that is, in degrees and parts of degrees, How the spaces or arcs Z Æ, PR, ZP, HÆ in the heavens are to be measured by a spectator at the spot A on the surface of the earth, is now to be explained. This is done by means of observations made by the spectator at A, upon some heavenly body, with an instrument adapted for the purpose of measuring circular arcs: by these observations, which are made when the heavenly body is either upon the meridian of the place or not, the angular distance of the body from the zenith or from the horizon is ascertained. Thus ifs be the sun (see fig. 5.) on the meridian, its angular distance s Z from the zenith, (called its zenith distance,) or its angular distance s H from the horizon, (called its altitude,) is measured, and ascertained in degrees and parts of degrees. As, however, A is the place at which these observations are made, the angle Z As is all that can be determined from observation; but this angle is not the measure of the arc Zs, because A is not the centre of the sphere of the heavens ; but the angle ZOs is the proper measure of this arc, since, by the supposition, the meridian circle and the corresponding • From a Greek word, and thus applied, signifies simply a change of place. tive arcs Zs and Hs may be accurately measured. The parallax is computed and given in astronomical tables, for the purpose of making this requisite correction. CHAPTER IV. On the Methods of Determining the Latitude. THS following methods are those which are in use for finding the latitude of places on land: 1st. By the altitudes of those stars (called circumpolar stars), which never go below the horizon of the place the latitude of which is required. 2ndly. By the greatest and least alti tudes of the sun above the horizon of the place in the course of the year. 3rdly. By the observed altitude or zenith distance of a star or other heavenly body when on the meridian. 4thly. By the zenith distances of stars, which pass the meridian near to the zenith of the place. 5thly. By various altitudes of a star, observed when it is near to the meridian, and then reduced to the meridian by computation. 1st Method. By the altitudes of circumpolar stars. Suppose (fig. 6.) that Z PRQ H Æ, is that imaginary circle in the con P the north pole, and s the circumpolar star on which the observations are to be made. The little semicircle drawn through s and s', parallel to the equator, will represent the apparent path of the star in its motion caused by half a daily revolution of the earth. It is evident from a mere inspection of fig. 6, that the star's greatest and least altitudes above the horizon will be when the star is on the meridian; its greatest when it is above, its least when below the pole P. Let s be its position in the first case, and s' in the other; then Rs is the star's greatest altitude, Rs its least altitude. By means of either of the instruments called an astronomical quadrant, or an astronomical circle, R's and Rs may be observed and measured, and the number of degrees and parts of degrees contained in it be ascertained. Then as the halfcircles, which the star has described in its apparent motion from s to s, is parallel to the equator, (for the motion of the earth, which is the cause of this apparent motion of the star, is perpendicular to the axis of the earth, so that the path of the star is also perpendicular to the axis, and therefore parallel with the equator;) and as the equator is every where at the same distance, viz., 90° from the pole P, the half-circles is also every where at the same distance from P; therefore Ps' is equal to P s. Now Rs, which is known from observation, is equal to PR + Ps; and Rs', which is also known from observation, is equal to PR - Ps', or PR - Ps. Adding these two quantities, Rs and Rs together, we have Rs + Rs', equal to 2 PR; therefore PR, or the height of the pole above the horizon, (which has already been proved to be equal to the latitude of the place ZE,) is equal to of Rs+ Rs', or one half the sum of the greatest and least altitudes of a circumpolar star, which altitudes being known from observation, the latitude of the place is found. This mode of finding the latitude does not require any correction to be applied to the observed altitudes on account of parallax, as the body observed is a fixed star; but a correction of these altitudes is required, in consequence of the refracting power of the air and vapours which surround the earth and have effect upon the apparent places of heavenly bodies, contrary to the effect of parallax,-parallax making bodies appear lower in the heavens; whereas, a ray of light passing through the atmosphere becomes refracted and bent downwards, and the body from which the ray proceeds, appears above its true place in the heavens. The space through which a body is raised by refraction (and which is different for different altitudes), is given in tables computed for various altitudes; this correction must, of course, be subtracted from the apparent observed altitudes. 2dly. By the greatest and least altitudes of the sun above the horizon in the course of a year. The path in which the sun's apparent yearly motion in the heavens takes place (called the ecliptic) is, at one point of it, about 23° 28' on the north side of the equator; and, at the exactly opposite point, it is the same number of degrees and minutes on the south side of the equator. These two points of the ecliptic are the farthest off from the equator, and are exactly 90° distant from the two points where the ecliptic and equator cut each other, which are called the equinoctial points. The sun is in the former point on or about the 24th of June, and in the latter on or about the 24th of December. To all persons, therefore, living between the north pole and latitude 23° 28', it will, on the 24th day of June, when it comes on the meridian, be the highest above the horizon, or have its greatest altitude, compared with its altitude on every other day in the year; and, in like manner, it will, on the 24th of December, have its least meridional altitude. Let S (fig. 6.) be its position in the former, and S' in the latter, of these two cases. Then, as is the point in the equator from which S and S are both distant 23° 28', Æ S and ÆS' are equal. The altitudes of the sun's centre in both positions are to be observed with an instrument, which observation, when corrected for parallax and refraction, will give HS and H S', the greatest and least meridional altitudes of the sun in the course of the year. Now, HÆ=HS ÆS, and HÆ is also =H S'+Æ S' or ÆS, and therefore 2 HÆ=HS+HS'; and H E, the height of the equator above the horizon, or the co-latitude of the place, is equal to the sum of the greatest and least meridional altitudes of the sun in the course of the year. As, however, it seldom happens in practice that the sun, when it comes upon the meridian of the observer, is exactly at that point of its path where it is farthest from the equator, but has either already passed that point, or has not yet quite reached it, certain corrections upon the observed altitudes become necessary, in order to allow for this circumstance. 3dly. By the observed altitude, or the observed zenith distance of a star or other heavenly body, when on the meridian. This method of finding the latitude is that which is generally employed for common geographical purposes. It is the most simple in practice, as depending only upon one observation, and is also, on this account, the most immediate in its result. It is also adapted for nautical purposes, the only difference between the modes of conducting the operation on land and at sea being in the instruments employed for making the observations, and also that, at sea, the heavenly body selected for observation is either the sun or moon, because, from the motion of the vessel, it is difficult to obtain a correct observation of the meridian altitude of any body having so small an apparent magnitude as a star. A few remarks will be made in a subsequent page, explanatory of some of the peculiarities of the modes of finding the latitude at sea: we shall, therefore, in the present instance, confine ourselves to the supposition, that the observer who is about to adopt this method of ascertaining his latitude, is on land. Suppose (fig. 6.) S or S' to be the star or other heavenly body which is selected, S being a heavenly body above the equator, S' being a heavenly body below it; the observation is to be made when the body is on the meridian. Let ZS H represent a portion of the meridian in the heavens, and Ee Qq represent the equator: SH or S'H is then ascertained from observation, if the altitude be taken; or S Z or S' Z, if the zenith distance be taken; which it is more usual to take, as, from the inequalities of the earth's surface, it is difficult to obtain on land a true horizontal boundary. These observations must be corrected for parallax and refraction, if the body be either the sun or moon; and for refraction only, if it be a fixed star. Now, the object being to ascertain either EH, the height of the equator above the horizon (which has been already shown to be equal to the co-latitude), or EZ, the zenith distance of the equator, (which is the latitude,) it is evident that 4th. By the zenith distances of stars When this method is adopted, the observations are generally made at two places having different latitudes; and the latitude of one of the places is supposed to be previously known. It is immaterial whether both places are or not situated upon the same meridian; the star must be one which passes near the zenith of both places. The observations are generally made at both places on the same day; if they happen to be made on different days, various corrections become requisite, which it is as well, if possible, to avoid. The instrument employed on this occasion is one called a zenith sector, by which small zenith distances can be measured with great exactness. Let Z, Z', (fig. 7.) be the zeniths of any two places, Q the equator; and suppose that the latitude of the place whose E Fig. 7. H zenith is Z (that is, ZÆ,) is known, the 5th. The remaining method is by making several successive observations upon the same star at several and successive altitudes above the horizon, when it is near the observer's meridian. The various altitudes thus obtained are made the basis of a computation by which the star's actual meridional altitude is obtained. This is called reducing the observed altitudes to the meridian. It is a process too intricate to be introduced in this place. The star's meridional altitude is thus obtained with great exact ness. The latitude is then very easily ascertained by the application of the third method. This mode of computing the latitude, by which it may be obtained to within the fraction of a second, is that which was employed by the French astronomers in their last operation of measuring an arc of the meridian. At sea none of the preceding methods, except the third, are ever employed; the first and fourth are founded upon observations made with instruments requiring some nice adjustments by means of the plumb line and the spirit level, in order that the instrument may be placed exactly in the plane of the meridian and in a horizontal position-these adjustments cannot be made at sea, owing to the unsteady motion of the vessel: the fourth method is moreover not applicable to all places; the second method would clearly be useless, since at sea an immediate result is required; and the fifth is too complex in its calculations to be fitted for nautical purposes. The instrument used at sea is Hadley's sextant, by which any angles whatever may be measured, and it does not require any previous adjustments, being held in the hand of the observer. The horizon being well defined at sea, the altitudes of heavenly bodies are taken. The observations are taken when the sun or other body is near the meridian, and are continued until it is found that its altitude has attained its greatest quantity and begins to decrease: at its greatest altitude the body is on the meridian of the observer at $ (fig. 6.); the complement of the latitude is, therefore, obtained, as in the third method, by adding or subtracting the distance of the observed body from the equator, according as it is below or above it, or has a south or north declination. The sun or moon is commonly the object observed, and the Nautical Almanack gives the declination of these bodies for every day in the year. The corrections for parallax and refraction must be made upon the observed altitudes. Besides these corrections, another is rendered necessary, in consequence of the observer being elevated above the surface of the sea. This elevation causes a correspondent depression or sinking of the horizon, and gives a greater apparent altitude to the observed body than it really has. This correction is called the dip. But as it frequently happens that, at the time when the sun or moon is on the meridian, clouds prevent the observing of its meridional altitude, the latitude may then be obtained by observing two altitudes out of the meridian at different times, and noting the interval of time which elapses between the times of observation. ZP (fig. 6.), or the co-latitude, is then computed by the resolution of three spherical triangles, a mere mathematical process, which we need not stop to investigate. CHAPTER V. Longitude-Mode of Measuring Time -Sidereal Time Apparent Solar Time-Mean Solar Time-Equation of Time. HAVING by one of the foregoing methods ascertained the parallel of latitude in which any particular place is situated, the next inquiry is directed to the finding of the longitude, or the position which a place occupies in the parallel with respect to what is called the first meridian. In this country the meridian of the observatory at Greenwich is generally taken for the first meridian. Various are the methods which have been proposed for finding the longitudes of places; in every point of view the subject is one of very considerable interest, not only on account of its great importance in commerce and science, but also because the attempts which have for so many years been made, in order to determine the longitude with the same accuracy with which the latitude of places is found, have hitherto been unsuccessful. Since the time of Queen Anne it has been regarded as an object of great national importance; and a board, called the Board of Longitude, consisting of various official and scientific persons, was then established for the purpose of encouraging and directting attempts to determine it. All the methods for finding the longitudes depend upon the manner in which time is measured; and in order to attain a clear notion of them, it will be proper to explain at some length how a measure of time is obtained. Properly considered, time is, in itself, without parts, and indivisible; the flow or lapse of time is, however, capable of being measured by means of events happening in time, and which, when compared one with another, are of different continuance, taking up more or less time in their completion. Time and space are in one respect similar; space is in its nature indivisible, it does not contain within itself any marks or circumstances of division; but by means of bodies which are situated within it, we are able to consider space as though it were divided into parts. What bodies |