On 22 August 1704 James Stenning sat down in Ockley, Surrey, and began to write in a new exercise book. It had 34 leaves, and was about 16 cm by 20 cm: the size of a decent-sized hardback book, but slimmer and with only a paper cover.

"If a city being besiged hath provision for to maintain 1760 souldiers 10 months how many will it maintain for 16 months?"

"In a family Consisting of 7 Persons there are Drunk out 100 kilderkins of beer in 12 Dayes; how many kilderkins will be Drunk out in 8 Days by another family Consisting of 14 persons?"

James Stenning was twelve. His parents were in all probability farmers in what was basically an agricultural village.

James Stenning and the pound of Virginia tobacco.

He used his book to record his work on the subject of arithmetic over the remainder of the year 1704. He signed his name at the beginning and end, and gave a few other dates along the way, so that today we can chart his progress. This wasn't his first arithmetic book: he already knew how to add and subtract, multiply and divide.

The book would have cost something, since it has nice marbled covers at the front and back. Young James started off using it very neatly: he signed his name in various styles of calligraphy, and set out his mathematical rules and working quite beautifully for the first few pages. After that, things deteriorated a bit, and by the end of the book he was using it as much as anything for rough working and practice related to the exercises he was being asked to do.

"If 44 Acres of Grass be mowed by 8 men in 9 Days how many Acres shall be mowed by 24 men in 30 Dayes?"

So what was it like to learn arithmetic in eighteenth-century Britain? Perhaps the surprising thing is that we can find out at all. Some schoolchildren kept their exercise books and passed them on to their children and grandchildren, so that occasionally they turn up in book sales today. As we'll see, some people even kept using their school exercise books as adults: there are examples where the owner went back to the book as an adult and corrected some of the examples or added new ones. So it seems that this material could be really useful in everyday life in the eighteenth century.

What were they learning?

An eighteenth-century child would begin arithmetic by learning to read and write numbers. Numeration was sometimes called the first part of arithmetic (the other four were addition, subtraction, multiplication and division). Or it was sometimes counted as part of literacy, and taught and learned alongside reading and writing letters and words. Either way, most of those who learned to read — and some of those who didn't — learned to read Arabic numbers and understand how the place value system — units, tens, hundreds — works. And they went on through a study of arithmetic tailored to the systems of units for weights, measures, and money that were in use in eighteenth-century Britain. Even adding and subtracting could be quite tricky:

26 lb, 14 oz plus 24 lb, 13 oz.

If that was in metric units it would be easy. But here you have to know that there are sixteen ounces (oz) in a pound (lb). It took a while to learn all the information you needed about the different units: exercise books often contained tables of all the different weights and measures. And then it took a while again to learn how to manipulate actual numbers in those systems of weight and measure.

Multiplication and division came in if you wanted to know things like how many pence there are in twelve pounds and six shillings. If you got more advanced you might learn how to use multiplication and division to convert from one system of weights and measures to another. Or to work out problems like this one:

"If 1 Pound of Virgina Tobacco cost 10s 1/2d what Cost 3 hogsheads each weighing 17 cwt 2 qtr 12 lb?"

There are 12 pence (d) in a shilling (s) and 20 shillings in a pound. And there are 28 pounds (lb) in a quarter (qtr) and 4 quarters in a hundredweight (cwt). Can you do it? What strategies would you use?

That brings us on to the second big topic that James Stenning, and many like him, studied: the rule of three. What's that? At its simplest, it deals with questions posed in this form:

"If 4 yards of cloth cost 12 shillings, what will 6 yards cost at that rate?"

Today we might solve it using algebra, but there are other ways to approach it. Here's one

"Multiply the second number by the third, and divide the product thereof by the first … and the quotient thence arising is the fourth number … and is the number sought, or answer to the question."

That's from one of the popular textbooks of the time: Cocker's Arithmetic. James Stenning was using it: he copied some of his examples from it. And this is the rule Stenning used to solve this type of problem. It was called the golden rule or the rule of three. Let's apply it to our problem.

The second number is 12; the third is 6. Multiplying them together makes 72, and dividing by the first number – 4 – makes 18. So the answer is 18 shillings.

You can check, if you like, that this is the same answer you get by modern methods.

The rule of three only deals with one type of problem: problems where A is to B as C is to D, and we need to find D. If the problem is of a different type, you need a different rule. There was a whole series of different rules with different names. The rule of three inverse; the double rule of three; the compound rule of three. The rule of fellowship used a similar method to find out how the profits of a joint enterprise should be shared out to the investors; the rule of alligation showed how to find the correct price of a product made by mixing together two or more ingredients.

So, for example, if you have a problem like this one:

"A farmer mingled 20 bushels of wheat, at 5s. per bushel, and 36 bushels of rye, at 3s. per bushel, with 40 bushels of barley, at 2s per bushel; now I desire to know what one bushel of that mixture is worth?"

James Stenning and the besieged city.

Then what you need is the rule of alligation. It goes like this: add together the total value of the ingredients. And add together their total quantity. That gives you a price and a corresponding quantity; what we want is the price that corresponds to one unit in quantity. In the case of this example the total quantity is 20 + 36 + 40 = 96 bushels. And the total value is 100 + 108 + 80 = 288 shillings. Now we can use the rule of three. If 96 bushels of the mixture cost 288 shillings, what does one bushel cost? Multiply the second number (288) by the third (1), and divide by the first (96). The result is 3 shillings, the correct price for one bushel of this mixture of wheat, rye and barley.

It's an unfamiliar way of doing things, to us, but it has its advantages. You never have to learn algebra — which would have been using a hammer to crack a nut — and you never have to manipulate abstract symbols, just numbers. It helps you learn to think about and manipulate ratios. And it works.

For all those reasons, the rule of three and its relatives made up a long-standing tradition in school arithmetic, that lasted well into the second half of the twentieth century. The different rules were learned and applied by many generations of schoolchildren, like James Stenning.

His exercise book shows that he learned the rule of three in five different forms over the course of four months in 1704. By the end of the book he knew how to solve quite a wide variety of practical problems, from mowing grass and transporting weights to feeding armies and funding businesses.

What was it all for?

The problems we've seen give some ideas. Arithmetic can help you convert currency and keep track of money — your own or your family's. It can help you make sure you're being paid what you should be, and that you're not spending more than you can afford. The rule of three could be applied to practical questions in accounting, in trade, and in business or military life.

During the eighteenth century mathematics was coming to permeate more and more of the trades that a person might wish to train in. Sailors needed more mathematics if they wished to know how to navigate; merchants needed to perform ever more complex currency calculations on an increasing range of goods. The state was gathering more data and interpreting it in more complex ways. The Royal Military Academy employed two mathematical professors not just to teach mathematics to cadets but to do research, for instance on the mathematics of projectile motion.

The examples in exercise books and textbooks were often tailored to these kinds of practical uses. And they often sent a rather un-subtle message to the student. Consider this example:

"A goldsmith bought a wedge of gold which weighed 14 lb. 3 oz. 8 pw. for the sum of 514£. 4s. I demand what it stood him per ounce?"

(Gold was weighed using Troy weight, where there were 12 ounces (oz) in a pound (lb) and 20 pennyweight (pw) in an ounce.) The mathematics is quite simple, but the numbers are huge. Is there a hint here that if you learn your arithmetic properly, you'll grow rich? I think so.

And finally, where were children learning all this? James Stenning's village of Ockley had no school until he was 39, so he presumably learned his mathematics somewhere else: at home, perhaps, or at the home of a teacher, who might have been the local vicar.

Some part-time mathematics teachers took their role very seriously. There are some wonderful examples where a teacher has taken a printed book and added extra handwritten bits to it, to make a personalised version for a particular student. There are also lots of examples where a textbook has been written in: the examples worked out, mistakes corrected or notes added, by a student or a teacher. Sometimes it looks as if the book was being used to self-teach, without the benefit of a school or a teacher: the dedication that shows is quite moving, and it gives a real sense of how important some Georgian people thought it was to learn mathematics.

(They were right: in the 1730s Thomas Simpson, a weaver working in Spitalfields in East London, joined the Spitalfields mathematical society, a kind of self-help group where men of all social classes could learn and improve their mathematics. By 1743 he was Professor of Mathematics at the Royal Military Academy in Woolwich, one of the most prestigious mathematical jobs in the country. In the 1780s Mary Edwards could support herself after the untimely death of her husband because she was able to make calculations on behalf of Nevile Maskelyne's longitude project, becoming part of a network of so-called "calculators" that stretched across Britain.)

A hundred years on

Ann Mohun decorates her exercise book.

Let's look at another example, an exercise book that was used at a school. Ann Mohun learned her arithmetic about a hundred years later than James Stenning, in 1808. But she was learning the same kinds of things. Her book looks very different, though. What does it tell us?

Ann Mohun lived in the village of Ingleby in North Yorkshire, and she went to the school there. Teaching mathematics to girls became a lot more common during the eighteenth century, but Ann was still probably lucky to be taught as much as she was. Her exercise book included tables of the weights of wool and hay, practice in subtraction, multiplication and division, and exercises in the handling of dates and money.

"A comet will appear in the Year of our Lord one Thousand eight hundred eighty-eight, how many years to that time?"

"What come 10 Sheep to at £1, 17s, 2d per Sheep?"

We saw the messiness of the later pages of James Stenning's exercise book. Ann's showed a similar tension between neatness and mess. Some pages had neat headings, coloured in using watercolours, illustrations in multiple colours, and working set out the way a teacher or parent would have wished to see. Other pages were more chaotic. If she got bored, she doodled, drawing pictures of flowers and trees at the sides of the pages. If she had something to say to another student, she wrote it down, since it seems she wasn't allowed to speak in class:

"Run and Jump and talk alound in the fields but in the house and among your parents and friends Be quiet and Sedate [sit down] William."

And when the sums came out right she jubilantly wrote the answer across he page at an angle, making it stand out but contributing to the impression of wonderful disorder. Other students, and perhaps her siblings, wrote their names from time to time (maybe they were recording the fact that they had helped her with a particular example). It doesn't seem as if anyone was supervising her work very closely: by contrast with James Stenning's experience, perhaps.

The mathematics she learned hadn't changed much in the hundred years since James Stenning, though her exercise book was more focussed than his on learning different systems of weight and measure and performing tricky conversions between them.

"How many pence, shillings and Pounds are there in 17280 Farthings?"

"In 27 Acres, how many Roods and Perches?"

(There are 4 roods in an acre and 40 perches in a rood.)

But the uses she made of her mathematics might have changed since James Stenning's day. The world was quite a different place, and everyday finance and budgeting, for instance, had become more complicated as trade had increased and more goods were available at more prices. The world was a smaller place, too.

"How many Barley Corns will reach round the World, which is 360 Degrees, each Degree 69 Miles and a half?"

(There were supposed to be three barley-corns to an inch.)

Intriguingly, we know Ann kept this exercise book by her as an adult. She signed her name in it again after she had married — Ann Elcoat — and she made note of the places she lived: Darlington, and Stockton-on-Tees. She added a few notes on other subjects: there's something about a German dance called the polka. And she added a few corrections and extra notes to the mathematical examples, and even in a few places some new examples.

"How often will the wheel of a coach 17 feet circumference turn in 100 mile?"

So her exercise book wasn't just a family heirloom, or a reference book. It was a living exercise book, still used and added to year after it was first written. She passed it on to her descendants: a signature on one page calls it "Nancy Elcoat's book" — presumably her daughter. And it still exists today, so that we can know something of the world of mathematics in Georgian England.

Further reading

There's a great little book about English schoolbooks: John Dennis, Figuring it out: children's arithmetical manuscripts 1680-1880 (Huxley Scientific Press, 2012). For America there's also Nerida Ellerton and Ken Clements' Rewriting the history of school mathematics in North America 1607–1861 (Springer 2012). For more about everyday mathematics in Georgian England see Benjamin Wardhaugh's Poor Robin's Prophecies: A curious Almanac, and the everyday mathematics of Georgian Britain (Oxford, 2012). A sister article to this one has appeared in Convergence; it discusses how geometry was learned in Georgian England.

Problems

1. How many pence, shillings and Pounds are there in 17280 Farthings? (From Ann Mohun's exercise book, 1808.)

There are four farthings in a penny, twelve pence in a shilling, and twenty shillings in a pound.

Solution: 17,280 ÷ 4 = 4320, so 17,280 farthings make 4320 pence. 4320 ÷ 12 = 360, so 17,280 farthings make 360 shillings. 360 ÷ 20 = 18, so 17,280 farthings make 18 pounds.

2. How many Barley Corns will reach round the World, which is 360 Degrees, each Degree 69 Miles and a half? (From Ann Mohun's exercise book, 1808.)

There are three barleycorns in an inch, twelve inches in a foot, three feet in a yard, and 1760 yards in a mile.

Solution: 691/2 × 360 = 25,020, so the distance round the world is 25,020 miles … which is 25,020 × 1760 = 44,035,200 yards … which is 44,035,200 × 3 = 132,105,600 feet … which is 132,105,600 × 12 = 1,585,267,200 inches … which is 1,585,267,200 × 3 = 4,755,801,600 barleycorns.

3. If a city being besiged hath provision for to maintain 1760 souldiers 10 months how many will it maintain for 16 months? (From James Stenning's exercise book, 1704.)

Hint: first work out how many soldiers the city could feed for one month.

Solution: If the city can feed 1760 soldiers for 10 months, it can feed 1760 × 10 = 17,600 soldiers for one month. So it can feed 17,600 ÷ 16 = 1100 soldiers for 16 months.

4. In a family Consisting of 7 Persons there are Drunk out 2 kilderkins of beer in 12 Dayes; how many kilderkins will be Drunk out in 8 Days by another family Consisting of 14 persons? (From James Stenning's exercise book, 1704.)

A kilderkin is 18 gallons.

Hint: first work out how much beer is drunk by one person in one day.

Solution: If 7 people drink 2 kilderkins in 12 days, then each person drinks 2 ÷ 7 = 2/7 of a kilderkin in 12 days. So each person drinks 2/7 ÷ 12 = 1/42 of a kilderkin each day. So 14 people, in one day, would drink 1/42 × 14 = 1/3 of a kilderkin. And 14 people, in 8 days, would drink 1/3 × 8 = 8/3 or 22/3 kilderkins. Since a kilderkin is 18 gallons, 2/3 of a kilderkin is 12 gallons, and the answer is 48 gallons, or 2 kilderkins and 12 gallons.

About the author

Benjamin Wardhaugh is a writer and historian, based in Oxford. His research is mainly about the history of everyday mathematics, as well as the ways maths has been learned and used in the past. He's the author of How to Read Historical Mathematics, A Wealth of Numbers: An anthology of 500 years of popular mathematics writing and Poor Robin's Prophecies, mentioned above. And he was once a member of the NRICH team (Plus's sister site), back in 2000–2002.