Gunpowder and Geometry Read online

  The eighteenth century was a period of heroic manual calculation (as the Nautical Almanac illustrated), and well-judged, accurate printed tables were of real use to those involved in such work. Hutton rightly saw that the genre offered a route to bring himself once more to public notice. The Board of Longitude accepted his project and agreed to grant him two hundred pounds to see it through the press: nearly as much as his annual salary at Woolwich. Five hundred copies were printed, and as well as being sold they were adopted as a standard part of the Nautical Almanac computers’ equipment.

  The table of powers showed once again that Hutton loved rapid, accurate calculation and was extraordinarily good at it. The preface displayed a virtuosic facility at getting the tables to do more than they at first sight seemed capable of: using a table of squares to compute square roots, using interpolation or repeated calculation to multiply numbers larger than those allowed for in the tables. One reviewer dryly remarked that ‘but little trouble will be saved’ by using the tables in such complex cases, but that wasn’t entirely the point: as before, Hutton was telling the reader something important about who he was and what kind of mind he had.

  There was more (with Charles Hutton there was always more). One of the natural goals for a mathematician interested in tables was to work on tables of logarithms. Judiciously deployed, logarithms speeded many types of calculation, and tables of them to six or eight or even more decimal places had been in print since early in the seventeenth century. Computing them was laborious, and getting them right in detail – and printed correctly in detail – was famously hard; the standard work, Sherwin’s Tables, was notorious for its inaccuracy by the time of the 1771 fifth edition. Hutton himself had compiled a list of several thousand errors in that book, and he reckoned the time was right to begin again from scratch.

  He conceived a project to print new, freshly calculated logarithm tables to seven places, together with supplementary smaller tables, enabling the user to find the logarithm or antilogarithm of any number to twenty or to sixty-one places. Through the early 1780s he worked on these: they represented thousands of hours of work on top of everything else he was doing. Calculate, recalculate. Check the calculation. Transcribe into a fair copy. Check the transcription. Quiet, endless scratch of the quill against a background of ordnance testing, cadets drilling (or rioting) and military bands practising. You wonder how he found the discipline, the energy, or even just the time.

  In fact, he didn’t, or not all of it. The manuscript of his logarithm tables survives, and it clearly shows he got his family to help him. The rough work and the fair copies are in at least two different hands: Margaret’s almost certainly, and quite possibly those of one or more of his older children. A biography that appeared later during Hutton’s lifetime, indeed, acknowledged that the labour and calculation for the product tables was ‘chiefly owing to the industry’ of Margaret, who assisted him with other, unspecified laborious calculations too.

  It wasn’t an unusual arrangement for women to work as unpaid assistants in support of their husbands’ work. Acknowledgement was rare, so we don’t know exactly how common it was. Margaret’s hand also possibly appears making comments and revisions in some of Hutton’s scientific manuscripts from the 1780s. His daughter Isabella would later become Hutton’s main amanuensis; when old age made his own handwriting wobbly she wrote nearly all his letters for him. The youngest, Charlotte, was not as yet old enough to be involved, but her turn would come.

  So, rather than heroic solitary labour on the tables – or perhaps on anything else – we should imagine a sort of bustling family workshop, in which drafts were passed from hand to hand and might receive work from two or three different people. The name of Charles Hutton alone appeared on the title pages. But others played a role, as he occasionally acknowledged, and without them he would almost certainly have been unable to complete the volume of work he did.

  The point perhaps deserves pressing slightly. Within the philomath world, women were visible, though they were not as numerous as men. In The Ladies’ Diary some women hid behind male or neutral pseudonyms; only a few appeared under their own names. In the world of the Nautical Almanac, too, one woman was involved as a calculator in this period. Male-dominated but not male-exclusive worlds, then, as far as the evidence goes. The glimpse Hutton’s manuscripts provide of the participation of his family suggests that women’s roles may have been more substantial than we think, more than we can usually see. That, when we read a publication by ‘Charles Hutton’, we’re not hearing his individual voice alone but a composite, in which other voices from his household, both male and female, are involved.

  Meanwhile, the calculations were done. Hutton added an enormous preface, setting out the history of logarithms and their calculation over the previous two hundred years and detailing how his own tables had been calculated and how to use them. More than anything he had done so far, the preface attempted to establish him as more than a mere technician; someone who was not only good at mathematics but knowledgeable about the subject in a humane, humanistic way. A vast range of reading was displayed; information from two centuries was synthesised and formed into a narrative. Judgements were passed, credit was assigned or reassigned. Hutton, here, was reinventing himself as an authority on the history of mathematics, on its nature; and perhaps even an authority on where mathematics should go next, what it was useful for, why it mattered.

  There were delays in the calculating work, for reasons we shall hear about in Chapter 6, and there were delays in printing the book, when Hutton insisted on a demanding programme of checking that entailed comparing each proof sheet with the manuscript several times. Almost certainly, other members of his household were involved once again. He was evidently confident of their work; and in a printed list of errata he admitted to only seven mistakes in the logarithm tables when they finally appeared.

  The tables were admired by reviewers; it was hard to see how one could do much else, since time alone would determine whether they were really as accurate as Hutton said. On the whole it seems they were, and they remained in print until 1894, through thirteen editions. Like Hutton’s table of powers they became part of the standard set of books loaned by Maskelyne to Nautical Almanac computers and comparers. Reviewers noted the immense erudition of the preface and the authoritative style:

  a judicious arrangement of the subjects treated of, and a clear unambiguous statement of propositions and their demonstrations, are the leading excellencies in subjects of this kind; and these, we will venture to affirm, are no where more conspicuous than in the work before us.

  At the same time, in a different mathematical world from that of the Royal Observatory and the Board of Longitude – but certainly an overlapping one – Hutton was also networking; working his earlier connection with the world of the Georgian philomaths and The Ladies’ Diary in particular. His collection of material from the Diary had continued to appear in separate numbers across the period of his move to London, and when it was collected together in volume form in 1775 it received some favourable notices in the London papers. The Diary itself continued to appear each year, of course, but during 1773 its editor, Edward Rollinson, died. There was certainly no more natural choice to replace him than Charles Hutton, and within days of his arrival in Woolwich, later that summer, he was approached by the Stationers’ Company of London – which published the Diary – and asked to do so. By 1775 he was established as editor of the Diary, and he held a sale of Rollinson’s books from his house in Woolwich. In 1781, following the death of another editor, he signed an agreement with the Stationers’ Company to compile not just the Diary but several other annual almanacs too; he was in charge, at his peak, of at least nine different titles, although a few remained in other hands.

  The Diary work chiefly involved dealing with the substantial numbers of letters it received, containing solutions to the previous year’s problems as well as proposing new problems. Most arrived in a rush before the deadline of 1 May.
It was not just mathematics: there were also riddles in verse, questions on various topics. Regular notes from the editor to keep contributions short, and in some years the drastic expedient of printing some parts in almost unreadably small type, indicate the flood of material that was coming in. At times Hutton kept material back for two years before he could find space for it.

  There was sifting of solutions, checking their correctness and choosing which were well enough expressed to be printed; there was judgement of the literary merit of the poetic compositions and questions. There was the occasional act of censorship; in 1775 Hutton noted that ‘The enigma on a Candlestick is too indelicate for insertion.’ There were complex queries from contributors to deal with, and accusations of plagiarism or false attribution. There were proof sheets to check.

  If all this gives the impression of an immense and perhaps an uncontrolled busyness, it should. Hutton was developing quite a habit of taking on mathematical odd jobs. For Maskelyne and to some degree for the Board of Longitude he was becoming a preferred choice for computation, mathematical proof-reading, and even mathematical translation work. With his almanac work and the new editions of his early books (there were at least five during the decade from 1775 to 1785), it begins to be surprising he had time for his teaching. He also reviewed mathematical books – anonymously – for the monthly periodicals. His letters comment on how busy he was, and his prefaces on the degree of interruption some of his projects received from others.

  It was relief from teaching elementary geometry to unruly cadets, and it was visibility and status of a kind. But much of it left him rather out of sight, in danger of seeming a mere technician. Computers and comparers were not credited in the Nautical Almanac; the editor of The Ladies’ Diary was well paid but nowhere named in its pages. The production of a book of mathematical tables made one a ‘compiler’ or an ‘editor’, not an ‘author’. But Hutton also aspired to a higher level of visibility and status (and reward). There were wider worlds than that of the Board of Longitude and more prestigious ones than that of the philomaths.

  Indeed, as early as March 1774, less than a year after his arrival in Woolwich, Hutton had been proposed for fellowship of the Royal Society, the premier scientific society in Britain. Colonel J. Phipps, who plays no recorded part in Hutton’s life besides this, seems to have been the moving force, though ten other Fellows signed the proposal. They inevitably included Nevil Maskelyne as well as Samuel Horsley, another of the examiners for the Woolwich job; also named was the influential botanist Joseph Banks, one of the rising men at the Society. The certificate stated that Hutton was ‘well acquainted with Mathematical and Philosophical Literature’, a claim with which few would have argued, ‘a Gentleman’ and ‘likely to become an useful Member’. After a lumbering process, involving delays during the summer recess, he was elected in June and formally admitted to fellowship on 10 November.

  Hutton used his status as FRS when advertising his books, and he usually attended one meeting of the Society out of two: as many as he felt he could make time for. He published a few papers in the Society’s Philosophical Transactions, though these were modest affairs: a brief solution of geometrical problems; a six-page suggestion to construct trigonometrical tables using a new set of units (radians rather than degrees); a new algebraic trick for finding infinite series whose value was equal to that of π; a lengthy, pedestrian article surveying the known methods of solving cubic equations. Attention-grabbing these weren’t, and they remained firmly within the sphere of technical mathematical work rather than front-rank natural philosophy.

  But Nevil Maskelyne had a specific project in hand which would ultimately shed glory on both himself, his country, and his friend Charles Hutton. In 1772 he had proposed to the Royal Society that it use some money left over from another project to undertake experiments on the gravitational attraction of mountains. Part of the point was to demonstrate that there was such an attraction; a French experiment in the 1730s had been inconclusive, and it would be an important confirmation of the Newtonian theory, which stated that every object in the universe exerted a gravitational attraction on every other, proportional to its mass. The experiment would ‘make the universal gravitation of matter palpable’, as Maskelyne put it. It would also, if the experiment was done carefully, enable the experimenters to work out the strength of the gravitational attraction of a given mountain, and work back from that to deduce the mass of the earth, a quantity that was so far a matter of mere vague guesses.

  How to do it? Simple enough: find a reasonably isolated mountain and set up a plumb line next to it. By comparing the line’s direction with the background of stars, you can see how far the mountain is deflecting it from vertical. Delicate work, but quite feasible with eighteenth-century equipment.

  Charles Mason (one of the men who measured the Dixon Line) worked on the selection of a site in 1773; he considered several mountains and chose one in central Scotland. With a mile-long ridge running east–west, and fairly isolated from other hills, bleak Schiehallion he reckoned was as close to ideal as the experiment was going to get. Reuben Burrow, once an assistant at the Royal Observatory and an acquaintance of Hutton, went north and surveyed the site with a local man named Menzies in 1774 and 1775, and after a certain amount of foot-dragging Maskelyne himself went up to run the astronomical observations. They lived in tents and huts and suffered greatly from bad weather (it was rumoured that the name Schiehallion meant ‘constant storm’ in Gaelic). But eventually the work was done: 337 observations of 43 different stars from two different locations. When Maskelyne worked through the numbers, it turned out that the plumb lines in the two places, on either side of the mountain, really did point in slightly different directions, beyond what would be expected from their difference in latitude.

  This was success insofar as it showed that the hill exerted a gravitational attraction, and the divergence of the two plumb lines (three thousandths of a degree or so) enabled Maskelyne to deduce that the density of the earth was about double that of the mountain. Read to the Royal Society in July 1775 and published in the Philosophical Transactions later that year, his paper won Maskelyne the Society’s prestigious Copley Medal. As he put it, everyone was now obliged to be a Newtonian, at least as far as believing in the universal mutual attraction of matter.

  But there was more work to be done. Detailed processing of the survey data could potentially yield a much more accurate estimate of the density of the earth. It would require a mass of calculation, determining the shape of the mountain from the survey and then working out precisely its expected gravitational effect at the two observation points.

  Maskelyne himself declined to do it: too laborious. It can’t have taken him long to think of a suitable man for the job; indeed, he had already discussed with Hutton some of the computational techniques he might use. The Royal Society agreed to pay Hutton, and even met his rather cheeky request for a new set of quality drawing instruments to help him make plans and diagrams.

  For about a year in 1777–8 Hutton devoted much of his working time to the project; an ‘immense labour’. The easy part was taking the survey data and using it to prepare a detailed map of the site. As far as we know this was only the second or third actual map Hutton had prepared, but the mathematical techniques were well known; indeed, they were detailed in his own textbook on mensuration. The quantity of work, nevertheless, was huge. The surveyors had taken more than seventy sections through the Schiehallion site: most vertical, some horizontal and some neither. In all they described nearly a thousand points, each fixed in position by taking angles from other points. Several thousand trigonometrical calculations were needed to turn the data into a map four feet square covering the mountain and its surroundings.

  To one of his versions of this map Hutton added contour lines, ‘connecting together by a faint line all the points which were of the same relative altitude’. Although people had described the idea before, Hutton seems to have been the first person actually to make a
contour map, and it’s a pity the map itself has not survived.

  The really hard part came next: given the contour map of the mountain, to work out the expected gravitational attraction of the hill at each of the two observation sites. After at least one false start, and with the help of some ideas from fellow FRS Henry Cavendish, Hutton came up with a feasible way of doing it. Instead of dividing the map into a grid of squares, he divided it up using circles and parallel lines. For each of the two astronomical observation points Maskelyne had used, he drew twenty rings, crossed by twelve parallel lines in each quadrant, making a total of nearly a thousand irregularly shaped patches of land. Or rather, nearly a thousand irregularly shaped pillars of rock exerting their separate, determinable gravitational effects on the observatory. Adding up all the gravitational effects involved a specially constructed slide rule and a computational trick suggested by Cavendish.

  The end of it all was that the hill should be expected to attract the plumb line just under ten thousand times less than the earth did. The astronomical observations showed that in fact it attracted it nearly eighteen thousand times less, so Hutton concluded that the hill was less dense than the earth by a ratio of eighteen to ten, or nine to five. To put it another way, the earth, taken as a whole, was 80 per cent more dense than Schiehallion. Supposing the mountain was made of common rock weighing 2,500 kilograms per cubic metre, it followed that the earth, taken as a whole, had a mass of about 4,500 kilograms per cubic metre of its volume. (Kilograms were not yet invented; Hutton expressed everything in relation to the density of water, which comes to much the same thing.)

  Hutton’s paper was a virtuoso performance: certainly the most spectacular piece of work he had yet done. To his death ‘weighing the earth’ remained one of his proudest achievements. In the Philosophical Transactions the paper filled a hundred pages, with tables of data, example computations, maps, endless details about the site and the mathematics. As an additional reward for those who made it to the end, he deduced the densities of the other planets from that of the earth (the Newtonian model of how things move in the solar system makes it possible to do this) and speculated about the internal structure of the earth, inferring the existence of ‘great quantities of metals, or such like dense matter’ deep inside the planet. Roughly half the matter in the earth would need to be metal to account for his proposed overall density.