Gunpowder and Geometry: The Life of Charles Hutton, Pit Boy, Mathematician and Scientific Rebel. Benjamin Wardhaugh
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