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of a place on the globe. The first merely shows that the place is sumewhere on a small circle (a parallel) at a definite distance from the equator; the second merely shows that the place is somewhere on a great circle (a meridian) that makes a definite angle with another great circle which passes through a fixed conventional place of reference. To know the position of a place the point of the intersection of these two circles of the meridian with the parallel—must be determined; but this cannot always be done at sea, at any given or required instant, by any of the ordinary rules of nautical astronomy; though it may be done by a combination of rules; or partly by computation and partly by projection; or wbere a good point cannot be ascertained as that on which the ship is, a line may be found on or near which she is known to be, and this at the time may be priceless. The position of the sbip is thus determined by a method of utilising parts of circles which, in their completeness, would be oblique to the parallels and meridians.

When the declination of a celestial object coincides in amount and name with the latitude of a place on the terrestrial sphere, it must, at some time during the earth’s rotation on its axis, appear on the zenith of that place ; it will do so when the object's hour. angle for the place is 0 h., that is, when it is on the meridian. When this occurs, the Greenwich time by chronometer being known, let it be taken as granted that the object is above the horizon of another place; that its altitude is observed, and its zenith-distance consequently known. In Fig. 1 the object is vertical to the point S of the globe; with S as the pole, and the observed zenith-distance, Sa, as a polar-distance, describe

'S the small circle a a' a" a' : this is a circle of

ta' position, on some point of which the observation has been made, for from

Fig. 1. very point within or without this small circle, a less or greater

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zenith-distance than Sa would be observed at the instant of the object being at S. “ If, then, the navigator can project this small circle upon an artificial globe, or chart, the knowledge that he is upon this circle will be just as valuable to him in enabling him to aroid dangers as the knowledge of either his latitude alone or his longitude alone; since one of the latter elements only determines a point to be in a certain circle, without fixing upon any particular point of that circle."-CHAUVENET.

The altitude of another celestial object, s', taken at the same time as the former, gives a second circle of position (Fig. 1, dotted circle). The observer being in the circumference of each of these circles, must be at one of their points of intersection, P or P': there will be no difficulty in ascertaining which is to be taken, as it will generally be indicated by the dead reckoning.

The circles of which we have spoken are such as they would appear when

2 represented on the spherical surface of a globe, and they illustrate the principle of the problem. On a Mercator's chart, where the distance between the parallels is considerably augmented in the higher latitudes—in order to preserve the proportion that exists at different parts of the earth's surface between the meridians and the parallels

Fig. 2. circles of position would be represented as elliptical figures (Fig. 2); perhaps we had better say, as curves of position, which, to delineate properly, would require to be computed for every 5 or 10 degrees : happily, in the projection of the problem, we only require a very small part of these carves, for which we assume two latitudes a few miles on each side of the latitude by D.R. We may take the tangent (T) to the curve, or the chord (C); but our computation and projection will be the more perfect the more closely the chord and the tangent coincide—in fact, we shall then have the best line of position.

It is not our purpose to write the rule for Sumner's Method ; we take it that the majority of our readers already know it; those


who do not, cannot too soon make its acquaintance: our intention is simply to comment on its value.

The data for the problem are the correct Greenwich date by chronometer; simultaneous altitudes of two stars, or of a star and planet—by far the best objects to give the ship's position ; when the sun is the object there must be an earlier and later altitude, with the sun's bearing at the first sight, and course and distance carefully noted in the interval of the observations; two assumed latitudes, the basis of which must be the latitude by D.R.; and, finally, the elements from the Nautical Almanac; of the latter we have not a word to say beyond this—there is no excuse for taking them out inaccurately.

About the chronometer:–Everything depends upon this instrument, unless we are satisfied to merely get the correct latitude by means of the hour angles. But, really, there is no difficulty in verifying the error and rate. We know that all our regular lines of passenger steamers carry at least three chronometers—some, more; and of those that traverse the Atlantic to the States, the West Indies, Brazil, and the Cape, equally with those that take the route through the Red Sea to India, in which the voyages are short, or at regular intervals, if the Greenwich time is not known on board within four seconds, or less, under every variation of temperature, then all we can say is, that it ought to be; and it is certain that all the instruments cannot break down at once; therefore, for these voyages, and for the purpose of making good landfalls, if sights are to be had at all, the problem in part, or in its entirety, is everything that could be desired. The same remarks apply, with almost as much force, in the case of our large colonial traders, when at least two chronometers are carried, since a voyage is rarely accomplished without some well-known spots being sighted, which should be so many landmarks for the verification of the error and rate of the Greenwich timekeeper. But the case is immeasurably different when there is but one chronometer on board, and its accuracy has never been tested during the whole voyage—not perhaps from unwillingness, but from sheer inability, on the part of the master, and towards affecting which his examination for master's certificate never gave

him the slightest clue or help. While we write, such an one “made tolerable landfalls,” and yet his chronometer, with the given error and rate applied, we found to be 3m. 17 8. slow on M.T. at Greenwich. Who can doubt the danger in which a vessel so navigated might be placed ?—for all else being taken to be approximately correct, the error of the chronometer places every part of this problem too far east, or too far west, bodily; and a vessel, put on a line of positivn, might unsuspectingly be brought into danger on the west coast of Ireland, when supposed to be making for St. George's channel. While expressing our own unbounded confidence in the problem under discussion, we have considerable doubt as to its value in the last case : a master so circumstanced should pay special attention to his latitude, and to the famous three L's generally—therein lies safety ; but to take the indications of an unverified chronometer, as if it gave a tolerable line of position, is only to go in search of that which would surely not be found.

Simultaneous altitudes of two celestial objects are unquestionably the best for determining the position of a ship,—which is thus got at once without any change of place or interval of time for which to allow. With a good knowledge of the stars and planets, two objects can be selected at pleasure, and in such relation to each other that the angle between their verticals shall be the best possible—something between 60° and 120°, and so develope a good point of intersection. If there be any doubt, a third star will give, with the two others, a space or triangle of certainty, within which the ship must be. Taken in the twilight—and how often may this be done when no sun has been visible all day—the altitudes, by a practised hand, ought to be obtained within a limit of 2 to 3—less rather than more. Attend to the remarks of Raper : “ The observation of stars at night is a very different observation from other altitudes by day; and, to ensure success, the observer should make it a matter of special practice. It is, however, during the twilight that stars and planets may be most advantageously observed at sea, as the horizon at that time is strongly marked, and, when not sufficiently so, may be rendered distinctly visible by the inverting telescope."

When assuming the two latitudes it is generally sufficient to

select them about 30' or less on each side of the latitude by D.R.; this will much depend upon what length of time has elapsed since the ship's position had been previously determined. If the altitudes are simultaneous, and the lines of position (when computed and projected) intersect considerably beyond one or other of the assumptions, then take another latitude a little beyond that of the intersecting point, compute anew for this, and so project again. The position will be more accurately determined in this manner, for the latitude is an important element in the computation of the hour-angle. You will of course reject, as outside the requirements of the problem, in fact as erroneous, all that portion of the computation based on the most distant assumed latitude.

It is not, however, essential that the same assumed latitudes should be used in computing both lines of position; it is only more convenient to do so, as it saves some logarithms. In the case of two altitudes of the same object, as of the sun, where a course and distance have been made in the interval, if the course has been nearly north or south, it would be better to assume two latitudes differing from those used for the first observation, and such that they may be more in accordance with the altered position of the ship.

It is scarcely necessary to warn the intelligent navigator against making an assumption of latitude that shall render the computation of the hour-angle impossible; the sum of the altitude, latitude, and polar-distance can never exceed 180°—when it is equal to that quantity, the object is on the meridian. Still it may happen that no opportunity of observing must be lost; the altitude is taken, and must be used for what it is worth ; we may, however, be able to show that when there is a doubt as to the side of the meridian on which it is, it may be advantageously treated as if it were a meridian altitude, and its hour-angle of 0 h. will give one point of the circle of altitude, and a less assumed latitude give the other point; the navigator is lucky if, in very uncertain weather, having got his morning sight, and the provisional observation near noon, he can obtain another and better sight later in

the day.

(To be continued.)

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