Gunpowder and Geometry

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 the 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.

Hutton remained capable of a misstep. Late in 1771 the bridge at Newcastle was partly destroyed by a flood, massively disrupting travel in the city and the area. While an inquiry was taking place and before a replacement had been decided on, Hutton rushed into print – it’s hard to see how he got the text through the press as fast as he did – with a 102-page book on the theory of bridge building. It was an able, even a learned work, full of geometrical diagrams and cutting-edge mathematical analysis using some very clever calculus; the voice was recognisably the same self-confident mathematician of the Mensuration. The chief problem tackled was to find what shape the upper curve of a bridge should be for a given lower curve – or vice versa – in order to ensure its theoretical equilibrium. Hutton found some neat results and expressed a decided preference for a curve of his own invention, related to the catenary (the shape of a hanging rope or chain).

All this was very well, but as the reviewers were quick to point out, few people could have been less qualified than Charles Hutton to lay down strictures about how a bridge should be built. He had no practical knowledge of structural engineering, and his mathematical analyses both contained elementary technical errors and displayed total ignorance even of the actual shapes of bridges. His formulae produced absurd results, demanding a bridge of infinite thickness in some cases. ‘We are at a loss to account for the author’s design in presenting this work to the public,’ concluded one. If that stung, and if Hutton regretted his rush into print, his reputation by this point could stand the damage.

Another large project was already in hand, too: a cheeky scheme to reprint The Ladies’ Diary itself. What better way to appeal to the philomaths of Britain? Hutton proposed a five-volume collection featuring every puzzle, problem, question and answer that had appeared in the Diary since its foundation in 1704.

It was a finely judged plan. Older copies of the original Diary were hard to come by – some issues extremely so – and as long as the price was right most of the hundreds of British philomaths could be relied on to purchase this collected edition of their favourite reading. Yet it would involve very little work: simply transcribing and very lightly rearranging the text from seventy slim pamphlets. Hutton even engaged an assistant to do the actual editing. (George Coughron’s was an interesting story in itself, but a tragically short one. Son of a farmer, good at mathematics, he won a national mathematics competition sponsored by the British Oracle and moved to Newcastle, where he fell in with Hutton, perhaps a little in awe of the older man – he was twenty to Hutton’s thirty-five. Their collaboration was not to continue past the Ladies’ Diary project, and Coughron fell victim to smallpox in 1774.)

Hutton added more extra matter to his Diarian Miscellany than might have been expected. It was issued in quarterly ‘parts’, and to each he appended a selection of new mathematical essays and correspondence. In true philomath style he also included a set of questions each time: readers’ answers to be printed in the following issue.

Hutton’s associates Coughron and Fryer themselves made contributions, as did a range of big names in the philomath world and a number of other people who were probably Hutton himself: ‘Nauticus’, ‘Geometricus’, ‘Astronomicus’ and ‘Analyticus’. He felt confident enough to admit to some errors in his Mensuration. But the problems column never really took off, perhaps because the market was already so crowded.

Nevertheless, Hutton found plenty of buyers for the Miscellany: it came out in slim ‘parts’ between July 1771 and July 1775. His publisher once again pushed the project hard, taking out advertising space in the national papers for every one of the thirteen separate parts.

So good was the idea that someone tried to copy it, and Hutton found himself with a rival. Samuel Clarke launched a Diarian Repository in 1774, printing just the mathematical items from The Ladies’ Diary and leaving out the other puzzles, and there was an undignified flurry of adverts and counter-adverts in the press. Hutton thought the work defective and took to calling it the ‘Repository of Errors’, but he was badly rattled by the incident, and matters were not helped when a couple of the numbers of his Miscellany were delayed at sea by bad weather; printing in Newcastle for distribution from London had its perils. But Hutton won the day. Clarke’s Repository abruptly ceased to appear, some months before reaching completion.

When Hutton’s Miscellany was printed complete in 1775, the collection made a handsome set of five neat volumes for which the publisher asked one pound nine shillings. The new matter took up an extra volume by itself, and was separately titled – a little pompously – Miscellanea Mathematica. Well indexed, the Miscellany became for some decades the standard reference for The Ladies’ Diary and its problems and contributors, and it incidentally contributed much to the Diary’s accessibility to modern historians.

Up to this time – the early 1770s – it’s not clear that Hutton’s relentless self-promotion in print had any very definite object. There were a number of relatively prestigious mathematical appointments available in Great Britain, including teaching jobs at institutions like the Royal Mathematical School at Christ’s Hospital, or the public lecturing posts at Gresham College in London. The chairs at Oxford and Cambridge were not realistically open to someone without a university degree, but a position in one of the ancient Scottish universities might have been possible with the right use of Hutton’s personal network. If he explored any of these possibilities, no trace of it has survived.

What happened in fact was a little more surprising.

Hutton was in London on Mensuration business when he heard from a friend – Edward Williams of the Royal Artillery – that an unusual job was open. John Lodge Cowley, professor of mathematics at the Royal Military Academy at Woolwich, had retired due to declining health. The job attracted two hundred pounds a year and a house at the Academy. Plenty of people wanted it, and quite probably some of them knew some mathematics. The official in charge – George, Viscount Townshend, Master-General of the Ordnance – reckoned it would be a good idea to fill the post with a man of ability rather than a well-connected nonentity. He had taken the rare step of announcing a public competition, giving it to be understood that ‘merit alone’ would decide the result. Advertisements had appeared in the newspapers.

Williams urged Hutton to present himself as a candidate. Back in Newcastle his supporter Robert Shaftoe did the same, and Hutton let himself be persuaded. In May 1773 he travelled back to London to take part in the supposedly impartial competition.

Merit alone? ‘Merit is useless,’ wrote Horace Walpole, ‘it is interest alone that can push a man forward. By dint of interest one of my coach-horses might become poet-laureate, and the other, physician to the household.’ There were increasingly meaningful public examinations at the British universities during the eighteenth century, and it was not absolutely unknown for public appointments to be made on a competitive basis. Expectations about public life and the right use of patronage were changing, after all, but they were changing slowly.

So the candidates for Townshend’s ‘impartial’ examination turned up armed with the usual array of recommendations from noblemen and politicians: testimonials, promises of favour or reminders of favours owed. Hutton would later claim that he himself competed ‘without any interest’. But in reality he too took his precautions. He obtained a recommendation from his acquaintance the Duke of Northumberland, to whom he had dedicated his Mensuration. Northumberland was a prominent Tory, a colliery owner and a Fellow of the Royal Society, and he controlled seven votes in Parliament. Through Robert Shaftoe, Hutton also obtained a recommendation from the Earl of Sandwich, a Whig (the party was in power at the time) who had been First Lord of the Admiralty; to Master-General Townshend his was probably a louder voice than Northumberland’s. Having covered both political bases, Hutton finally equipped himself with a letter from the prominent northern mathematician William Emerson, testifying to his intellectual abilities. Thus protected, he faced the examiners.

Theirs were names he knew, but none was even an acquaintance. Three, however, were men you had almost certainly heard of if you had any contact with the British scientific and mathematical scene. Nevil Maskelyne was the Astronomer Royal; Samuel Horsley a member of the Council of the Royal Society; John Landen a highly regarded mathematical author and Fellow of the Royal Society. The fourth examiner, representing the military side, was Henry Watson, a military engineer with experience from Havana to Bengal as well as a contributor to The Ladies’ Diary and a friend of and literary executor to its former editor.

They started by examining the candidates viva voce. It was a stiff competition. Other candidates included Benjamin Donne, Master of Mechanics to the King, and Hugh Brown, translator of major works on military mathematics. (There were at least six candidates in all, but Hutton’s telling of the tale seems to have grown over the years. The obituaries have him facing down a field of nearly a dozen.)

The questions ranged across the entire field of mathematics, its history, what the best books were on certain subjects, and how best to teach it. When the first day was over the examiners handed the candidates a set of deliberately abstruse written problems in mathematics and natural philosophy, telling them to come back at the end of the week with whatever answers they could manage.

A persistent rumour – it was still circulating in 1825, after Hutton was dead – said that Hutton, away from his books and his friends, struggled with the written questions, and visited the Duke of Northumberland in London in some despondency. The Duke, so the tale goes, got the examiners to provide a new, fairer set of written questions. Even if that is true, it must have been a tough week, wrestling with the problems all day for several days before finally writing up whatever answers one had been able to obtain and turning them in at the end of the week. And Hutton must have spent a tense weekend in his London lodgings waiting for the result.

It came on Monday. The examiners reckoned that most of the candidates were sufficiently well qualified to do the job and had given complete satisfaction as far as the written and verbal questions were concerned. But they felt compelled to single out Hutton for particular recommendation, on account of the exceptional strength of his performance. The Board of Ordnance ratified their decision on 25 May. And so patronage and merit worked together (as they sometimes did) and Charles Hutton became Professor Hutton, of the Royal Military Academy at Woolwich.

For Hutton it would be the biggest single change since he left the coal pits. And it all happened with furious speed. In less than a fortnight the school in Westgate Street was to let; in six weeks it was taken by John Fryer, Hutton’s former assistant. He advertised that he intended to keep up the teaching ‘in the same Manner as practised by Mr Hutton’, and had hopes of retaining Hutton’s students as his own.

And Hutton himself, after a brief visit to put his affairs in order, was gone from Newcastle. He would never visit the region again.

The coach journey – three days and two nights – had nearly shaken him to pieces on previous trips, so this time at least he made the journey to London in the different style offered by sea travel. It might take a couple of weeks, and he perhaps reflected that this was the same route that took so much of Newcastle’s other produce to the capital. Almost a million tons of coal sailed from Tyne to Thames annually, the ships crowding the two rivers. In June 1773, so did Charles Hutton.

4

Professor

Woolwich. South of the Thames, east of the City of London. The ‘Warren’. Rabbits were bred there in the Middle Ages, and by 1773 it’s an apt name once again. The teeming site is now Britain’s biggest munitions manufactory, its largest ordnance store.

Five hundred people working in a hundred acres. The closest thing in the Georgian world to a modern factory. Warehouses, workshops, laboratories, furnaces. A canal, a barracks, a parade ground. The Royal Artillery is quartered there; so is its cadet company. So is its band.

It’s loud; it’s dirty, dusty, smoky. The smells of gunpowder, its smoke and its ingredients hang heavy in the air: sulphur, saltpetre and charcoal. A harsh, confusing landscape, security-conscious and alienating to outsiders. Not so different from the collieries, perhaps.

There were two sides to the change in Charles Hutton’s status. He was no longer a member of that despised class the provincial schoolmaster, with a merely de facto high standing among British philomaths. His status was no longer based solely on his own furious efforts at self-promotion. He was no longer vulnerable to catastrophe the moment the fashion in Newcastle schools changed or a rival published a cheaper textbook. He was now a man who had received public recognition for his talents and his hard work, in the form of a teaching appointment in an institution of national significance.

On the other hand, Hutton had left a situation as absolute lord and master of a thriving business and had become an employee. He had changed from headmaster to ordinary teacher, and, what was more, he was becoming a civilian employee in a military establishment. He would have to work with a new set of colleagues and teach a new kind of pupil. He would have been less than human had he not experienced moments of doubt about what he was getting into during the summer of 1773.

First impressions were certainly discouraging, and in some ways Hutton retained mixed feelings about his new situation for years, as his letters show. The Royal Military Academy occupied a few rooms in the bustling military installation at Woolwich, downstream from the city of London and on the opposite side of the river. Docklands territory today, it was then barracks, ordnance factory, munitions depot and more. The dockyard went back to Henry VIII’s time, ordnance testing over a century. There was a laboratory for making gunpowder and a foundry for making ordnance and shot. By the mid-eighteenth century the ever-increasing collection of buildings and activities was straining the limits of possibility. Laboratory, arsenal stores and ordnance testing were much too close together, and from time to time there were fires or unplanned explosions. The latrines stank. There was too little water to go around. The four battalions of the Royal Artillery would move out of the site in 1777, but the Academy – despite sporadic complaints and a serious attempt to find a new site in the 1780s – would remain at the Warren until the next century. Contemporary guidebooks tried to talk the place up, with varying success:

The Royal Military Academy at Woolwich.

In the warren, or park, where trial is made of great guns and mortars, there are some thousand pieces of ordnance for ships and batteries; with a prodigious number of shot, shells, and grenadoes, heaped in large piles of various forms, and which have a very striking and pleasing appearance.

Even today, with the mounds of weaponry gone and the site sleepy, and with a good map in your hand, it takes an effort to find the building in which the Royal Military Academy held its schoolroom. The two imposing floors by Vanbrugh had windows east and west to let the light all the way through, with one big high-ceilinged room on the top floors. There were also a barracks for the cadets and dedicated houses for the two principal masters, of whom Hutton was now number two. His house needed ten pounds’ worth of repairs, hastily carried out during the summer he arrived.

The human landscape was not much more encouraging. Hutton’s family didn’t at first accompany him to Woolwich, and his new colleagues were a somewhat sorry crew. The Academy had existed for over fifty years, with a formal king’s warrant since 1741, but it was proving hard to get the setup right. Reform after reform had taken place. Initially it was meant as a sort of regimental university, providing lectures to anyone in the Royal Artillery who chose to attend: officers, NCOs, cadets, all those associated with the regiment. A series of mid-century reforms had given it a more manageable role as a sort of superior school for the Royal Artillery cadet corps. There was now an inspector-of-studies, with military rank, to look after the daily running of the Academy, to oversee the curriculum and make sure it was being delivered. Above him stood the Lieutenant-Governor, who reported to the Board of Ordnance and its Master-General. (The Ordnance, by the way, was not strictly speaking part of the Army: it was structured and governed separately, which goes some way to account for the uniqueness of its Academy.)