Do you still count on your fingers?

I fell in love with numbers some time before I began going to school. I loved arithmetic from day one. I was one of those horrible people who were good at maths at school and actually enjoyed doing it, ending with me doing maths A-level. I studied maths at university to about BSc level before changing to  philosophy with an emphasis on history and philosophy of science, where I spent several years working in a research project into history of mathematical logic. During my student years, as a mature student, I worked as evening manager in a culture centre, where amongst other things I was responsible for stocking up the change in the admission cash boxes for various areas and then counting the takings at the end of the evening and preparing the payment for the bank. All of my life I have been very good at mental arithmetic and it should be clear from all of the above that I have spent a lot of time in my life calculating but at times I still find myself counting on my fingers to check something I’ve calculated in my head! 

Finger counting is one of the earliest forms of human calculation as Jessica Marie Otis tells us in her By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England.^[1] As soon as I read the title and description of this book, given my very strong interest in all aspects of mathematics in the Early Modern Period, I knew that I wanted to read it. I just hoped that my expectations would not prove excessive. Exactly the opposite occurred, my expectations were comprehensively exceeded by Jessica Otis’ excellent book, which instantly acquired a slot on my list of all time favourite history of maths books. 

The Introduction opens by explain the religion in the title, not a term one would normally expect in a history of arithmetic:

In an almanac written for the year 1659, George Wharton mentioned a biblical passage that would have been familiar to early modern “well-wishers,” “lovers,” and “students” of mathematics: “For indeed all things were made by God, in Numbers, Weights  and Measures.” An unknown reader found this passage so interesting that they underlined it in their copy. The implications of this simple statement were enormous. If numbers were indeed the building blocks of God’s Creation, then everything in the world could be understood through the use of numbers, particularly the concrete numerical exercises of weighing and measuring. Furthermore, a knowledge of numbers would be necessary for those who wished to study any aspect of God’s inherently numerical Creation. This biblical passage was not a common subject for Sunday sermons, and the Book of Wisdom in which it can be found is technically part of the Apocrypha. Nonetheless, Wisdom 11:21–God’s creation of the world by numbers, weight, and measure–formed a rallying cry for people who wished to encourage the wider study of mathematics among the general population of early modern England.

We normally think of the history of mathematics in terms of the big names and the major development in the abstract disciplines that cause so many school kids to moan and to ask, when will I ever need this in real life. Names such as al-Khwārizmī and Newton and the disciplines with which they are associated algebra and calculus. Otis’ book deals with that branch of mathematics that almost everybody needs and regularly uses in their everyday lives, simple arithmetic. She examines in depth the transformations that this discipline went through in England in the early modern period, placing those transformations firmly in their cultural, political, social, and religious contexts. This is contextual history at its finest. Otis encapsulates the transformation she describes in her book on the second page of the introduction:

This transformation in numerical practices was complex and wide-ranging in its impact on early modern society and thought. In part, it was a transformation in symbolic systems–the culturally agreed upon symbols and syntax used to represent numbers. It was also a transformation in mathematical education, enabled by increasing literacy rates, and the printing revolution. Most important, it was a transformation in technologies of knowledge, specifically the way the people of early modern England conceived of and used numbers in their daily lives. 

Having laid out the path that her research and book take, Otis delivers up six chapters each one of which deals with a different aspect of the modes of presentation, manipulation, methods of calculation of numbers and how they changed and why over the roughly two hundred years the book covers. 

Chapter one, “The Dyuers Wittes of Man: The Multiplicity and Materiality of Numbers,  delivers up the raw material, the presentation of numbers, which Otis introduces thus:

The main symbolic systems employed during the sixteenth and seventeenth centuries can be roughly divided into three categories based on their most prominent material characteristics: performative, object based, and written.

Performative is both spoken number words, children learning to count verbally intuitively and almost everyone expected to have a least a basic grasp of numbers and counting. The earlier mentioned finger counting is also performative, both in the simple variant, ten fingers-ten numbers, and the more complex systems using body parts to count up to twenty or complex figure manipulations to represent numbers up to one thousand. 

Object based can simple be using any objects in place of fingers–peas, stones, twigs–to count and manipulate numbers. However, it also cover the much more sophisticated tallying, with tally sticks, and the use of counting boards. These two methods dominated in the early part of the period covered and were still present at the end. Otis gives an in depth analysis of the uses of both forms of number manipulation and their significance in the period. 

There were two written systems, Roman and Arabic numerals. The former dominated at the beginning of the period but they were only used for recording but not for calculation. Otis explains that Arabic numerals were gradually introduced initially only for specific purposes but like the Roman numerals not used for calculations.

Having been introduced to the methods of numerations, in the second chapter, “Finding Out False Reckonings”: Thrust and the Function of Numbers, Otis takes us through the use of numbers in creating accounts, their principal use in the sixteenth century. We first learn about the problems of creating accounts that could be trusted and also could not be altered after the fact. Tally sticks were the only system that was safe from falsification and became dominant. Otis describes the various strategies employed to prevent people altering written accounts, whether in words or numerals. The least safe were those written in Arabic numerals, so they were, on the whole, avoided. We then get taken into the world of creating accounts, where the counting board was the dominant method, with the results of the calculation being mostly written in words or Roman numerals. Otis, however, explains that it was not unusual to find accounts with a mix of the use of the counting board, Roman numerals and Arabic numerals.

If you are doing your accounts using arithmetic then you first have to learn how to do it. Otis’ third chapter, “Set Them To the Cyphering Schoole”: Reading, Writing, and Arithmetical Education,  takes the reader on a journey through the world of teaching arithmetic in England in the sixteenth and seventeenth centuries. She describes the fascinating world of the printed, early modern, arithmetic textbooks. Their authors, their scope, their contents, and their methods and how these all change and evolved during the period. She also introduces us to the users of the textbooks through the analysis of their often extensive marginalia. Having learnt about the books, we now learn about the schools, starting  with the so-called petty schools, private institutions that taught young pupils the basics of writing and arithmetic and moving on to the grammar schools and apprenticeships for older children Apparently the grammar schools, which concentrated on Latin, Greek and reading classical literature often employed extra teachers to teach their pupils the basics of writing and arithmetic. Otis explains that can be assumed that apprentices were taught  the basics of writing and arithmetic as part of their training. She also briefly touches upon the lack of interest in mathematics at the Oxbridge universities.

All of the topics that Otis analyses in her, I repeat, excellent book interest me but I was particularly pleased by her fourth chapter, “According to Our Computations Here”: Quantifying Time. The main quantification of time analysed in detail in this chapter is days, months and years in the form of calendars and in particular the problems engendered by the calendar reform, which took place in the middle of the time period she covers. This was naturally very much a religious problem and more that justifies the religion in the book’s title. This is a topic that I have devoted much effort in studying and I can happily report that I learnt quite a lot of new things through the aspects of the topic that she chose to bring to the fore. 

However, in this chapter I also detected the only serious error in her book that I’m aware of. Concerning the date of the Feast of Easter, the principle cause or reason for the calendar reform she writes the following:

In AD 325, the Council of Nicaea had officially fixed Easter to the Sunday following the first full moon after the vernal equinox of March 21.^[2]

This is simply factually wrong, to quote Wikipedia^[3]:

In 325 an ecumenical council, the First Council of Nicaea, established two rules: independence from the Jewish calendar, and worldwide uniformity. However, it did not provide any explicit rules to determine that date, writing only “all our brethren in the East who formerly followed the custom of the Jews are henceforth to celebrate the said most sacred feast of Easter at the same time with the Romans and yourselves [the Church of Alexandria] and all those who have observed Easter from the beginning.

It is historically important that it did not provide any explicit rules to determine that date, as there followed a, at time bitter, dispute, largely between Alexandria and Rome, who employed different systems, over the correct way to determine the date. A dispute involving religion, astronomy, and mathematics and which was first settled by about the tenth century. 

In my opinion the key quote in this chapter is:

During the early modern period, temporal locations were defined using three different conceptions of time–linear, episodic, and cyclical. Linear time fixes a single important event as the center of its chronology and measures duration forward and backward from the event along an imagined time line. Episodic time is similar to linear time but relies on multiple events, which are usually of the same type, and reckons duration by each event only until the next event occurs. That next event then becomes the new chronological reference point, and the process repeats itself. Finally, cyclical time consists of an event or events that repeat in a regular pattern without a clear beginning or end. 

She notes that all three systems were used in conjunction with each other during this period and goes into a detailed analysis of the various and varied ways of dating that the people employed. I particularly like the emphasis that she gives to regnal calendars, that is dating according to the year of the reign of a specific monarch, a widespread dating system, right up to the modern period, that usually gets ignored in discussion on calendars and dating. 

She includes a long and detailed discussion of the problems engendered by the acceptance by some countries and rejection by other, in particular England, of the calendar reform and how various people, traders for example, dealt with the problem of multiple dates for the same day. A problem exacerbated by the use of different dates for the start of the year. For a nice example of this see my blog post on the dates of Newton’s birth and death. I found totally fascinating the use of fractional notation in dating to cope with the problems, such as “19/29 March 1638/9”,^[4] which was new to me. 

She devotes a fair amount of space in this chapter to the topic of almanacs, another favourite topic of mine, and the tables of calendars, important dates, regnal calendars, astronomical events and, and that they contained to save people the trouble of calculations and conversions. 

The chapter closes with a look at the introduction of the mechanical clock and with it a new type of clock time and the effects that this had. I warmly recommend the whole book, but I would also  recommend it for chapter four alone. 

In chapter five, “It is Oddes of Many to One”: Quantifying Chance and Risk, we turn away from the world of calendars, dates, and almanacs to address the early modern world of gaming, bets and odds. What we have here is the very gradual emergence of probability theory in gambling but still at the stage of guestimates, not mathematical calculations. Otis takes the readers through the complex and oft heated discussion on God’s providence contra natural order and fortune, which at times led to the condemnation of gaming by puritans. Leaving the world of gaming we enter the world of insurance also based on estimated rather than calculated odds. This progressed from marine insurance against losses at sea, through life insurance, the first lives insured being those of slaves, who were after all maritime trade goods, to fire insurance following the Great Fire of London. These early ventures into insurance being closer to wagers in gaming that anything mathematically calculated. However, towards the end of the period under discussion insurance based on statistical analysis of the mortality bills began to emerge, a process driven by the plague and the list of deaths it engendered.

Chapter six “David’s Arithmetick”; Quantifying the People, takes us further down the road of the newly crystalising statistical analysis. David is King David, who in the Bible conducted a census of his people, and censuses is the topic that open this chapter. After a brief survey of censuses from antiquity Otis takes us through the early use of them in early modern England, mainly for tax purposes, which made them unpopular. She follows the development of censuses to answer specific single questions leading up to the work of John Gaunt and William Petty, who founded modern demographic analysis, initially under Petty’s name Political Arithmetic. Here Otis’ journey through the story of numbers in early modern England ends. 

With academic books I usually comment on the academic apparatus, notes, bibliography, etc. The book has them in spades, the text is a mere 160 pages, attached to which are 100 pages of endnotes, bibliography and index. The endnotes are extensive, detailed and highly informative and definitely worth reading parallel to the main text. The forty page bibliography of primary and secondary sources is exhaustive and reading through it, I had the feeling that I would require at least ten years, if not considerably more, to read my way through all of the listed texts. A comprehensive index finishes off the whole.

There are only a handful of greyscale illustrations but it is not a topic that lends itself to lots of pictures. Otis writes fluidly with an easy to grasp style and I found myself ploughing through her excellent text smoothly at high speed, consumed by all the wonderful titbits it contains.  

My brief sketches of each chapter barely scratch the surface of the in depth research and detailed analysis that Otis brings to her topic. Research and analysis on the highest academic level but here presented in a way that the non-expert can read and comprehend with ease. 

In the Epilogue Otis summarises her own work perfectly:

Numbers were a socially pervasive technology of knowledge, with the power to shape modes of thought. As much as possible, By the Numbers has reconstructed a cognitive element of English culture by looking at not just how people used numbers, but how people though about numbers and how they used them to think about events in their day-to-day lives. In analysing numbers this way, it gives ordinary people a stake in philosophies and technologies of knowledge, and in doing so reconnects some of the revolutionary changes in early modern science with the changes occurring in everyday numeracy. This period witnessed not only the mathematization of elite natural philosophy but also the increasing use of numbers by ordinary men and women to interpret the world around them. Many later eighteenth-century developments in economics and politics, among other fields, had antecedents in early modern ways of thinking about numbers. The men and women of seventeenth-century England understood the potential of numbers as a technology of knowledge, even if they had not yet developed the mathematics to completely explain the world via numbers. 

I can only repeat what I’ve already said, this is a truly excellent piece of academic research brought skilfully and lucidly to paper. If you have any interest in numbers, their usage and the evolution of that usage over a wide range of fields in the early modern period then I can’t recommend this book highly enough. I would also recommend it, even for those more interested in the social, religious and political developments in early modern England for the contextual history of the period that it provides. 

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^[1] Jessica Marie Otis, By the Numbers: Numeracy, Religion, and the Quantitative Transformation of Early Modern England, OUP, 2024

^[2] Otis p. 93

^[3] I know one shouldn’t quote Wikipedia when criticising an academic work but I know that the Wikipedia statement is correct and it saves me having to search through my books on the topic looking for a passing quote:

^[4] Otis p. 97