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CHAP. III. GEOGRAPHICAL LATITUDES DETERMINED. 133

(202.) The latitude of a station on a sphere would be merely the length of an arc of the meridian, intercepted between the station and the nearest point of the equator, reduced into degrees. (See art. 86.) But as the earth is elliptic, this mode of conceiving latitudes becomes inapplicable, and we are compelled to resort for our definition of latitude to a generalization of that property, (art. 95.) which affords the readiest means of determining it by observation, and which has the advantage of being independent of the figure of the earth, which, after all, is not exactly an ellipsoid, or any known geometrical solid. The latitude of a station, then, is the altitude of the elevated pole, and is, therefore, astronomically determined by those methods already explained for ascertaining that important element. consequence, it will be remembered that, to make a perfectly correct map of the whole, or any part of the earth's surface, equal differences of latitude are not represented by exactly equal intervals of surface.

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(203.) To determine the latitude of a station, then, is easy. It is otherwise with its longitude, whose exact determination is a matter of more difficulty. reason is this: as there are no meridians marked upon the earth, any more than parallels of latitude, we are obliged in this case, as in the case of the latitude, to resort to marks external to the earth, i. e. to the heavenly bodies, for the objects of our measurement; but with this difference in the two cases- to observers situated at stations on the same meridian (i. e. differing in latitude) the heavens present different aspects at all moments. The portions of them which become visible in a complete diurnal rotation are not the same, and stars which are common to both describe circles differently inclined to their horizons, and differently divided by them, and attain different altitudes. On the other hand, to observers situated on the same parallel (i. e. differing only in longitude) the heavens present the same aspects. Their visible portions are the same; and the same stars describe circles equally inclined, and

similarly divided by their horizons, and attain the same altitudes. In the former case there is, in the latter there is not, any thing in the appearance of the heavens, watched through a whole diurnal rotation, which indicates a difference of locality in the observer.

(204.) But no two observers, at different points of the earth's surface, can have at the same instant the same celestial hemisphere visible. Suppose, to fix our ideas, an observer stationed at a given point of the equator, and that at the moment when he noticed some bright star to be in his zenith, and therefore on his meridian, he should be suddenly transported, in an instant of time, round one quarter of the globe in a westerly direction, it is evident that he will no longer have the same star vertically above him: it will now appear to him to be just rising, and he will have to wait six hours before it again comes to his zenith, i. e. before the earth's rotation from west to east carries him back again to the line joining the star and the earth's centre from which he set out.

(205.) The difference of the cases, then, may be thus stated, so as to afford a key to the astronomical solution of the problem of the longitude. In the case of stations differing only in latitude, the same star comes to the meridian at the same time, but at different altitudes. In that of stations differing only in longitude, it comes to the meridian at the same altitude but at different times. Supposing, then, that an observer is in possession of any means by which he can certainly ascertain the time of a known star's transit across his meridian, he knows his longitude; or if he knows the difference between its times of transit across his meridian and across that of any other station, he knows their difference of longitudes. For instance, if the same star pass the meridian of a place A at a certain moment, and that of B exactly one hour of sidereal time, or one twenty-fourth part of the earth's diurnal period, later, then the difference of longitudes between A and B is one hour of time or 15', and B is so much west of A.

CHAP. III.

DETERMINATION OF LONGITUDES.

135

(206.) In order to a perfectly clear understanding of the principle on which the problem of finding the longitude by astronomical observations is resolved, the reader must learn to distinguish between time, in the abstract, as common to the whole universe, and therefore reckoned from an epoch independent of local situation, and local time, which reckons, at each particular place, from an epoch, or initial instant, determined by local convenience. Of time reckoned in the former, or abstract manner, we have an example in what we have before defined as equinoctial time, which dates from an epoch determined by the sun's motion among the stars. Of the latter, or local reckoning, we have instances in every sidereal clock in an observatory, and in every town clock for common use. Every astronomer regulates, or aims at regulating, his sidereal clock, so that it shall indicate Oh Om 0s, when a certain point in the heavens, called the equinox, is on the meridian of his station. This is the epoch of his sidereal time; which is, therefore, entirely a locai reckoning. It gives no information to say that an event happened at such and such an hour of sidereal time, unless we particularize the station to which the sidereal time meant appertains. Just so it is with mean or common time. This is also a local reckoning, having for its epoch mean noon, or the average of all the times throughout the year, when the sun is on the meridian of that particular place to which it belongs; and, therefore, in like manner, when we date any event by mean time, it is necessary to name the place, or particularize what mean time we intend. On the other hand, a date by equinoctial time is absolute, and requires no such explanatory addition.

(207.) The astronomer sets and regulates his sidereal clock by observing the meridian passages of the more conspicuous and well known stars. Each of these holds in the heavens a certain determinate and known place with respect to that imaginary point called the equinox, and by noting the times of their passage in

succession by his clock he knows when the equinox passed. At that moment his clock ought to have marked Oh Om Os; and if it did not, he knows and can correct its error, and by the agreement or disagreement of the errors assigned by each star he can ascertain whether his clock is correctly regulated to go twenty-four hours in one diurnal period, and if not, can ascertain and allow for its rate. Thus, although his clock may not, and indeed cannot, either be set correctly, or go truly, yet by applying its error and rate (as they are technically termed), he can correct its indications, and ascertain the exact sidereal times corresponding to them, and proper to his locality. This indispensable operation is called getting his local time. For simplicity of explanation, however, we shall suppose the clock a perfect instrument; or, which comes to the same thing, its error and rate applied at every moment it is consulted, and inIcluded in its indications.

(208.) Suppose, now, two observers, at distant stations, A and B, each independently of the other, to set and regulate his clock to the true sidereal time of his station. It is evident that if one of these clocks could be taken up without deranging its going, and set down by the side of the other, they would be found, on comparison, to differ by the exact difference of their local epochs; that is, by the time occupied by the equinox, or by any star, in passing from the meridian of A to that of B: in other words, by their difference of longitude, expressed in sidereal hours, minutes, and seconds.

(209.) A pendulum clock cannot be thus taken up and transported from place to place without derangement, but a chronometer may. Suppose, then, the observer at B to use a chronometer instead of a clock, he may, by bodily transfer of the instrument to the other station, procure a direct comparison of sidereal times, and thus obtain his longitude from A. And even if he employ a clock, yet by comparing it first with a good chronometer, and then transferring the latter

CHAP. III.

LONGITUDES FOUND BY CHRONOMETERS. 137

instrument for comparison with the other clock, the same end will be accomplished, provided the going of the chronometer can be depended on.

(210.) Were chronometers perfect, nothing more complete and convenient than this mode of ascertaining differences of longitude could be desired. An observer, provided with such an instrument, and with a portable transit, or some equivalent method of determining the local time at any given station, might, by journeying from place to place, and observing the meridian passages of stars at each, (taking care not to alter his chronometer, or let it run down,) ascertain their differences of longitude with any required precision. In this case, the same time-keeper being used at every station, if, at one of them, A, it mark true sidereal time, at any other, B, it will be just so much sidereal time in error as the difference of longitudes of A and B is equivalent to: in other words, the longitude of B from A will appear as the error of the time-keeper on the local time of B. If he travel westward, then his chronometer will appear continually to gain, although it really goes correctly. Suppose, for instance, he set out from A, when the equinox was on the meridian, or his chronometer at Oh, and in twenty-four hours (sid. time) had travelled 15° westward to B. At the moment of arrival there, his chronometer will again point to Oh; but the equinox will be, not on his new meridian, but on that of A, and he must wait one hour more for its arrival at that of B. When it does arrive there, then his watch will point not to Oh but to 1h, and will therefore be 1h fast on the local time of B. If he travel eastward, the reverse will happen.

(211.) Suppose an observer now to set out from any station as above described, and constantly travelling westward to make the tour of the globe, and return to the point he set out from. A singular consequence will happen he will have lost a day in his reckoning of time. He will enter the day of his arrival in his diary, as Monday, for instance, when, in fact, it is Tuesday.

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