Читать книгу Gunpowder and Geometry: The Life of Charles Hutton, Pit Boy, Mathematician and Scientific Rebel - Benjamin Wardhaugh - Страница 10

Woolwich: the Royal Military Academy. Charles Hutton’s study. 13 January 1779. He’s in a gloomy mood.

‘I am here almost as recluse as a hermit,’ he writes to a northern acquaintance, ‘being almost single in my studies & manner of thinking in this place, my nearest neighbour in these respects being the Astron. Royal at Greenwich with whom I have the honour to be on very good terms.’

As recluse as a hermit? What about his family?

Hutton’s wife and children did not accompany him to Woolwich. His relationship with Isabella had broken down some time after the birth of their fourth child in 1769, and she never left the North-East.

Hutton and his family were always coy about this, and most of his obituarists and early biographers did all they could to avoid telling the story; so it’s hard to say just what really happened. One Newcastle historian was indiscreet, though, and related in the 1820s that Hutton was initially accompanied to Woolwich by a different woman: an officer’s widow named Maxwell. Apparently he soon found himself obliged to dismiss this lady on account of her extravagant habits. That sounds slightly fanciful, and it was written many years after the fact by a writer whose sources of information are far from clear. It is certain, though, that within a few years Hutton was living with another woman. At least some of his friends knew Margaret Ord as ‘Mrs Hutton’ – although his first wife was undoubtedly still living – and in 1778 she bore him a daughter, Charlotte Matilda.

Who was Margaret Ord? Born around 1752, she was in her twenties during Hutton’s first decade in Woolwich: eighteen years younger than Isabella. It would be interesting to know something about her background: how she compared with the first Mrs Hutton in terms of social rank, for instance. Unfortunately for the curious historian, Ord was not an uncommon name. There were Ords in the Royal Artillery and a prominent family of the same name in Newcastle; the latter, indeed, were part owners of the Long Benton colliery where Hutton, once, had worked. There were three Fellows of the Royal Society named Ord around this time. Margaret could have been related to any of them; the fact is that we know nothing of her origins, nothing of where and how she and Charles Hutton met.

Isabella remained in Newcastle and took to styling herself a widow, but by March 1776 Hutton’s son Harry was in Woolwich as a cadet at the Royal Military Academy. He graduated about a year later and went into the Royal Artillery. Hutton’s other children initially stayed with their mother, but by the early 1780s they too had moved to Woolwich. Just what had passed between them, their father and their mother we will never know.

Not quite a hermit, then: by the end of his first decade at Woolwich Hutton headed a household consisting of himself, Margaret, and his four daughters ranging from twenty-one-year-old Isabella down to six-year-old Charlotte.

But still, Woolwich was not London, and it could indeed feel isolated. You can walk to Woolwich from the City of London, but it takes half a day. You can shorten the time by riding, or take a boat; to row from the Tower of London down to Woolwich took a couple of hours. In time, Hutton took to renting a set of rooms in the city, at one of the Inns of Court, and spent a couple of days there every fortnight, judging that that made the best use of his time.

Much of Hutton’s time was in fact his own, since his teaching filled only three afternoons each week. He took every opportunity he could find to fill that time by doing extra work and making new professional connections. He was, and would always remain, a superb networker.

Nevil Maskelyne.

A couple of miles up the river, at Greenwich, lay the Royal Observatory. This was the domain of the Astronomer Royal, Nevil Maskelyne, who was a member of the committee that examined the candidates for the Woolwich job and selected Hutton. And as Hutton put it in his letter of 1779, Maskelyne was his closest scientific neighbour. As early as September 1773 Hutton was corresponding with Maskelyne’s assistant Reuben Burrow, lending books back and forth, and soon Hutton was on good terms with the Astronomer Royal himself. They would remain close until Maskelyne’s death; an obituarist reckoned Hutton among Maskelyne’s ‘most intimate friends’.

Honest and popular, Maskelyne was a key member of the London scientific world. By the mid-1780s Hutton was sending him drafts of his papers to look at, and on occasion detailed comments from Maskelyne found their way into the published versions of Hutton’s books. Hutton acknowledged Maskelyne’s ‘generous advice and assistance’ with a dedication to him in 1785.

One of the projects in which he involved Hutton arose from his role on the Board of Longitude. The board existed to assess – and potentially to reward – schemes for finding the longitude at sea, a problem whose unsolved state was leading to losses of life for Britain and for every nation engaged in more than coastal seafaring. Maskelyne had a scheme of his own for finding the longitude: to use the moon’s predictable motion across the background of stars – or relative to the sun – as a sort of clock. Starting in 1767, with the blessing of the Board of Longitude, he oversaw the printing of tables of the moon’s position in the sky up to several years in advance, under the title of The Nautical Almanac and Astronomical Ephemeris. If you observed the moon’s position and compared it with the table of predictions, you could deduce exactly what time it was. Knowing the exact time, an accurate look at the apparent position of the sun or the stars would tell you where you were.

The annual books of tables cost two shillings and sixpence; you also needed an instrument for observing the moon – a ‘Hadley’s quadrant’ costing eight pounds or so – and a two-shilling book of extra tables. The calculations could be reduced to a feasible, if laborious, recipe that took about half an hour. The Nautical Almanac was distributed at ports around Britain, Europe and America, and during the final third of the eighteenth century ‘lunars’, so called, became an accepted method of finding your position at sea, and much the cheapest. An alternative way to determine the exact time and hence your position was to carry a really good clock; but a clock accurate and reliable enough – like the chronometers built and promoted by John Harrison – cost dozens of guineas, and unlike books of lunar tables they broke if you dropped them. Maskelyne took some criticism for his suspicion of the chronometer method, but frankly he was right; for most sailors it was still an inaccessible and impractical answer to the ‘longitude problem’.

An issue of the Nautical Almanac contained the moon’s position for every three hours, night and day, of the whole year. Making the tables in the first place was laborious, and far beyond the power of one person, even if Maskelyne had had nothing else to do (he did) and had been paid to work on the Nautical Almanac (he was not). Instead he outsourced the work on the cottage-industry model, to a network of human ‘computers’ around the country. Maskelyne’s computers were teachers, surveyors, minor mathematical authors: much the same kind of people who contributed to philomath journals like The Ladies’ Diary. Indeed, they were sometimes recruited directly from the ranks of the Diary’s problem-solvers.

They worked not with the equations and geometry that described lunar theory, but rather from a set of computational instructions prepared by Maskelyne. Calculating a single lunar position typically involved looking up about a dozen figures in printed tables and carrying out a similar or larger number of seven- or eight-figure arithmetical operations, all done in base 60. It was demanding, meticulous work.

The computers were (mostly) good at what they did, but errors had the potential to lead to large losses of life, and a good deal of careful checking was needed to make sure no disastrous mistakes found their way into the printed tables. So a ‘comparer’ kept an eye on things, standing between Maskelyne and the computers. And here Hutton got involved. It was unglamorous work involving liaison with the computers as well as with Maskelyne. And it was a deeply picky process. The computers worked in pairs, without communicating with each other; one found the moon’s position for every midnight and one its position for every noon. The comparer merged the tables and checked that the moon’s predicted motion contained none of the implausible jumps that would signal a mistake in someone’s calculations. If there was a problem, the comparer redid the calculations himself until everything was right and he had a full month’s table of lunar positions for both noon and midnight.

Then he selected some stars that lay close to the moon’s path, and sent the complete, correct table back to the computers so they could both, independently, compute tables of the moon’s predicted distances from those stars through the month. When the comparer received this information, he checked that the tables drawn up by the two computers were identical and, once again, sorted out any discrepancies by repeating the work himself if necessary.

He also prepared various other pages of the Nautical Almanac such as the initial explanation of symbols and a chart showing the positions of Jupiter’s satellites. And finally, when the almanac was being printed, he corrected the proof sheets: yet more checking of long tables of numbers that were supposed to be identical.

On and off during 1777–9 Hutton did all this, covering the comparing work for a total of twelve months’ worth of Nautical Almanacs; he also performed some extra tasks such as checking the predictions for eclipses of Jupiter’s moons that were printed in one almanac. He was paid (a total of about seventy-five pounds), but the money was far from being the point. ‘Comparer’ was a position of significant trust, and Maskelyne did not give it to just anyone. Hutton was very possibly doing Maskelyne a favour by filling in for months when no other comparer had been found or was available. And by doing so, and doing it well, he significantly increased his credit in the network around the Astronomer Royal. He was establishing himself as part of Maskelyne’s mathematical/astronomical circle, and confirming his valuable relationship with the Astronomer Royal.

There was more. His work for Maskelyne gave Hutton the opportunity to make contact with the Board of Longitude itself, and to do more work for it on an occasional basis. Through 1779 he corrected proofs of mathematical publications for the board – at a guinea a sheet – and for one book he was paid to translate a preface from Latin into English. In 1781–2 he provided lunar computations apparently outside the normal cycle of work on the Nautical Almanac, for which he was paid ten pounds ten shillings ‘for my Trouble’. And by 1780 he was writing to the Board to present a work of his own: a book of mathematical tables.

Mathematical tables had long been one of Hutton’s interests. His very first book, the School-master’s Guide, had ended with a little table of the first twelve powers of each of the nine digits, and the 1770 Mensuration similarly closed with a thirty-page table of the areas of segments of a circle. During the 1760s and 1770s he had found himself repeatedly making computations involving roots and reciprocals of numbers,

and as it seemed probable that this might be the case with me for many years longer, I formed the resolution of preserving all such roots and reciprocals as I should occasionally produce in my calculations, that I might have them always ready on any future occasion; which I did by entering them always in a little book, ruled for the purpose, till I have at last collected to the number of 1000.

He published the resulting table in the 1775 Miscellanea Mathematica.

His next venture of the kind was similar, but larger: a standalone table of the products and powers of numbers: products up to 1000 times 100; powers up to 100¹⁰. It could well have been collected together over a period of years like the table of roots and reciprocals, and when it was complete it enabled the rapid looking-up of over a hundred thousand different products or powers.

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