Gunpowder and Geometry Read online

  It was said, too, that there was no water on the site, but Hutton obtained a set of iron boring rods and by drilling down fifty feet found a spring and dug wells. All this cost money, and Maskelyne lent him a hundred pounds in August 1793.

  There was a certain, perhaps inevitable eccentricity to what resulted. Echoing the geometrical folly that was Severndroog Castle, Hutton built for himself and his family what came to be known as the Cube House. Its foundation was a forty-foot square, and it was forty feet high. It was not wholly austere, however: Ionic pilasters and bow windows both north and south made it a handsome villa, and he stuccoed it.

  He built more, in more sober style: pairs of semi-detached houses in two rows, some of four storeys and some of five. By 1805 almost twenty of the houses that Hutton built stood in a sort of suburban park estate, a private village. Hutton and his family lived on the site for a number of years. And of course, after the initial outlay of capital, the rents brought him a comfortable independence. By the time he was paid back, Maskelyne had made a profit of 30 per cent on his loan; Hutton certainly made more, and it got about that the canny mathematician had made his fortune in the building business.

  8

  A Military Man

  Woolwich Common, 13 September 1787. A fine dry day.

  Sponge your gun. Load with cartridge; with ball; ram. Run up the gun; traverse it to aim. Prime and fire.

  Bang.

  When the smoke clears the assembled class of cadets groan, as perhaps do Major Blomefield and Professor Hutton. Not for the first time, the firing crew has missed its mark, failing to hit the ballistic pendulum from seventy-eight feet away. Eighteen ounces of iron, travelling slightly faster than the speed of sound, have buried themselves harmlessly in the earthwork beyond.

  Reset the pendulum and try again. Sponge; load; ram; aim; fire.

  Hutton taught at the Royal Military Academy for twenty more years after the end of the American war and the shipwreck of his career at the Royal Society, and he achieved a great deal there. It was being said, and it would continue to be said, that he had raised the standard of achievement at the Academy quite strikingly. ‘Our British university for the military science’, one magazine commentator called it, indicating the degree to which it now exceeded what was expected from a mere school. Admiring quotes are cheap, but a clearer indication of the high reputation of the Woolwich teaching was its use as a model by other military schools. Private academies adopted its textbooks and in some cases imitated its weekly schedule of classes. The foundation of the Royal Military College, initially at Great Marlow and High Wycombe in 1801 – it later moved to Sandhurst – was avowedly on the model of the Academy at Woolwich, with three departments corresponding to Woolwich’s three ‘academies’, and a curriculum based from the first on the study of mathematics. It was in a sense a satellite of the Royal Military Academy, and the same was true of the East India Company’s training college, founded in 1809 at Addiscombe. There the Woolwich Academy was a model in respect of uniform and badge as well as in the style of teaching; there had long been links between the training of East India and Royal Artillery cadets, the former actually being accommodated at Woolwich for a period.

  Thus, Hutton’s mathematisation of the syllabus was no local phenomenon but the beginning of a change in British military training that would have profound effects on the mental attainments and habits of mind possessed by British officers, not just in the Royal Artillery and the Engineers but across the entire Army. In time, his ideas would spread also to North American military training and to British India.

  To teach well was a fine thing, but to export one’s ideas on this scale involved more, and Hutton was once again engaged in translating his knowledge into textbooks during the 1780s. As with his round of education-based writing in the 1760s and 70s, this would support his teaching, removing the need to bring several different printed sources into the classroom; it would also, of course, consolidate Hutton’s own reputation. And the captive market represented by the Royal Military Academy and the other military schools meant that such ventures could be confident of making a profit, too.

  First came a sort of junior version of his Mensuration, the 1786 Compendious Measurer. ‘Brief, yet comprehensive’, it was intended for school and practical use; as a result it adopted a more familiar manner and was much smaller and cheaper. There were no proofs; this was simply a compendium of rules and examples suitable for study in the classroom. Definitions were cut down to a minimum; each rule was given in a form suitable for copying into the student’s own book; each had a worked example and a diagram, and each had questions with answers.

  The Measurer was, for much of its length, a curiously bland book. The arithmetical part had no real-world examples except where absolutely necessary: there was none of the colour and little of the idiosyncrasy of Hutton’s earlier textbooks such as the Guide, and much less sense that Hutton’s own personality was on display. Very many cases of measuring solid shapes were worked through without any indication of when you might meet them in reality or how you would know if you did. There were no prisms that resembled hatboxes. But the book did swerve towards practicality at the end, with sections on a wide range of real-world applications: measuring timber, brickwork, marble slabs; carpenters’ and joiners’ work, slaters’ and tilers’. Plasterers, painters, glaziers, pavers, plumbers: you can almost hear the roll-call of people who had worked or were working on Hutton’s own building projects up on Woolwich Common, and it’s tempting to wonder whether any sections of the Measurer used, unacknowledged, advice from Hutton’s workmen at home. In a telling slip, he referred in one problem to ‘my plumber’. There was a military flavour too, with questions about the weight of a pint of gunpowder or the best way to pile up spherical cannon balls. And, again, a keen sense of the physical space in which Hutton worked: ‘How long, after firing the tower guns, may the report be heard at Shooters-Hill, supposing the distance to be 8 miles in a straight line?’ (the answer is 37⅓ seconds).

  There was a section in the Measurer on conic sections, those shapes you get when you take a slice through a cone (I always imagine a conical piece of cheese cut on a slant by a knife): ellipse, hyperbola, parabola. After a brief interlude in which he published a Key (a book of worked answers) to his Guide, Hutton worked up this material into a book-length treatment of the Elements of Conic Sections. Which version came first – the long one in the Conics or the short one in the Measurer – is hard to say; they had a lot in common in the way they were organised, and quite a lot of straightforward repetition of questions; quite obviously they had their common source in Hutton’s teaching at Woolwich.

  The Conics was a hard book – perhaps Hutton’s hardest – and it was written in a definition–theorem–proof style that he elsewhere usually avoided. Proofs ended with ‘QED’. He set out steps of argument on separate lines for ease of view, but by doing so he added to the sense of almost oppressive formality in this presentation of difficult material. He also, as far as possible, reused the same wording in the three major sections of the book – devoted to ellipses, hyperbolas and parabolas – heightening the impression of an almost hieratic formality and stateliness of presentation. For me it conveys less the delight in his own dexterity of his other books than mere slog. It doesn’t help that the odd mistake can be found, resulting from the reuse of text without all the necessary changes. It’s hard to imagine what the cadets made of it. Admittedly conic sections, notably the parabola, had some importance to the theory of artillery, but the students can surely have felt little real enthusiasm for it. The austerity was leavened somewhat at the end of the book with – as in the Measurer – a random selection of practical exercises, many of them taken from the Mensuration or the Measurer and some of them quite fun. Repeat the working that led Archimedes to cry eureka. If you’re above London in a hot air balloon and can see Oxford, how high are you? How fast must a 32-pound cannon ball move to do the same damage as Vespasian’s celebrated battering ram?
r />   For all that, the Conics was notably well received. It circulated as a manuscript for a while, and was taught from in that form; eventually it received the commendation of Charles, Duke of Richmond, now the Master-General of the Ordnance. Ordered to be printed, it appeared in 1787. Mathematicians admired it: ‘I am much pleased with your Conicks’, wrote John Playfair, ‘and expect much advantage both from them and from the collection of Problems at the end.’ And, presumably at the instigation of Richmond, Hutton received a perhaps singular honour for a mathematical author; he was presented at court and allowed to kiss the King’s hand. It’s a pleasure to imagine the pit boy at the court of St James, and the scientifically minded king, patron of Sir Joseph Banks, meeting, however briefly, the pariah of the Royal Society. The two men were near-contemporaries, Hutton just a year older than his sovereign.

  A man who could be presented to George III for writing a two-hundred-page textbook about conic sections was a man who was good at more than just mathematics. The incident showed Hutton at his very best as a skilled manipulator of networks and user of personal connections. At a less frenetic pace than in the 1770s, perhaps, and with less of the heroic incaution of his younger days, he was building up a rich world of people who liked and admired him, people who owed him something from having been taught, helped or advised; and prestige followed, as indeed did the more tangible rewards of money and employment.

  As well as teaching and publishing, Hutton had never abandoned his dreams of being taken seriously as an experimenter, a natural philosopher, and he returned to the appropriately military subject of ballistics with new seriousness after the American war was over. He had continued to make experiments since his 1778 Copley Medal, with a new long account appearing in the 1786 Tracts while the experiments were still going on. Ultimately he staked his hopes of scientific greatness on the subject, and expended a great deal of effort on making experiments over the years from 1775 to 1791. By the end he had blown up nearly a ton of gunpowder in the pursuit of knowledge.

  The problem was an old one. A body fired in a constant gravitational field travels in a parabola if air resistance is ignored, and you can describe the parabola and figure out exactly where it will land if you know the speed and the angle of elevation with which it left the gun. Range tables calculated from this theory had been appearing in print since the sixteenth century, and gunnery teaching that used it still throve in private and public academies across Europe.

  Unfortunately, air resistance cannot be ignored. Working gunners knew perfectly well that the parabola theory therefore provided no description at all of where real cannon balls really fell, with the result that these large, expensive, dangerous pieces of equipment were being employed on a trial-and-error, rule-of-thumb basis. If the target was close enough you pointed your gun straight at it (in other words, point-blank range). If not, you pitched the gun up a few degrees and adjusted until you hit it. Range tables, to be any use, had to be worked out empirically for the individual gun. For the purpose of firing mortars or the new exploding shells at high elevations, say into a besieged town, this simply wasn’t good enough, and the knowledge that the theories they were teaching and learning were incorrect was demoralising, to say the least, for students and staff in institutions like the Royal Military Academy.

  Several things were going wrong, and Hutton hoped to elucidate all of them by a sufficiently rigorous experimental study of cannon and the projectiles they fired. For a start, muzzle speeds simply weren’t known. Hutton’s work with the ballistic pendulum, extending that of Benjamin Robins, had begun to change that. His experiments enabled him to say with increasing confidence how muzzle speed depended on the weight of the powder charge, the weight of the shot, and the length of the gun.

  So far, so good. Another problem for the practical gunner was that the strength of gunpowder was variable. With a given charge, sometimes the ball would blaze out of the muzzle at a thousand feet per second and sometimes it would do rather less, emerging in extreme cases with a mere pop. So Hutton invented – improved might be a fairer way to put it – a machine to test and quantify the strength of gunpowder, and in his own experiments he put in place a system of sifting out the coarser, less effective grains from the powder until what was left had the uniform strength he required. Another problem solved, and Hutton’s ‘eprouvette’ found practical use later on for testing gunpowder at the Royal Arsenal.

  A third issue was air resistance, and its complicated dependence on the speed of a projectile – which itself was constantly changing in flight – as well as its shape. As well as firing balls at pendulums, Hutton also measured ranges directly by firing balls down the Thames at elevations of fifteen degrees. He stationed two parties of cadets to note the splashes where the balls fell. (No one seems to have warned the shipping.) The results were of some use, although a mathematical law would continue to be somewhat elusive. But the work was profoundly demoralising in another way, because it made it quite clear that firing balls at long ranges could not be an exact science. In a mile’s flight the shot were wandering anything up to a few hundred feet to the left or the right. The skilled gunners of the Royal Artillery were unable even to hit the river consistently. What was going wrong?

  Hutton blamed, in part, the quality of ‘windage’, which he had already shown was wasteful as to the use of powder: the difference between the size of a ball and the size of the bore, which for safety’s sake was often set as large as a twentieth of the calibre of the gun: a tenth of an inch or more. With the ball rattling from side to side in the barrel by that much, he reasoned, perhaps it was no surprise it didn’t always go where expected. He wondered whether cylindrical shot might fly straighter, as well as advocating a reduced windage to control the problem.

  But, once in the air, there might be wind to deflect the ball, and as Robins had already hinted there seemed to be a distinct tendency for spinning balls not to fly straight but twist away to left or right even in calm conditions (the phenomenon would eventually be studied under the name of the Magnus effect; it’s real, and it makes spherical cannon balls irretrievably inaccurate at long ranges). All this tended to point to the conclusion that however many experiments Hutton did and however good the mathematical laws he derived, they wouldn’t enable gunners to hit anything they couldn’t hit already.

  Yet still he persisted. Direct measurement of ranges proving less conclusive than he might have hoped, Hutton investigated the resistance of the air more closely, hoping to discover how much speed a given ball lost in travelling a given distance. There were some mathematical models and some good guesses in print, but Hutton hoped he could do better. It was a problem Newton himself had worked on, and it would have been a coup indeed to solve a problem on which the Newtonian theory was known to be incomplete.

  So he fired balls at a pendulum from a cannon that was itself mounted on a pendulum. By measuring how much the freely swinging cannon recoiled, he could deduce the ball’s muzzle speed; by measuring how much the ball-struck pendulum was displaced he could deduce the ball’s speed at impact. The difference between the two told him how much speed the ball had lost due to air resistance.

  The method worked, and Hutton pursued a lengthy series of experiments enabling him to say in some detail how the air’s resistance depended on the ball’s size, weight and speed. Not satisfied with this, he also found some of Robins’s old apparatus and used it to do more work on air resistance. The ‘whirling machine’ placed an object at the end of an arm, and used the force from a descending weight to whirl it around in a big circle. Descending weights behave predictably, and by observing how fast the weight was falling compared with freefall, Hutton could deduce how much the object on the arm was resisted by the air. It was a neat, delicate apparatus, and it helped Hutton fill in gaps in what he knew about the resistance exerted by air.

  Other people were also interested in ballistics experiments, and Hutton received detailed reports of similar work to his, done by officers at Landguard Fort and at Gib
raltar; he also saw reports in the Philosophical Transactions of experiments by Count Rumford, of which he did not think much. He briefly attempted a synthesis of some of the data that had come his way from these various sources, but quickly abandoned it, perhaps reasoning that he did not wish to rely on experiments done outside his own meticulous control. The work at Woolwich went on.

  But by the end of summer 1791 it was all too clear that, overall, the results were disappointing. Air resistance obeyed no clear mathematical law. Yes, it depended on the surface area of the moving body, and different shapes (a cone or a hemisphere, say) produced consistently, quantifiably different degrees of resistance. But the key question was how resistance depended on speed, and here Hutton was unable to make things work. He massaged the data to ‘regularise’ them (equivalent to drawing a straight line through a scatter plot); he laboriously fitted equations to the data once regularised. Because of the physical model of air resistance he had in mind, he thought that above a certain high speed one component of the resistance would reach a maximum and remain constant. But because he wanted a single equation to work with, he instead sought one law that would cover all speeds. He found it in the rule that resistance was proportional to the two-and-one-tenth power of speed; but because that was intractable to work with (in calculus terms, he was unable to integrate it) he abandoned it in favour of a different rule, a sum of multiples of the speed and its square, that fitted the data much less well: one of the saddest moments in the whole work.

  Again and again Hutton’s frustration was obvious in his writing about the ballistics experiments. Now he had fixed on a mathematical law describing the resistance, he would have liked to be able to use it to derive a description of the shape of a projectile’s path. But that proved intractable for purely mathematical reasons. He drew a picture and gave a description in words, but he could find no way to describe it quantitatively, try as he might. And he would have liked to make some theoretical predictions for the range of different balls at different speeds; but, similarly, the equations wouldn’t or couldn’t be solved.