at the place being 7h 40m A.M., the true lunar distance 61° 52′ 35′′-4, and the distances registered in the Connaissance des Temps being on 5th at 21h = 61° 6′ 22′′, and on the 6th at Oh 62° 45′ 51′′. Ans. 2h 43m 38s W. 4. Find the longitude for the data in the sixth example of the preceding problem, the true distance of the moon's centre from Regulus being 57° 47′ 12′′-4, and their registered distances in the Connaissance des Temps being on the 17th at 15h 59° 2' 7", and at 18h = 57° 9′ 45′′. = Ans. 2h 0m 11s.1. 5. Find the true apparent time at the place, and the longitude for the true lunar distance 83° 20′ 55′′ in the seventh exercise of the preceding problem, the latitude being 10° 16′ 40′′ S., and the sun's declination 23° 22′ 48′′ N.; also the next less and greater registered lunar distances being at 15h app. time = 83° 6' 1", and at 18h 84° 28′ 26′′. Ans. Time 1h 39m 38s-3, long. =10h 7m 4s.3. NAVIGATION. 481. The department of navigation that belongs to Practical Mathematics, consists in the solution of the problems of determining the direction and distance of the intended port from the port left, or from the place of the ship at any time, and also the determining of the ship's place at any instant during the voyage. The principles of plane trigonometry, modified in their application according to circumstances, are sufficient for the solution of these problems. The ship is navigated, as nearly as possible, by the path which is the shortest distance between the two places, but, from contrary winds and intervening land, it is generally necessary to sail in a track of a zig-zag form; the distance sailed in each direction being known, as also the direction, the method of ascertaining which is afterwards explained. DEFINITIONS OF TERMS. 482. When a vessel is obliged to sail to the right or left of the direction of the intended port, it is said to tack. When the ship is tacking towards the left, and the wind consequently on the right, it is said to be on the starboard tack; and when it is tacking towards the right, it is said to be on the larboard tack. 483. A ship does not sail exactly in the direction of her keel or longitudinal axis, but deviates towards the side that is opposite to the wind; and the angle contained between the apparent and real direction is called lee-way. The real direction is observable by the track of the vessel in the water, called the ship's wake, or by the direction of the logline; and the lee-way can therefore be estimated. 484. The angle formed by the meridian and the direction of the ship's track is called the course, and its complement the co-course. 485. A line, cutting all the meridians at the same angle, is called a rhumb-line; which, when continued, approaches nearer and nearer to the pole, in a spiral form, but without ever reaching it; it is also called a loxodrome; whereas the arc of a great circle, which is the shortest distance between two places, is called the orthodrome. 486. The portion of a rhumb-line, intercepted between wo places, is called their nautical distance. 487. The distance of a ship from the meridian left, reckoned on the parallel of latitude of the ship's place, is called its meridional distance. When the nautical distance between a place left and one arrived at is very small, the meridional distance will be very nearly the numbers of miles that the vessel has sailed east or west of the place left, and is called easting or westing, as the distance that it has sailed due north or south is called northing or southing. 488. If the nautical distance is supposed to be divided into an indefinite number of minute equal parts, the sum of all the meridional distances belonging to these parts is called the departure. 489. The difference of latitude of two places is an arc of a meridian, intercepted between the parallels of latitude passing through their places. 490. The difference of longitude of two places is an arc of the equator, intercepted between their meridians. INSTRUMENTS USED IN NAVIGATION. 491. The mariner's compass is the instrument by which the course is measured. This compass consists of a circular card suspended horizontally on a point, and having for one of its diameters a small magnetised bar of steel called the needle. The circumference of the card is divided into 32 equal parts called points of the compass, and each point is divided into 4 equal parts called quarter points. The point of the card which coincides with the north end of the needle, is called the magnetic north; the opposite point, the magnetic south; and the middle points between these, on the extremities of the diameter perpendicular to the needle, are called the magnetic east and west. These are called the cardinal magnetic points, and the other points are named from their situation in reference to these points. The true cardinal points have been already explained in article 384. Since there are 8 points in each quadrant, therefore a point is an angle of 11° 15'. direction for many years, but in different places its direction is not towards the same part of the horizon. The angular difference between the magnetic and true north is called the variation or declination of the compass, being west or east, according as the magnetic north is towards the left or right of the true north. The compass needle may be affected sensibly by the attraction of iron placed near it, and even by a great mass of iron at a considerable distance, as in a ship of war by the guns. When the metal is symmetrically distributed in reference to the longitudinal axis, the needle is not affected when the direction of this axis coincides with the magnetic meridian or vertical plane passing through the needle; and its local attraction produces the greatest error in the true variation when the direction of the axis of the ship is perpendicular to the former direction. The variation of the needle at London is at present about 241°. The points of the compass are seen in the foregoing figure. The middle point between N. and E. is called NE.; that between N. and NE. is called NNE.; and so on. 492. The log is a piece of wood of the form of a circular sector, which is nearly quadrantal; and the arc of it is loaded with lead, so that it floats vertically with the central point uppermost. The line called the log-line is so attached to the log that, when the line is drawn gently, the log turns its flat side towards the ship, so that it remains nearly immoveable while the line is unwound from the reel. The log-line is about 100 fathoms long, and is divided into equal parts called knots; each of which is generally subdivided into fathoms. A knot is the 120th part of a nautical mile, or 6079 feet, and ought therefore to be 50 feet 8 inches. In practice, however, 50 feet is usually made the length of a knot, for the log being drawn a small way towards the vessel during the operation of estimating the ship's rate, or, as it is called, of heaving the log, the distance given by this line is nearer the truth, and, besides, it is safer that the reckoning should be in advance of the ship, or ahead of it, as it is termed. The time, when observing the ship's rate by the log-line, is estimated by a sand-glass, which measures half minutes, that is, it runs out in 30 seconds. Since 30 seconds is the same part of an hour that a knot is of a mile, the number of knots run out in 30 seconds shows that the rate of the vessel is just the same number of miles per hour. Sometimes the sand-glass and log-line, from various causes, become incorrect, and therefore the rate measured by them, or the distance sailed, must be corrected. 493. The angular instruments used in navigation are Hadley's quadrant or sextant. The principles on which this instrument is con structed will be under stood from the adjoining figure. The graduated arc AB is the limb of the instrument, CM an index, moveable about an axis at M, with a vernier at its extremity C. M is a small mirror attached to the index I CM, and placed perpendicularly to the plane ABM of the instrument; N is a similar small plate of glass, called the fore horizon glass, one half of which is a mirror; and it is placed parallel to the mirror M when the index coincides with MB, or rather with the zero point at B, and is fixed in this position. When the angular distance between two objects, as two stars, at S and I is to be measured, the plane of the instrument is first placed in the same plane with the objects, and in such a position that one of them, I, is visible through the glass N to the eye situated at E, and then the index CD is moved till the image of S, after two reflections from M and N, appears to coincide with I, seen directly through the plate; and the angle, subtended by their distance, namely, angle E, is then measured by double the arc BC. The ray SM proceeding from S is reflected in the direction MN by the mirror M, and then at N by the mirror N, |