Gunpowder and Geometry

There were three more children: Isabella, Camilla and Eleanor (known to her family as Ellen). They were baptised in the nonconformist chapel at Hanover Square.

At the time of writing Hanover Square is a demolition site tucked away near the line of the old town walls. Trains rattle past over the nearby viaduct, and it’s hard to get a sense of what was once Newcastle’s only open square. The chapel there was established in 1727; from 1767 it had a school, and by 1810, after some modification, it was large enough to hold an organ and six hundred people. Its congregation included prominent local poets, newspaper proprietors and politicians.

Hutton had remained a zealous Methodist for some time after his childhood conversion: one report says he wrote sermons and preached them, though if this is true they have – sadly – not survived. His connection with Hanover Square may mean he had now left behind a movement which at this date still aimed to reform the Church of England from within, not from without.

In fact the Hanover Square chapel would later acquire a reputation for Unitarianism, and Hutton’s presence there likely signals that he and his wife had come to be interested in more radical kinds of Protestant nonconformity, mixing in circles which questioned even such traditionally core doctrines as the Trinity, the atonement, and the divinity of Jesus. His private commitments are nowhere recorded, but a wider circle of Unitarians and those with radical sympathies – religious and political – would shape Hutton’s professional development long after he left Newcastle.

This could have had serious practical consequences. Probate courts, marriage, schools and universities all potentially discriminated against non-Anglicans, and the ill-named Toleration Act of 1688 specifically excluded deniers of the Trinity (as well as Catholics) from its provisions. Meanwhile the Blasphemy Act of 1698 threatened them with up to three years’ imprisonment and loss of civil rights. Enforcement was patchy, but the risk of penalties was real, as was that of the loss of friends. Hutton’s clerical benefactor Ivison is conspicuous by his absence from Hutton’s life after his move to Newcastle.

Work continued, and at a remarkable pace. Materially, Hutton was certainly prospering. His trajectory culminated for the moment with a final move to Westgate Street, one of Newcastle’s wealthier residential spots. He met the recurring problem of inadequate premises by acquiring a plot of land and building his own house and school. It was quite the elegant Georgian pile, with cellars and other conveniences. The Huttons could now avail themselves of all that the prosperous, growing city had to offer.

The old town walls (they had started to come down in 1763) enclosed an area of less than 200 acres, but those acres held the north of England’s capital and a good proportion of the northern counties’ population. There were tall elegant buildings and wide open spaces by the several churches. By the early 1770s the town had three hundred street lamps and a well-organised night watch. If the lower town tended to be smoky from house fires, and if the riverside was dominated by busy warehouses and the bustle of shipping on the Tyne (not to mention a growing concentration of poor tenements near the Black Gate), there were open fields just a little further up the hill, some under conversion into elegant pleasure gardens.

Newcastle in 1745.

The town offered the full range of mid-Georgian amenities. Subscription concerts, both at the assembly rooms and outdoors in the summer. Visits from famous musicians en route from London to Edinburgh, who would often stop and give a performance or two to offset the costs of travel. Charles Avison, Newcastle’s own resident composer and concert promoter, a man who enjoyed national fame. A literary club; theatres. Scientific lectures: Newcastle was the first provincial town to have them, with both local talents and the nationally famous visiting on tour. A popular press: for most of the century there were two weekly papers. Newcastle had local histories and local poets and balladeers. There were shops, inns, clubs, societies; fashion, food, wine.

Ultimately it all rested on coal; you could hardly forget that, if you lived anywhere near either the river or the coalfields, and Charles Hutton was never likely to forget it either. The sale of coal was so lucrative, contemporaries reckoned, that despite all the goods it imported Newcastle made a net gain every year, and had more money per head than anywhere else in the kingdom.

Mathematically, too, Newcastle and its environs were a rich world. There was William Emerson, a nationally famous mathematician who lived in nearby Hurworth. His studied eccentricity of manner and dress (home-made linen, big floppy hats and shapeless old jackets) earned him a local reputation as a wizard. Hutton corresponded with him and they became acquainted, though unsociable Emerson and ambitious Hutton did not really hit it off. There was John Fryer, who assisted Hutton with his surveying work; he was also Hutton’s teaching assistant at Westgate Street.

The school had its own separate entrance. Advertisements, briefer and more sober now, stated that there ‘Youth are qualified for the Army, Navy, Counting-house’; they could also be ‘compleatly instructed in the Theory and Practice of Land Surveying, with the use of the necessary Instruments’. Newcastle Grammar School took to sending its students to Hutton for specialist mathematics teaching. Not an altogether unusual arrangement – everyone knew that private academies did mathematics and science better than the grammar schools – but a gratifying endorsement of Hutton and his work.

His new-found middle-class status also meant that Hutton could work as a private tutor to the local gentry. As one biographer put it, Hutton’s ‘manners, as well as his talents’, now ‘rendered him acceptable’ in this role. Robert Shaftoe was one such patron, his home at Benwell Hall one of the more impressive local mansions (the Bobby Shaftoe of the popular song was a relative). Hutton’s tuition of his children impressed Shaftoe so much that he took to attending the lessons himself, revising the mathematics he had perhaps learnt at college. And he gave the young man the run of his impressive library. Newcastle had at least a couple of subscription libraries, and no shortage of booksellers, but access to a large, private book collection was a boon for Hutton, who remorselessly continued to improve himself. Over the years he added a reading knowledge of French, Italian and German to his early-acquired Latin. By 1772 he had read enough on geography to offer public lectures in the subject (to ‘gentlemen and ladies’) at half a guinea for the course. How many takers he found is not recorded.

Indeed, Hutton’s school was becoming something of a centre for learning. During the Christmas vacation of 1766–7 Hutton taught mathematics to other schoolteachers there, and at about the same time external lecturers began to use it as a venue. Caleb Rotherham covered geography, astronomy and other scientific subjects; likewise the popular James Ferguson. Ferguson was a house guest, and gave private performances to Hutton’s friends and family in the evenings, though Hutton was shocked when he discovered how little geometry the man knew. Hutton’s school was in a way a forerunner for the Literary and Philosophical Society that would be founded in Newcastle thirty years later, its first paid lecturer the same Caleb Rotherham.

His arrangement with Newcastle Grammar School brought Hutton himself two of his most celebrated pupils, and certainly his most colourful. John Scott was the son of a coal merchant; Bessie Surtees a wealthy banker’s daughter. Both went to Hutton for their mathematics lessons, but it was presumably not under his eye that their youthful romance blossomed. Scott went up to Cambridge in 1772, but the calls of love proved stronger and in November he came back and eloped (ladder, first-floor window) with Bessie. The scandal – or the romantic adventure, depending on your perspective – ran and ran through the nineteenth century, and Bessie Surtees merchandise is still to be had in her Newcastle home town.

3

Author

A country clergyman, anywhere in England, any time in the eighteenth century. At his desk, in his study. April. Open windows; sunlit air. An open manuscript of mathematical work. Solutions to puzzles from magazines, copied fair.

He copies this year’s set of solutions fairer still, adds a couple of suggestions for problems the magazine could ask this year. Mends his pen and adds a covering letter. Dear Sir. I enclose my mite for this year’s Diary, hoping you will find it worthy of notice. Your humble servant. Dusts the sheets, folds, seals.

A scene repeated many times – many hundreds of times – across Britain every year of the Georgian period. A few months later, in about October, the annual magazines went on sale: The Ladies’ Diary, The Gentleman’s Diary, The Mathematical Repository, and more. Some readers had the thrill of seeing their names, their mathematical work in print – a few won small prizes. Others looked in vain for their names, their work in the magazine, and concluded, humiliated, that their solutions had been wrong.

If Hutton had done no more than succeed as a provincial schoolteacher, his story would be a striking but not a very unusual one. The majority of mathematics teachers in Georgian Britain, indeed, were working-class lads who had made good: self-made men who had themselves attended private academies or bettered themselves by private reading. There were schools right across the United Kingdom that bore witness to their success in attracting students, providing them with high-level instruction and sending them out to work in the burgeoning literate and numerate trades.

Hutton was not satisfied with this. He wanted the wider recognition and the promise of greater rewards that publication would bring. And he approached publication through that remarkable Georgian institution, the mathematical periodical.

Now that they have disappeared, it’s hard even to imagine them, but in their heyday there were a dozen or so monthly or annual magazines whose purpose, or one of whose purposes, was to print mathematical problems and readers’ solutions to them. Construct a triangle given its base, one adjacent angle, and the line bisecting the opposite angle. Find a fraction with the property that, if you subtract its reciprocal, you get a square number. How many ways can you make fifteen from a pack of cards? This was not Sudoku, and it was not elementary arithmetic. The problems could be hard, using lots of algebra and geometry and sometimes even calculus. Since both problems and solutions were sent in by readers, an inevitable show-off effect meant that over time the problems tended to grow harder, the solutions more elaborate.

Despite the difficulty of the mathematics, the magazines sold plenty of copies: thousands, even tens of thousands. At any one time there were probably several hundred readers contributing their problems and solutions to them: it’s hard to say because, intriguingly, anonymity was the norm. Your name appeared in print only if you specifically said it was all right to print it. The writers were teachers, practitioners, gentlemen and women enthusiasts, schoolboys. They called themselves ‘philomaths’, and they loved mathematics for aesthetic and moral reasons as well as because it was, for many of them, a lifeline to a wider world of culture and ideas than they would ever reach otherwise: a world in which mathematical competence was everything. The austere language of mathematics was a very good place in which the shy, the modest and the provincial might both hide and shine, while allowing working-class and female mathematicians to contribute, perhaps anonymously, without being labelled. It enabled them to interact in a controlled way with people all across the country, to display what they were good at, improve their skills, lighten their countrified boredom.

Hutton approached this printed world under the tolerably transparent anagram of Mr Tonthu. Close to home, there was a mathematics column in the weekly Newcastle Courant, but he didn’t touch it. Instead, starting in December 1761, he sent in a string of able, elegant solutions to problems in Martin’s Magazine of the Sciences. (Benjamin Martin was a schoolmaster, lecturer, optician, seller of mathematical instruments, author, editor and tireless self-promoter: he issued his magazine monthly from 1754 to 1763.) He also appeared as the proposer of four questions of his own.

After two years Hutton/Tonthu became more ambitious: five of his solutions were printed in the much more prestigious Gentleman’s Diary, an annual compilation devoted to mathematical and other puzzles, and reputedly the home of the hardest of the philomaths’ problems. Hutton’s questions included equations to solve, geometrical constructions, and formulae for trigonometric expressions. Then he felt it was time for a fresh start under his own name, and the world heard no more of Mr Tonthu after 1763.

This time he aimed right at the centre of philomath culture: The Ladies’ Diary. Set up in the first decade of the century to contain charming little anagrams and easy mathematical problems in verse, the Diary had become under a series of editors the queen of the philomath journals, its four dozen pages containing – as well as an astronomical almanac for the year – problems as many and as hard as any of them. Some of the contributors were women, or pretended to be, but it had become primarily a place not to engage in genteel discussion of mathematics but to display high-level, up-to-date skills. Indeed, as an attempt to make mathematics a subject of polite public discourse it had by mid-century failed; like philomath culture as a whole it had become another instance of mathematics’ tendency to exclude and therefore to be anything but polite.

Visibility in this world was a prize worth having, and Hutton seized it. Over a decade from 1764 to 1773 he sent The Ladies’ Diary a total of fifty-seven correct solutions, of which twenty were printed in full. He answered the prize question correctly on five occasions: prizes were determined by lot and he won a total of twenty-eight free copies of the Diary for his pains. He also proposed four questions for solution by others, though two of them proved to be rather too hard and no correct solutions were received except his own. Still, if you were a British philomath in this period, you most certainly got to hear about Mr Charles Hutton of Newcastle.

Hutton would later write that an advantage of the mathematical correspondence promoted by The Ladies’ Diary and its sisters was that ‘considerable additions are made to the stock of mathematical learning in general, as well as to the particular knowledge of individuals’. Behind the scenes, he was finding other ways to add to his own stock of mathematical learning. Having already attended the schools of Hugh James in Newcastle and Mr Robson at Delaval, once in Newcastle he embarked on a systematic, historically motivated programme of mathematical reading, covering the Greeks, Romans, Spaniards, French and Germans as well as British mathematical writers.

During 1763 he distilled the fruit of his reading and his teaching experience – all six years of it – into a short textbook on arithmetic. The School-master’s Guide was published in Newcastle on 3 March 1764.

Its subject was just what Hutton had been teaching: elementary arithmetic, beginning with addition, subtraction, multiplication and division. The book continued with proportional reasoning in all its diversity and how to find square roots and cube roots. There was little more: one of the selling points of the book was its spare, uncluttered approach. Yet the careful control with which Hutton increased the complexity of his examples, and his penchant for introducing new tricks, rules or exceptions midway through what looked like routine series of examples, certainly kept things interesting. We gain a sense of what Hutton’s teaching was like in person: agile, thoughtful, tremendously well organised. Whatever exercise is being done, there’s always a slightly harder version of it just over the page.

Indeed, one of the reasons for the Guide’s success was the clarity with which it presented Hutton himself as a safe, sure, capable guide to the tricky territory of beginners’ arithmetic. Here was a man who loved calculation, who was almost preternaturally good at it. A man for whom common sense would unproblematically tell whether an answer was reasonable or not, for whom number sense was – as a matter of course – good enough to use obvious simplifications when the numbers in a calculation suggested them. For whom long division could be done largely in your head after a bit of practice: ‘when you are pretty ready in division, you may, even in the largest divisions, subtract each figure of the product as you produce it, and only write down the remainders.’

There were a few missteps in the Guide, indeed, when things were evidently clear to Hutton but he was unsuccessful in setting them out lucidly in words. Some of his attempts to give verbal equivalents of algebraic rules would have been scarcely comprehensible without the help of an able teacher. Some of his special tricks complicated more than they simplified: if a multiplier is itself a product, multiply by its factors separately. If it’s not a product, find a nearby number, multiply that, and then correct the answer by adding or subtracting.

But ultimately the aim of all his rules, tricks and practice examples was to impart to students something of his own feel for numbers, to help them develop a number sense and be able to select the right calculatory process even in an unfamiliar situation. And in that he appears to have succeeded.

Hutton moreover took pains to come across as a humane man, one who knew that children would get things wrong, that ‘calculations of the same accounts made at different times will sometimes differ’, that some pupils were simply not fitted for difficult calculation or found it off-putting. He drew on a wide range of personal knowledge to help the mathematics mean something to his students. Examples adopted almost every imaginable viewpoint: the workman who must get his quantities of material right; the factor who must manage multiple accounts dextrously; the substantial landowner who would redesign his bowling green or compute the value of his shipping interests discounted against time or loss.

Not surprisingly, it was the perspective of the merchant that returned again and again, and international trade was seldom far from view: 30 barrels of anchovies, 71 hundredweight of tobacco, 5 chests of sugar, 3 barrels of indigo. You can almost hear Hutton telling his students (and their parents): See how useful mathematics is, how rich it can make you, how much it can transform your life.

Writing a book was a much bigger step than sending in problems and solutions to magazines, and it demanded much more care. Blunders now would be costlier than wearing an embarrassing garment in a pit village. For the first edition Hutton paid for the printing himself, meaning that he alone bore the financial risk in case the book failed to sell. His patron Robert Shaftoe in fact contributed to the cost in return for the book’s dedication to him. The book was produced by a local print shop, with Hutton reputedly cutting his own type with a penknife when the shop didn’t have the fractions or algebraic characters the book needed.

The Guide was advertised in a number of newspapers, but of direct reaction there was practically none: no reviews, no comment in the press. It faced stiff competition. Even within Newcastle there were other mathematics textbooks being promoted, and other mathematical authors longer established and better known. The Banson dynasty, who dominated the city’s Free Writing School, had been publishing their own arithmetic books since 1709, most recently in 1760. Another northern author had an ‘easy introduction’ to mathematics out in 1763.

Despite that, the Guide found a market. We don’t know how many copies were printed, but a decent stock had sold out within a year or so, and Hutton managed to interest a London publisher in bringing out a second edition. This was good news, and ensured a much wider circulation for the book, this time at no financial risk to the author. His growing reputation was doing its work. By the time of the third edition, in 1771, the advert could say that the little book had ‘been found … useful in schools all over the kingdom’. The Guide, in fact, would run and run: it was still in print in the 1860s. The name of Charles Hutton was becoming harder and harder to avoid if you were interested in mathematics and its teaching.

Contacts in London made a huge difference. After the Guide Hutton devised a new, more ambitious publication project: a book on mensuration. This could have been a subject for another slim textbook on the model of The School-master’s Guide. But Hutton had something much grander in mind. Not a little book of practical rules but a veritable encyclopedia covering every aspect of geometry and its practical use. Hutton took to riding over to the village of Prudhoe at weekends to consult with the schoolmaster there, a Mr Young, who coached him in advanced geometry and mensuration and, it was said, worked over drafts of his new book with him.

Announced in the Newcastle papers in December 1767, Hutton’s Treatise on Mensuration appeared in twenty-eight instalments between March 1768 and November 1770. His publisher diligently promoted it in a range of national and local newspapers. Hutton undertook his own publicity campaign, writing personally to a long list of philomaths culled from The Ladies’ Diary and elsewhere. He obtained permission to dedicate the book to the Duke of Northumberland.

The results were spectacular. When the Mensuration appeared as a single collected volume at the end of 1770, the list of subscribers contained more than six hundred names. Probably amounting to more than half the active mathematicians and lovers of mathematics in the United Kingdom, from Penzance to Dundee, they included two dukes, one earl, and astronomers from Oxford University and the Royal Observatory. Both the English universities and most of the Scottish ones were represented, as were surveyors and instrument makers, schoolmasters and country curates, surgeons, excise officers and Fellows of the Royal Society.

This was self-publicity on a scale rare in Hutton’s century, or indeed in any. A good deal of money was involved – 600 subscriptions at fifteen shillings a book were not to be sneezed at – but the visibility had a value that could hardly be measured. It was rapidly becoming impossible to do mathematics, to like mathematics, to be aware of mathematics in Great Britain without knowing Charles Hutton’s name. Hutton was well advanced on the road from provincial schoolmaster with a taste for mathematical puzzles to national celebrity. He was aware of the change himself, of course. Throughout the 1760s he called himself ‘schoolmaster’ or ‘writing master’, but by the 1770s his title pages proclaimed him ‘author’; in 1772 he would switch to ‘mathematician’. And while Tonthu had been ‘of Newcastle’, Charles Hutton could call the town coldly ‘that part of the country in which I reside’, implying choice, impermanence, a lack of decisive ties to his provincial life.