CHAPTER II. APPLICATION OF SPHERICAL TRIGONOMETRY TO NAUTICAL ASTRONOMY. 103. IN our chapter on Navigation we have laid down several methods of determining the place of a ship at sea, by help of the account kept on board of its progress through the water, that is, of the course and distance sailed ; and, if confidence could be placed in this account, even when kept with the utmost care, the art of Navigation would be perfect. Such perfection, however, it is hopeless to expect; for it does not seem possible to measure, with strict accuracy, either a ship's rate or the direction in which she moves, both of which may indeed be continually varying. In order, therefore, to determine the place of a ship at sea, with that accuracy which the safety of navigation requires, it is absolutely necessary that we be furnished with methods entirely independent of the dead reckoning, and these methods it is the business of Nautical Astronomy to teach. "It must not, however, be understood that the dead reckoning is without its value; on the contrary, when combined with astronomical observations, it is of considerable utility in detecting the existence and velocity of currents, and is indispensably necessary to fill up the short intervals which may occur in unfavorable weather between celestial observations: But the too general practice of relying exclusively upon it cannot be sufficiently deprecated, and numerous instances might be adduced of the fatal consequences of this reliance, in the loss of vessels, from errors of such magnitude that they might have been detected by the most superficial knowledge of nautical astronomy, and the aid of even a good common watch." (Capt. Kater's Nautical Astronomy in the Ency. Met.) 23 Definitions. 104. 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, becausethe applications of which we speak are not affected by the inquiry, whether the motions which the heavenly bodies present to an observer on the earth are really as they appear or not. At Art. 79, we defined several lines, some of which, geographers have found it convenient to consider as described on the surface of the earth; but which astronomers extend to the heavens. We give here some additional matter on the same subject. The position of any point on the celestial sphere, like the position of a point on the terrestrial sphere, is marked out, as we have said, 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 follows: 1. Aries. 14. Cancer. [7. Libra. 3. 10. VS Capricornus. 11. Aquarius. Gemini. 6. Virgo. 9. Sagittarius. 12. Pisces. The first six of these signs are on the north of the equi noctial, 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. Parallels of latitude on the terrestrial sphere correspond to parallels of declination on the celestial. 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, (Art. 79,) 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 ; one of which is directly over the head of the observer, and called his zenith, and the opposite one his nadir. That vertical one 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 has been already defined to be 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. a The amplitude of a celestial object is the arc of the hori. zon, 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. On the Corrections to be applied to the observed Altitudes of Celestial Objects. (105. The true altitude of a celestial object is always understood to mean its angular distance froin the rational horizon of the observer. This is not obtained directly by observation ; but is the result of certain corrections applied to the observed altitude.* These we shall now enumerate and explain. The observed altitude is obtained by means of an instrument called a quadrant of reflection, or simply a quadrant. This instrument is a frame of wood in the form of a sector of a circle, the arc of which is graduated to degrees and parts of a degree. This frame is suspended so that the plane of the circle shall be vertical. It has an arm, one extremity of which is attached to the centre of the circle, and which is moveable about this point; upon this arm is a small mirror, and opposite to it is a plane glass half of which is mirror and half transparent. When a heavenly body, seen by double reflection in these two mirrors, is brought by the movement of the arm, upon which one of the mirrors is placed, to coincide with the line of the horizon at sea as seen through the transparent part of the opposite glass, the outer extremity of the arm points out upon the graduated arc the number of degrees of altitude of the heavenly body above the horizon. The construction of this instrument depends upon the optical principle that the angle of incidence is equal to the angle of reflection. The angular movement of the image of the heavenly body is double the angular movement of the arm, so that to measure the greatest altitudes the limit of which is 90°, the gradą. ated arc need be but the eighth of a circumference; the degrees upon it are however numbered as if it were a quadrant to save the trouble of doubling them. The instrument takes its name from the amount which it measures, instead of from the magnitude of its arc. There are colored glasses attached, which can be interposed so that the rays of light, coming from the heavenly body to the eye, can be made to pass through them when taking the altitude of the sun. Of the Dip or the Depression of the Horizon, (106.) Let E represent the place of the observer's eye, and s the situation of any celestial body; the first object is to obtain. its apparent altitude above the horizontal line EH; that is, the angular distance SEH. Now, as to the observer, the visible horizon is EBH', the altitude given by the E H B A instrument is the angle SEH'; hence we must subtract from this observed altitude the angle HEH', called the Dip or Depression of the Horizon, in order to obtain the apparent altitude SEH. The angle HEH', or its equal c, is calculated for various elevations, AE of the eye above the surface of the sea from the proportion, CE; EB VEC2 CB2: rad. : sin. c ; and the results are registered in a table, (Table IX.) Of the Semidiameter. (107.) When the foregoing correction for dip has been applied, the result will be the apparent altitude of the point observed above the horizontal plane through the observer's eye. If this point be the uppermost or lowermost point of the disc of the sun or moon, a further correction will be necessary to obtain the apparent altitude of the centre; that is, we must apply the angular distance due to the semidiameter; by which we mean, the angle at the eye of the observer subtended by the semidiameter or radius of the sun or moon. This quantity, which is continually varying, both for the sun and moon, in consequence of the variation of the distance, is given in |