Shadwell’s Manual for the Accurate Determination of Meridian Distances by Chronometer



After a long struggle with the Board of Longitude, English woodworker John Harrison was finally awarded, in 1773, the prize for meeting the requirements of the English Longitude Act of 1714 for finding a practical and useful means of determining longitude at sea (the Board of Longitude having being specifically set up to administer the Act). Nevertheless, it was not until the late 19th century that reliable marine timekeepers or chronometers started to emerge in quantity to enable general maritime use for finding longitude.

Also, by this time in the 19th century the (electric) telegraph had been invented and telegraph lines were starting to connect populous places. As described in this 2014 paper Australian Longitudes by “Wire and Wireless”, longitude by telegraph became an acceptable and accurate method of obtaining longitude as it allowed time differences to be measured…over any distance spanned by the telegraph.

Figure 1 : Depiction of Sydney Cove, circa 1831, showing the shipping in Circular Key and built-up Rocks area with Fort McQuarrie (left) the present site of the Sydney Opera House. The longitude of Fort Macquarie became the first principal meridian in Australia. This depiction was made during the voyage of Cyrille Pierre Théodore Laplace (1793–1875) a French navigator famous for his circumnavigation of the globe on board La Favorite (Brochure image).


While indeed the technology for finding longitude now existed, in 1855 then Captain Charles Shadwell first pointed out that unless a rigorous regime was maintained the whole methodology of marine timekeeping on any vessel might be flawed rendering any derived longitude(s) worthless.

Shadwell’s 1861 publication titled Notes on the Management of Chronometers and the Measurement of Meridian Distances, showed that the use of the new timekeepers or chronometers for finding longitude was, for the most part, unreliable (Shadwell’s 1861 edition was a revised version of his earlier 1855 first edition). Thus, he concluded that information from voyages regarding longitude is kept in a state of perpetual fluctuation, from which it is impossible that universal precision can ever be obtained.

The aim of Shadwell’s book was to condense the dispersed information about the application of chronometers to the accurate interpretation of differences of longitude and endeavour to supply naval officers, and others entrusted with the care of chronometers, with a manual of instruction how best to use them, and how to furnish systematic results in recording the meridian distances of the several places visited during their voyages.


Charles Frederick Alexander Shadwell

Admiral Sir Charles Frederick Alexander Shadwell, KCB FRS, was born the fourth son of Sir Lancelot Shadwell in 1814 and joined the Royal Navy in 1827.

A Lieutenant by 1838, in Castor he was present during operations off Syria in 1840. Between 3 December 1841 and 4 July 1846, Shadwell was a Lieutenant in Fly, commanded by Francis Price Blackwood, surveying the Torres Straits and northern coast of Australia. A Commander by 1846, in 1850 he was commanding HMS Sphynx and took part in the Second Anglo-Burmese War of 1852. Promoted Captain in February 1853, from 1 August 1856 to 2 January 1860, in Highflyer (from commissioning at Portsmouth), was present during the Second Anglo-Chinese War including the capture of Canton in December 1857, and attack on Peiho or Taku Forts on 25 June 1859, when a wound rendered him permanently lame. From 1861 he commanded HMS Aboukir and then from 1862 HMS Hastings (from commissioning) the flagship of Rear-Admiral Lewis Tobias Jones.

Appointed Captain-Superintendent of Gosport victualling yard and of the Royal Haslar Hospital in June 1864 Shadwell was promoted Rear-Admiral in January 1869 and later became Commander in Chief, China Station 1871-1874. On his posting to China he vacated his appointment as Naval aide-de-camp to the Queen, which he had received in March 1866. Promoted Vice-Admiral in April 1875, in 1877 he received the flag officers' pension for meritorious service.

Shadwell had been elected a Fellow of the Royal Society (FRS) in 1861 and in 1873 was gazetted a Knight Commander of the Bath (KCB) and later in 1878 made President of the Royal Naval College, Greenwich.

After Shadwell retired in 1879, he lived at Meadow Bank in Melksham in Wiltshire, dying unmarried in March 1886 at age 72 (his tribute from the February 1887, Royal Astronomical Society is at Appendix A).


Longitude from the observation of Lunar Distance

While John Harrison (1693-1776) was perfecting his timekeeper his nemesis, depending on whom you believe, Nevil Maskelyne (1732-1811) the then Astronomer Royal, accepted the chair of the Board of Longitude in 1765. With his background Maskelyne believed longitude could be found from the heavens with astronomy, as was latitude. This belief and being chair of the Board is reported to have clouded his judgement when it came to assessing other options for finding longitude. In Maskeleyne’s defence it is reported that while using Harrison's time piece H4 was considered more accurate for finding longitude, Maskelyne's method combined with his Tables was far cheaper. Maskelyne’s tables at then a few shillings against the many pounds for a timekeeper saw Maskeleyne’s method predominate for the next century.

Maskeleyne’s method for finding longitude from astronomy, came in the form of an observation called Lunar Distances. To determine his longitude by this method, an observer had to measure the angle between the centre of the Moon and a listed star (the lunar distance) along with both their altitudes. Next, he had to calculate his own local time and correct the Moon’s position for the twin effects of parallax and refraction. From the Nautical Almanac the time at Greenwich when the Moon was at the same calculated position was then extracted. The difference between the Greenwich time and the observer’s local time, then gave the difference in longitude from the meridian of the Royal Observatory at Greenwich. To make the whole process as simple as possible, Maskelyne proposed the Board of Longitude publish a Nautical Ephemeris. What came to be called The Nautical Almanac and Astronomical Ephemeris was first prepared for the year 1767 and published in 1766. Some 2,000 copies (price 2 shillings and 6 pence) were printed. At the same time Maskelyne also published his Tables Requisite to be Used with the Astronomical and Nautical Ephemeris to serve as a handbook for navigators using lunar distances. These Tables Requisite, were published from time to time, and it is said that the 1781 edition sold 10,000 copies immediately on publication.

When published, The Nautical Almanac contained the lunar tables of German mathematician Tobias Mayer (1723-1762). In 1755 Mayer had submitted to the British government an amended body of manuscript tables, which James Bradley (1693-1762), Astronomer Royal from 1742-1762, compared with the Greenwich observations. Bradley found Mayer’s work to be sufficiently accurate to determine the moon's position to 5 seconds of arc, and consequently longitude at sea could be determined to within around half a degree of arc (50 kilometres approximately). In consideration of Mayer’s contibution to the finding of longitude at sea, his widow received from the Board of Longitude £3,000 (equivalent to £408,000 in 2018).

The Nautical Almanac for 1906 was the last to publish tables for the Lunar Distance method for finding longitude.


Shadwell’s Introductory Observations

The Lunar Distance method for finding longitude in conjunction with the tables of The Nautical Almanac had existed for some 100 years when Shadwell’s publication became available. Nevertheless, timekeepers had now become more reliable and readily available and Maskeleyne’s cheap alternative was now being challenged by the more accurate method. In this changing world, Shadwell’s opening chapter provided an overview of the then state of play with what he terms maritime geography.


The application of chronometers to the accurate determination of "Meridian Distances," or the differences of longitude of distant stations, has usually formed an important object in the scientific voyages of modern times, from those undertaken in the last century by the illustrious Cook, to those performed in our own day by more recent navigators.

The happy invention of the Electric Telegraph, the successful accomplishment of its submarine connexion, and its application to astronomical purposes, would seem to have completely and successfully solved the problem of differences of longitude of stations which are either situated on the same continent, or, if occupying insular positions, only separated by narrow seas; and there can be little doubt that, before long, the various observatories in the British Islands and on the Continent will, by its means, be accurately linked together, and their relative positions consequently determined to the last degree of mathematical correctness. Considered as base stations, they may then be viewed as forming salient points in a network of triangles described on the surface of our globe, to which minor places can afterwards be conveniently referred by geodetic means, so as ultimately to combine them in one comprehensive whole, hitherto unexampled for accuracy in the annals of geographical science.

From the very nature of its invention, however, and from the probable limits to its use, interposed by the difficulties of its submarine connexion, except in narrow seas, the application of the Electric Telegraph to the question of Terrestrial Longitude will doubtless be comparatively very limited; and it must still be to the successful appliance of the ordinary means at the disposal of the navigator that we must continue to look for the final solution of questions relating to the relative longitudes of outlying stations on the ocean, and their connexion with the fixed points on the great continents.

Fortunately, at a period when the successful application of the system of galvanic signals to astronomical purposes has given a great impetus to the final solution of the questions concerning the relative longitude of stations on land, the gradual improvements effected of late years in the construction of marine chronometers, and the yearly increasing extension of the application of steam to ocean navigation, seem at the same time to afford increased facilities to the Navigator for the improvement of Maritime Geography, as he will thus be enabled also to maintain an honourable rivalry with the Astronomer, and, like him, to contribute his fair share towards the ultimate perfection of geographical science.

In examining the earlier history of modern hydrography, and on inquiring into the circumstances which have hitherto impeded its progress towards final perfection, notwithstanding the zealous and useful labours of numerous scientific navigators, and the voluntary contributions of many intelligent commanders, two causes will, we apprehend, be found to have tended, although accidentally, to retard its satisfactory development; first, the practice of mixing up in one indiscriminate combination the astronomical data for the positive settlement of disputed positions, and the relative evidence afforded by chronometric measurements; and, secondly, a want of clearness of comprehension of the relative values of absolute and differential longitudes.

For instance, nothing is more common than to find, on examining even comparatively recent works on hydrography, that the data quoted for the settlement of a given position are of the most miscellaneous and incommensurable character : lunar observations, eclipses of Jupiter's satellites, occultations of stars by the moon, solar eclipses, and chronometric measurements by various authorities, from adjacent and independent points, all blended together in one crude and inharmonious result; or again, than to find chronometric determinations of a purely relative character, and often measured from two or more independent stations, confounded with absolute results.

Places are fixed absolutely by astronomical observations, or relatively by chronometers. This distinction must be clearly kept in view if ever we wish to arrive at final and conclusive results, and if we desire to avoid the perpetual oscillation of ideas which a mixture of the two principles is sure to entai [first person singular past historic of ‘enter’] on us.

Much has been done of late years towards simplifying the conclusions of maritime geography, by collecting from the records of astronomical and chronometric observations satisfactory details for the final establishment of several important fundamental positions.

…Many concurrent circumstances of a favourable nature, and peculiar to the present time, seem to be conducive to the more systematic application of chronometers on board ship to the objects of science in the measurement of differences of longitude. The mechanical construction of chronometers has attained a high and unexampled degree of perfection, these improvements having at the same time been accompanied by a very considerable reduction in their cost; so that chronometers are now no longer rare instruments only within reach of the wealthy; and in lieu of one solitary chronometer as formerly, it is now not unusual to find three or more good chronometers on board every ship. The more general diffusion of a good practical education among young officers, both in the royal and mercantile navies, at the same time renders them more capable of applying chronometers to the accurate purposes of science and better able to appreciate their results. The increasing application of steam machinery, moreover, to men of war, whereby the average duration of passages will be much shortened and return voyages greatly facilitated, and the extension of lines of ocean steam navigation by the great mercantile companies to all parts of the globe, seem to afford, simultaneously with the abovementioned circumstances, increased facilities for the systematic and careful measurement of chains of meridian distances.

If this view of this important question be correct, and the writer's partiality for a favourite subject has not caused him to take a too favourable view of present circumstances, it is important to consider whether the time has not arrived when it might be advisable to attempt to concentrate and condense in a practical form the necessary instructions for the guidance of those, who may be desirous of undertaking the accurate and systematic measurement of chronometric differences of longitude.

…Speaking generally, we apprehend that, unless from peculiarly favourable circumstances of previous service in a surveying ship, on a scientific voyage, or under a scientific chief, officers in general have no organised knowledge of the minutiae and details requisite to be attended to in the accurate measurement of meridian distances. No single book supplies the required information, and an officer furnished with some good chronometers, and desirous of advantageously employing them, has probably to grope his way as best he may, and devise a system for himself.

This want we propose to endeavour to remedy. With this object in view, in the following pages we shall endeavour to collect together and arrange various hints relating to the custody and management of chronometers on board ship, at present existing only in a traditional form, or to be found scattered amid many books often not easily accessible. The questions relating to errors and rates next engage attention; and subsequently, the formulae for the determination of meridian distances are discussed and arranged in an organised and systematic manner, worthy, it is hoped, of the present advanced state of hydrographic science.

…The utility of chronometric determinations, and the value of their results, as well as the possibility of their advantageous incorporation with the previous labours of others, much depends on the degree of care bestowed on the minute details of their manipulations. Many series of observations have often had their value materially impaired by the accidental neglect of some small particulars; and doubtless a few measurements executed with a due regard to accuracy of detail, are many times more valuable than much larger masses of observation reduced and recorded in a loose, uncertain, and unsystematic manner.

Chronometric determinations obtained with no special regard to accuracy, reduced approximately, and recorded vaguely, although perhaps formerly valuable contributions to our then knowledge, are unsuited at present to the existing condition of maritime geography.…As a contribution towards the improvement of geographical science, and in furtherance of the development of the above views, these pages have been undertaken : and if in the hands of scientific officers or intelligent travellers they are found at all conducive to the systematic realisation of this important subject, they will not have been written in vain, or the author's labour lost.


By contrast in an emerging Australia, at the time of Shadwell’s 1855 publication the name of Van Diemen's Land had just been changed to Tasmania, the Colony of Victoria had only been separated from the Colony of New South Wales for a few years, and the Colony of Queensland was yet to be determined.


Meridian Distances

The Meridian Distance or difference of longitude in time between any two places, is obtained chronometrically by comparing the errors on local mean time shown by a chronometer at the two places in succession; the error at the first place being corrected by the known rate of the chronometer in the interval, so as to give the state of the watch at the moment of the second observation. The errors of the chronometers being thus known simultaneously at the two places, their difference represents the meridian distance or difference of longitude in time, between them.


Figure 2 : Example of using a marine chronometer to find longitude.

A meridian is another name for a line of longitude. Meridians run from pole to pole and are notated from 0 (zero) degrees thru Greenwich England to 180 degrees east and west. A Meridian Distances is just the difference in longitude between two meridians of longitude. Meridians are related to time by the spinning motion of the earth such that 360 degrees equals 24 hours. As depicted in Figure 2 above, the Meridian Distance between longitudes 0 and 30 degrees west is two hours of time as there are 15 degrees of arc per hour of time. From this relationship between time and longitude it can be seen that time needed to be measured to decimals of a second as 0.1 second of time equated to 1.5 seconds of longitude or about 40 metres of distance, if very precise longitude was required. 

A ship would generally carry several chromometers from different makers, scientific vessels many more. On Cook’s second voyage when he departed in July 1772, Larcum Kendall's No.1 (K1) and John Arnold's No.3 timekeepers went with him on Resolution, and John Arnold's Nos.1 and 2 timekeepers went in Tobias Furneaux's Adventure (K1 was a copy of John Harrison's fourth timekeeper (H4)). Cook on his third voyage of 1776-1780 again took K1 in his own ship while Larcum Kendall's No.3 (K3) went in Discovery. Table 1 below lists the twelve chromometers aboard HMS Fly employed surveying the coast of Australia from March 1842 to April 1846. Circa 1825, Heinrich Christian Schumacher (1780-1850) is reported to have carried eighty-six (86) ship’s chromometers over European roads of the time in an effort to establish the longitude differences between Greenwich, Copenhagen, Altona and other places.


Table 1 : Chronometers of HMS Fly employed surveying on the coasts of Australia, during a period of four years,

from March 1842 to April 1846.


Serial Number

























Parkinson & Frodsham



Porthouse (Pocket)









Parkinson & Frodsham (Pocket)


All these chronometers were the property of Her Majesty's Government, except Chronometer C, which belonged to Captain Blackwood.

Z was employed as the standard; C was also frequently taken on shore for the purposes of astronomical observation.

The other chronometers (excepting A and B) were never moved from the chronometer room, from the time they were received on board at Devonport, in March 1842, till the ship's return there in June 1846, except when the Fly was hove down for repairs at Sydney in October 1845.

Chronometers A and B having stopped on various occasions were repaired at Hobart and Sydney. The continuity of their performances was thus interrupted.

The rates were always determined by observations, made on shore, with the artificial horizon, and usually by the method of equal altitudes.


The chronometer considered most reliable in running consistently ie not fast one day and slow the next or gain or lose time unpredictably would be selected as the standard against which all the other chronometers would be rated (compared). These rates would be recorded and used to adjust the time interval given by a particular chronometer. The adjusted time interval was thus considered to be equal to the time interval given by that chronometer had it been able to be theoretically read at both places at the same instant. The time interval or meridian distance obtained from all chronometers should then be the same. It was from this time interval that the meridian distance and hence longitude was derived.

Importantly the chronometers were used to measure time intervals rather than tell absolute time. While simple mathematics could be used to determine average rates of time gained or lost, Shadwell suggested that when opportunity allowed the rates of time gained or lost should be calculated using the Method of Least Squares. The value so derived by this method would minimise the sum of the squares of the differences between this value and the observed values for every single observation.

Additionally Shadwell noticed, as happened on scientific and related voyages, observations on the sun, stars, moon and planets were used to modify the timed meridian distance. Unless all the pertinent data was provided regarding such observations then the validity of the resultant longitude could be difficult to verify. 

Perhaps the major issue Shadwell had with existing procedures was that the chronometers were not protected against changes in temperature nor was temperature a consideration when Meridian Distances were extracted. Shadwell provided an example from Harrison's timekeeper after its second trial voyage to the West Indies in 1764. Harrison had provided the British Admiralty before the voyage, with the equivalent of a calibration certificate for his timekeeper. The certificate showed the expected rate of the timekeeper at every ten degrees of temperature from 42° to 82°F. After the voyage Harrison’s timekeeper was found to have gained 54 seconds during the 156 day journey and thus it met the Admiralty’s criteria. If, however, as Shadwell points out allowance be made for the variation of the thermometer, as stated by him [Harrison] before his departure, it will be found to have lost only 15 seconds. Shadwell goes on to say that English navigators appear to have been as equally unmind­ful of the refinements of temperature corrections…. No allusion is made to the subject in the account of Cook's voyages, nor does the matter seem to have engaged the attention of the commanders of our various scientific voyages, and hydrographic expeditions, during the present century, in any practical degree; although the failures and anomalies which sometimes presented themselves in chronometric measurements, have often been attributed to excessive or irregular fluctuations of temperature.


Figure 3 : Relative sizes of Harrison’s H1 to H4 with far right Kendall’s K1; overall approximately a 90% reduction in size.


The Determination of the Errors and Rates of Chronometers

Having discussed accommodating the chronometers aboard ship, their winding and the minimisation of environmental effects, Shadwell moves on to describe how the errors and rates of the ship’s chronometers can be determined. The three modes, summarised below he stated, are available to officers serving on board ship, and within compass of the means ordinarily at their disposal.


Transit observations

Such observations were generally only undertaken by ships on scientific voyages or surveying duties where they were supplied with the appropriate instrumentation. Thus such instrumentation will very rarely be found among the apparatus employed on board ships engaged in the ordinary duties of the service.

Those requiring a sextant

At sea, altitudes observed with a sextant using the sea horizon, even under the most favourable circumstances, were liable to be impacted by; the inaccuracy of the contact with the celestial object; the uncertain effects of refraction on the appearance of the horizon; and errors arising from an inaccurate estimation of the correction for dip (of the horizon). Thus while the simplest and most convenient kind of astronomical observations available for the determination of time were those which could be made with a sextant, the observations must be made on land using an artificial horizon.

Observations made on shore with a sextant and artificial horizon allow contact with the celestial object to be made with great precision, while the effects of refraction and estimation of dip are negated. Please refer to Figure 4 below where the use of an artificial horizon is shown.

The best artificial or mercurial horizon comprised a shallow trough filled with mercury, and screened from the wind when necessary, by a glass cover. The mercury was stored in a bottle and decanted into the trough for use and then poured back into the bottle when the observations were completed. While it was best to avoid using the glass cover, if its use was unavoidable, error due to imperfection in, or non-parallelism of, the surface of the glass had to be minimised. This was achieved by marking the glass cover and trough so that the cover could always be placed over the trough in exactly the same position. Any errors introduced by the cover would then be equal for all observations. Alternatively, the cover could be reversed during the obser­vations; half the observations being made with the cover in one position, and half in the other position, thus neutralising any errors.

The sextant used for such observations should not be indifferently hacked about for other purposes; it is of great importance that an observer should be able to place reliance on the trustworthiness of his instrument, and on the stability of its index error. If, therefore, he has the command of more than one instrument, the best may be suitably reserved for the finer kinds of observation, such as lunars at sea, and determining the latitude and time on shore; and the other one to the common observations required in the ordinary navigation of the ship at sea.

Care should be taken, especially in tropical climates, that the sextant is not unnecessarily exposed to the action of the sun's rays, and, when not actually in use during the pauses between the observations, it should be kept in the shade, or screened with a handkerchief : nothing tends to ruin sextants more, than unnecessarily roasting them in the sun.

The methods of observation for the determination of time was either equal or corresponding altitudes or single or independent altitudes. Equal or cor­responding altitudes observations were preferred as any errors in knowing the latitude of the place of observation, the declination of the celestial object observed, and its altitude as recorded by the observer and his sextant were negated.

The celestial objects observed was either the sun or stars. The sun was the most convenient, and generally preferred.

Preconcerted signals

Such signals refer to where the precise moment of mean time at a place was indicated by the fall of time-balls, the display of signal-flags, the flashes of guns, etc as well as any special measures which may occasionally be devised between different ships for the intercomparison of their chronometers.

As well as understanding the signal(s) and what it/they represented, regarding the local time, arrangements must be in place aboard ship to transmit that instant from where the phenomenon was observed on the ship’s deck to its chronometer room.


Figure 4 : Diagrams showing the use of an artificial horizon;

(top) at sea the altitude of the celestial body with reference to the sea/sky horizon must be corrected for the dip of the horizon;

(bottom) the observer views the celestial body both directly and reflected from the artificial horizon

removing the need to account for dip and being able to determine the sea/sky horizon.


After discussing all these issues and then resolving them with procedures, relevant mathematical formulae and worked examples, Shadwell then sets out his Manual.


Shadwell’s Instructions for the Management and use of Chronometers

Reception and Stowage.— When chronometers are received on board ship, it is of importance that they should be at once stowed in the place prepared for their reception, in the position which it is intended they should permanently occupy; and when once suitably located, they should on no account be subjected to subsequent removal or displacement. The possible contingency of the ship being docked for extensive repairs, in which case their temporary removal would be unavoidable, is the sole exception to this general rule, excluding, of course, the accidental necessity for sending any particular chronometer on shore for the purposes of repair.

In selecting a place for their reception, much must of course depend on the size and accommodation of the ship, the par­ticular nature of the service on which she is employed, and on the guidance of other circumstances; but, if possible, under favourable auspices, the following general conditions should be attended to:—

The chronometers should be placed low down in the ship (both because there is there less motion, and because the tempe­rature is more equable); amidships; as far from the extremities, and as near the centre of motion, as convenient; not near the chain cables or other large masses of iron, so as to ensure freedom from the possible disturbance of magnetic influence; not in drawers, where the tremor caused by opening and shutting them acts injuriously on their balances, nor suspended from the deck in cots or swinging tables, which has been proved by experience to be objectionable.

The master's cabin in men-of-war is also objectionable, because there the chronometers are not amidships. So also the casing of the rudder-head in the captain's cabin, where we have known them to be placed, and where, doubtless, they looked very ornamental and scientific, is likewise to be deprecated, because they are there subjected to the more violent motion of the extremity of the ship, to the possible vibration of the rudder, and to the injurious influence of draughts of cold or damp air from the stern-windows: we believe, however, that this practice has now very properly been discontinued.

In vessels fitted with screw-propellers, the chronometers should be placed sufficiently far forward, as not to be affected by the vibration of the screw when under steam.

In Her Majesty's ships a suitable place is now generally fitted for the reception of the chronometers in the after-cockpit or orlop-deck. In frigates, the after-part of the steerage, while unobjectionable in other respects, would appear to be both acces­sible and convenient. In flush-deck vessels the fore-part of the captain's cabin, amidships, would seem to be the most appro­priate place; and in merchant-ships, the master's cabin, amid­ships, and as far forward as possible, perhaps offers the only eligible position.

The best mode of stowing them seems to be as follows:—

A box, divided into as many partitions as there are chrono­meters to be stowed in it, should be securely attached by screws to a solid block of wood, about thirty inches in height, and firmly bolted to the beams of the deck below. Each partition should in depth be about equal to that of the largest box of the chronometers to be placed in it, and in length and breadth about two inches longer than the sides of the box it is intended to receive; the partitions, moreover, should be furnished either with separate lids, or there should be one general lid to close the whole box. Great care should be taken that the block and partitioned box thus prepared should be entirely detached from all contact with contiguous stanchions or bulkheads; and the block and box, moreover, should be surrounded with a strong external casing, the sides and lid of which should on no account be permitted to touch it, a clear space of at least two inches being left all round.

Each chronometer in its box thus prepared, and moving freely in its gimbals, is then to be placed in the space allotted to it on a bed of horsehair, cotton, or shreds of bunting, about three inches thick, the interstitial spaces around its sides being stuffed with the same material to within half an inch of the top of the inner box. Of the three substances named above, we prefer horsehair.


Effects of Magnetism.— Undoubted instances are on record of the performance of chronometers being affected by the action of magnetic influences, chiefly owing to the fact of the balances having accidentally acquired polarity. Since attention has been called to this subject and greater caution exercised in the con­struction of the balances and balance-springs so as to avoid this source of error, instances of this kind are no doubt now extremely rare, and quite exceptional, in modern chronometers of good construction. We should, however, be taking a very limited view of the subject, were we solely to confine our attention to the consideration of permanent polarity in chronometer balances, and lose sight of a possibly more prolific source of error, which, in all probability, is caused by the induced magnetism of considerable masses of iron on ship-board, in various stages of development, varying in different positions of the ship, and change of magnetic latitude.

Owing, no doubt, to the difficulty of arriving at definite con­clusions on a matter so subtle and minute, we think, requires us to recommend precautionary measures to obviate them. The following precautions are therefore suggested:—

The marks on the dial-plates of the chronometers should all occupy the same relative positions; that is, the line joining the twelfth and sixth hour-marks should all be parallel to one another, in order that, retaining the same invariable position with reference to the "fore-and-aft" line of the ship, they may be similarly affected by the possible local magnetic attraction of the ship's iron, and that in cases where the balances of the chrono­meters have accidentally acquired any degree of polarity, their mutual influences on each other may be reciprocal.

Proximity to vertical iron stanchions or knees, arm-stands of muskets or rifles, any large masses of unmagnetised iron, and in iron-built ships vertical iron bulk-heads, should also, in all cases, be very carefully avoided.


Effects of Temperature.— Ever since, in scientific voyages of modern times, the value of the chronometer has been recognised, as an instrument capable of being employed in the accurate determination of differences of longitude, changes of temperature have been universally admitted to be one of the principal causes of marked changes of rate. Great attention has in consequence been paid by the makers of these instruments to the mechanical details of their construction, to the improvement of their balances, and to the adjustment of their compensations. Considerable ingenuity has been employed, and with much success, in en­deavours to eliminate or control the fluctuations of rate produced by variations of temperature.

The general tendency of alteration of temperature in an uncompensated chronometer is, that an increase of temperature causes the rate to be retarded, or to alter in a losing direction; a decrease of temperature, on the contrary, accelerates the rates, or produces an alteration in a gaining direction. The object of the compensation is to correct this defect, and to produce uniformity of rate in spite of alteration of temperature.

If the size and arrangements of the ship permit, a separate cabin may be advantageously appropriated for the reception of the chronometers, a solid block and fittings, as already described, being placed within it. Such an arrangement affords the means, moreover, of adopting measures for maintaining in the chrono­meter-room "uniformity of temperature" which, under the circum­stances already discussed, is a point of great importance. When the temperature of the chronometer-box was maintained at a uniform standard by means of a lamp; the admission of air being regulated by an aperture, the size of which could be adjusted at pleasure by means of a slide, the temperature of the interior air being shown by a thermometer, the watches performed their functions with extreme regularity.

Another precaution to protect the chronometers from sudden draughts of cold air and to prevent the radiation of heat was its covering with a coarse woollen (fearnought) coat over the partition containing the "standard" chronometer. A flap was made in the coat, so that at any moment a comparison with the "standard" could be obtained without uncovering the rest.


Winding-up Chronometers.— The chronometers having been received on board, and stowed in the manner previously described in the place appropriated for their reception, it is of importance at once to commence and adopt a uniform and systematic manner of winding them up, and comparing them daily with one another.

Methodical arrangements in these particulars favour the stability of rate of the chronometers, assist in the detection of irregularities, and diminish the probabilities of their being accidentally allowed to run down; while in the reduction of chronometric observations for the determination of meridian distances, systematic plans of comparison are indispensable to accuracy of computation, and very influential in diminishing the amount of labour required, if the number of chronometers employed is at all large.

It is of course a matter of purely arbitrary convenience what time is selected for the daily winding and comparison of the chronometers; the only point of importance is, that what­ever arrangement be adopted, it be uniformly and systematically adhered to.

On the whole, we think 8 A.M. to be a time conveniently adapted to this duty, and consistent with the general arrange­ments of a man-of-war. The officers generally charged with this important function are not, under ordinary circumstances, likely to be then absent on detached duty, nor are they usually engaged in the customary morning evolutions. In cases, moreover, where it is proposed to take sights for time, during the forenoon, the usual morning comparison may be made subservient to that end, and taking the place of the first comparison before the observa­tions, may save the necessity for an extra special comparison.

The chronometers should be wound first and compared after­wards. In winding them they should habitually be attended to, in the same order, from day to day, one by one, as they lie in their places; so that the mechanical habits of regularity in this particular may be a safeguard against the caprice of memory or accidental distraction from any disturbing cause. From want of system in this particular, we have known instances where the attention of the officer engaged in this duty being accidentally distracted during its performance, the chronometers have been compared, but some or all of them not wound up, that operation being forgotten, and the omission not detected till the chrono­meters ran down.

In winding up chronometers the turns of the key should always be counted, and the last turn made gently and carefully, until it is felt to butt. It is said, that it has sometimes happened to persons over careful, that they have let their chronometers run down by having counted the number of turns, and never winding up close for fear of injury to the chain or works, by which they have always lost a little of the chain each day; so that, after a time, the chronometer is found to stop just at the time it should be wound up.

In winding up box-chronometers, the chronometer should be inverted carefully in its gimbals, held firmly in the left hand, and the key pressed close home with the right; after the operation is performed, and the key withdrawn, care should be taken that the keyhole is again covered with the slider to secure it from dust or damp, and then the chronometer should be gently eased down into its natural position without violence or jerk.

Many chronometers are fitted with permanent keys at their backs, which in some respects seem preferable to moveable ones, especially if they are fitted with rack-work to prevent the ill effects of attempting to turn them the wrong way.

In winding up pocket-chronometers, the watch should be held firmly in the left hand with the wrist pressed gently against the breast, the key should be turned equably and steadily with the right, care being taken to avoid giving the watch any circular motion upon the key. The common but vicious practice of turn­ing the left as well as the right hand is injurious, for two reasons: first, because the circular motion affects the regularity of the balance; and, secondly, because the compound motion of the two hands doubles the velocity of winding, and increases the chances of straining or snapping the spring from the jerk at the conclusion of the operation.

Chronometers and watches which are wound at the back usually require the key to be turned from right to left. Those having their keyholes on the dial-plate, on the contrary, from left to right.

The chronometers should be wound daily, whether con­structed to go for one or two days; eight-day chronometers once a week — say on Sunday, a day easily remembered. If the number of chronometers is large, the precaution should be taken of examining them after winding, to see that none have been accidentally forgotten, either by looking at the winding index, if furnished with that apparatus, or by trying the key.

Standard Chronometer.— The chronometers having been all wound up, they should next be compared. To facilitate their systematic comparison, and to organise more easily the reduction of chronometric observations, it is customary to select one chrono­meter as a "standard" to which all observations for the determi­nation of the time are in the first instance referred; the indication of the other chronometers at the same moment being subsequently obtained by means of the comparisons. The chronometer selected to perform the duty of the "standard" should be one of first-rate character, and by a maker of established repute : it will be convenient that it should have a clear and distinct beat to half-seconds; also, that its dial-plate be well marked; and it is advis­able, although not indispensable, that its rate (at any rate at starting on the voyage) should be small. Stability of rate, how­ever, is at all times much more important than the smallness of its amount. For the facility of comparison with the other chrono­meters it will be found convenient to arrange the chronometers in their box in such a manner that the standard shall occupy a central position among them. The XII and VI hour marks should all be parallel to one another, and to the "fore and aft" line of the ship.

Distinction of Chronometers.— As the description of chrono­meters by means of their makers' names and numbers is tedious and troublesome, it is often customary to denote each chronometer emphatically by a single letter, A, B, or C, etc., an arrange­ment sufficiently distinctive to the persons who have to use them, and at the same time brief and concise when a series of results obtained from them has to be recorded; care being taken that in the commencement of the chronometric journal, and also in the records of their performances transmitted at any time to head­quarters, a preliminary notation is made of the actual maker's name and number belonging to each chronometer.

In accordance with this arrangement, it will be found con­venient emphatically to denote the "standard chronometer" by the letter Z, the final letter of the alphabet, the other chrono­meters being represented by the initial letters, A, B, C, D, etc., in order of alphabetical sequence, or by the initial letter of their makers' names, if preferred: such as A, for Arnold; D, for Dent; F, for Frodsham ; M, for Murray, etc.; these little arrange­ments being, of course, matters of mere fancy and convenience.


Comparison of Chronometers.— It is very desirable, when practi­cable, that the comparisons of the chronometers with the standard should all be made by one person. The effects of personal equa­tion are thereby much simplified, and the chances of small contin­gent errors materially diminished.

The mode of proceeding is as follows : the observer, with book in hand, containing a ruled form for the entry of the time shown by the several chronometers takes a beat from the standard five seconds before the arrival of the second hand at any five or ten seconds mark, and then quickly casting his eye on the chrono­meter to be compared, counts with his ear the tent beats which elapse before the second-hand arrives at the mark selected; at the completion of the interval he reads the other chronometer, and the comparison is effected. The operation being repeated a second time to correct the first judgment, the final observation is recorded. It will be found convenient to make the first comparison at 50s, or at 20s, and again at the minute or half minute for the final result.

This is done in succession for each chronometer. Half a minute elapsing between each comparison affords ample time to record the last one and to prepare for the next, while the fact of the standard being only noted at the minute and half-minute, makes the notation of its time a mere matter of form, and allows the writer's attention to be concentrated on the "second" of the other chronometer. When the round is completed, he takes the precaution of seeing that the "minute" of the standard has been duly noted, so as to ensure that no constant error affects all the comparisons.

Since, as has been previously observed, change of temperature is supposed generally to be the principal cause of marked changes of rate, it is desirable that the temperature of the chronometer-box be duly recorded, and accordingly means should be devised for placing within it a maximum and minimum thermometer, the indications of which, at the time of winding, should be duly noted daily in the comparison book, so that if decided fluctuations of rate are observable in any of the chronometers, its cause may be sought among the records of the changes of temperature. The mean of the readings of the maximum and minimum thermometer is to be taken as the mean temperature for the preceding 24 hours. The height of the barometer may also with propriety be noted at the same time.

As a matter of technical convenience, and as favouring uniformity of treatment, it is advisable that, for the purposes of com­parison, the "standard" be always considered fast on the other chronometers, whatever their actual indications may be, 12 hours being added to the time shown by the standard in taking their difference when necessary.

In Her Majesty's Navy it was formerly customary to furnish every ship with one chronometer. If the captain supplied a private one in addition, then the Government gave another, so as to make three chronometers in all. It was argued that if a ship had but one chronometer, it would be unwise implicitly to trust it; and, therefore, great caution was necessary in navigation. If the ship had two, and they happened to differ, it would be impossible to tell which was right. If, however, she had three, the coincidence of any two of them would throw a strong probability on the truth of their results; while the mean of the three could probably be more safely relied on than any one of them taken singly; added to which, the examination of their intercomparisons gave the means of detecting which of the three was irregular.

By a recent more liberal regulation, the Admiralty now furnish all sea-going ships with three chronometers.


Chronometer Journal.— A book (styled the "Chronometer Journal") should be kept for the daily record of the comparisons of the chronometers. The form No.I, seems con­veniently adapted for all practical purposes.

The sole use of the chronometer journal is to enable the persons engaged in the management of the chronometers to detect irregularities, and to estimate the value of their results; and also to obtain, when necessary, by interpolation, their comparisons with the standard for any required moment when special compa­risons have been omitted to be made.

Besides the "Chronometer Journal" it will also be found advisable to devote specially to the service of the chronometers, and to the computation of the meridian distances, a "Sight-Book" and a "Chronometer Work-Book". In the former are to be entered all observations of "single or equal altitudes" for the determina­tion of time, errors, and rates, together with all occasional special comparisons and memoranda connected therewith. The latter should be exclusively appropriated to the calculations and reduc­tions of the observations, for the determination of the errors and rates, and the computation of the meridian distances.

The "Chronometer Sight-Book" and "Work-Book" being as above recommended wholly and exclusively reserved for their own specific uses, all miscellaneous memoranda of common obser­vations connected with the keeping of the ship's reckoning, or the calculations relative thereto, as well as generally all extraneous matter of any kind whatsoever, are to be rigidly excluded from them, and made in other books destined particularly for those uses. It may often happen, amid the multiplicity of affairs which engage an officer's attention on board ship, that the reduction of the observations relating to the chronometers cannot he made till a leisure opportunity offers, sometime after they have been taken. It is then a great satisfaction to find all the necessary data clearly arranged, and recorded in such a manner that they may be confidently relied on.

Some of these remarks and recommendations may appear at first sight rather dogmatical and unnecessary; and, perhaps, only those who have been much engaged in operations of this nature can fully appreciate the advantages of methodical and systematic habits of comparison, observation, and computation, or can ade­quately realise the saving of time and labour involved in a careful attention to these minutiae.

Assistant-Watch.— We have already established, as a funda­mental maxim, that when the chronometers have been received on board and once stowed in their places, they should on no account be. subsequently removed. The practice, therefore, of taking a chronometer on deck or on shore, for the purpose of observation, should never be resorted to; a pocket-chronometer, or a good pocket-watch with a second-hand, should on these occasions be used as an assistant. The actual time of any observed phenomena being then, in the first instance, taken by the assistant-watch; the corresponding times shown by the chrono­meters at the moment are subsequently obtained by applying the comparison.

In comparing the assistant with the standard it will be found convenient, as in other cases, to consider the standard always fast.

It will be advisable, if possible, to appropriate a pocket-chronometer to the purpose of performing the duty of assistant-watch; if a pocket-chronometer is not available, or cannot be spared, a good pocket-watch with a second-hand must be substituted: in either case, in order to give the watch so employed a fair chance of maintaining the stability of its rate, it should be kept in the chronometer-box, except when actually in use, and subjected to the same careful usage and delicacy of treatment as the other chronometers. Pocket-chronometers, or good watches employed as such, notwithstanding their name, should never be worn in the pocket. When carried in the pocket, they are placed in a vertical position, and subjected to the warmth arising from contact with the body. When removed from the pocket for use, they are held in a horizontal position, and probably subjected to a con­siderable alteration of temperature. The alteration of position is liable to disturb the regularity of the balance, and the varia­tion of temperature to produce fluctuations of rate. A chro­nometer is merely a machine, after all; and a perfect chrono­meter, like a perfect circle or a perfect straight line, only a happy mathematical accident; and it is therefore unreasonable to suppose that it can be expected to perform its functions accu­rately, unless delicately treated and subjected to uniformity of physical conditions.

It will be advisable, therefore, to keep the assistant-watch in a small box, which, when carried on shore or on deck for the purpose of observation, should be maintained in a horizontal position, and, in its passage to and fro, should always be carried by hand. Care should also be taken that, when the box is open for the purpose of noting time, the watch should be kept in the shade, and not exposed to the direct action of the sun's rays, which, especially in a tropical climate, is likely to be very injurious.

Miscellaneous Observations.— When it may be necessary to move box-chronometers from the shore to the ship, or on other occasions, they should be locked in their gimbals, so as to prevent an unusual oscillatory motion and to retain their horizontal position, under which condition alone their balances act with perfect fairness. In carrying them by hand, the best way is to sling them in a kerchief, passing the bight under the bottom of the box and the ends through the handles, tying them at the top in a reef-knot; care should be taken not to give the chronometers a vibrating or oscillating motion, and to carry them as steadily as possible.


Shadwell’s Manual was then reduced to a series of Instructions which as stated below forms part of the new code of instructions for the government of the fleet. These instructions meant that chronometric observations were obtained and recorded in a standard way and could be supplied to the Admiralty as necessary.



The following summary of instructions for the management and use of chronometers has recently been prepared at the Admiralty, and forms part of the new code of instructions for the government of the fleet :—


On Embarking.

Whenever chronometers are to be transported, clamp the catch of their gimbal rings and carry them by hand, or slung with a line, or in a handkerchief, taking care not to give them any shock or circular or oscillating motion.

When embarked, stow them in the case prepared for them on beds of horsehair, shreds of bunting, or raw cotton, padding them around and between with the same soft material, so as to prevent the possibility of motion or concussion. The line joining their XII. and VI. hour marks should be all in the same direction, and parallel to the keel. Never use sawdust or wood shavings.

Release the catch of the gimbal rings; see that the gimbals work freely, but not too easily.

The chronometers being once secured in their places are on no account to be subsequently moved or displaced until relanded at home.

The chronometer case should be covered over with a covering of coarse woollen cloth to guard them from sudden changes of temperature.


On Winding and Comparing.

Wind up daily at the same hour (8AM), counting the turns, and winding carefully until the key is felt to butt; eight-day chronometers on Sundays.

Immediately after winding, compare each of the chronometers with the "standard" (the best chronometer, previously selected,) and note the comparisons in the chronometer journal, Form No.I.

Note also at the same time the thermometer (which should be kept within the chronometer box) and also the barometer.


Errors and Rates.

Chronometers being virtually intended for determining the "difference of longitude" or "meridian distance" between places visited by the ship, as well as for her safe navigation, their errors and rates must be very carefully ascertained. The ERRORS are best found by "equal altitude" observations of the sun, or in default of them, by single altitudes A.M. or P.M.: all such observations should be made with an artificial horizon on shore.

The RATES are to be ascertained by comparing their errors on local mean time, obtained consecutively at convenient intervals of not less than five or more than ten days; seven days is a convenient average interval. The difference of the errors divided by the number of days elapsed between them gives the mean daily rate of the interval.

At places where time signals are established for giving daily "local or Greenwich mean time," they may be taken advantage of, and the errors and rates deduced from them: but independent astro­nomical observations are most to be preferred. The errors should be recorded, with the rates deduced from them, as shown in Form No.II.


On Meridian Distances.

The chronometric longitudes of places visited are not required, but the "meridian distance," or difference of longitude in time, as shown by each chronometer between those places from the observations, is to be carefully recorded, as in Form No.III.


On Position of Place of Observation.

It is very important that the exact site of the observations should be distinctly specified. At a well frequented place, it is advisable that they be made on the site previously selected by former observers, as great confusion arises from the unnecessary multipli­cation of sites of observation.

If it be a new station, its situation should be carefully described, its latitude and approximate longitude be given, or its connexion with some adjacent known position.


The Chronometer Journal.

The chronometer journal, kept in Form No.I, is to contain the record of the daily comparisons at the time of winding.

Notice should be taken in the column of remarks of any circum­stances likely to affect the performance of the chronometers, such as gales of wind, violent motion of the ship, her striking the ground, heavy firing of guns in action or for exercise, accidents to or stoppage or removal of any of the chronometers, storms of thunder, lightning, and general direction of the ship's head, etc.

The daily rates of the chronometers, determined from time to time by observation, should also be noted as a record, and for the purpose of occasional comparison with the column of second differences.

The chronometer journal, being neatly and methodically kept in the manner above described, there will be no necessity for keeping a fair copy of it, the original being carefully preserved in readiness for transmission to the Admiralty, if required.


Official Returns.

As the results arising from the good management of chrono­meters are only permanently required for the purpose of settling longitudes, Form No.III, when filled up, is to be transmitted annually to the Secretary of the Admiralty, with the ship's remark book. But as there is much probability of the others, No.I and No.II, being also required, officers are strictly enjoined to pay especial attention to all the foregoing particulars, both as to the precepts they contain and the forms directed by them to be kept, in order that when these are applied for, on any occasion, for the investigation of results in longitude, they may be immediately forthcoming, in a satisfactory condition, for reference in the Hydrographic Office.

On the ship being paid off the chronometer journal is to be forwarded, with the other returns, to the Secretary of the Admiralty.


(The abovementioned forms :

Form No.I : Chronometer Journal for the record of Daily Comparisons of Chronometers.

Form No.II : Return of observed Errors and Rates of Chronometers.

Form No.III : Meridian distance………………………………to………………………………

may be viewed at :


A review of Shadwell’s 1855 publication from the Royal Astronomical Society may be read at Appendix B.


Closing Remarks

Not unlike today technology then was impacting how things were done. It is difficult today to know if Shadwell’s work improved the future establishment of longitude from maritime observations. Nevertheless, his manual is as complete and perhaps even better than many User Guides produced today!



Compiled by Paul Wise, 2019.




Her Majesty’s Nautical Almanac Office (2019), The Nautical Almanac, accessed at :


Shadwell, Admiral Sir Charles Frederick Alexander (2018), accessed at :


Shadwell, Charles Frederick Alexander (1861), Notes on the Management of Chronometers and the Measurement of Meridian Distances, newly revised 1861 London, accessed at :




Appendix A



February 1887, Fellows and Associates deceased : Shadwell, Sir C. F. A.

Monthly Notices of the Royal Astronomical Society, Vol.47, p.140

ADMIRAL Sir CHARLES FREDERICK ALEXANDER SHADWELL K.C.B., F.R.S., was born on January 31, 1814, and was the fourth son of the late Right Hon. Sir Lancelot Shadwell, Vice Chancellor of England. He entered the navy May 3, 1827 from the Royal Naval College, and passed his examination in 1833. On June 28, 1838, he obtained his lieutenant's commission, and was appointed in the following July to the Castor, Captain Edward Collier, with whom be served on the coast of Syria in 1840, and assisted at the operations against Caiffa, Jaffa, Tsour, and St. Jean d'Acre. His next appointment was as first lieutenant of the Fly, in the East Indies, from December 1841 until June 1846, when he was promoted to be commander. In February 1850 he commanded the Sphynx, for service in the East Indies, and held this command through the Burmese War of 1851-1853, for his services is which he was on in February 7, 1853, promoted to post rank, was gazetted to the Order of the Bath, and received the Burmese war medal with clasp. His next appointment was, in August 1856, to the Highflyer, which he commanded in the last war with China, taking part in the operations in the Canton River and in the capture of Canton, and being wounded in the attack on the forts at the mouth of the Peiho River in July 1859. For his services on this occasion Captain Shadwell was mentioned in despatches for " his valuable assistance." After the China War he was appointed to the Aboukir, which he commanded in the Channel and North America and West Indies in 1861 and 1862. His next appoint­ment was that of flag-captain, in the Hastings, to Rear-Admiral Sir Lewis F. Jones at Queenstown, which he held till June 1864, when he was made Captain Superintendent of the Royal Hasler Hospital and of the Royal Clarence Victualling Yard. In this post he remained till he became Rear-Admiral in January 1869, vacating at the same time his appointment as Naval Aide-de-Camp to the Queen, which he had received in March 1866. He was commander-in-chief on the China station from 1871-1874. In 1873 he was gazetted a Knight Commander of the Bath, and in 1877 received the flag officers' pension for meritorious service. His last appointment was that of President of the Royal Naval College at Greenwich, which post he held from March 1878 till March 1881. He died at Meadowbank, Melksham, Wilts, on March 1, 1886.

In the year 1861 Sir Charles F. A. Shadwell was elected a Fellow of the Royal Society. He was the author of the following works : “Formulae of Navigation and Nautical Astronomy” (for the use of naval officers and students of nautical astro­nomy), “Tables for Facilitating the Approximate Prediction of Occultations and Eclipses”, “Tables for Facilitating the Determination of the Latitude and Time at Sea by Observations of the Stars”, “Tables for Facilitating the Reduction of Lunar Observations”, “Notes on the Management of Chronometers and the Measurement of Meridian Distances”, “Notes on Interpolation, Mathematical and Practical” (intended to form Ch. VI. of a work on navigation and nautical astronomy, which the author left incomplete at his death).

He was elected a Fellow of this Society on January 8, 1847.


Appendix B


Notes on the Management of Chronometers and the Measurement of Meridian Distances.

By Captain Charles F. A. Shadwell, R.N. C.B. &c. London, 8vo. 1855.


The author states that this work is mainly designed for the use of officers of the Royal Navy, but he expresses a hope that it may be perused with advantage by intelligent individuals of the Mercantile Marine. His object has been not so much to show how Chronometers may be rendered subservient to the ordinary purposes of Navigation as to exhibit their application to the advancement of Maritime Geography.

The work is divided into nine chapters, the first three of which are devoted to various preliminary details relative to the management of chronometers and to the determination of local time. In the fourth chapter the author explains the mode of ascertaining the rates of chronometers by two observations of a similar kind taken at a convenient interval. On this part of his subject he remarks : —" For the determination of the error on local mean time, it has formerly generally been the practice in the measurement of meridian distances, or on taking a de­parture from a port, to adopt as the starting-point the error shown by the last observation used in the determination of the rate; but inasmuch as in the deduction of the latter, we necessarily place equal confidence in both the observations, there seems to be no good reason why we should not do the same in assuming the error and adapting as our working error the mean of the two errors on which we have already agreed to make the rate depend.

" The mean error thus introduced may probably be assumed to be more accurate than either of the single elements on which it depends; while this mode of proceeding, moreover, will have the advantage of referring both the error and the rate to the same epoch."

The author remarks that this mode of viewing the subject has the advantage of simplifying the process for finding the corrections to be applied for the changes of rate which may have taken place in the transit from port to port.

The principles explained in this chapter are illustrated by a copious collection of examples. In one of these the error and rate of the chronometer are determined from observations of a dissi­milar kind made in two places whose difference of meridian is known. " This example," says the author, " affords an illustration of an important remark of Raper's, ‘that as the longitudes of the several places approach to precision, ships will employ the differ­ence of longitude as a means of obtaining directly the sea-rates of their chronometers, instead of waiting to obtain harbour-rates; thus exemplifying one of the most important ends to which the perfection of hydrography can serve.' "

Upon the question with respect to the best interval for deter­mining the rates of chronometers, the author justly remarks that if the stability of the rate could be relied upon, its value would be obtained with greater accuracy the longer the interval of time included between the observations for ascertaining the values of the error on mean time.

" In practice, however, this theoretic view is limited in its ap­plication by the impossibility of depending confidently on the steadiness of the rate over long periods, and by the consequent necessity for checking the performances of chronometers by fre­quent determinations of their errors, and thus breaking up the intervals on which the rates depend into short periods.

" As a matter of practice, therefore, it seems advisable when circumstances permit, that the rate of a chronometer should not depend on observations made at an interval of less than five or more than ten days. Seven days will be found a convenient ave­rage interval; and in the case of eight-day chronometers, more­over, it embraces the period affected by the whole weight of the chain.

" With the above limitations it may be laid down as a maxim that chronometers cannot be rated too often when time and oppor­tunity permit.

" It seems advisable, moreover, when the measurement of me­ridian distances is in contemplation, that in so far as may be prac­ticable, the two rates employed should depend on observations made at equal intervals of time; since, when the intervals are very unequal, the small errors of observation do not exercise an equal influence on the final results, and their values are unduly affected by the errors of observation attendant on the rate deter­mined at the shorter of the two periods."

In the fifth chapter the author explains the mode of obtaining the errors and rates of chronometers by a combination of several observations taken within a convenient interval of time. The accumulations of the rate for the partial intervals form the groundwork of a series of equations of condition, which are solved by the method of least squares. On the question with respect to the advantage attending the application of this method, the author remarks, that " although in most cases the mean arithmetic error corresponding to the mean of the times of observation may be considered as sufficiently accurate and convenient for practice; yet in eases where an examination of the observations seems to indicate considerable instability or fluctuation of rate during the period of rating, and when the computer does not object to the additional labour involved in the latter more elaborate process, there is no doubt its results will usually be more satisfactory, and certainly more correct, while it will fully repay the extra trouble employed in its manipulation."

In the sixth chapter the author treats of the chronometric de­termination of meridian distances, explaining the different hypo­theses which have been adopted for finding the variation of the rate during the voyage between the two places where the rate has been ascertained. The following chapter contains a copious col­lection of examples illustrative of the principles laid down in this chapter.

In the eighth chapter, the author shows how the difference of longitude between two places may be obtained from observations giving the errors of the chronometers at the two stations, inde­pendently of a knowledge of their preceding or subsequent rates. The error of the chronometer is ascertained : first, upon starting from the first station; secondly, upon arriving at the second station; thirdly, upon quitting the second station; and, lastly, upon arriving again at the first station. From these data the difference of longitude between the two stations, and the rate of the chronometer for the interval included between the departure from and subsequent arrival at the first station, termed the travelling rate, is deduced by a formula for which the author expresses his obligation to the Rev. George Fisher.

The ninth chapter of the work is devoted to a series of pre­cepts on the mode of recording the results of chronometric mea­surements.

It may be remarked, in conclusion, that the subject-matter of the work is arranged in a very lucid order, that the explanations are concise and simple, and that they are in every instance illustrated by numerous examples. Possessing such claims to consideration, the work cannot fail to prove a most valuable manual to the naval profession in general, and more especially to the class of persons whose instruction the author had mainly in view while engaged in preparing it for publication.




Royal Astronomical Society pp.196-199, Provided by the NASA Astrophysics Data System, downloaded from at West Virginia University on July 18, 2015, accessed at :