daughter Hypatia is the first female mathematician whose life is reasonably well recorded. An astronomer and Neoplatonist philosopher, she was murdered by a mob of Christian monks in 415 AD.