in the case of the Sun, is the time from apparent Noon; if a star has been observed, it must be combined with the right ascension of the sun and star. At sea, however, seldom happens that the quantities necessary for the calculations are known with precision, various causes tending to render their determination inaccurate. The knowledge of the altitude depends on the skill of the observer, the goodness of his instrument, and the distinctness of the horizon; and moreover, when obtained, requires to be reduced by various corrections; one of which, the depression of the visual ray, or, as it is commonly called, the Dip, is very variable, owing to the uncertainty of refraction near the horizon. The latitude originally deduced from altitudes, subject to similar errors, has to be brought up to the place of the ship, by estimating the run from the time of the last observation. Finally, the declination, taken from the Nautical Almanac, depends on the accuracy of the estimated longitude. It is clear, if we suppose that the assumed values of either or all three of these quantities be different from the truth, that we must expect a corresponding error in the deduced time; and such is the case in practice. The object of the following remarks, is to point out means by which the effect of probable errors in the three quantities mentioned may be calculated. The table given at the conclusion is taken from a French work, in which the subject has been fully considered; and if this, or similar works, were more generally in the hands of British seamen, the following sketch would be superfluous. Putting for the hour angle, a for the altitude, & for the latitude, & for the declination, for the azimuth or bearing from the meridian, and for the angle at the Sun or star included by the Polar and Zenith distances, the partial errors in for a change in a, and & are expressed as follows: The demonstration of these expressions may be found in most works on trigonometry. do de table, the arguments for each being the azimuth at the top, and the latitude at the side. The azimuth may be either observed with the compass at the time when the sights are taken, or it may be calculated from the expresSin. 0. cos. d Cos. a sion, Sin. n = and calculate the azimuth, at the same time that we learn what confidence is to be placed in the time deduced, we shall obtain a series of values of the variation of the magnetic needle; good observations of which are much wanted, independently of its use in merely working the dead reckoning, or shaping a course. At sea, in the day time, the probable error in altitude ought not to amount to more than two or three minutes, even in extreme cases, but at night it may greatly exceed that quantity. The latitude, however, if some days have elapsed without an observation, may be erroneous to the extent of a degree or more. If the practice be established, both to observe The value of de d& will always be small in comparison with the other two, for even at the equinoxes, when the sun's declination varies most rapidly, an Memoire sur l'Astronomie Nautique, M. Mazure Duhamel, Paris, 1822. 2 G NO. 5.-VOL. I. error of one hour in the estimated Greenwich time would be required to produce an error of one minute in the declination. We may, therefore, safely neglect it at sea. On shore, however, when observations for time are made, either for the measurement of differences of longitude, or for the purpose of rating the chronometers, and consequently when more accuracy is required, it may be proper to calculate the effect of a small change in the declination; in Sin. 0. cos. such cases, after calculating from the expression Sin. ↓ = Cos. a de d d and being similar, enter the table with at the top, and at the side, and in the column of do will be found the value of de d 8. But when observations have been made on shore, it will perhaps be better to calculate all the values of de directly from the expressions given for them, than to take them from the tables, which, however, are quite sufficient for the purpose when at sea. de de do Having obtained the values of da' d' d' they must be multiplied by the values of d a, dø, and d 8, in minutes, and the sum of the products, attending to their signs, will be the whole error in 0. It depends, of course, entirely on the experience of the observer, guided by the circumstances under which the observations are made, to determine what values are to be given to d a, &c. There is nothing, however, to guide us to the knowledge, whether the altitude, latitude, and declination, be greater or less than the truth; we therefore cannot know with what signs these partial do errors of are affected. Consequently (leaving out of the question) we d & must take the sum of the two other partial errors, as the greatest possible error, and their difference as the least probable error; and these should be registered in the journal with each observation-their mean being, of course, the mean probable error. de From an examination of the table, it will be seen, that the greater the azimuth, the less are the values of d 0. This holds true until the bearing from the meridian amounts to 90°, when their values are a minimum, but they again increase as the body moves from that position. The angle is also a maxi. mum when is; and as depends on the angle, it is a minimum when ŋ is 90°. Hence we may perceive the reason of the rule given in most works on navigation, which directs sights for time to be taken when the body is in or near the prime vertical; in other words, bearing east or west. This rule should be carefully followed; and in cases when the body is in that position when below the horizon (which is the case when the declination is of a contrary denomination to the latitude) the altitudes should be taken as near the horizon as possible, provided they exceed 3 or 4 degrees, below which greater errors might be introduced, owing to the uncertainty of refraction, than those we wish to avoid. Most works on navigation give tables of the altitude of the sun, and the time from noon, when it is in the most favourable position for observing. If not furnished with such tables, they may be computed from the following expressions::Sin. Sin. Cos. Cotan. p. tan. 8, Sin. a = When rating chronometers by sights observed on shore, care should be taken that the observations are made from day to day, when the sun has the same azimuth; as in that case the error in 0, from an erroneous assumption of , will be constant. A day or two before leaving port, sights should be obtained both before and after noon, at equal azimuths, in order to determine the true time and the error of the watch; but the rate must be collected from the daily sights, which are comparative one with another, although the absolute time of each may be erroneous. The error in an altitude taken with a sextant and artificial horizon, may be estimated at 20" or 30", more or less, according to the ability of the observer, and the perfection of the instrument; that in the latitude must depend on circumstances, which may vary in every case. While on the subject of chronometers, we may offer a remark on the very imperfect manner in which the determination of differences of longitude by them are registered in naval remark-books. We find it simply set down, that Cape was found to be in longitude -°--", but not a word to inform us at what place the error of the chronometer on Greenwich time, had been last ascertained, or what was its assumed longitude; of course, any error in the longitude of that place will also exist in the longitude of Cape determined. The practice should be, to mention that such a place was found to be so many degrees, &c. East or West of the place where the error of the chronometer (on mean time at that place) had been last ascertained, and in addition should be noted, the sights from which the error and rate had been ascertained, with a description of the instruments, the latitude used in the computation, and the effect of any probable errors in the elements, determined as shewn above. An example will serve to point out how useless the reports, made in the usual form, may become. Suppose a vessel determines the error of her chronometers on Greenwich time, at a place, A, supposed to be in 56o 35′ W., and sails to an unknown port, C ; arriving there, she finds it to be in longitude 61° 24′ W., which information is sent home. Another ship sails from a place, B, supposed to be in 68° 3'W., and going to the same port, C, places it in 61° 38′ W., which is also transmitted to the proper quarter. In the mean time, the Commander-in-chief thinks it highly necessary to determine the true positions of A and B. He therefore detaches a ship for the purpose, which, after a long series of observations at A, places it in 56° 25′ W., and measuring the difference of longitude betwixt A and B, finds it to be 11° 15'. B is consequently in 67° 40′ W. Now, had the two ships first mentioned, instead of transmitting the absolute longitude of C, sent their determinations of its distance west and east of A and B, viz., 4° 49′ W. and 6° 25′ E., the hydrographer, by combining those determinations with the true longitudes of A and B, determined specially as mentioned, would find that the one ship placed C in 56° 25′+4° 49′ 67° 40 presumption in favour of the truth 61° 14′ W., and the other in 61° 15′ W.; and there would be a strong 4 H |