I now enter upon the consideration of the fourth topic. As has been already stated, the British government in 1714 offered a reward of £20,000 to any one who should discover the longitude at sea within thirty miles.
Two methods were proposed for the solution of the problem.
Morin proposed what is now substantially the lunar method; and Maskelyne devoted all his energies to the solution of the problem by observing astronomical phenomena, such as the eclipses of Jupiter's satellites. Maskelyne made a trial of this method during a voyage to Barbadoes, with an apparatus suspended from the rigging of a ship, called Irwin's marine chair. But the attempt was not successful, Offing to errors occasioned by the motion of the ship.
On the other hand, mechanicians devoted every energy to the mechanical solution. Huygens had, as early as 1665, in a voyage to the coast of Guinea, made the trial of the method by watches, without success. Trial of this method was now renewed with watches made by this maker, but it was soon found that the method would be of little value till some contrivance was devised for correcting errors produced by the variation of the temperature. At last Harrison, by the very perfection of his workmanship, produced a chronometer with which a trial of the coveted prize was made in a voyage to Jamaica. As the longitude of the island differed from that shown by his chronometer by only ten and three-quarter miles, he claimed the reward, which, after a successful second trial, was awarded to him.
We have, then, two essentially different methods for the determination of the longitude at sea:
(a) By lunar distances, occultations, and eclipses of Jupiter's satellites, &c.
(b) By chronometers, assuming a constant rate throughout the, entire voyage.
The latter method has for a long time been regarded as far more reliable than the former. Let us examine into the grounds of this opinion. We are, at the outset, met with the difficulty that in observations at sea we have no means of comparison with the truth. It is not sufficient to say that several observations agree with each other. Agreement inter se may be quite a different thing from agreement with the truth. We are obliged to have recourse to observation made on land, and then find, if possible, some means of comparing the value of these with those made at sea.
Even in the determination of the positions of fixed, observatories, in which appliances for the utmost refinement are at hand, the values derived often vary widely from the truth. As late as 1755, a century after the establishment of the observatory at Greenwich, the difference of longitude between that station and the Paris observatory was assumed to be 9m. 16s. Gen. Roy, in the progress of the trigonometrical survey of England, obtained the value 9m. 18.8s. In 1796 9m. 20s. was assumed to be the correct value. In 1830 it was found to be 9m. 21.5s. from one thousand transits of moon: and stars, while the actual difference, as determined by the telegraphic method, was found to be 9m. 20.6s. We have here a variation, of 5.5s., or one and one-quarter miles.
The range of the earlier determinations of the difference between the longitude of Greenwich and Brussels is ten miles. A thorough discussion from moon culminations in 1836 gave a result which differs from that since found by the telegraphic method over Is. To indicate the magnitude of the errors of longitude of the old State House, Boston, from transits of planets over the sun and from eclipses, I give the results of the several determinations by Dr. Bowditch:
| State House from Greenwich | Greenwich from Harvard Col’ge | Deviation f’m truth | ||||
h | m | s | h | m | s | s | |
From transit of Mercury … 1743 | 4 | 44 | 19.2 | 4 | 44 | 31.6 | -0.7 |
Eclipse of the sun … 1766 |
|
| 17.5 |
|
| 29.9 | +1.0 |
From transit of Venus … 1769 |
|
| 18.0 |
|
| 30.4 | +0.5 |
Eclipse of the sun … 1778 |
|
| 14.9 |
|
| 27.3 | +3.6 |
Eclipse of the sun … 1791 |
|
| 18.0 |
|
| 30.4 | +0.5 |
Eclipse of the sun … 1806 |
|
| 11.8 |
|
| 24.2 | +6.7 |
We have here an absolute error, of 2.2s., or more than one-half a mile, with a range of 10.3s. In passing, I may say that these results are a remarkable approximation to the true value, but they were deduced by one of the greatest and most practical mathematicians then living. They are, however, hardly comparable with any single observation, either on land or sea. During the time it would take an ordinary computer to "work up" these observations, a fast-sailing ship would be well on her journey around the world.
Perhaps the best illustration Of the uncertainty resting upon the early 'determinations of longitude is furnished by the various values of the longitude of Washington, which have been given.
In 1822 Lambert found for the longitude of the Capitol:
| h | m | s | |
Occultations in 1793 | ? = | 5 | 7 | 5.2 |
Occultations in 1804 | ? = |
| 7 | 37.7 |
Eclipse in 1811 | ? = |
| 7 | 21.8 |
Eclipse in 1813 | ? = |
| 8 | 43.5 |
Value adopted | ? = |
| 7 | 42.0 |
Elliott in the same year found ? = 5h. 8m. 7.2s., but his value was rejected from the mean given above, on account of its supposed discordance. We have here a range of over nineteen miles, and the final value is nearly six miles in error.
Even in the determination of the longitude of the present observatory there is comparatively a wide range between the different results.
Adopting ? = 5h. 8m. 12.1s., we have:
| Value Found | Correction | |||
h | m | s | s | ||
Bond chronometer expedition … 1849-50 | 5 | 8 | 12.0 | +0.1 | |
Bond chronometer expedition … 1851 |
|
| 12.3 | -0.2 | |
Bond chronometer expedition … 1855 |
|
| 13.4 | -1.3 | |
From moon culminations | Loomis | 5 | 8 | 7.4 | +4.7 |
Gillis |
|
| 10.0 | +2.1 | |
Bache |
|
| 9.9 | +2.2 | |
Newcomb 1860 |
|
| 11.6 | +0.5 | |
Newcomb 1862 |
|
| 9.8 | +2.3 | |
Eclipses | Bache |
|
| 11.1 | +1.0 |
Pierce |
|
| 11.6 | +0.5 | |
Occultation of Pleiades | Pierce |
|
| 11.4 | +0.7 |
We have here a range of 6s., or one and one-half miles, while the mean is in error 1.3s.
Two hundred and six moon culminations gave a longitude for San Francisco which has since been found to be 4s. in error.
The telegraphic determination of the longitude of Lisbon, by Lt. Comdr. F. NE. Green, U. S. N., giving a result 8s. different from the accepted value, is the most recent instance of an erroneous value.
These examples are sufficient to show that it is no easy task to determine accurately the geographical position of a point on the earth's surface, even under the most favorable conditions and with the most perfect instruments.
Resuming our investigation, we begin with moon culminations. These, though essentially the same in principle, are probably somewhat more accurate than lunars. With the exception of the Willet's Point observations of lunars, we are obliged to limit our, investigation to observations of nearly the same class, viz, moon culminations. These are more accurate when the longitude depends upon observations at both of the stations to be determined, since the errors of the tables of the moon are thus for the most part eliminated.
In the case of a fixed observatory whose position has been ascertained by other and more accurate methods, we have the data for testing the accuracy of this method.. Before the application of the telegraphic method of determining longitude, many observations of this class were made. I have collected and reduced the most important of these with the following results. All the observations made during a given year were grouped according to the limb of the moon observed. The number of observations for each limb and the number of years during which the observations were continued are given in the following table:
Places of observations | 1st Limb. Limiting years | No. of observations | 2d Limb. Limiting Years | No. of observations |
Washington .. Edinburgh | 1839-41 | 67 | 1839-41 | 23 |
Washington .. Greenwich | 1839-42 | 61 | 1839-42 | 42 |
Washington .. Hamburg | 1839-42 | 65 | 1839-42 | 12 |
Washington .. Oxford | 1840-41 | 39 | 1840-41 | 12 |
Washington .. Cambridge, En. | 1839-42 | 58 | 1839-42 | 33 |
Greenwich .. Hudson | 1839-42 | 57 | 1839-40 | 16 |
Cambridge .. Hudson | 1839-44 | 74 | 1839-42 | … |
Edinburgh .. Hudson | 1838-43 | 86 | 1840-42 | 15 |
Oxford .. Hudson | 1840-43 | 30 | 1840- | 5 |
Hamburg .. Hudson | 1839-44 | 58 | 1842-43 | 19 |
As an illustration of the error of Mr. Main's statement that the lunar problem' is completely solved, I add the details from the Willet's Point observations for 1871:
From Moon Culminations
Date | Corr. By British Ephemeris (s) | Corr. By American Ephemeris (s) |
January 30 | -5.2 | -17.1 |
September 25 | -24.2 | -9.1 |
September 26 | +14.9 | +1.6 |
September 27 | +11.0 | +1.4 |
September 27 | +2.2 | -8.4 |
September 28 | -2.9 | … |
September 28 | -16.2 | -12.1 |
September 29 | -10.1 | -10.0 |
October 23 | -14.6 | +11.9 |
October 27 | +7.8 | +0.9 |
October 27 | +18.9 | +14.2 |
October 28 | +50.2 | +0.0 |
October 28 | +52.7 | +0.1 |
From Lunar distances for 1870
With Sextant
Date | Corr. (s) |
January 27 | +19.1 |
September 3 | +47.0 |
October 19 | -18.4 |
October 29 | -1.0 |
October 30 | -0.2 |
November 1 | -38.0 |
November 6 | -96.0 |
November 29 | -82.3 |
Finally, we have fortunately on this point positive and conclusive testimony from the observations of Mr. Fisher, astronomer on Capt. Parry's second voyage. He found that the mean of 2500 observations in December differed ten miles from the mean of 2500 in the following March, and that the mean of a still larger number made on both ships differed ten miles from those in March and twenty-four miles from those in December. It is the testimony of Capt. Heywood that any set of lunars may be expected to differ 6' or 7' from any other set equally good, taken at a different time of the year, and this independently of accidental errors.
Collecting results, we have:
Place and Circumstances of Observation | Average error (s) | Rage bet. Greatest and least (s) | Coefficient |
Greenwich-Edinburgh. Observations at each station, compared final result with truth | 2.4 | 21.8 | 9.0 |
Washington-Greenwich. Observations at each station, compared final result with truth | 2.1 | 21.6 | 10.3 |
Hudson-Greenwich, compared with mean result for each station | 6.3 | 17.4 | 2.9 |
Brussels-Greenwich, compared with mean result for each station | 5.0 | 17.6 | 3.5 |
Places in Turkey with Paris, compared with mean result at each station | 6.7 | 27.8 | 4.1 |
Camp Riley, comp’d of mean with tabular places | 14.0 | 55.0 | 3.9 |
Willet’s point, compared with transit observations - |
| ||
From American Ephemeris | 9.3 | 43.0 | 4.6 |
From British Ephemeris | 14.0 | 55.0 | 3.9 |
Lunars compared with known positions | 40.7 | 97.1 | 2.4 |
For fixed observatories, with the most perfect instruments, we must therefore expect from the lunar method of moon culminations an absolute error of 2.2s. within a range of 21.6s. as the result of any number of observations. These results correspond in a general way with an investigation made by Professor Pierce. He found the ultimate limit to be ±0.55s. when one limb of the moon was observed. He says: "Beyond this it is impossible to go with the utmost refinement. By heaping error upon error, it may crush the influence of each separate determination, but does not diminish the relative height of the whole mass of discrepancy. But this discrepancy between the results for different limbs of the moon often amounts to 10s. in the mean determination of a year." The assumption, that the ultimate limit of accuracy is as great as 1s. seems to be a very moderate widening of the limits. I find it to be 2.2s.
For fixed observatories, comparing with the mean result at any station, we must expect a relative error of 6.0s., with a range of 20.9s.
For fixed observatories, using the moon's tabular places, we must expect an error of 12.4s. with a range of 51.0s.
For lunar distances, with the sextant, on land, we must expect an error of 40.7s., with a range of 97.1s.
For single lunar observations at sea these quantities should have a coefficient at least as great as three additional units.
It will be noticed that the coefficient is much larger for comparisons with known values than for comparisons with the mean of a given series.
I now take up the subject of chronometers, and make a similar investigation.
The sources of error are:
(a) Variations of rate, arising from the action of magnetism.
The only observations under this head that I can find are those made by Mr. Fisher and by Professor Airy. Mr. Fisher made a very elaborate series of experiments, from which he found that the earth's magnetism changed the daily rate in one case 4.5s., in another 3.2s., and in another 4.1s.
Professor Airy made similar experiments and found similar results, the extreme variation on account of the influence of terrestrial magnetism being 5.8s. He found the following rates for Brockbanks, No. 425:
When the figure—
XII was north, rate = 4.64s
XII was east, rate = 8.70s
XII was south, rate = 9.61s
XII was west, rate = 5.75s
He also found that the action of terrestrial magnetism could be eliminated by placing the chronometer on the top of a compass-box whose needle was perfectly free, provided the elevation was properly adjusted. When this adjustment was made he found for—
XII N rate = 9.24s
XII E rate = 9.41s
XII S rate = 9.75s
XII W rate = 10.03s
This corresponds with my own observations in the case of clocks. I have long noticed that the Cambridge and Boston clocks gain and lose together, due to some kind of a sympathetic action between the two.
(b) When chronometers are swung on the same support it is probable there is a sympathetic action between them similar to the results recently found with the transit of Venus clocks.
(c) Variation on account of change of barometric pressure. This varies between .3s. and .8s. per day for every inch of change in the barometer.
(d) Variation between land and sea rates. The elder Professor Bond made a full investigation of the average difference between the land and sea rate, and found them essentially the same. It had been previously assumed that there was an average gain of the latter over the former. But in individual cases there are, without doubt, great changes of the rate, arising mostly from careless handling in transportation to the ship.
Almost every chronometer will change its rate when its circumstances, either of rest or motion, are changed. The Boston standard clock of Messrs. Bond & Son almost invariably has a different rate on Sunday from any other day of the week. So, also, it takes a new rate when the streets are covered to any considerable depth with snow.
The jar of machinery affects the rate; hence the rate of a sailing vessel is more steady than that of a steamer.
(e) Variation of rate at sea, on account of change of temperature. Mr. Hartnup, of Liverpool, was the first to give the chronometer rate for different temperatures. So far as I can find he is the only one who does so now, and yet failure in this respect occasions errors of enormous magnitude. In one case which Mr. Hartnup instances, if the navigator had relied upon his average rate he would have been nearly sixty miles in error. This is a reform in rating chronometers which is imperatively demanded, and it is one easily accomplished.
Let us now attempt to determine the limits within which a chronometer can be depended on to give the longitude at sea. Here we are limited in a great measure to the performances of chronometers on land, inasmuch as at sea we have no means of comparison with a normal standard. The chronometer tests at the Greenwich observatory, however, afford abundant facilities for ascertaining the performance of chronometers under varying conditions of temperature. I select for discussion three series—the first from 1842 to 1852, the second from 1853 to 1862, and the third from 1863 to 1871. The quantities to be determined are the greatest difference between the rates during a given trial, which usually extended over a period of about six months, and the greatest difference of rates between one week and the next following.
The only rule followed in the selection of the chronometers chosen for these tests has been the preference given to makers who entered their chronometers the greatest number of times during the entire interval from 1842 to 1871. The method of proceeding was as follows: Having selected for this test, e. g., the chronometers of Frodsham, the quantities sought were obtained from the chronometer tests recorded in the volumes of the Greenwich observations between the years 1842 and 1871. Inasmuch as the value given for each group is a mean value, derived from the number of chronometers entered for trial between the limiting dates, I have added the greatest value for a given series. The addition of these columns is important, since the liability to error is as great as the greatest range of error.
The temperature at which the comparisons were made varies from about 35° to about 95° Fahrenheit. The 'variation of rate under nearly the same temperature will appear from the column "Greatest range in daily rate between one week and the next following."
In order to exhibit still further the variations due to a change of temperature I have selected from the Greenwich reports for the years 1867-68-69-70-71 five chronometers showing the best performance, and five showing the poorest performance. The observations cover a period of four weeks, during which time the chronometers were subjected to a high temperature (about 95°), and during the period of four weeks immediately preceding, under ordinary temperature.
Summary
Year | Mean diff. of daily rate between high and ordinary temperatures (s) | Greatest variation of daily rate (s) | Range of daily rate under ordinary temperature in 1 mo. (s) | Range between different chronometers (s) | Range of daily rate under high temperature in 1 mo. (s) | Range between different chronometers (s) |
Best | ||||||
1867 | .36 | .66 | .44 | .88 | .56 | .51 |
1868 | .25 | .39 | .25 | .22 | .47 | .53 |
1869 | .31 | .56 | .44 | .49 | .21 | .42 |
1870 | .39 | .76 | .45 | .55 | .46 | .39 |
1871 | .28 | .49 | .56 | .50 | .43 | .25 |
Means | .32 | .57 | .43 | .53 | .43 | .42 |
Poorest | ||||||
1867 | 2.03 | 4.08 | .80 | .91 | 1.15 | 1.87 |
1868 | 1.55 | 2.25 | .94 | 1.45 | 1.43 | 2.93 |
1869 | 1.94 | 3.39 | .56 | .54 | .90 | 1.48 |
1870 | 2.18 | 3.46 | .81 | 1.54 | 1.70 | 2.01 |
1871 | 3.28 | 4.47 | .40 | .51 | .78 | .78 |
Means | 2.20 | 3.51 | .70 | 0.97 | 1.19 | 1.81 |
From these tables I draw the following conclusions:
(a) There was a very marked and steady improvement in the construction of chronometers between 1842 and 1871. The greatest range of daily rate during the entire trial fell from 3.40s; in 1842-52 to 2.41s. in 1863-71. The greatest range of the daily rate between one week and the next following was reduced from 1.91s. in 1842-52 to 1.19s. in 1863-71. These were the palmy days of chronometer construction. I have made only a partial discussion of the trials since 1871, but the evidence is pretty conclusive that the, averages are larger than for the period 1863-71.
(b) Under the most favorable conditions for the excellent performance of chronometers, the average change of the rate between one week and the next following exceeds 1.2s. Of course, with the newly acquired rate the accumulation of error will be proportional to the time until another change takes place.
(c) The average liability to error in the daily rate between one week and the next following is not far/from 3 seconds.
(d) The ratio-value between the average of first-class chronometers and the average of poor ones is nearly as 1 to 7.
On the Magnitude of the Errors of Chronometers Employed in Short and Regular Sea Voyages.
It will be allowed on every hand that a discussion of the actual errors of the chronometers used on the Cunard line of steamships will exhibit results considerably better than the general average. The well-known skill of the navigators in the service of this company, the perfection of every appliance, the general sameness of the conditions of each voyage, the number of chronometers employed, and the accurate rating of each chronometer at the beginning and end of each voyage by comparison With the time from fixed observatories, all conspire to give the most favorable results which can be expected from the performance of chronometers in the present state of chronometer construction. With the assistance of Mr. Aug. McConnell, I have made a discussion of the chronometer errors of the Cunard steamers sailing between Liverpool and Boston during the years 1871-1872 and 1873.
Average Results
Steamer | No. of voyages | Av. Length of voyage | Av. Error of daily rate (s) | Av. Error at end of voyage (s) | Range between greatest and least value (s) | Range between three chronometers (s) | Greatest value of series (s) |
Palmyra | 7 | 14.7 days | +0.57 | 8.83 | 32.0 | 19.83 | 52.8 |
Olympus | 9 | 15.2 | 0.35 | 5.03 | 11.3 | 11.20 | 25.2 |
Samaria | 13 | 14.9 | 0.44 | 6.77 | 13. | 16.27 | 48.3 |
Hecla | 11 | 14.7 | 0.37 | 5.29 | 10.8 | 16.65 | 39.9 |
Parthia | 3 | 13.0 | 0.41 | 5.47 | 6.9 | 5.43 | 12.1 |
Tripoli | 4 | 15.8 | 0.30 | 4.60 | 11.1 | 12.52 | 19.2 |
Siberia | 13 | 14.9 | 0.40 | 5.84 | 19.5 | 13.77 | 46.2 |
Batavia | 7 | 15.2 | 0.30 | 4.59 | 9.3 | 12.16 | 23.8 |
Malta | 6 | 15.2 | 0.86 | 12.16 | 24.0 | 28.97 | 58.5 |
Means | 8 | 15.4 | 0.44 | 6.52 | 15.42 | 15.20 | 36.22 |
Chronometer Rates on Sailing Vessels.
The chronometers in this discussion were selected without regard to rates, except when the variation exceeded 2s., and the chronometers needed repairs at the end of the voyage. The materials for this part of the discussion were furnished by Mr. McConnell.
Collecting results, we have:
Maker | Av. Length of voyage | Av. Error of daily rate (s) | Av. Error at end of voyage (s) |
Bond & Sons | 84 days | 0.19 | 16.1 |
Bliss & Creighton | 80 | 0.92 | 73.7 |
Hutton | 75 | 0.29 | 21.6 |
Negus | 86 | 0.42 | 36.5 |
Berraud | 127 | 0.47 | 59.7 |
Frodsham | 125 | 0.54 | 67.9 |
Poole | 98 | 0.40 | 39.3 |
Tobias | 78 | 0.38 | 29.5 |
Adams | 83 | 0.58 | 47.8 |
10 other makers | 71 | 0.61 | 43.6 |
Means | 91 | 0.48 | 43.6 |
The longitude expeditions undertaken by Professor Wm. C. Bond in 1949-50, and by Professor George P. Bond in 1855, for the accurate determination of the longitude between Cambridge and Greenwich, furnish the most reliable data available for a correct estimate of the degree of accuracy to be expected from the average performance of a chronometer during a short sea voyage. On these expeditions the rates of all the chronometers employed were carefully determined at each station.
In this discussion only those chronometers are selected which made three or more voyages. The longitude given by each chronometer is compared with ? = 4h. 44m. 30.9s. The numbers given are current numbers, and not the numbers given to the chronometers by the makers. The values given under each number are the separate corrections to the assumed longitude given by the same chronometer on different voyages. The exceptional value of these determinations justifies the publication of the details.
Combining the mean corrections given by each, chronometer on different voyages, without regard to weights, we have the following results:
Expedition of 1849-50 + .63s
Expedition of 1855 - 1.18
Final Correction - 0.27
In order to form an estimate of the liability to error in any chronometer, we must compare the greatest and the least corrections given by the same chronometer on different voyages. The average range for 1849-50 is 19.5s., and for 1855 it is 10.8s. The extreme range is shown in the following table:
Limits | s | s | s | s | s | s | s | s | s | s | s | S |
55 … | 50 to 55 | 45 to 50 | 40 to 45 | 35 to 40 | 30 to 35 | 25 to 30 | 20 to 25 | 15 to 20 | 10 to 15 | 5 to 10 | 0 to 5 |
Number of Cases
Expedition |
| |||||||||||
1849-50 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 8 | 4 | 4 | 4 | 5 |
1855 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 2 | 6 | 8 | 19 | 10 |
The values given on pages 370 and 371 represent absolute corrections from which the error of rate has been eliminated. It is not possible, therefore, to determine the average error of the daily rate directly from these series. It can be obtained, however, somewhat indirectly from the table given on page 368. It appears from this table that the average range between the greatest and the least values of the average errors at the end of a voyage of 14.8 days is 15.42s., and that the coefficient by which this average error must be multiplied in order to produce the latter value is 2.37. The mean of the average range for the expedition for 1849-50 and that for 1855 is 15.2s., from which we may with tolerable safety assume the average error at the end of a voyage of 15 days to be not far froth 6.4s., giving for the average daily error of rate the value ±0.43s., which is almost identical with the values already found on pages 370 and 371.
I add, without details, the results of two other longitude expeditions:
| No. of Chronometers | Length of Voyage (days) | Average Error at End of Voyage (s.) | Range between greatest and least Value (s.) | Coeff. |
Cambridge, New York and Chagres | 5 | 12 | 7.6 | 26.4 | 3.5 |
Eclipse Expedition to Labrador | 5 | 20 | 3.7 | 13.1 | 3.5 |
Here we have from a limited number of observations the value ±0.41s. for the average error of the daily rate.
It must be remembered that the results so far obtained are average results obtained from a large number of observations made under the most favorable conditions. They indicate that a chronometer of average quality is at any time liable to change its daily rate by an amount as great as 0.5s. That this value is probably too small will appear from the following table given by Mr. Hartnup, who made the investigation in 1863. The quantities given represent the error at the end of a voyage of the given duration.
Length of Voyage No. Mos. | Average error from 1700 Chro. | Best 10 in 100 | 2d best 10 in 100 | 3d best 10 in 100 | 4th best 10 in 100 | 5th best 10 in 100 | 6th best 10 in 100 | 7th best 10 in 100 | 8th best 10 in 100 | 9th best 10 in 100 | Worst 10 in 100 |
| Miles | Miles | Miles | Miles | Miles | Miles | Miles | Miles | Miles | Miles | Miles |
1 | 6 | 0 | 1 | 1 | 2 | 3 | 4 | 5 | 7 | 9 | 25 |
2 | 14 | 0 | 2 | 4 | 5 | 7 | 9 | 11 | 15 | 24 | 62 |
3 | 23 | 1 | 3 | 6 | 9 | 12 | 15 | 18 | 25 | 41 | 101 |
4 | 33 | 1 | 4 | 8 | 13 | 17 | 22 | 28 | 36 | 61 | 149 |
5 | 44 | 1 | 5 | 10 | 17 | 22 | 29 | 39 | 49 | 84 | 187 |
6 | 56 | 2 | 6 | 13 | 21 | 28 | 37 | 50 | 64 | 108 | 233 |
7 | 69 | 2 | 8 | 16 | 25 | 34 | 46 | 62 | 80 | 134 | 280 |
8 | 82 | 3 | 10 | 19 | 30 | 41 | 55 | 74 | 98 | 159 | 328 |
9 | 95 | 3 | 12 | 22 | 35 | 48 | 65 | 86 | 117 | 184 | 376 |
10 | 108 | 4 | 14 | 26 | 40 | 56 | 75 | 98 | 137 | 208 | 425 |
11 | 122 | 4 | 16 | 30 | 46 | 64 | 86 | 111 | 157 | 233 | 474 |
12 | 136 | 5 | 18 | 34 | 52 | 72 | 97 | 124 | 178 | 258 | 524 |
An error of 524 miles in a ship's position may seem to be rather too large an estimate, but Lord Anson, in a voyage around Cape Horn, did better than this. One of his ships was 500 miles out of her reckoning, and one actually made land on the wrong side of the continent, the error of position being over 600 miles.
In 1871 M. Hartnup made an additional investigation, obtaining the following results:
| Extreme Difference of mean daily rate between any 2 weeks (s) | Variation of daily rate due to change of temperature (s) | Mean of extreme difference between any 2 days of each week (s) |
Mean from 297 chronometers | 2.19 | 1.73 | 0.98 |
Mean from 40 poorest | 6.28 | 4.55 | 2.80 |
Mean excluding 40 poorest | 1.56 | 1.29 | 0.70 |
In order to show how largely chronometer errors may be diminished when the rate is given for different temperatures in the form adopted by Mr. Hartnup, I add the following table, given by him in this connection, derived from the actual performance of a given chronometer:
Mean Daily Rate
Date | Temp = 55° (s) | Temp = 70° (s) | Temp = 85° (s) |
April, 1967 | - 0.4 | + 1.4 | + 2.3 |
May, 1868 | - 0.6 | + 1.4 | + 2.2 |
April, 1869 | - 1.4 | + 1.7 | + 2.7 |
November, 1869 | - 0.2 | + 1.9 | + 2.8 |
February, 1871 | - 0.2 | + 1.5 | + 2.3 |
From the first column of Mr. Hartnup's first table I deduce the following values for the error of the daily rate from the mean of 1700 chronometers, assuming the months to be calendar months:
No. months | Average error of daily rate (s) | No. months | Average error of daily rate (s) |
1 | + 0.80 | 7 | 1.31 |
2 | 0.93 | 8 | 1.37 |
3 | 1.02 | 9 | 1.41 |
4 | 1.10 | 10 | 1.44 |
5 | 1.17 | 11 | 1.48 |
6 | 1.24 | 12 | 1.51 |
I am not sure that I have given the correct interpretation to Mr. Hartnup's values, since it is difficult to understand why the, error should increase with the time.
From Mr. Hartnup's second table we have the value ± 0.98s. Adopting the mean between the latter value and ±0.46s., the value derived from this discussion, we have finally ±0.72s. as the average daily error of the rate of an average chronometer.
The average coefficient of safety for chronometer errors only derived from this discussion is 3.2. If, therefore, the navigator has a chrononzeter of average excellence, he must at the end of twenty days expect from it an average error of 3.6 miles, and he must look out for an error of 3.6m. X 3.2, or 11.5 miles.
It must be borne in mind that these results are independent of the errors of observation with the sextant, which are still to be added.
In estimating the limits within which it is possible to locate the position of a ship at sea by astronomical observations, it is necessary to take into account all the errors fo which such observations are liable. I shall consider only the method usually employed, viz. the measurement of the altitude of the sun with a sextant at a given time before it comes to the meridian for longitude, and the measurement of its altitude at culmination for latitude.
First of all we must estimate the magnitude of the errors to which sextant observations are liable.
They are as follows:
(a) Instrumental errors, such as excentricity, errors of graduation, index error, &c. In a first-class sextant errors of this class often exceed one minute of arc.
(b) Error in noting the time. No observer at sea pretends to note the time closer than is. In fact it is impossible for him to do so. If we assume the low limit of is. and multiply by the coefficient 3.5 we find an error from this source amounting to nearly one mile.
(c) Error arising from an imperfect sea-horizon. I am convinced that errors from this cause may amount to several miles. In a series of observations made to test the value of Lieut. Beecher's artificial horizon, the range of error was six miles. Nor does the use of an artificial horizon mend the matter, for there is always an undetermined constant between the two methods which ranges from two to eight miles.
(d) Errors arising from the use of approximate data. It is almost universally the practice to take data to the nearest minute of arc, e. g. it is the practice to take the sun's diameter as 30'. Many navigators lump all corrections together and call the sum 12'. The error from this source may amount to no less than five miles. Another source of error is the failure to use the value of the refraction corresponding with the actual condition of the atmosphere indicated by the thermometer and barometer at the time of observation. The only observations which I can find bearing upon this point were made by Commander Bayfield at Quebec in 1832, the thermometer reading 11° Fahrenheit. He found that the error arising from the use of the mean instead of the actual refractions amounted to three miles.
(e) Errors affecting the longitude which depend upon the latitude of the ship and upon the declination and the observed altitude of the sun. The errors arising from this source may be much larger than is generally suspected, and they are the more important because, for the most part, they escape the attention of the navigator.
Let,
A = the observed altitude of the sun
? = the declination of the sun
? = the latitude of the place of observation
? = the observed hour angle
n = the observed azimuth
? = the angle at the sun
We obtain sin½? = [√sin(s-(90°-?))sin(s-(90°-?))/cos?cos?]
Where
s = ½[(90°-?)+(90°-A)+(90°-?)]
sin? = sin?cos?/cosA = cos?sinn/cos?
sinn = sin?cos?/cosA
In order to compute the values of the various errors which may affect ?, we differentiate the general equation:
sinA = sin?sin? + cos?cos?cos?
From which
cosAdA = (sin?cos?-cos?sin?cos?)d?
+ (sin?cos?-cos?sin?cos?)d?
- cos?cos?sin?d?
But
cos?sin?-sin?cos?cos? = cosAcos?
sin?cos?-cos?cos?sin? = - cosAcosn
cos?sin? = cosAsin?
By substitution we easily find:
d? = dA/cos?sinn + d?/cos?tang? – d?/cos?tangn
In the proceedings of the American Association for 1881, p. 134, Mr. S. C. Chandler has given the following convenient form to this equation:
Dividing both members of the equation,
cos?sin? = cos?sinn
by cos?, we have:
cos?tang? = cos?sinn/cos?
whence d? = (1/cos?sinn)[cos?d?-dA-cosnd?]
From this equation it appears:
(1) That the effect of an error in the observed altitude upon the time will be minimum when ? = 0° and n = 90°, or when the sun is in the prime vertical.
(2) That the effect of an error in ? will be minimum when ? = 0° and ? = 90°.
(3) That the effect of an error in ? will be minimum when ? = 0° and n = 90°.
As an example we will assume that in a given observation of the sun or of a star the following small errors have been made, viz.:
d?=+1’
dA=+1’
d?=-1’
We compute the effect of these errors upon ? in three assumed cases:
| I | II | III |
? = | 39° 54’ | 51° 30’ | 70° 0’ |
? = | + 17° 29’ | - 6° 37’ | + 44° 0’ |
A = | 15° 54’ | 13° 40’ | 63° 30’ |
? = | 83° 37’ | 58° 8’ | 10° 10’ |
n = | 80° 16’ | 60° 15’ | 16° 32’ |
? = | 52° 26’ | 32° 58’ | 7° 46’ |
- dA/cos?tangn = | - 1.32’ | - 1.85’ | - 10.27’ |
- d?/cos?tangn = | - 0.22’ | - 0.92’ | - 9.85’ |
+ d?/cos?tang? = | - 0.81’ | - 1.55’ | - 10.10’ |
d? = | - 2.35’ | - 4.32 | - 30.31’ |
= | - 9.4s | - 17.3s | - 121.2s |
Ordinarily there will be some elimination between d?, dA and d?, but the errors may all act in the same direction, as in the examples given, The third example may perhaps be considered an extreme case, but it will serve to show the necessity of a computed table of coefficients which shall serve as a guide in estimating the value of any given observation. An appropriate name for them would be local coefficients of safety.
(f) Errors arising from the error in the estimated run of the ship between the morning and the noon observations. The morning or afternoon observations give one co-ordinate of position, viz. the longitude; and the noon observations, the other, viz, the latitude. To get both co-ordinates for the same instant requires an allowance for the run of the ship during the intervening time. It is impossible to give any definite estimate of the magnitude of the errors thus introduced, but I suspect they will in general be found to exceed all the other errors combined.
In order to determine the degree of accuracy which may be expected from the sextant considered as an instrument of observation, when used by trained and skillful observers, I have collected such observations of this class as were available, both for time and for latitude. In most cases the average error has been found by comparison of the observations inter se. The observations made at Willers Point have an exceptional value, since they were compared with transit observations made at nearly the same time.
For Latitude
Observer | Place of Observation | Average Error | Range of Error | Coeff. |
Williams, 1793 | Cambridge | 3.6” | 14” | 3.5 |
Paine, 1831 | Norfolk, Va | 9.6 | 40 | 4.2 |
| Richmond | 7.2 | 31 | 4.3 |
| Washington | 4.6 | 18 | 3.9 |
| Baltimore | 5.0 | 20 | 4.0 |
| Providence | 10.1 | 50 | 5.0 |
| Washington Territory | 7.2 | 46 | 6.4 |
| Washington Territory | 10.9 | 41 | 3.8 |
1869 | Willet’s Point | 7.0 | 24 | 3.4 |
1871 | Willet’s Point | 7.8 | 29 | 3.7 |
1872 | Willet’s Point | 9.0 | 35 | 3.9 |
Hall and Tupman | Malta | 10.5 | 59 | 5.6 |
| Malta | 10.2 | 37 | 3.6 |
| Malta | 8.0 | 34 | 4.2 |
| Syracuse | 5.0 | 30 | 6.0 |
| Syracuse | 9.2 | 36 | 3.9 |
| Syracuse | 7.7 | 56 | 7.3 |
| Syracuse | 7.1 | 28 | 3.9 |
| Syracuse | 6.8 | 28 | 4.1 |
| Syracuse | 6.7 | 30 | 4.5 |
Harkness | Des Moines | 6.4 | 47 | 7.3 |
Means |
| 7.6” | 35” | 4.6 |
For Time
Observer | Place of observation | Average Error | Range of Error | Coeff. |
|
| s. | s. |
|
Hall and Tupman | Malta | 1.6 | 7.3 | 4.6 |
| Syracuse | 1.0 | 4.3 | 4.3 |
Newcomb | Des Moines | 0.6 | 2.6 | 4.3 |
Harkness | Des Moines | 1.1 | 4.7 | 4.3 |
Means |
| 1.1 | 4.7 | 4.4 |
For the mean of any number of observations with the sextant, therefore, we must expect an average error of 7.6" in latitude and of 1.1s. in longitude, and the average liability to error of a single observation will be about 35" in latitude and 4.7s. in longitude. For sea observations an additional coefficient of 3 units will be none too great. No instrument of precision is so liable to constant errors as the sextant. In all the cases given above these constant errors were thoroughly investigated and the resulting corrections were applied. It is not to be expected that this can be done with sextants used at sea. Even with observations on shore this instrument often acts in a most perverse and unaccountable way. The observations of Williams for the determination of the latitude of Harvard College in 1782-3 are so instructive on the point of agreement inter se and disagreement with the truth that I copy his separate results:
Latitude of Harvard College
From the Sun | From Fixed Stars | From the Pole Star |
42° 23’ 22.3” | 42° 23’ 25.1” | 42° 23’ 28.6” |
25.3” | 22.9” | 27.7” |
20.5” | 34.1” |
|
| 28.7” |
|
| 36.5” |
|
| 24.2” |
|
| 28.6” |
|
The mean adopted was 42° 23' 28.5". By comparison with the latitude of the observatory this result is 64" too great, and yet the average deviation from the mean is only 3.6".
In 1825 Mr. R. T. Paine, perhaps the most skillful observer with the sextant in this country, if not in the world, found the value of the latitude of the Old State House,. Boston, to be 42° 20' 30", a value which differed 118" from that given on page 297 of the Transactions of the American Academy, which was also the result of a sextant determination. In 1828-9 194 observations with sextant "Ramsden 1403" gave 20' 57.8", and 390 observations with "Ramsden 1375" gave 20' 57.9". Yet the mean of 442 observations between 1833 and 1873, divided equally between northern and southern stars, gave 21' 23", with a range between greatest and least for 37 dates amounting to only 7.5". This result is 7" different from the value derived by Borden in his Trigonometrical Survey of Massachusetts. It gives for the latitude of Harvard College 42° 22' 15", the true value being '22' 48.6". It is, therefore, 34" in error, while the earlier value is 59" in error. Mr. Paine's subsequent observations for the latitude of the Unitarian Church in old Cambridge gave 46.7", or within 2" of the truth.
It is well known that there is in general a well defined difference between results derived with the sextant firm observations of the sun and of fixed stars, though Mr. Paine always got substantially identical results. At Syracuse in 1870 Professor Harkness found the latitude from Polaris 7.1" to be less than from the sun. Professor Hall found a difference of 24.7". Struve, in his survey of Turkey, found that Alpha Aquilae gave latitudes 13" too large, and that this correction sometimes ran as high as 30.7".
It is obvious from this brief discussion that there is danger of placing too high an estimate upon the accuracy of the ordinary sextant considered simply as an instrument of observation. It is hardly necessary to say that agreement of repeated readings on a given part of the limb is not necessarily an indication of accuracy. The errors which are usually introduced in the graduation of a sextant belong to a class known as periodic errors. They are formed by successive increments of very minute errors, which it is perhaps impossible to measure individually, but which, by continual additions, may, at a given point on the limb, amount to a quantity several times greater than the error of observation.
The data at hand for assigning a limit to the actual errors of observation with the sextant at sea, taking into account all the errors to which such observations are liable, are exceedingly meagre. For the present I limit myself to the following discussions:
(a) Scattered through the volumes of the Nautical Magazine will be found records of the determinations of the longitude of various stations, chiefly in the West India Islands, made by various British naval expeditions. The data required are in many cases wanting, but sufficient are available to furnish a fair estimate of the average range of error. The chronometers employed were rated at the Greenwich Observatory at the beginning of the voyage, and the observations for time at the terminal stations were made in the usual way with the sextant. They were probably made on shore, though I can find no definite statement to this effect. Evidently more than usual care was taken with the observations and reductions. It is probable also that a sufficient number of observations were made at each station to eliminate accidental errors in a large measure.
(b) During the spring and slimmer of 1880 Officer W. H. Bacon, of the Cunard steamer "Scythia," kindly undertook for me a series of systematic observations from which the relative errors .could be determined with considerable certainty. A complete series for a single day consisted of five sights at intervals of fifteen minutes, about 8 o'clock in the morning, five sights in the neighborhood of 11 o'clock, and five sights at the corresponding hours in the afternoon. Observations were also made when the ship was in known positions as often as possible.
This series of observations has an exceptional value on account of the conscientious fidelity with which the programme was adhered to and of the skill with which they were made. The relative errors were determined by comparing each position with the mean of the series, the rate being determined both from the morning and afternoon observations and from the log.
The results obtained are found in the following table:
Limits in Miles | Average Error from Observations at 9h and 3h | Average Error from Log at 9h and 3h | Average Error from Observations at 11h and 1h | Average Error from Log at 11h and 1h | Difference between Observation and Log at 9h and 3h | Difference between Observation and Log at 11 and 1h |
| No. Cases | No. Cases | No. Cases | No. Cases | No. Cases | No. Cases |
| 1 | 0 | 0 | 0 | 7 | 6 |
0.5 … 1.0 | 0 | 6 | 2 | 3 | 1 | 2 |
1.0 … 1.5 | 8 | 13 | 3 | 5 | 3 | 3 |
1.5 … 2.0 | 4 | 5 | 3 | 3 | 3 | 2 |
2.0 … 2.5 | 6 | 4 | 6 | 5 | 2 | 3 |
2.5 … 3.0 | 2 | 1 | 3 | 4 | 1 | 0 |
3.0 … 3.5 | 2 | 2 | 6 | 5 | 7 | 2 |
3.5 … 4.0 | 4 | 1 | 4 | 5 | 1 | 2 |
4.0 … 5.0 | 1 | 3 | 6 | 5 | 4 | 4 |
5.0 … 6.0 | 0 | 0 | 2 | 1 | 1 | 5 |
6.0 … 7.0 | 0 | 0 | 2 | 1 | 2 | 2 |
7.0 … 8.0 | 1 | 1 | 0 | 1 | 1 | 1 |
8.0 … 9.0 | 2 | 0 | 1 | 1 | 0 | 2 |
9.0 … 10.0 | 0 | 1 | 0 | 0 | 1 | 2 |
10.0 … 11.0 | 0 | 0 | 0 | 0 | 1 | 1 |
11.0 … 12.0 | 0 | 0 | 0 | 0 | 2 | 1 |
12.0+ | 1 | 1 | 0 | 0 | 0 | 0 |
It will be seen that the results from these two investigations do not materially differ from those previously found. In fact, whatever the line of investigation pursued, we reach substantially the same conclusion, viz, that the average error of a single observation at sea is not far from three miles, and that the average coefficient by which this number must be multiplied in order to provide for every contingency of danger is 3.5.