**a guest post by Ray Schrire**

It is time for an unofficial Crocodile Mystery.

*Humphrey Walcot’s grocery bill. Folger, L.f.196*

These are a few of my favorite items from the merchant Humphrey Walcot’s shopping list of May 8, 1601 (a bill I picked up semi-randomly from the Folger’s digital image collection):

*One pound and a quarter cherries 3 shillings 4 pence*

*2 pounds three quarters damsons 7 shillings 4 pence*

*One pound and a half gooseberries 4 shillings*

*One pound barberies 2 shillings 8 pence*

*2 pounds and a quarter pear plums 6 shillings*

*4 pounds paste of plums 16 shillings*

*4 pounds candied spices 24 shillings*

*5 pounds and a half marmalade* [yummy!] *7 shillings 4 pence*

*3 pounds ginger 5 shillings 6 pence*

*One pound ambergris 5 shillings*

*4 marchpanes* [a thin waffle with decorative marzipan on top] *13 shillings 4 pence*

*3 marchpanes 15 shillings* [how is this a good deal?]

*Half a biscuit bread 6 pence*

*28 dozen glass plates 14 shillings*

So, how much did all these items come to? Or here is an easier one, how much does half a pound of cherries cost?

Fear not, dear not so mathematically inclined *Collation* reader, these questions are not the real mystery. The real one is this: how did Humphrey Walcot (or more likely one of his stewards) reckon these numbers at the turn of the seventeenth century?

To be sure, while Walcot’s list is quite unique for some of its items—whatever did he do with so much plum paste?—writing lists of items and prices was very common. The Folger’s digital image collection is packed with accounts, bills, and receipts, demonstrating how people from all walks of life needed to do things with numbers every day. Given the folding marks, holes, and dirty eighth of a page, it is easy to imagine Walcot’s bill filed on a thread in some merchant’s office as he was engrossed in bookkeeping activities (thank you Heather Wolfe for walking me through this!). So a more general form of this mystery is: how did early-modern merchants, housewives, and apprentices add numbers, convert currencies (at least some of the items on Walcot’s list needed to come from afar), or divide a sum of money in order to know if they had found a good bargain on their pear-plums?

In England circa 1600 there were several quite different ways in which the numbers on such a bill could have been calculated; however, two methods were more likely than others: using counters or by pen. The former is a method of calculation requiring the manipulation of tokens (known as counters) on a table, placed in lines to represent different, usually decimal, orders of magnitude (1s, 10s, 100s, etc.). The latter is a method of calculation involving the manipulation of written numbers, much in the same way taught today in most Western schools. As we can see from the title of the anonymously written, *An Introduction for to Learn to Reckon with the Pen & with the Counter*, these two methods were the subject of many vernacular arithmetic textbooks, which in the sixteenth and seventeenth centuries were readily available from the London presses.^{1}

Before we try to solve our mystery by dealing with both these methods, we should not miss the important clue hidden in the title of this popular textbook: namely, early moderns usually learned to reckon numbers *with* something. This is important since such a tangible approach to numbers seems to be at odds with most accounts of early modern mathematics.

The standard narrative for the development of mathematics underscores the Renaissance as a transitional period in which European mathematics finally became a truly abstract field of knowledge (in fact, we might say that since the Renaissance, mathematics became the paradigm for an abstract field of knowledge). There is no denying that the early modern was a watershed period in the history of mathematics, with important developments in number theory, algebra breaking free from its subordination to the concrete spaces of geometry, and geometry going through a revolution due to Descartes and Fermat, which paved the way for the development of modern calculus by Newton and Leibniz. And in face of all these lofty theoretical developments, there were merchants such as Humphrey Walcot who learned to manipulate numbers with material instruments, be it counters or pens.

Indeed, while the differences between reckoning numbers with counters or with the pen are related to some of these high intellectual shifts, the larger point here is that the history of popular arithmetic is not simply derived from the history of mathematics. (Walcot probably wouldn’t have cared that just a few decades before he bought three pounds of ginger, the solution to the quartic equation had been published by Gerolamo Cardano in what is one of the most important milestones in the development of algebra since antiquity.) While the history of mathematics is an established field of inquiry with a long tradition, endowed chairs, textbooks, courses, and journals, the history of popular arithmetic is still a mostly obscure field of study.

So how did Walcot perform the necessary arithmetic required for his grocery shopping? On first inspection, judging by the numerals used in the bill, the use of counters seems the more likely option. Since the use of counters in early modernity justifies its own *Collation* post, I will not discuss it at length now, but will only say that Roman numerals evolved in close relationship to the counters (or the abacus, which is mathematically, but not materially, the same thing). “The Roman numeral system […] is simply the easiest way to record the result of an abacus operation,” writes Reviel Netz in his thought provoking essay, “Counter Culture.”^{2} Reckoning with tokens on a table (or with beads strung in an abacus) doesn’t leave much of a paper trail. For this reason, paying careful attention to numerals is one of the best ways for historians to uncover past arithmetical practice. In this sense, the Roman numerals used to note the sums of each item on Walcot’s bill seem to point to the counters as the medium of calculation.

More careful attention to the numerals in Walcot’s bill, however, suggests that the counters method was probably not the one used for this calculation. “Now thou art an O without a figure. I am better than thou art now. I am a Fool. Thou art nothing,” Lear’s fool famously celebrated the arrival of the zero on the arithmetic stage (1.4.197-199). Humphrey Walcot’s bill commemorates the same event in much less poetic terms, with a Hindu-Arabic zero creeping in among the Roman numerals.

A mishmash of Roman numerals with Hindu-Arabic numerals was not uncommon in the sixteenth century. This indicates the extent to which both methods of numeration, and perhaps both methods of calculation, coexisted in early modernity. Indeed, this blend of numerals is a useful warning against simply inferring from the type of numerals the method of calculation. (After all, we still today sometimes use Roman numerals, but most of us don’t use coins to reckon our bills).

While using a (Roman) *i* instead of a (Hindu-Arabic) *1* doesn’t change the arithmetic, using a zero does. On the counters you always count something (tokens, beads, stones) and there is no way of representing what isn’t there. For this reason, Roman numerals don’t have a sign for zero. On the other hand, in the pen method zeros play an important part, since they help remind the user of an order of magnitude when adding numbers from top to bottom. Indeed, offering further conformation of the hypothesis that Humphrey Walcot’s bill was reckoned with the pen rather than with counters is the second handwriting seen on it. This second hand was perhaps proofing the numbers written down by the steward, converting the Roman numerals to Hindu-Arabic ones.

And here we arrive at the heart of the mystery: if Humphrey Walcot’s bill was calculated with the pen, where is the paper involved in the calculation? This is a general problem for the history of popular arithmetic: the paper used in making the calculation is by and large missing from our collections. We know the pen required paper. In fact, if we follow the examples in printed textbooks and manuscript notebooks, early-modern arithmetic needed more paper than we use today (for an example, look at this *Collation* post about Sarah Cole’s arithmetic notebook). We even encounter the odd sheet of paper used for calculations in the flyleaves of books.

But what we don’t find are sheets and sheets of raw calculation. When I tried to calculate the sum of Walcot’s bill according the method detailed in *An Introduction for to Lerne to Recken with the Pen, or with the Counters*, I ended up using half a sheet of paper (and a bit less than half a sheet to calculate how much was half a pound of cherries). And if this is only one bill, we can just imagine the sheer volume of paper needed by a professional merchant who reckoned his yearly incomes with pen in hand. We might think that somewhere, in some collection, sheets and sheets of paper full of boring calculations are stored, but to the best of my knowledge, this isn’t the case (and if you think otherwise, please do drop me a line with a shelf number!). If ours was a crime mystery, we would still be lacking the murder weapon.

No doubt much of the paper used for calculation hasn’t come down to us simply because it was discarded. Paper full of the raw calculations of last year’s shopping is of very little use. But perhaps there is another way out of this mystery: erasable writing surfaces. After all, allegorized personifications of *arithmetica* are often depicted with an erasable writing surface in hand.

There are also visual and textual clues to suggest that chalk was used sometimes to calculate bills directly on the table in the pen method. But marks of chalk on a table, like the movement of tokens, is not something historians can study directly.

Luckily, there is one type of erasable surface that did make its way into our collections. A few years ago, Peter Stallybrass, Roger Chartier, J. Franklin Mowery, and Heather Wolfe argued that when Hamlet stated that “from the table of my memory / I’ll wipe away all trivial fond records” (1.5.105-106), he was actually referring to writing tables—an erasable writing technology which came in from the Low Countries and became a best-selling product in seventeenth-century London.^{3} These small writing tables were made from ass skin or from treated paper and were intended to be used with a stylus, pencil or pen. They were often bound together with printed calendars, pictures of gold coins, and lists of distances between various cities. As this printed material suggests, the intended clients of writing tables were merchants and shopkeepers, just like Humphrey Walcot.

So what kind of trivial records do we find on erasable writing tables, and more importantly, do we encounter actual raw calculations? I was lucky enough to be granted a Folger Research Fellowship to examine the surviving erasable writing tables up close and study their contents (that is, if there is any content that was not wiped away clean). While the full findings of this survey are still on their way, preliminarily results do indeed suggest that one of the uses of erasable writing tables was to perform arithmetic calculation in the pen method.

The Folger’s V.a.480, for example, is a writing table bound in a block-stamped and gilt calf-skin binding with a stylus used to lock the clasps together. Its slightly yellow erasable tables are covered with notes in Italian as well as with arithmetical problems, dating to the seventeenth and eighteenth centuries. The arithmetic involves the addition and multiplication of rather large numbers, leading, understandably, to some minor mistakes. (The pen method, as we all remember from school, was not foolproof).

Another example of erasable tables covered with calculations—an example that brings us much closer to Humphrey Walcot’s bill—is Harvard’s Houghton Library, STC 26049.8. This copy of *Writing tables with a kalender for xxiiii yeres*, bound in a tooled brown calfskin binding typical of the English stationer and binder Frank Adams, includes 20 erasable writing sheets. Many of these sheets are currently empty, but some contain what at first seem like addition of numbers that don’t quite add up.

These numbers, however, begin to make sense when we see that they are not pure numbers as much as they are sums of money, written in pounds, shillings, and pence. (Hence: it is not that 1106+296 are equal to 400, it is that 1 pound, 10 shillings, and 6 pence added to 2 pounds, 9 shillings, and 6 pence are equal to 4 pounds, 0 shillings, and 0 pence). Could it be that Humphrey Walcot used such writing tables when he tried to calculate his expenses?

Humphrey Walcot’s bill opens a window onto a world of early modern quotidian mathematical practice. In the sixteenth and seventeenth centuries—as England’s role in global trade grew—merchants, housewives, shopkeepers, seamen, and numerous others increasingly relied on numbers to do their daily business. Yet as Walcot’s bill demonstrates, there are many aspects of this activity still unknown to us. Scholars still don’t fully understand the materiality of daily arithmetic, the use of different calculation techniques and numerals and the relationship of these activities to developments in higher mathematics. The social history of early modern popular numeracy has not yet been fully written (was it Humphrey Walcot who reckoned the bill? His steward? His wife?). And neither are the ways in which the daily arithmetic practices of men and women like Humphrey Walcot fed into the larger stories of early modern economic history, such as the rise of credit, commerce, capitalism, and indeed, slavery and colonialism. The history of popular numeracy no doubt entails many methodological pitfalls, but it also has the potential to add a vital layer to our accounts of early modernity.

Oh, those items came to 6 pounds, 6 shillings, and half a pound of cherries cost 1 shilling, 4 pence.

**Dr. Ray Schrire** is a Fellow at The Polonsky Academy for Advanced Study in the Humanities and Social Sciences at The Van Leer Jerusalem Institute. His work brings together cognitive science and book history in order to uncover how early moderns thought while facing challenges such as learning Latin in grammar schools, reckoning their shopping bills, or taking note of the weather.

- Jessica Otis, “’Set them to the Cyphering Schoole’: Reading, Writing, and Arithmetical Education, circa 1540–1700,”
*Journal of British Studies*56, no. 3 (2017): 453-482. - Reviel Netz, “Counter Culture: Towards a History of Greek Numeracy,”
*History of Science*40, no. 3 (2002): 327. - Peter Stallybrass, Roger Chartier, J. Franklin Mowery, and Heather Wolfe, “Hamlet’s Tables and the Technologies of Writing in Renaissance England,”
*Shakespeare Quarterly*55, no. 4 (2004): 379-419.

It is worth remembering, too, that pounds, shilling, and pence were the standard currency units in Great Britain until decimalisation in 1971, and the mental arithmetic required to operate in the base 12 of pence, the base 20 of shillings, was standard arithmetic for school children learning their numbers. The calculation of Lsd did not require any more special equipment than basic arithmetic requires today in a base 10 system. Merchants on the market could perform these calculations rapidly and without paper or error: it just takes practice!

Ray, you might want to check out the Bagot papers. Walter Bagot seems to have been pretty bad at maths, so the address leaves of letters he received often contain calculations, not all of which add up. A couple of particularly interesting examples that I can remember off the top of my head are L.a.536 1v (in which, unless I’m very much mistaken, he gets his sums wrong – not for the first time either), as well as L.a.490 1v and 2r, where he appears to have set himself actual maths problems involving the price of eggs and sheep. In one case he seems to have got to the right result after several attempts, in the other his attempt at a solution seems a bit… misguided.