latitude by D. R., and from which may be found the middle latitude; with the middle latitude find the difference of longitude corresponding to the departure, apply this to the longitude of last position, and the result will be the longitude by D. R. The employment of the tabular form will be found to facilitate the work and guard against errors. It will be a convenience to include in that form columns showing the hour, together with the reading of the patent log (if used) each time that the course is changed or the dead reckoning worked up. The employment of minutes and tenths in dead reckoning rather than minutes and seconds is recommended. Example: A vessel under sail heading NE. f E. (on which course deviation is i pt. Easterly) takes departure from Cape Henry lighthouse (see Appendix IV for position), bearing SSW. i W. per compass, distant 1.4 miles. She then sails on a series of courses, with errors and distances as indiqated below; wind about SE. by E. Required the position by dead reckoning; also the course and distance made good by dead reckoning. Example: A steamer's position by observation at noon, patent log reading 27.3, is Lat. 49° 15' N., Long. 7° 32' W. Thence she steers 262° (per compass), the total compass error on that course being 20° W., until 12.30, at which time, patent log reading 33.9, the course is changed to 260° {p. c), same error. At 4.12, patent log 80.5, sights are taken from Which it is found that the true longitude is 8° 46' W., and the compass error 19° W. At 6.15, patent log reading 6.1, a sight is taken from which it is found that the true latitude is 48° 34' 30" N. At 8 p. m. the patent log reads 27.5. Required the positions by D. K. at each sight and at 8 o'clock. 206. Allowance Foe Cubbent.—When a vessel is sailing in a known current whose strength may be estimated with a fair degree of accuracy, a more correct position may be arrived at by regarding the set and drift of the current as a course and distance to he regularly taken account of in the dead reckoning. Example: A vessel in the Gulf Stream at a point where the current is estimated to set 48° at the rate of 1.8 miles an hour, sails 183° (true), making 9.5 knots an hour through the water for 3h 30TM. Middle latitude 35°. Required the course and distance made good. 207. Finding The Cubbent.—It is usual, upon obtaining a good position by observation (as the navigator usually does at noon), to compare that position with the one obtained by dead reckoning, and to attribute such discrepancy as may be found to the effects of current. It has already been pointed out that other causes than the motion of the water tend to make the dead reckoning inaccurate, so that it must not be assumed that currents proper are thus determined with complete correctness. Current is said to have set and drift, referring respectively to the direction toward which it is flowing and the velocity with which it moves. It is evident that, in calculating current by the method of comparing positions by observation with those by account, the navigator must limit himself to the periods during which the dead reckoning has been brought forward independently, without receiving any corrections due to new points of departure. In case it is desired to find the current covering a period during which fresh departures have been used, as from noon to noon, find the algebraical sums of all the differences of latitude and longitude from the table, and apply these to the latitude and longitude of original departure—that of the preceding noon; this gives the position from the ship's run proper, and the difference between this and the position by observation gives the set and drift for the twenty-four hours • if an allowance has been made for current, as explained in the preceding article, that must be omitted in bringing up the position which is to take account of the run only. 208. Day's Run.—It is usual to calculate, each day at noon, the ship's total run for the preceding twenty-four hours. Having the positions at noon of each day, the course and distance between them is found as explained in article 175, Chapter V. The position by observation is used in each case, if such has been found; otherwise, the position by dead reckoning. Example: At noon, January 22, the position of a vessel by observation was Lat. 35° 1(K N., Long. 134° 01' W. During the next 24 hours, the run by account was 60.1 miles north and 153.2 miles east. At noon, January 23, the position by observation was Lat. 36° 03' N., Long. 131° 14' W. Required the position by D. R. at the latter time; also the run and current for the 24 hours. CHAPTER VII. 209. Nautical Astronomy, or Celo-Navigation, has been defined (art. 3, Chap. I) as that branch of the science of Navigation in which the position of a ship is determined by the aid of celestial objects—the sun, moon, planets, or stars. 210. The Celestial Sphere.—An observer upon the surface of the earth appears to view the heavenly bodies as if they were situated upon the surface of a vast hollow sphere, of which his eye is the center. In reality we know that this apparent vault has no existence, and that we can determine only the relative directions of the heavenly bodies—not their distances from each other or from the observer. But by adopting an imaginary spherical surface of an infinite radius, the eye of the observer being at the center, the places of the heavenly bodies can be projected upon this Celestial Sphere, or Celestial Concave, at points where the lines joining them with the center intersect the surface of the sphere. Since, however, the center of the earth should be the point from which all angular distances are measured, the observer, by transferring himself there, will find projected on the celestial sphere, not only the heavenly bodies, but the imaginary points and circles of the earth's surface. The actual position of the observer on the surface will be projected in a point called the zenith; the meridians, equator, and all other lines and points may also be projected. 211. An observer on the earth's surface is constantly changing his position with relation to the celestial bodies projected on the sphere, thus giving to the latter an apparent motion. This is due to three causes: First, the diurnal motion of the earth, arising from its rotation upon its axis; second, the annual motion of the earth, arising from its motion about the sun in its orbit; and third, the actual motion of certain of the celestial bodies themselves. The changes produced by the diurnal motion are different for observers at different points upon the earth, and therefore depend upon the latitude and longitude of the observer. But the changes arising from the other causes named are independent of the observer's position, and may therefore be considered at any instant in their relation to the center of the earth. To this end the elements necessary for any calculation are tabulated in the Nautical Almanac from data based upon laws which have been found by long series of observations to govern the actual and apparent motion of the various bodies. 212. The Zenith of an observer on the earth's surface is the point of the celestial sphere vertically overhead. The Nadir is the point vertically beneath. 213. The Celestial Horizon is the great circle of the celestial sphere formed by passing a plane through the center of the earth at right angles to the line which joins that point with the zenith of the observer. The celestial horizon differs somewhat from the Visible Horizon, which is that line appearing to an observer at sea to mark the intersection of earth and sky. This difference arises from two causes: First, the eye of the observer is always elevated above the sea level, thus permitting him a range of vision exceeding 90° from the zenith; and second, the observer's position is on the surface instead of at the center of the earth. These causes give rise, respectively, to dip of the horizon and parallax, which will be explained later (Chap. X). 211. In figure 29 the celestial sphere is considered to be projected upon the celestial horizon, represented by NESW.; the zenith of the observer is projected at Z, and that pole of the earth which is elevated above the horizon, assumed for illustration to be the north pole, appears at P, the Elevated Pole of the celestial sphere. The other pole is not shown in the figure. 215. The Equinoctial, or Celestial Equator, is the great circle formed by extending the plane of the earth's equator until it intersects the celestial sphere. It is shown in the figure in the line EQW. The equinoctial intersects the horizon in E and W, its east and west points. 216. Hour Circles, Declination Circles, or Celestial Meridians are great circles of the celestial sphere passing through the poles; they are therefore secondary to the equinoctial, and may be formed by extending the planes of the respective terrestrial meridians until they intersect the celestial sphere. In the figure, PB, PS, PB', are hour circles, and that one, PS, which contains the zenith and is therefore formed by the extension of the terrestrial meridian of the observer, intersects the horizon in N and S, its north and south points. 217. Vertical Circles, or Circles of Altitude, are great circles of the celestial sphere which pass through the zenith and nadir; they are therefore secondary to the horizon. In the figure, ZH, WZE, NZS, are projections of such circles, which being at right angles to the plane of projection, appear as straight lines. The vertical circle NZS, which passes through the poles, coincides with the meridian of the observer. The vertical circle WZE, whose plane is at right angles to that of the meridian, intersects the horizon in its eastern and western points, and, therefore, at the points of intersection of the equinoctial; this circle is distinguished as the Prime Vertical. 218. The Declination of any point in the celestial sjphere is its angular distance from the equmoctial, measured upon the hour or declination circle which passes through that point; it is designated as North or South according to the direction of the point from the equinoctial; it is customary to regard north declinations as positive (+), and south declinations as negative ( —). In the figure, DM is the declination of the point M. Declination upon the celestial sphere corresponds with latitude upon the earth. 219. The Polar Distance of any point is its angular distance from the pole (generally, the elevated pole of an observer), measured upon the hour or declination circle passing through the point; it must therefore equal 90° minus the decimation, if measured from the pole of the same name as the declination, or 90° plus the declination, if measured from the pole of opposite name. The polar distance of the point M from the elevated pole P is PM. 220. The Altitude of any point in the celestial sphere is its angular distance from the horizon, measured upon the vertical circle passing through the point; it is regarded as positive when the body is on the same side of the horizon as the zemth. The altitude of the point M is HM. 221. The Zenith Distance of any point is its angular distance from the zenith, measured upon the vertical circle passing through the point; the zenith distance of any point which is above the horizon of an observer must therefore equal 90° minus the altitude. The zenith distance of M, in the figure, is ZM. 222. The Hour Angle of any point is the angle at the pole between tbe meridian of the observer and the hour circle passing through that point; it may also be regarded as the arc of the equinoctial intercepted between those circles. It is measured toward the west as a positive direction through the twenty-four hours, or 360 degrees, which constitute the interval between the successive returns to the meridian, due to the diurnal rotation of the earth, of any point in the celestial sphere. The hour angle of M is the angle QPD, or the arc QD. 223. The Azimuth of a point in the celestial sphere is the angle at the zenith point; it may also be regarded as the arc of the horizon intercepted between those circles. It is measured from either the north or the south point of the horizon (usually that one of the same name as the elevated pole) to the east or west through 180°, and is named accordingly; as, N. 60° W.( or S. 120° W. The azimuth of M is the angle NZH, or the arc NH, from the north point, or the angle SZH, or the arc SH, from the south point of the horizon. 224. The Amplitude of a point is the angle at the zenith between the prime vertical and the vertical circle of the point; it is measured from the east or the west point of the horizon through 90°, as W. 30° N. It is closely allied with the azimuth and may always be deduced therefrom. In the figure, the amplitude of H is the angle WZH, or the arc WH. The amplitude is only used with reference to points in the horizon. 225. The Ecliptic is the great circle representing the path in which, by reason of the annual revolution of the earth, the sun appears to move in the celestial sphere; the plane of the ecliptic is inclined to that of the equinoctial at an angle of 23 27£', and this inclination is called the obliquity of the ecliptic. The ecliptic is represented by the great circle CVT. 226. The Equinoxes are those points at which the ecliptic and the equinoctial intersect, and when the sun occupies either of these positions the days and nights are of equal length throughout the earth. The Vernal Equinox is that one at which the sun appears to an observer on the earth when passing from southern to northern declination, and the Autumnal Equinox that one at which it appears when passing from northern to southern declination. The Vernal Equinox is also designated as the First Point of Aries, and is used as an origin for reckoning right ascension; it is indicated in the figure at V. 227. The Solstitial Points, or Solstices, are points of the ecliptic at a distance of 90° from the equinoxes, at which the sun attains its highest declination in each hemisphere. They are called respectively the Summer and the Winter Solstice, according to the season in which the sun appears to pass these points in its path. The Summer Solstice is indicated in the figure at U. 228. The Bight Ascension of a point is the angle at the pole between the hour circle of the point and that of the First Point of Aries; it may also be regarded as the arc of the equinoctial intercepted between those circles. It is measured from the First Point of Aries to the eastward as a positive direction, through twenty-four hours or 360 degrees. The right ascension of the point M' is VD'. 229. Celestial Latitude is measured to the north or south of the ecliptic upon great circles secondary thereto. Celestial Longitude is measured upon the ecliptic from the First Point oi Aries as an origin, being regarded as positive to the eastward throughout 360°. 230. Coordinates.—In order to define the position of a point in space, a system of lines, angles, or planes, or a combination of these, is used to refer it to some fixed line or plane adopted as the primitive; and the lines, angles, or planes by which it is thus referred are called coordinates. 231. In figure 30 is shown a system of rectilinear coordinates for a plane. A fixed line FE is chosen, and in it a definite point C, as the origin. Then the position of a point A is defined by CB = x, the distance from the origin, C, to the foot of a perpendicular let fall from A on FE; and by AB=y, the length of the perpendicular. The distance x is called the abscissa and y the ordinate. Assuming two intersecting right lines FE and HI as standard lines of reference, the location of the point A is defined by regarding the distances measured to the right hand of HI ana above FE as positive; those to the left hand of HI and below FE as negative. An exemplification of this system is found in the chart, on which FE is represented by the equator, HI by the prime meridian; the coordinates x and y being the longitude and latitude of the point A. 232. The great circle is to the sphere what the straight fine is to the plane; hence, in order to define the position of a point on the surface of a sphere, some great |