and then proceeding as directed in the latter part of the last article. 53. A. Sidereal Day is the interval between two consecutive passages of a fixed star over the same meridian. It is about four minutes shorter than the common day. Sidereal Time is time reckoned by sidereal days. A clock that is so adjusted as to move through 24 hours in a sidereal day is called a sidereal clock, or is said to be regulated to keep sidereal time. The point of time at which the sidereal day commences according to the present usage of astronomers, and also a very slight change in the definition of a sidereal day, will be noticed in a subsequent chapter. For the present it may be regarded as commencing when any fixed star, selected at pleasure, passes the meridian of a place; the clock being regarded as being regulated to sidereal time when it is so adjusted as to mark and continue to mark 0 h. 0 m. 0 sec. at the instant that star passes the meridian. The numbering of the hours of the sidereal day, is continued from 0 hours to 23 hours. 54. The Terrestrial Meridian of a place is the intersection of the plane of the meridian of the place with the earth's surface. It is very nearly, though not exactly, a circle. The terrestrial meridian of a place is usually considered as only extending from pole to pole. Thus pAp'a, Fig. 1, being the intersection of the plane of the meridian of the place A, with the earth's surface, the half pAp' is called the terrestrial meridian of the place A; the other half being called the opposite meridian. 55. The Latitude of a place, or as it is sometimes called the Geographical Latitude, is the arc of the meridian intercepted between the zenith of the place and the equator. It is said to be north or south according as the zenith is north or south of the equator. Thus ZQ, Fig. 1, is the latitude of the place A, to the north. It follows, from the definition and article (27), that the latitude. of a place is the same as the declination of the zenith of the place. It is also evident that, regarding the earth as a sphere, and consequently a terrestrial meridian as a circle, the latitude of a place * The details of this and other methods of adjusting these instruments to the plane of the meridian are given in treatises on practical astronomy. is its distance from the terrestrial equator, measured in degrees and parts of a degree, on the terrestrial meridian through the place. For the arcs ZQ, and Aq, being measures of the same angle ZCQ, contain the same number of degrees. 56. A Parallel of Latitude is any small circle on the earth's surface, parallel to the terrestrial equator. Thus Ana is one half of a parallel of latitude through the place A. 57. The Latitude of a place is equal to the altitude of the pole at that place. For the sum of ZQ and ZP, is equal to the sum of PH and ZP, each sum being equal to a quadrant. Hence ZQ= PH. But ZQ is the latitude of the place A, and PH is the altitude of the pole at that place; this altitude, in consequence of the extreme minuteness of CA, in comparison with CH (13), being the same whether observed from A or C. 58. The Latitude of a place is equal to the half sum of the greater and less meridian altitudes of a circumpolar star, at the place. Let the circle FGIK, meeting the meridian of the place A, in F and I, be the circle described by a circumpolar star in its diurnal motion. Then will PI=PF (15), and HI and HF will be the greater and less meridian altitudes of the star. Now, HP = HI — PI, and HP = HF + PF HF + PI. Hence by adding, 2 HP = HI+HF; or HP = 1} (HI + HF). But (57), HP is equal to the latitude of the place. Remark. This proposition assumes the two culminations to be on the same side of the zenith. If they are on different sides, the supplement of the greater altitude must evidently be substituted for the altitude itself. It may further be remarked that, in applying the proposition to find the latitude of a place, the observed altitudes require small corrections. These will be noticed in the chapter on refraction. 59. First Meridian. The first meridian is the meridian of some place arbitrarily selected, to which the positions of the meridians of other places are referred. The place selected for a first meridian, is commonly that of some noted Astronomical Observatory.* 60. Longitude. The Longitude of a place is the angle contained between its meridian and the first meridian; it is measured by the arc of the equator intercepted between the two meridians. Longitude is reckoned east or west, according as the meridian of the place is east or west of the first meridian. Thus, assuming the meridian PSP' of a place s, to be the first meridian, the angle DPQ, or the arc DQ, will be the longitude of the place A, to the east; and the angle DPD' or the arc DD" will be the longitude of a place s', to the west. As two terrestrial meridians form the same angle as their corresponding celestial meridians, the angle between the terrestrial meridian of a place and the terrestrial first meridian, expresses the longitude of the place. 61. Selection of the first Meridian. The selection of the first meridian being arbitrary, different first meridians are used in different countries. The English take for theirs, the meridian of their celebrated Observatory at Greenwich, near London, and the French, that of their public Observatory at Paris. In the United States, longitude is usually reckoned from the meridian of Greenwich. 62. Position of a place. The longitude and latitude of a place, determine its relative position on the earth's surface. For the longitude gives the position of the terrestrial meridian of the place, and the latitude gives the position of the place on that meridian. When the longitude and latitude of a city, or place of considerable extent, are given, they are generally to be understood as applying to the central part, or to some prominent public edifice in the place. 63. Difference of time under different Meridians. As the diurnal motion of a fixed star, in the circle it appears to describe from east to west, is uniform (15), and the time of its complete * An Astronomical Observatory is a building furnished with instruments for the especial purpose of making astronomical observations. revolution is twenty-four sidereal hours, it follows that, in each sidereal hour, it must move through the twenty-fourth part of 360°, that is, through 15°; and in the same proportion for other times. Hence the star in its westwardly motion is on the meridian of a place in east longitude, earlier, and on that of a place in west longitude, later, than on the first meridian, by intervals of time at the rate of a sidereal hour for each 15° in the longitude. Thus, if LUMV, Fig. 1, be the circle described by the star in its diurnal motion, and PSP', the meridian of a place s, be the first meridian; and if the longitudes of the places A and s', whose meridians are PMP' and PS'P', be 30° east and 30° west; then will the star be at M two sidereal hours earlier, and at S' two sidereal hours later, than at S. It therefore follows, that if we suppose sidereal clocks at the places A, s, and s', to be adjusted to mark 0 h. 0 m. 0 sec., when this star is on their meridians respectively, then at the instant the star is at S on the meridian of s, and consequently the clock at that place marks 0 h. 0 m., the clock at the place A in 30° east longitude, must mark 2 h. 0 m., the star having been on its meridian two hours previously; and the clock at the place s', in 30° west longitude, must mark 22 h. 0 m. of the preceding sidereal day, the star not arriving at the meridian of that place till two hours later. The same relation must exist among the times marked by the clocks at those places at any other instant of time. For instance, if when some other star is on the meridian of the place s, the clock at that place marks 7 h. 10 m. 30 sec., the clock at the place A, must at that instant mark 9 h. 10 m. 30 sec., and the clock at the place s', 5 h. 10 m. 30 sec. Hence the sidereal time reckoned at a given instant at a place in east longitude is later, and at a place in west longitude is earlier, than that reckoned at the first meridian. It is the same with time reckoned in the usual way, that is, by the diurnal motion of the sun, when allowance is made for some little inequalities in that motion. Thus, when it is nine o'clock in the evening at Greenwich, it is only 59 m. 20 sec. past three o'clock, in the afternoon, at Philadelphia, the longitude of which is 75° 10' west. 64. Expression of longitude in time. In consequence of the connection between the longitudes of places and the times reckoned at them at the same instant, longitude is frequently expressed in time; one hour corresponding to 15°, one minute to 15', and one second to 15". Thus, long. 75° 10′ W., and long. 5 h. 0 m. 40 sec. W., are synonymous expressions. 65. To find the longitude of a place by a chronometer. Let a chronometer, which keeps time accurately, be carefully adjusted to the time at a place, the longitude of which is known. Then being carried to the place of which the longitude is required, let the time shown by it at any instant be compared with the correct time reckoned at the place at that instant, and let the difference be marked east or west according as the time at the place is later or earlier than that shown by the chronometer: that is, than the time, reckoned at the same instant, at the place of known longitude. Then (63) by adding this difference to the known longitude, expressed in time, if it is of the same name with that longitude, or subtracting, if it is of a different name; the longitude of the required place will be obtained. It is not requisite that the chronometer should be so regulated as neither to gain nor lose any time. This would be difficult, if not impracticable. It is only requisite that its rate (41) should be well ascertained, as allowance can then be made for its gain or loss during the time of its transportation from one place to the other. Other methods of finding the longitude of a place will be noticed in a subsequent chapter. CHAPTER IV. FIGURE AND DIMENSIONS OF THE EARTH-GEOCENTRIC LATITUDE OF A PLACE. 66. By the figure of the earth is meant the general form of its surface, supposing it to be uniform, or that it corresponds with the surface of the ocean. This excludes the consideration of the irregularities in its surface, proceeding from mountains and valleys; which are indeed very minute in comparison with its whole extent. |