We shall here terminate the present chapter on the principles of Navigation, having now discussed the several cases of sailing which actually occur in practice. But the student who is desirous of prosecuting his inquiries on this very important branch of practical science to greater extent, will, of course, consult works expressly devoted to the subject. Of these the most elaborate in our language is the valuable "Elements" of Robertson, in two octavo volumes. The Treatise of Mr. Riddle is also an excellent work, abounding with practical examples very accurately solved, and upon the whole better adapted to modern practice, as well as more compendious, than Robertson's. Mr. Norie's Navigation is also a good practical book, and so is Dr. Bowditch's edition of "Hamilton Moore." CHAPTER II. APPLICATION OF SPHERICAL TRIGONOMETRY TO (71.) The solution of Astronomical Problems forms one of the most useful and agreeable applications of the theory of spherical Trigonometry. To such inquiries the theory itself, no doubt, owes its origin, as well as many of the successive improvements which it has gradually received, so that a specimen of its use in the solution of astronomical problems may reasonably be looked for in a book on Trigonometry. For the purpose of measuring the angular distances of the heavenly bodies from each other, and from the horizon, it is convenient to suppose them all situated as they really appear to an observer on the earth, viz. in a spherical concave surrounding our earth and concentric with it.. This imaginary concave is called the celestial sphere, or the apparent heavens; in it all the apparent motions of the heavenly bodies are, for the convenience of trigonometrical application, supposed actually to take place; and the entire celestial sphere to revolve daily round the earth as if this were at rest in its centre. All this is allowable, because the applications of which we speak are not affected by the enquiry, whether the motions which the heavenly bodies present to an observer on the earth are really as they appear or not. At the opening of last chapter we defined several lines which geographers had found it convenient to consider as described on the surface of the earth; most of these astronomers extend to the heavens. Thus the plane of the earth's equator, when extended to the heavens, marks on the celestial sphere the great circle called the equinoctial, and in like manner, the meridians being extended to the heavens, mark out the celestial meridians; also the axis of the earth, about which its real motion takes place, when extended to the heavens, is the axis about which the apparent motion of the celestial sphere takes place: this axis marks out the north and south poles of the heavens. As the sun performs its apparent revolution about the earth in 24 hours it passes over 15° in an hour; if then we consider, as astronomers do, that the day at any place commences at noon, or when the sun is on the meridian of that place, the time shown by the sun in any position will be expressed in degrees by the arc of the equinoctial, intercepted between the fixed meridian of the place, and that passing through the sun, or it will be expressed by the angle included by these meridians. Celestial meridians are, therefore, also called hour circles, and the angle between the meridian of the place and that through the sun is called the hour angle, or the horary angle. That meridian which is at right-angles to the meridian of the place is the six o'clock hour circle, since the sun obviously reaches it when half way between noon and midnight. Besides these lines, thus transferred from the earth to the heavens, there are others peculiar to the celestial sphere, which must be mentioned; these are, 1st, the ecliptic, which is the great circle path described by the sun among the fixed stars in its apparent annual motion about the earth in reality it is the path of the earth moving in a contrary direction about the sun. This circle crosses the equinoctial at an angle subject to an exceedingly small variation, determinable by observation and computation; its inclination to the equinoctial is about 23° 28', but it is always given with the minutest attainable accuracy in the Nautical Almanack. The points where the ecliptic crosses the equinoctial are called the N equinoctial points: the sun enters these points about the 21st of March and the 23d of September; the former being called the vernal equinox, and the latter the autumnal equinox. These names are given because at such times the nights are equal in length to the days all over the world; for as the two poles of the earth are at these times symmetrically situated with respect to the sun, the circular boundary, which separates the enlightened hemisphere from the darkened, must pass through both poles; and hence any point on the earth will be as long in being carried, by the earth's uniform rotation, through the enlightened part as through the dark part. The meridian through the equinoctial points is called the equinoctial colure. The position of any point on the celestial sphere, like the position of a point on the terrestrial sphere, is marked out by its latitude and longitude. On the celestial sphere the circle of longitude is the ecliptic; and perpendiculars to this, passing, therefore, through the poles of the ecliptic, are the circles of celestial latitude; the point from which longitude is measured is the vernal equinoctial point. Commencing at this point, too, the ecliptic is divided into twelve parts, called signs; a sign is therefore 30°. The twelve signs are named, and symbolically expressed, as follow: The first six of these signs are on the north of the equinoctial, the others on the south, and the vernal equinoctial point is called the first point of Aries. The longitude is measured from this point in but one direction, viz. in the order of the signs. Besides the above method of marking out the position of a celestial body, by means of its latitude and longitude, there is another way, viz. by means of its Right Ascension and Declination. The right ascension is measured on the equinoctial from the first of Aries, in the order of the signs, and the declination is measured on the perpendicular to this, or circle of declination passing through the object. We see, therefore, that what on the terrestrial sphere is latitude and longitude, is on the celestial sphere declination and right ascension; and parallels of latitude on the one correspond to parallels of declination on the other. Of these the two which are 23° 28′ from the equinoctial, one on each side, and which therefore touch the ecliptic in the first points of Cancer and Capricorn, are called the tropics of Cancer and of Capricorn. These first points of Cancer and Capricorn are respectively called the summer and winter solstice; because for a day or two before and after the sun enters them he appears to be stationary, and the days to be of equal length, so slowly does his declination at those times change, for his motion is obviously very nearly parallel to the equinoctial. The meridian, through the solstitial points, is called the solstitial colure, and that through the equinoctial points, the equinoctial colure. Having described the principal circles and points of the celestial sphere which are considered as permanent, or which do not alter with the situation of the observer on the earth, we come now to describe those which change with his place. The principal of these is the horizon, which has been defined already (63), and vertical circles which are perpendicular to the horizon, and on which the altitudes of celestial objects are measured. These vertical circles all meet in two points diametrically opposite, viz. the poles of the horizon; that one which is directly over the head of the observer is called his zenith, and the opposite one his nadir. That vertical which passes through the east and west points of the horizon is called the prime vertical; it necessarily intersects the meridian of the place (which passes through the north and south points) at right-angles. The azimuth of a celestial object is an arc of the horizon, comprised between the meridian of the observer and the vertical circle through the object, and hence vertical circles are sometimes called azimuth circles. The amplitude of a celestial object is the arc of the horizon, comprised between the east point and the point where the object rises, or between the west point and that where it sets; the one is called the rising amplitude, the other the setting amplitude. These definitions and remarks will suffice to render the following problems intelligible. 557 557 559 PROBLEM I. (72.) Given the sun's right ascension and declination to determine his longitude and the obliquity of the ecliptic. E R Let n EsQ represent the celestial meridian through the first of Cancer and Capricorn, that is, let it be the solstitial colure, ns the axis of the sphere, EQ the equator, eC the ecliptic, and nSs the declination circle, passing through the sun S; then ARS is a right-angle, and in the right-angled spherical triangle ARS there are given the right ascension AR, and the declination RS to find the longitude AS, and the obliquity SAR, which is an easy operation in right-angled spherics. It is necessary, however, to remark that as celestial longitude and right ascension are measured from A, the first point of Aries in the direction AS of the signs quite round the celestial sphere, when, of the four quantities in the problem, the obliquity and the declination are given to find the others, we must know on what side of the equinoctial the sun is, that is, whether the declination is north or south, for if the sun have the north declination RS, the longitude will be AS; but if it have the equal south declination R'S', the longitude being measured in the direction ASC round the globe to S', will be, instead of A' S', 360°-A'S'. It is moreover necessary to know not only on which side of the equinoctial the sun is, but also on which side of the tropic; for the sun, in passing from a tropic to the equinox, descends through the same gradations of declination as it ascended through in passing from the preceding equinox to the tropic, although its longitude and right ascension goes on increasing; in addition, therefore, to knowing whether the declination is north or south, we must also know whether it be encreasing or decreasing, in order to determine the longitude and right ascension without ambiguity; and these particulars will be known from knowing the time of the year when the proposed declination is observed; thus from the 21st of March to the 21st of June, during which time the sun is in the first quadrant of the ecliptic, the sun's declination is north and increasing; it afterwards continues to decrease, still remaining north, |