Gunpowder and Geometry Read online

  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 aga
in, 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.

  What the subscribers to the Mensuration got for their money was a fat book which, despite a title that associated it with practical matters, was in fact a comprehensive treatment of theoretical as well as practical geometry – a substitute for Euclid’s Elements, indeed, as far as geometry was concerned. It began with definitions (‘A Line is a length conceived without breadth’), and it ran all the way up to the volumes of polyhedra and the areas contained under algebraically defined curves. Most of the book consisted of increasingly complex geometrical problems with rules for solving them: to find the area of a semicircle; to find the volume of a segment of a sphere; to find the surface area of a hyperboloid. Compared with the Guide, the emphasis was much less on carefully graded examples and much more on comprehensiveness, on a solid, gap-less treatment of a large body of material. It explained how to find the areas and the volumes of certain shapes and solids; how to construct certain curves and surfaces such as those arising from slicing a cone, or from rotating the conic sections that resulted (‘A conoid is a solid conceived to be generated from the revolution of the parabola or hyperbola about the transverse ax[is]’).

  There was elegance and beauty here; there was also a tremendous display of learning. Hutton regularly succumbed to the temptation to exhibit his own cleverness at the expense of relevance or logical structure. Some of the more advanced, more difficult or more important solutions were backed up by proofs given in footnotes – and sometimes the footnotes were long, elaborate, even showy. Attached to a problem about right-angled triangles he indulged in not merely a demonstration but a ‘General Scholium’ (the term irresistibly recalls Isaac Newton’s use of it in his magnum opus, the Principia Mathematica) with ‘some new theorems concerning the relations of the sides and angles of triangles’. These derived from infinite series – never-ending algebraic formulae – for the sine of an angle. A problem aimed at finding the circumference of a circle from its diameter and vice versa occasioned a note which set out the history of infinite series for the circular ratio from the seventeenth century onwards. Finding the length of part of an ellipse became the occasion for another, typical burst of complexity, with the full apparatus of geometry, algebra and calculus deployed to provide five different ways of solving the problem. Passages like this give an impression of extraordinary authorial dexterity: partly because they seem slightly out of place, obtruding upon the reader’s notice.

  So this book, like the Guide, carefully fashioned Hutton himself. As seen in the pages of the Mensuration, he was a technically dextrous, extremely well-read mathematician, who could quote thinkers of the calibre of Newton as easily as obscure practical manuals of gauging and surveying. He was entitled to expect a lot of his readers: introducing calculus without explanation, stating rules without proofs because they were ‘too evident’ to need it, demanding faultless skill in imagining three-dimensional shapes and their manipulations.

  But this was also someone who knew all about mathematical practice. The final section of the book turned to the practical applications of geometry, and worked through lengthy rules and examples for surveying, gauging liquid volumes, and measuring roofs, windows and chimneys. Hutton went to some trouble to emphasise his experience as a surveyor; in fact his first, last and only published survey – a map of Newcastle – appeared in the same year, 1770. He suggested improvements to practice, dismissed some instruments and praised others, and casually remarked on the best way of operating in certain situations. He urged t
he use of decimal, not duodecimal arithmetic by measurers, and he lambasted sellers of timber for their sharp practices, at rather petulant length (Hutton must have had a bad experience on this score; he wrote a letter to one of the Newcastle papers about it too).

  This practical section culminated with the imagined exercise of planning and building a house. With columns at the front door and over six hundred feet of plaster mouldings, maybe the imagined house, a smart, ambitious Georgian edifice, bore some resemblance to the real one Hutton had planned and built for himself on Westgate Street.

  Not everyone was convinced by Hutton’s posturing. There were some poor reviews in the London papers, by anonymous authors whose objections centred on his pretensions to practical knowledge. By and large, though, the reaction seems to have been positive. And the Mensuration stood the test of time; it would run to four editions, remaining in print until 1812. As late as 1830 a supporter would call it the work of a ‘masterly hand’, still ‘by far the best treatise on mensuration’ published in any country. One reader in Shropshire broke forth in verse to express his admiration:

  O Science! trade and commerce are thy end,

  By thee we import, and by thee we vend;

  By thee we build our houses, till our lands,

  And weigh and measure with unerring hands.

  What art or rules could never yet display,

  Nor all the rules of Science till this day

  Were able to disclose [by] genius’ force,

  Thy true-born son hath traced to the Source.

  Newcastle bridge in ruins.

  Hutton’s daily round of work must have been frenetic during these years in Newcastle around 1770. To teaching and administering a school, together with a continuing programme of reading and self-improvement, were now added the demands of writing and seeing through the press a steady flow of publications. Mathematics is hard to proof-read, and a geometry book like the Mensuration also required hundreds of diagrams to be commissioned and checked. A young Newcastle engraver named Thomas Bewick, subsequently a celebrity himself for – particularly – his engravings of birds, did some of his first work on the Mensuration, and he remembered that Hutton frequently came into his work room to inspect what he was doing. Despite Hutton’s increasingly assertive public persona, he found him ‘grave or shy’ in this private setting.