greater than ZS, that is, the zenith distance is least when on the meridian, and hence the meridian altitude is greatest. In the same way it can be shown that the depression of a body below the horizon is greatest when on the meridian. EXERCISES. 1. What are the altitude and hour angle of the zenith? 2. What are the declination and latitude of the celestial pole? Ans. 90°: 0. Ans. 90: 66° 32′ (90 — 23° 28′) 3. How far is the pole of the ecliptic from the celestial pole, or, in other words, what is the magnitude of the arc PP' in fig. 4? Ans. 23° 28'. 521 4. What are the declination, right ascension, latitude, and longitude of? Ans. 0 180°: 0; 180°. 5. What point in the heavens has its declination, right ascension, latitude, nd longitude each equal to zero? Ans. First point of Aries (Y). 6. If a certain star cross the meredian at 11 o'clock P.M. to-night, at what o'clock will it cross the meridian-(1) to-morrow night; (2) 15 days hence, assuming the sun's change of right ascension throughout the year to be uniform? See Arts. (5) and (6).) Ans. (1) About 10.56 P.M. (2) About 10 P.M. 7. At what hour will the same star cross the meridian a year hence? Ans. 11 P.M. again. 78. A star is in the meridian 10° above the pole at midnight to-night, where will it be at midnight-(1) six months hence; (2) a year hence, supposing the sun's apparent motion in the ecliptic to be uniform ? 9. What is the sun's right ascension on 21st March, 21st June, 23rd Sepember, 21st December. Ans. 0 90°: 180° : 270°. 10. Calculate what would be the declination and right ascension of the sun on 21st April if the changes in these quantities were uniform throughout th year. Ans. 7° 49′ 20′′ N.: 30°. 11. Making the same assumption as in the last question—(1) Find at what ime the sun's right ascension should be 120°; (2) at what time should his éclination be 15° 38′ 40′′ N. Ans. (1) 21st July. (2) 21st May or 21st July. N.B.-The reader can, by reference to a celestial globe or the Nautical Almanac, see that the results obtained in Examples (10) and (11) are not the correct values of the right ascension and declination of the sun on the dates mentioned, which shows that the changes in these quantities throughout the year are not at all uniform. 12. What is the time of sunrise and sunset at any place during the equinoxes? Ans. About 6 A.M. and 6 P.M. 13. What is the hour angle of the sun at sunrise on 21st March? Ans. 90°. CHAPTER II. THE EARTH. 16. THAT the earth's shape is approximately spherical has been known from the earliest times. It will not here be necessary to do more than mention the different reasons which lead us to this conclusion. They are: (1) The hull of a ship disappears first, which shows that the ship is sailing on a convex surface. . (2) The outline of the earth's shadow, as seen on the surface of the moon during an eclipse, always seems an arc of a circle, and no body but a sphere can project a circular shadow in all positions. (3) The most conclusive proof, however, depends on the fact, which is found by observation, that equal distances gone over by the observer due north or south produce almost equal variations in the meridian altitude of any chosen star (or of the celestial pole). This could not happen except on the supposition that the earth is nearly spherical. Celestial Pole Constant in Direction. 17. The celestial pole being supposed to be situated at an indefinitely great distance away compared with any distance on the earth, therefore, as the observer changes his position on the earth's surface, the lines drawn from those positions in the direction of the celestial pole are practically parallel. Earth's Axis. Terrestrial Equator. Terrestrial Latitude and Longitude. That diameter of the earth which is parallel to the constant direction of the celestial pole is called the earth's axis. The earth's axis cuts the surface of the earth in two points called the north and south poles of the earth. That great circle drawn round the earth whose plane is perpendicular to the earth's axis is called the terrestrial equator. Great circles drawn through the poles of the earth are called terrestrial meridians. Therefore, every place on the earth's surface may be supposed to have its meridian. The meridian of Greenwich is called the first meridian. The latitude of a place is its distance north or south of the equator measured on the meridian through the place. The longitude of a place is its distance east or west of the first meridian, and is measured by the number of degrees in the arc intercepted on the equator between the meridian of the place and the first meridian. All places situated on the same parallel to the equator have evidently the same latitude, and situated on the same meridian have the same longitude. Latitude is measured north and south from 0° to 90°, and longitude east and west from 0° to 180°. Corresponding to the Tropics of Cancer and Capricorn on the celestial sphere, we imagine two small circles on the earth parallel to the equator, one north the other south, and distant from it about 23° 28′: these small circles are also called the Tropics of Cancer and Capricorn. The two small circles drawn round the north and south poles of the earth at a distance of 23° 28′ are called the arctic and antarctic circles respectively. The portion of the earth's surface enclosed between the two tropics is called the torrid zone, between the tropics and the arctic and antarctic circles the temperate zones, and between the arctic and antarctic circles and the poles the frigid zones. 18. The altitude of the celestial pole at any place is equal to the latitude of the place. For let O be the position of the observer; EOQ the meridian of the place, cutting the equator in E and Q. If OP represent the direction of the celestial pole as seen from O' then the line CP drawn from C, the centre of the earth, in the direction of the celestial pole, will be parallel to OP (the pole being so far distant). The horizon of the observer will be represented by a tangent plane OH drawn to the earth at 0. Then we have to prove that the angle ✪ which is the altitude of the pole = the arc EO or the angle p, which is E DIRECTION OF POLE. φ H the latitude of the place. Since OP is parallel to CP, the angle a = the angle ß; but 0 is the complement of a, and is the complement of ẞ; therefore = p, or altitude of pole latitude of place. From this it follows that the change in the altitude of the pole must equal the change in the latitude of the observer as he proceeds north or south. = Length of a degree of Latitude. Magnitude of Earth. 19. The measurement of the length of a degree of latitude on the earth is an operation of much practical difficulty. A position is chosen on the earth, and the altitude of the pole observed. Another station is chosen due north or south of the former position at such a distance from it that the altitude of the pole is increased or diminished by 1°, as the case may be. The length of the arc of the meridian between the two stations is then measured, and is found to have a mean value of about 69 miles, which must be the length of a degree. The length of a degree has thus been calculated at about twenty different places on the earth, and the results have not been found to differ to any very great extent, which is confirmatory evidence of the earth's approximate spherical shape. It has, however, been found that the length of a degree near the poles is somewhat greater than near the equator, which shows that the curvature of the earth is not so great at the poles as at the equator, or, in other words, that the earth is slightly flattened at the poles. In fact the figure of the earth is what is called an oblate spheroid, differing but little from a sphere. The lengths of a degree of latitude at different parts of the earth have been found to be as fol ows: The length of a degree being about 69 miles, an |