Sextant: A Voyage Guided by the Stars and the Men Who Mapped the World’s Oceans

Other prizes were later announced by the Dutch and Venetian Republics, by France and, eventually, by Britain. Under the terms of the British Longitude Act of 1714, a sum of up to £20,000 was offered as ‘a due and sufficient Encouragement to any such Person or Persons as shall discover a proper Method of Finding the said Longitude’. This would now be worth several million pounds.

The Longitude Act, however, imposed high standards of accuracy: to win the maximum amount the successful method had to be capable of determining longitude within a margin of error not exceeding half a degree of a great circle (equivalent to 30 nautical miles). Half the maximum prize would be payable when the Commissioners of the new Board of Longitude were satisfied that the proposed method extended to ‘the Security of Ships within Eighty Geographical Miles from the Shores, which are Places of the greatest Danger’, while the balance would be paid ‘when a ship … shall actually Sail over the Ocean, from Great Britain to any such Port in the West-Indies, as those Commissioners … shall Choose or Nominate for the Experiment, without Losing their Longitude beyond the Limits before mentioned’. Moreover, the reward would be paid only ‘as soon as such method for the Discovery of the said Longitude shall have been Tried and found Practicable and Useful at Sea, within any of the degrees aforesaid’. The words ‘Practicable’ and ‘Useful’ were to give rise to bitter disputes. Lesser rewards were available for proposals that the Commissioners judged of ‘considerable Use to the Publick’.

Though pendulum clocks coupled with the new ephemeris tables permitted land-based observers to determine their longitude accurately, nobody had yet managed to devise a time-keeper that could be relied on at sea. Existing spring-driven clocks and watches were hopelessly erratic, and despite valiant attempts it proved impossible to make pendulum clocks work reliably on board ship. Strenuous efforts were therefore made to find methods of determining longitude that did not rely on astronomical observations and which could therefore be employed without the need to know the time. Mapping the geographical variations in the direction of the earth’s magnetic field seemed to offer some hope, but in the end this line of enquiry proved abortive and the heavens became the exclusive focus of scientific attention among those seeking to solve the longitude problem. If sea-going clocks were not to be relied on, then perhaps the sailor could find the time from observations of the sun, moon and stars. The challenge was to identify a frequently occurring astronomical event the precise time of which could be both accurately predicted and easily observed on board ship, anywhere in the world. Published tables of the predicted times of such events would in principle enable the navigator to find the time at a given reference meridian (such as Greenwich or Paris) wherever he happened to be – providing the skies were clear. Comparison with the local time – derived from astronomical observations – would then reveal the observer’s longitude.

Various methods of achieving this goal had already been suggested. For example, in 1616 Galileo opened discussions with Spanish officials about the possibility of using observations of the appearance and disappearance of the moons of Jupiter (the four largest of which he had recently discovered) as a means of determining the time at the reference meridian. In return for a large fee for travelling to Spain to demonstrate his method to King Philip III, an annual royalty both for himself and his heirs, as well as appointment to the chivalric Order of Santiago, he proposed to draw up the necessary tables and update them annually; he even invented a telescopic device to be worn on the head that was supposed to permit making the necessary observations at sea.

But the Spanish lost interest and in 1635 an ageing Galileo turned to the Dutch, this time with improved tables and a mechanical device for representing the motions of the Jovian satellites that he called the ‘Jovilab’.

The Dutch States General responded enthusiastically and even appointed an astronomer to act as a technical go-between, but Galileo – who was by now going blind – was unable to generate the orbital parameters of the moons on which sufficiently precise predictions could be based.

In any case, a fairly powerful telescope was required to observe the moons of Jupiter and such an instrument could not be held steadily enough on board ship. And there was another problem lurking in the background: without an accurate shipboard time-keeper, how exactly was the navigator supposed to compare local time (derived most easily from sun sights) with the time obtained from the tiny Jovian moons – visible only after the sun had set? Galileo claimed he knew how to make a sufficiently accurate pendulum clock but he had not succeeded in doing so by the time he died, and anyway it would have been of no use at sea.

Jupiter’s moons were, however, very useful to land-based observers equipped with pendulum clocks – once the necessary tables had been produced at the Paris Observatory. In the 1680s a French expedition established the longitude of the Cape Verde Islands, Guadeloupe and Martinique using this technique,

and Picard and La Hire also employed it when making their map of France. Eclipses of the sun were among the other possibilities, but they were too infrequent to be of much use, and it was not until the invention of the sextant that it was possible to observe them with sufficient accuracy on board ship. Spanish navigators and astronomers had experimented with the technique in the sixteenth century, but the results, even at land-based observatories, were of no value.

So it is not surprising that when the Longitude Act was passed in 1714, few observers expected that anyone would soon succeed in claiming the big money. Many bizarre and frankly ludicrous proposals were put forward, and in consequence the quest became something of a standing joke. William Hogarth included a cheerful lunatic searching for a solution to the longitude problem in the background of the scene from the madhouse in the Rake’s Progress of 1735.

Such scepticism was misplaced. After a struggle lasting hundreds of years, two radically different solutions to the problem of finding the time on board ship emerged almost simultaneously in the 1750s – one mechanical, the other astronomical. Both, however, relied on accurate angular measurements made with a quadrant or, better still, a sextant. As we shall see, one method was based on a new kind of clock, while the other depended on the first accurate tables of the motions of the moon. In practice, however, the two techniques were to be mutually dependent for many years to come.

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The extraordinary story of the development by John Harrison (1693–1776) of the first accurate shipboard time-keeper – and his long struggle for official recognition of his feat – is by now well known. In 1759, after more than thirty years of experimentation, he produced a highly innovative ‘watch’ (known as ‘H4’). It was not regulated by a pendulum and exploited the ingenious principles of compensation he had developed in earlier experimental devices; it was also a great deal smaller and more practical. H4 easily passed the second (and possibly also the first)

of two rigorous sea trials – a voyage to Barbados and back in 1763. But the Longitude Board were cautious about recognizing Harrison’s remarkable technical breakthrough. Rigorously interpreting the wording of the Longitude Act they demanded to be convinced that the new watch’s exceptional performance had been more than a fluke and that the mechanism itself could be reliably replicated at an affordable price. Arguments about whether H4 and its maker had or had not satisfied the precise terms of the Act were to drag on for years. The elderly Harrison, vigorously supported by his son, William, was enraged by the apparently perverse delays in awarding him the full prize of £20,000, and by adjustments to the terms of the Act that – in his view – seemed designed to deny it to him. His tactless and explosively ill-tempered behaviour alienated many members of the Board, which had by the end of 1762 already funded his labours to the tune of £4,750 – a very substantial sum.

As guardians of public funds the Board were understandably anxious not to expose themselves to charges of waste. But their reluctance to reward Harrison in full was also influenced by the belief – which Newton had shared – that the only reliable solution to the longitude problem must be astronomical not mechanical. After all, watches could go wrong, and seemed very likely to do so in a damp and bumpy ship at sea – especially if the temperature varied a good deal, as it would on a voyage from Europe to the tropics. How would the navigator be able to tell if the watch started to misbehave? Who was going to fix it if it stopped? How could any errors be corrected?

These were perfectly fair questions, and as experience subsequently showed many chronometers did indeed perform poorly, often running irregularly or stopping altogether for no obvious reason. Even their own makers did not understand exactly what they were doing: they were artists as much as engineers, and they relied heavily on trial and error. The sun, moon and stars, by contrast, were the very embodiment of perfection – and indeed the basis of time itself. Until the invention of the ‘atomic clock’ in the mid-twentieth century, the movements of the sun and stars remained the fundamental indices of time. Harrison’s watch may well have seemed inelegant to the more mathematically minded members of the Longitude Board – a questionable, brute-force solution to a problem they regarded as essentially astronomical in nature.

Harrison’s great achievement was the invention of a radically new watch movement that could keep time accurately – not just in stable conditions on land, but also in the wildly variable environment of a ship at sea. He was certainly a difficult and irascible man, as was his son, but he was highly ingenious and extremely determined, and in 1773, following a powerful speech in the House of Commons by Edmund Burke, and a sympathetic intervention by King George III himself, Parliament (rather than the unbending Longitude Board) awarded him a further £8,750.

The practical marine chronometers (as these ‘time-keepers’ were eventually to be known) that relied on Harrison’s pioneering work were not, however, mere duplicates of H4. They owed much to the inventive skills of other watchmakers like Pierre Le Roy and Ferdinand Berthoud in Paris and Larcum Kendall, John Arnold and Thomas Earnshaw in London.

The chronometer we carried aboard Saecwen was a descendant of those developed in the last decades of the eighteenth century and probably differed little from them. It sat luxuriously in a pretty mahogany box with brassbound corners, secured by strong elastic cords in a safe corner of the cabin near the mast. Lifting the lid, a circular brass case was revealed, with a plain but elegant dial and thin, spear-shaped hands, the whole mechanism supported in a gimballed cradle that isolated it quite effectively from the motion of the boat. Colin alone undertook the delicate task of winding it, a ritual performed at the same time each day in order to maintain an even tension in the mainspring. The chronometer’s lovely, silky tick was like a breathless heartbeat. With the sextant, it was a thing of beauty.

The ‘PZX’ Triangle

The ‘PZX’ triangle is at the heart of celestial navigation and can be used to solve a variety of navigational problems.

The angle XPZ is the key to finding the ‘local time at ship’. P is the North Pole, X the ‘geographical position’ of the sun, and Z the position of the ship. The arc XA is the declination of the sun (tabulated in the Nautical Almanac); the arc ZB is the ship’s latitude (typically obtained from a ‘mer alt’). We can calculate the lengths of sides PX and PZ: PX is 90 degrees minus the sun’s declination, while PZ is 90 degrees minus the ship’s latitude. The third side, ZX, is equivalent to the ‘zenith distance’ of the sun, which is obtained by subtracting its altitude (observed with the sextant) from 90 degrees.

Using spherical trigonometry we can now derive the angle XPZ, which is the sun’s Local Hour Angle or LHA – in this case a measure of the time elapsed since the sun crossed the ship’s meridian. The time that has passed since the sun crossed the Greenwich Meridian (revealed by the chronometer) is its Greenwich Hour Angle or GHA. By subtracting the LHA from the GHA the navigator can obtain the required ‘local time’ and thereby the ship’s longitude. Similar calculations can be performed using other celestial bodies.

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Using our chronometer (duly set to Greenwich time) I learned from Colin a rough-and-ready method of determining Saecwen’s longitude. When the weather was clear I would time the moment of sunrise or sunset and compare the results with the times of these events tabulated in the Nautical Almanac. If, for example, the disc of the sun appeared over the eastern horizon at 0600 GMT according to the chronometer while the tabulated time of the same event at Greenwich was 0400, it followed that we were two hours or 30 degrees west of Greenwich. The results I obtained were – at best – accurate to about half a degree either way. In principle the same technique could be used to obtain the longitude by comparing the local and Greenwich times of a heavenly body’s transit across our meridian.

In practice, however, it is difficult to determine the exact moment of sunrise or sunset because atmospheric refraction, which is strongest at low angles, has the effect of ‘lifting’ the sun’s disc so that it remains visible for some time after it has actually dipped below the horizon. The timing of a meridian passage at sea is also problematical as heavenly bodies pause for a significant interval at the height of their arc. One way of doing so is to take two, timed ‘equal altitude’ observations of the relevant body on either side of the meridian and to halve the time difference between them. A major drawback of this method is that clouds may obscure the crucial second sight. It also requires a reasonably accurate clock. And precision is vital: an error of just one minute in the measurement of either local or Greenwich time can result in a positional error of as much as 15 miles.

How then is the navigator to obtain an accurate longitude, even if he or she has an accurate clock? Knowing the exact time at the reference meridian by itself is no help. There has to be something with which that time can be compared. The solution lay in discovering the local time at ship from sextant observations – usually altitudes of the sun in the morning or afternoon, though other heavenly bodies could be used. Mathematicians developed a variety of techniques for achieving this objective, all of which involved solving what came to be known as the ‘PZX’ triangle – see diagram above. These methods – which, as we shall see, relied on knowing the ship’s latitude – remained at the heart of celestial navigation until the emergence of the ‘new navigation’ in the 1870s.

Chapter 7 (#ulink_eed22e59-38f0-5808-9305-d519e5691c50)

Celestial Timekeeping (#ulink_eed22e59-38f0-5808-9305-d519e5691c50)

Day 8: Up again at 0400 and got a sunrise longitude fix of about 45° W at 0515. The same weather – force 5 from WSW with a fair bit of sunshine interspersed with low cloud and rain showers. Much rolling and rattling of crockery and cutlery. Not much speed – only 4 knots.

One week at sea. I tried to measure how far we had gone but failed to realize that on the small-scale North Atlantic chart the latitude scale is not uniform so I got it wildly wrong. Colin filled in the track chart. We have done 830-odd miles but there’s a long way to go.

The change of weather and the prospect of at least two more weeks at sea is depressing. At noon our position was 42° 30' N, 43° 57' W making a day’s run of 102 miles. Not all that fast.

While Harrison laboured, mathematicians and astronomers across Europe were also trying hard to develop a method of determining the time by the ‘lunar-distance method’. The theory underlying the use of ‘lunars’ was that the angular distance between the moon and the sun (planets and certain stars could also be used, with some loss of accuracy) changed so rapidly and predictably that it could be used like a celestial clock – a clock that told the same time anywhere in the world, though it was, of course, not always visible. But the complicated behaviour of the moon – powerfully influenced as it is by the gravity both of the sun and of the earth – made it much harder to predict its celestial coordinates with accuracy than those of the other heavenly bodies.

Although Newton had dazzled the world with the laws of motion that allowed the paths of the sun and its planets to be predicted with hitherto unimaginable precision, the moon had defeated him. But though his lunar tables were not good enough for the purposes of determining longitude, Edmond Halley (1656–1742) recognized that the errors in them recurred regularly every eighteen years and eleven days – in accordance with a well-known cycle of eclipses. This discovery enabled him to develop a rule for correcting the tables, which was later improved by the French astronomer Pierre-Charles Le Monnier (1715–99). There were still imperfections, but in 1750 another Frenchman, Alexis Claude de Clairaut (1713–65), published a new theory of the moon’s motion in response to a competition launched by the St Petersburg Academy in Russia. Though the predictions that accompanied it were incomplete, this represented a major step forward. The great Swiss mathematician Leonhard Euler (1707–83), building on Clairaut’s work, published his own theory of lunar motion in 1753, and it was this that enabled a young, self-taught German astronomer called Tobias Mayer (1723–62) to achieve a major breakthrough.

The effort to find an astronomical solution to the longitude problem was thus a model of scientific internationalism.

Appointed Professor of Mathematics at the University of Göttingen in 1751, Mayer closely analysed the available observational data and, with the help of Euler, prepared new tables of the moon’s motions that proved far more accurate than any previously available. His work first was already under discussion in England as early as 1754. In 1755 the Astronomer Royal, Dr James Bradley, reported to the Board on his examination of Mayer’s tables:

In more than 230 comparisons which I have already made I did not find any difference so great as 1½' between the observed longitude of the Moon and that which I computed by the tables … it seems probable that, during this interval of time, the tables generally gave the Moon’s place true within one minute of a degree.

A more general comparison may perhaps discover larger errors; but those which I have hitherto met with being so small, that even the biggest could occasion an error of but little more than half a degree of longitude, it may be hoped that the tables of the Moon’s motions are exact enough for the purpose of finding at sea the longitude of a ship, provided that the observations that are necessary to be made on shipboard can be taken with sufficient exactness.

With Mayer’s tables it seemed possible that a practical, astronomical method of determining the time accurately on board ship might at last be within reach. To make this lunar-distance method work successfully, however, a very accurate device for measuring angular distances was essential. The standard Hadley quadrant could measure angles only up to 90 degrees, and since the angular separation of the sun and moon often exceeded that amount a larger instrument would plainly be helpful. Mayer himself had proposed the use of a circular device – the ‘reflecting circle’ – but when Captain John Campbell of the Royal Navy was testing Mayer’s tables at sea in 1757 he found it inconvenient to use. Campbell then came up with the simple idea of enlarging Hadley’s quadrant, and so commissioned a leading instrument-maker, John Bird, to make the very first sextant.