To appreciate the degree of confidence to which these results may be ent.i.tled, it is necessary to know under what circ.u.mstances the observations were taken; also the method used in the calculations, and the corrections which have been applied beyond what is usual in the common practice at sea: of these the following is a general statement.
1st. The instruments used in taking the distances, were a nine-inch s.e.xtant by Ramsden, and three s.e.xtants of eight inches radius by Troughton, the latter being made in 1801, expressly for the voyage. On board the ship, the s.e.xtant was necessarily held in the hand, and the distances were sometimes so taken on sh.o.r.e; but in most of the latter cases, it was fixed on a stand admitting of the s.e.xtant being turned easily in any direction. The telescopes were of the largest magnifying powers which the motion of the ship, or state of the atmosphere could admit, and each longitude is the result of a set of observations, most generally consisting of six independent sights. They were taken either by lieutenant Flinders or by myself; those by him being designated in the column of _Observers_ by the letter F, the others by C.
2nd. Preparatory to the reduction of the apparent to the true distance, the four following corrections have been applied.
From the _sun"s semi-diameter_, as given in the nautical almanack, 3"
have been subtracted. In the almanacks of the years comprehending our observations, the semi-diameter was stated from Mayer"s tables, which gave it 3" too great; owing to the imperfection of the telescope with which Mayer observed.
The semi-diameters of the sun and moon being less in the vertical, than in the horizontal direction, on account of the differences in the refraction, they have been reduced proportionally to these differences and to the angles at the points of contact in measuring the distance.
This correction is called _contraction of the semi-diameter_.
Before using the _moon"s horizontal parallax_ in the nautical almanack, where it is calculated for the equator, it has been corrected (printed as "diminished", and corrected in the errata) by a number of seconds depending upon the lat.i.tude of the place, and upon this a.s.sumed position: that the earth is a regular spheroid, whose polar axis is to the equatorial axis, as 320 to 321. This, and the preceding correction are unnecessary, unless where great exactness may be required.
The _refraction_ of the heavenly bodies given in the tables, being calculated for a mean height of 50 of Fahrenheit"s thermometer, and 29.6 inches of the barometer, it has been corrected for the difference between these means and what was the state of the atmosphere at the time of observation.
3rd. In reducing the apparent to the true distance, Mr. Crosley has used the method of Joseph Mendoza de Rios, Esq., F. R. S., given with his _Nautical Tables_, second edition, 1809; and the tables from which the corrections were taken and the computations made, are those of the same valuable work.
4th. The reduced distance, found as above, has been corrected to the spheroidal figure of the earth, according to the theory explained in the _Philosophical Transactions_ of the Royal Society of 1797; and for doing which, rules are given by Mr. Mendoza with his _Nautical Tables_ of 1801.
This calculation is tedious, and the correction, more especially in low lat.i.tudes, too small to be necessary in common cases.
5th. In the nautical almanack the distances are given to every three hours, but the irregularities of the moon"s motion being such as to cause some inequality in the different parts of this interval, the distance at the hour preceding, and at the hour following the time of observation, was found by interpolation from the two nearest given on each side; and having the distances at Greenwich for each hour, the observed distance can never fall more than half an hour from one of them; and the moon"s inequalities do not then produce any sensible error in the corresponding time, as obtained from common proportion. The correction arising from this process is seldom so important as to be necessary in sea observations.
6th. The longitude deduced from a comparison of the true distance at observation with the hourly distances at Greenwich, is contained in the following tables under the head of _Longitude from Nautical Almanack_.
But as it frequently happened, that the observation was not taken exactly in the place which it is intended to fix, this longitude is reduced to that place by the application of the difference shown by the time keepers to have existed between the two situations. In ascertaining this difference, the rates of going allowed to the time keepers are generally those found at the place which is to be fixed; whether applied to observations taken before arriving, or after quitting that place. This, however, could be done only at those stations where rates had been observed; at the intermediate points, where the result of lunar distances is given princ.i.p.ally as an object of comparison with the time keepers, the rates allowed in the reduction are those found at the station previously quitted; but then the difference of longitude is corrected by the quant.i.ty consequent on the following supposition: that the time keepers altered their rates from those at the previous, to those at the following station, in a ratio augmenting in arithmetic progression. The difference of longitude, thus corrected when necessary, is given under the head of _Reduction by time keepers_; and the longitudes reduced by it to the place intended to be fixed, are taken to be of equal authority with those resulting from observations made in the place itself.
7th. But these longitudes, whether reduced to, or observed in, the place to be fixed, still require a correction which is of more importance than any of those before mentioned. The theories of the solar and lunar motions not having reached such a degree of perfection as to accord perfectly with actual observation at Greenwich, the distances calculated from those theories and given in the almanack become subject to some error; and consequently so do the longitudes deduced from them. The quant.i.ties of error in the computed places of the sun and moon, have been ascertained at Greenwich as often as those luminaries could be observed; and Mr. Pond, the astronomer royal, having permitted access for this purpose to the table of errors kept in the Observatory, Mr. Crosley has calculated the corresponding effects on the longitude, and proportioned them to the time when our observations were taken. The combined effect of the two errors forms a correction to the longitudes obtained from the sun and moon; but when the moon was observed with a star, then the moon"s error alone gives the correction. But it has sometimes happened, that there were many days interval between the observations of the moon at Greenwich, and that the errors preceding and following are so extremely irregular, that no accuracy could be expected in reducing them by proportion; in these unfortunate cases, that part of the error belonging to the moon has been taken absolute, such as it was found on the day nearest to the time of observation; but the sun"s error is always from proportion. These corrections, with the interval in the Greenwich observations of the moon, are given under their proper heads.
8th. The longitudes thus computed, reduced to the intended point, and corrected, are placed under each other; and the mean of the whole is taken to be the true longitude of that point, unless in certain cases where it is otherwise expressed. The mean is also given of the longitudes _uncorrected_ for the errors of the sun and moon"s places, that the reader may have an opportunity of comparing them; and some sea officers who boast of their having never been out more than 5", or at most 10", may deduce from the column of corrections in the different tables, that their lunar observations could not be ent.i.tled to so much confidence as they wish to suppose; since, allowing every degree of perfection to themselves and their instruments, they would probably be 12", and might be more than 30" wrong.
In the nautical almanacks for 1811 and 1815, the distances are computed from the new tables of _Burg_ for the moon, and of _Delambre_ for the sun; and it is to be hoped that the necessity of correcting for errors in the distances at Greenwich will have ceased, or be at least greatly diminished. Should the computed places of the sun and moon be happily found to agree with actual observation, and supposing that our results may be taken as the average of what practised observers with good instruments will usually obtain when circ.u.mstances are favourable, then lunar observations taken in 1814 and afterwards, may be ent.i.tled to confidence within the following limits:
From one set of distances, consisting of six independent sights, the error in longitude may be 30" on either side; but will probably not exceed 12".
From six sets on one side of the moon, each set consisting as above, the error may be 20"; but not probably more than 8".
Twelve sets of distances, of which six on each side of the moon, are not likely to err more than 10" from the truth; and may be expected to come within 5".
The error in sixty sets, taken during three or four lunations, and one half on each side of the moon, will not, I think, be wrong more than 5"; and will most probably give the longitude exact to 1" or 2", This degree of accuracy is far beyond what the hopes of the first proposers of the lunar method ever extended, and even beyond what astronomers accustomed only to fixed observatories will be disposed to credit at this time; but in thinking it _probable_ that sixty sets of lunar distances will come within 1" or 2" of the truth, when compared with correct tables, I conceive myself borne out by the following facts.
In Port Lincoln, I observed an eclipse of the sun with a refracting telescope of forty-six inches focus, and a power of about two hundred. It was recalculated by Mr. Crosley from Delambre"s and Burckhardt"s tables, the one made four and the other ten years afterwards. The longitude deduced from the beginning differed only 1" 31.5" from that at the end, and the mean of both only 1" 17" from _thirty_ sets of lunar distances corrected for the errors of the tables.
The Spanish admiral D"Espinosa observed emersions of the first and second satellites of Jupiter in 1793, at Port Jackson, and also an eclipse of the sun which he recalculated by the tables of Burg. He deduces from thence the longitude of Sydney Cove to be 151 12" 45"; and from forty-four sets of lunar distances by lieutenant Flinders, it would be 151 11" 49" east.
At Port Louis in the Isle Mauritius, the Abbe de la Caille observed an eclipse of the sun, the transit of mercury over the sun"s disk, and various occultations of Jupiter"s satellites; M. d"Apres also observed several occultations; and this place should therefore be well determined.
Its longitude in the Requisite Tables is 57 29" 15" east; and from _twenty-seven_ sets of distances taken whilst a prisoner there, I made it, when corrected for the errors of the tables, 57 29" 57" east.
In appreciating the degrees of accuracy to which a small or larger number of lunar distances may be expected to give the longitude, I suppose the observer to be moderately well practised, his s.e.xtant or circle, and time keeper to be good, and his calculations to be carefully made; and it is also supposed, that the distances in the nautical almanack are perfectly correct. As, however, there may still be some errors, notwithstanding the science and the labour employed to obviate them, it cannot be too much recommended to sea officers to preserve all the data of their observations; more especially of such as may be used in fixing the longitudes of places but little, or imperfectly known. The observations may then be recalculated, if requisite; the corrections found to be necessary may be applied; and the observer may have the satisfaction of forwarding the progress of geography and navigation, after having contributed to the safety of the ship, and benefit of the particular service in which he may happen to have been engaged.
The following tables, set out in the book, are not reproduced in this text version of _A Voyage to Terra Australis_--refer to the _html_ version, available from I. (Reference from Chapter III., "_Longitude_ from thirty-one sets of distances of the sun east and west of the moon, of which the particulars are given in Table I. of the Appendix to this volume") LONGITUDE OF THE OBSERVATORY IN PRINCESS ROYAL HARBOUR, KING GEORGE"S SOUND
Table II. (Reference from Chapter III, "The _longitude_ from sixteen sets of distances of the sun east and west of the moon, of which the individual results are given in Table II. of the Appendix to this volume") LONGITUDE OF THE SOUTH POINT OF LUCKY BAY, ARCHIPELAGO OF THE RECHERCHE
Table III. (Reference from Chapter V., "_Longitude_ of the point, deduced from twenty-two sets of distances (see Table III of the Appendix to this volume)") LONGITUDE OF THE ANCHORAGE IN FOWLER"S BAY
Table IV. (Reference from Chapter VI., "The _longitude_, from thirty sets of distances of the sun and stars from the moon (see Table IV. of the Appendix to this volume)") LONGITUDE OF THE TENTS, AT THE HEAD OF PORT LINCOLN
Table V. (Reference from Chapter IX., "Point Nepean is in _lat.i.tude_ 38 18" south. The _longitude_ from twelve sets of distances taken by lieutenant Flinders in the port, and six others by me ten days before arriving, the particulars of which are given in Table V. of the Appendix to this volume") LONGITUDE OF POINT NEPEAN AT THE ENTRANCE OF PORT PHILLIP
Table VI. (Reference from Chapter X., "_Longitude_ from forty-four sets of distances of the sun and moon, of which the individual results are given in Table VI of the Appendix to this volume") LONGITUDE OF CATTLE POINT, PORT JACKSON
Table VII. Variations in rate and errors in longitude, made by Earnshaw"s time keepers No. 543 and No. 520, between the Cape of Good Hope and Port Jackson.
END OF VOLUME I