National Meteorological Library and Archive C_1825–1832. Open Government Licence v3.0.
The series continued in all for over forty years. After about 1850, though, the gaps due to illness became more frequent, and other hands than Molesworth’s filled in an increasing proportion of the entries. The latter volumes of the series gradually petered out, the last full observation being made early in October 1867. A final, terse note came on Wednesday, 9 October 1867: ‘rain’.
Molesworth’s tables of data and of number symbols were the outcome of a variety of processes: observation, measurement, counting and calculation. At the end of each year, she prepared summaries which relied particularly on both computation and counting. At the fullest, there was a summary for each month of the year, for which she counted the number of days on which rain had fallen and the number of days on which the wind had stood in each of eight directions, as well as computing the month’s total rainfall and the mean and ranges for the barometer and three thermometers. There was also a summary for each whole year, for which Molesworth provided a similar set of totals and means. On occasion she even prepared multi-year summaries. Francis Bacon would have been proud.
Molesworth did not just collect data; she also corresponded with a number of other weather observers and meteorologists, owned and read books on the subject, and published in periodicals some of her summaries of the weather of Cobham. On her death, her collection of dried plants was presented to the herbarium at the Royal Gardens at Kew, and her weather diaries were presented, at her request, to the Meteorological Society in London. In 1880, a volume of summaries extracted from them was published. Their editor picked up on the statistical side of the work, writing that
Miss Molesworth’s labours will not have been … useless if they add anything to the amount of information which we may look forward to from the careful observations now being carried on in the same field of research, promising – it is not too much to say – to be of infinite value to the country agriculturally, by showing the bearing of weather influences on the growth of our food crops.
Caroline Molesworth’s work illustrated both the pleasures and the frustrations of systematically observing the natural world. On numerous occasions, observations had to be reported as missing or questionable because of breakages or other difficulties among her instruments. Every trip away from home occasioned a gap in the series of observations that could never be made up or substituted for. And the very process of dividing winds into eight points of the compass, rounding temperatures and pressures to the nearest whole number, and so on, emphasised that turning observations into numbers must always involve simplification and, therefore, loss.
The process of aggregation at the end of each year possessed something of the same ambiguity. On the one hand, it promised to transform a series of isolated observations into something bigger; a series of atoms into a coherent picture. On the other hand, though, the summing and averaging effected a real loss of texture and granularity compared with the original observations. And even on the scale of a whole year, the result could still feel like a series of details rather than a description of a climate. In 1827, Caroline Molesworth counted one hundred and forty-one days with rain or snow or frost, twenty-three of them in December; forty-two days of northerly wind, eleven of them in September. There were fifty-six days of southerly wind, spread across every month of the year. The most common wind over the year was from the northwest … It fell to her posthumous editor to note certain relationships between the dates at which plants flowered and variations of temperature and rainfall from year to year. Perhaps it would not have been possible to turn the numbers into a bigger picture during Molesworth’s lifetime.
By the nineteenth century, and still more by the twentieth, Arabic numerals were everywhere, carried to most of the countries in the world, their family tree now dense with branches. Generations of textbooks along the lines of Bhaskara’s and Al-Khwarizmi’s presented ‘numeration’ as the reading, writing and visualising of Arabic numerals on dust boards, blackboards, parchment or paper. It had become common in ordinary speech and writing to say ‘numbers’ and mean the Arabic number symbols, as though the two were the same: as though there were no conceivable or at least worthwhile other way of representing numbers. (The neat way to expose the paradox here is to observe that many people would say 2,543 is a four-digit number, but no one would say Mary is a four-letter girl. Arabic numerals become identified with the thing they represent far more readily than words do.)
Meanwhile, the number concept itself was being steadily expanded beyond the ‘natural’ numbers possible with beads, fingers or tallies. Fractions and ratios had been written down in the ancient Near East; irrational numbers like certain square roots – which cannot be expressed by any fraction – were already of interest in ancient Greece. The many writers of ciphering books studied negative numbers, and Molesworth used them to record temperatures. In the context of mathematical research, ‘real’, ‘imaginary’ and ‘complex’ numbers would be defined during the nineteenth century, and in the twentieth, ‘hyperreal’ and ‘surreal’ numbers too. Much of this was supported by the power and flexibility of the Arabic number symbols, steadily extended using negative signs, decimal points and other devices.
This may seem like an end point; it may seem, even, as though the rise of the Arabic numerals is the story of counting. But it is in reality only one branch among many. Writing numbers down as symbols is only one of many ways of counting; and indeed the distinctive structure of the Arabic numerals is only one of several ways a set of number symbols can be organised.
INTERLUDE
Number symbols
As the other examples in this book illustrate, counting symbols have displayed a range of different systems at different times and places. The basic choice is between having one symbol repeated several times (|, ||, ||| …) and having a set of different symbols to symbolise the different numbers (α, β, γ, …). The first system is that of the simplest tallies, whether ancient or modern; the second that by which the books of Homer’s Iliad and Aristotle’s Metaphysics are numbered to this day. But as with counting words – and number symbols very often imitate their structure from the counting words in the language of their first users, albeit frequently with some tidying up – a system with no more structure than this quickly becomes unusable for larger counts.
If a number base is used – say, a special symbol for the number ten – it can be combined with the smaller numbers simply by juxtaposing it: XI meaning ten-and-one, for instance. For multiples of the base, the choice must then be made again, between simply repeating its symbol the required number of times, and adopting a whole series of symbols to denote the different multiples. In Roman numerals, for instance, ten, twenty and thirty may be shown by X, XX and XXX; in Greek by Ι, Κ and Λ. Another alternative is to reuse the original set of number symbols with some sort of modification: a, b, c becoming a', b', c', say, to show that they now mean ten, twenty and thirty. Or, finally, you can make no modification, but rely on the relative position of the symbols to show that some mean units and others mean multiples of the base. This last is the system of the Brahmi numerals that became the dust numerals, the Toledan numerals, and finally the Arabic numerals.
Each of these ways of structuring a system of number symbols has been used over long periods; each is capable of remaining stable over hundreds or even thousands of years: none has in fact any very strong tendency to evolve into – or be replaced by – any of the others. The advantages and disadvantages of different kinds of system depend very much on what you want to use it for. Some are quicker to learn, some quicker to write; some use a smaller set of symbols overall, others tend to represent any given number with a shorter string of symbols. Says Stephen Chrisomalis, historian of number symbols, ‘There is no ideal numerical notation system; rather, each system is shaped by a set of goals that its users and inventors seek to attain, and that they can achieve only by compromising on other factors.’
About a hundred different sets of number symbols have appeared throughout history; from the dawn of writing until about 1500 CE, their number steadily increased. But number symbols – like alphabets – have a marked tendency to be reused by more than one culture; the successful ones have been particularly successful at travelling, perhaps particularly successful as tools for communication between different communities, cultures, languages and places. The cuneiform numbers, the Egyptian demotic and Greek systems, the Roman numerals, and of course the Brahmi numerals all travelled widely, as did many more. The Arabic numerals were eventually carried around the world with – mainly – the European languages and cultures; other systems meanwhile became extinct. In the last five hundred years the number of distinct systems has fallen, on the whole.
As recently as the 1990s, it was not uncommon to hear it suggested that the decimal place-value system effectively represented the end of history as far as number representations, counting and arithmetic were concerned, no significant improvement being likely or even possible. Yet such certainties are no sooner stated than they begin to crumble. There have always been other ways to count, and there will always be other systems of number symbols. And there was an element of chance in the meteoric rise of the Arabic numerals: place-value systems have been invented at other times and places without overrunning the world. For the specific purpose of doing arithmetic in writing, the Arabic numerals have real advantages: but spare a moment to consider how long it takes most children to learn the forty-five addition facts and thirty-six multiplication facts in the base-10 tables, and how long to become fluent at even a limited set of operations on larger numbers: addition, subtraction, multiplication and division. (The extraction of roots, present in many medieval and Renaissance textbooks, seems to have largely dropped out of view in modern classrooms.) How easy it is to make a mistake, after all, despite the vaunted advantages of the system: and what a relief it is nowadays to outsource the whole thing to a calculator. Similar effort devoted to analogous tasks with different ways of representing numbers – the counting board, say – can make them, too, feel efficient and seem natural. Chrisomalis again: ‘We do not stand at the end point of a linear historical sequence, but in the midst of a branching and complex yet patterned and explainable world of written numbers.’
So, not an end point after all; and at most times and places the history of counting has not been written at all, but has involved other ways of keeping track. The drift of the last century has been that way even in the parts of the world dominated by Arabic numerals. In Molesworth’s own lifetime, there came the first hints that the era of performing large tabulations, calculations and even counts by hand, using Arabic numerals on paper, would not last for ever. Machines were beginning to appear that could perform some of those functions automatically, and whose descendants would eventually transform the way humans related to counts, numbers and data. Over the last few decades, the prevalence of number representations in the world has increased exponentially, but the overwhelming majority of them are not Arabic numerals on paper but something quite different: binary representations encoded in electrical impulses.
Another hugely important branch on the tree of counting must surely be the story of physical devices – manual, mechanical and eventually electrical and electronic – that assist with the counting process: the story, in other words, of counting machines. Like the Arabic numerals, they have swept the world, making their story into a global one; but nowhere is their deep history more richly recorded than in East Asia.
6
Machines that count
: Around East Asia
From African roots, the story of counting spread its branches through the Fertile Crescent, Europe and India. East Asia is home to another branch, one that has undergone its own evolution for millennia. Here the number words are also decimal for the most part, as in the large language families further west. Here number symbols are powerfully in play even in some of the earliest extant texts. But here counting also involves a distinctive set of counting devices. Everybody has seen an abacus – the suanpan in Mandarin, the soroban to the Japanese and the jupan to Koreans – whose operators became famous far beyond East Asia for the skill and speed with which they worked.
After the abacus, electrical machines. Right up to the 1950s it was a matter of comment that they worked more slowly than the most skilled abacus operators. But eventually the machines overtook the humans, and it is now in the form of electrical and magnetic signals that most of the world’s counting is done: the binary representations deep inside modern digital devices. This is a global story, with early technical innovations coming from the USA in particular. Today, East Asia is a leader in both the manufacture and the consumer uptake of the products that make the digital revolution.
And before the abacus? Before the abacus, East Asia was home to a different technology for recording and manipulating counts, whose operators also attained astounding dexterity and efficiency. It was in use for centuries, and the patterns it made provided the shapes of the classical Chinese number symbols. That technology was the counting rods.
Hong Gongshou: Counting with rods
Zhili, Qimen district, Huizhou prefecture, China. The fourteenth year of the Chongzhen emperor (1641 CE). Hong Gongshou is being assessed for tax liability.
He reports to the assessor his own age (forty-seven) and the ages of his brother, son and wife. The assessment document also records the size of his fields, from a survey carried out a decade earlier, and their liability for the summer and autumn taxes. It goes into detail about the size of eight plots of land including a paddy field and a patch of marsh, and it closes with a note that the family’s house is constructed of straw and possesses three rooms.
Hong’s summer tax is given as 6 dou 4 sheng 4 ge 1 shao of wheat; the autumn tax 1 shi 6 dou 3 sheng 9 ge of husked rice.
The same scene was played out millions of times each year across Ming China, as tax assessors tried to get a grip on who owed what under the country’s complex system of assessments. Revenues were raised across a land area of 4 million square kilometres, from the Pacific to the Himalayas, and from a population of more than 150 million; the total collected amounted to perhaps 5 or 10 per cent of the country’s agricultural output.
For administrative purposes, the vast country was divided into thirteen provinces, then into prefectures, sub-prefectures and counties. At the lowest level, groups of 110 households were organised into li, each under a rotating lichang (chief, alderman) who performed, among other things, the collection of taxes. The emperor controlled it all from the capital at Beijing, in a system which relied heavily – at least in theory – on his direct control. Some of the questions brought to his attention were very small: ‘the relocation of a particular business tax station, or from which productive area a particular county was to draw its salt supply, or how many rolls of silk were to be awarded to a foreign tributary mission’.
Such matters were passed upwards from the village level and assembled by a pyramid of officials in the prefectures and provinces, headed by the minister of revenue. Those same structures also organised the immense task of moving the goods – after collection – across the country to where they were distributed. For instance,
as late as 1578 the imperial university in Nanking still received 3,500 piculs of husked rice from Ch’ang-chou prefecture, 100 piculs of wheat from Ning-kuo prefecture and 100 piculs of green lentils from Ying-t’ien prefecture, all in South Chihli, and also over 20,000 catties of dried fish from Hukwang province, all these items being charged to the tax quotas of the respective districts.
As well as direct levies on agricultural output, there was an obligation to provide labour services and various specialised kinds of goods – office supplies, military equipment, medical supplies, even brooms to sweep the palace – from time to time. Furthermore, the assessments were adjusted based on a ten-yearly survey of land and an annual census of the number of people and amount of property in each household. Though the system was simple in theory, it inevitably tended to burgeon into increasing complexity, with surcharges and additions to cover the transport of the goods, and an ever-increasing willingness to substitute silver bullion for the set of items nominally due. This led to complex calculations, with outcomes that sometimes had to be specified in thousandths of an ounce of silver. A modern commentator suggests that even the simplest cases might each have taken an hour and a half to calculate, giving the example of ‘a labor service payment of 0.0147445814487 taels of silver on each picul of grain in basic land tax assessment’, a situation which ‘provided a paradise for lower-echelon tax collectors and bookkeepers’. Attempts at consolidation and simplification were made in the sixteenth century, but they were only ever implemented patchily.
Even if there was a tendency for some of this to become a mere formality, with survey and census numbers being carried over from one period to the next unchanged or only mechanically updated, nevertheless the administrative effort involved was huge, as was the volume of tax records being generated. By Hong Gongshou’s time some were complaining that the whole exercise was a waste of paper and writing brushes. Four copies of each tax assessment were made: one each for the county, prefecture, provincial and imperial governments. The last was the subject of an elaborate storage regime on an island on lake Hsuan Wu near Nanjing. By the 1640s there were 1.7 million volumes in seven hundred storage rooms, very probably the largest archive the world had yet seen.
The largest archive, and therefore the largest collection of number symbols. Chinese, then as now, had multiple ways of recording the outcome of a count or a calculation, and the surveys and censuses certainly reflected some of that variety.
