Example IV. Two light-houses are observed from a ship sailing S. 38° W. at the rate of 5 miles an hour. The first bears N. 21° W., the other N. 47° W. At the end of two hours, the first is found to bear N. 5° E., the other N. 13° W. What is the distance of the light-houses from each other? Ans. 6 miles and 30 rods. IV. CURRENT SAILING. 87. When the measure given by the log-line is taken as the rate of the ship's progress, the water is supposed to be at rest. But if there is a tide or current, the log being thrown upon the water, and left at liberty, will move with it, in the same direction, and with the same velocity. The rate of sailing as measured by the log, is the motion through the water. If the ship is steered in the direction of the current, her whole motion is equal to the rate given by the log, added to the rate of the current. But if the ship is steered in opposition to the current, her absolute motion is equal to the difference between the current, and the rate given by the log. In all other cases, the current will not only affect the velocity of the ship, but will change its direction. Suppose that a river runs directly south, and that a boat in crossing it is steered before the wind, from west to east. It will be carried down the stream as fast, as if it were merely floating on the water in a calm. And it will reach the opposite side as soon, as if the surface of the river were at rest. But it will arrive at a different point of the shore. Let AB (Fig. 28.) be the direction in which the boat is steered, and AD the distance which the stream runs, while the boat is crossing. If DC be parallel to AB, and BC parallel to AD; then will C be the point at which the boat proceeding from A, will strike the opposite shore, and AC will be the distance. For it is driven across by the wind, to the side BC, in the same time that it is carried down by the current, to the line DC. In the same manner, if Am be any part of AB, and mn, be the corresponding progress of the stream, the distance sailed will be An. And if the velocity of the ship and of the stream continue uniform, Am is to mn, as AB to BC, so that AnC is a straight line. (Euc. 32. 6.) The lines AB, BC, and AC, form the three sides of a triangle. Hence, 88 If the direction and rate of a ship's motion through the water, be represented by the position and length of one side of a triangle, and the direction and rate of the current, by a second side; the absolute direction and distance will be shown by the third side. Example I. If the breadth of a river running south (Fig. 28.) be 300 yards, and a boat steers S. 75° E. at the rate of 10 yards in a minute, while the progress of the stream is 24 yards in a minute; what is the actual course, and what distance must the boat go in crossing? Cos. BAP: AP :: R: AB=310.6 (BC+AB): (BC-AB): : Tan. Sin. BAC (BAC+BCA): Tan. (BAC-BCA)=17° 33' 50". The angle BAC is 55° 3' 50" Then BC:: Sin. ABC: AC=879 the distance. And DAC-BCA=19° 56' 10" the course. Example II. A boat moving through the water at the rate of five miles an hour, is endeavoring to make a certain point lying S. 221 W. while the tide is running S. 7830 E. three miles an hour. In what direction must the boat be steered, to reach the point by a single course? Ans. S. 58° 33′ W. 89. But the most simple method of making the calcula tion for the effect of a current, in common cases, especially in resolving a traverse, is to consider the direction and rate of the current as an additional separate course and distance; and to find the corresponding departure and difference of latitude. A boat sailing from A (Fig. 28.) by the united action of the wind and current, will arrive at the same point, as if it were first carried by the wind alone from A to B, and then by the current alone from B to C. Example I. A ship sails S. 17° E. for 2 hours, at the rate of 8 miles an hour; then S. 18° W. for 4 hours, at the rate of 7 miles an hour; and during the whole time a current sets N. 76° W. at the rate of two miles an hour. Required the direct course and distance. The course is 21° 48′ 50′′, and the distance 42 miles. Example II. A ship sails SE. at the rate of 10 miles an hour by the log, in a current setting E. NE. at the rate of 5 miles an hour. What is her true course? and what will be her distance at the end of two hours? The course is 66° 13', and the distance 25.56 miles. V. HADLEY'S QUADRANT. 90. In the preceding sections, has been particularly explained the process of determining the place of a ship from her course and distance, as given by the compass and the log. But this is subject to so many sources of error, from variable winds, irregular currents, lee-way, uncertainty of the magnetic needle, &c. that it ought not to be depended on, except for short distances, and in circumstances which forbid the use of more unerring methods. The mariner who hopes to cross the ocean with safety, must place his chief reliance, for a knowledge of his true situation from time to time, on observations of the heavenly bodies. By these the latitude and longitude may be generally ascertained, with a sufficient degree of exactness. It belongs to astronomy to explain the methods of making the calculations. The subject will not be anticipated in this place, any farther than to give a description of the quadrant of reflection, commonly called Hadley's Quadrant,* by which the altitudes of the heavenly * See note H. bodies, and their distances from each other, are usually measured at sea. The superiority of this, over most other astronomical instruments, for the purposes of navigation, is owing to the fact, that the observations which are made with it, are not materially affected by the motion of the vessel. 91. In explaining the construction and use of this quadrant, it will be necessary to take for granted the following simple principles of Optics. 1. The progress of light, when it is not obstructed, or turned from its natural course by the influence of some contiguous body, is in right lines. Hence a minute portion of light, called a ray, may be properly represented by a line. 2. Any object appears in the direction in which the light from that object strikes the eye. If the light is not made to deviate from a right line, the object appears in the direction in which it really is. But if the light is reflected, as by a common mirror, the object appears not in its true situation, but in the direction of the glass, from which the light comes to the eye. 3. The angle of reflection is equal to the angle of incidence; that is, the angle which the reflected and the incident rays make with the surface of the mirror, are equal; as are also the angles which they make with a perpendicular to the mirror. 92. From these principles is derived the following proposition; When light is reflected by two mirrors successively, the angle which the last reflected ray makes with the incident ray, is DOUBLE the angle between the mirrors. If C and D (Fig, 29.) be the two mirrors, a ray of light coming from A to C, will be reflected so as to make the angle DCM= ACB; and will be again reflected at D, making HDM=CDE. Continue BC and ED to H, draw DG parallel to BH, and continue AC to P. Then is CPM the angle which the last reflected ray DP makes with the incident AC; and DHM is the angle between the mirrors. By the preceding article, with Euc. 29. 1, and 15. 1, And HDM-EDC=EDG+GDC=DHM+PCM. Therefore CPM=2DHM. ray Cor. 1. If the two mirrors make an angle of a certain number of degrees, the apparent direction of the object will be changed twice as many degrees. The object at A, seen by the eye at P, without any mirror, would appear in the direction PA. But after reflection from the two mirrors, the light comes to the eye in the direction DP, and the apparent place of the object is changed from A to R. Cor. 2. If the two mirrors be parallel, they will make no alteration in the apparent place of the object. 93. The principal parts of Hadley's quadrant are the following; 1. A graduated arc AB (Fig. 17.) connected with the radii AC and BC. 2. An index CD, one end of which is fixed at the center, C, while the other end moves over the graduated arc. 3. A plane mirror called the index glass, attached to the index at C. Its plane passes through the center of motion C, and is perpendicular to the plane of the instrument; that is, to the plane which passes through the graduated arc, and its center C. 4. Two other plane mirrors at E and M, called horizon glasses. Each of these is also perpendicular to the plane of the instrument. The one at E, called the fore horizon glass, is placed parallel to the index glass when the index is at 0. The other called the back horizon glass, is perpendicular to the first and to the index at 0. This is only used occasionally, when circumstances render it difficult to take a good observation with the other. A part of each of these glasses is covered with quicksilver, so as to act as a mirror; while another part is left transparent, through which objects may be seen in their true situation. 5. Two sight vanes at G and L, standing perpendicular to the plane of the instrument. At one of these, the eye is placed to view the object, by looking on the opposite horizon glass. In the fore sight vane at G, there are two perforations, one directly opposite the transparent part of the fore horizon glass, the other opposite the silvered part. The back sight vane at L has only one perforation, which is opposite the center of the transparent part of the back horizon glass. 6. Colored glasses to prevent the eye from being injured by the dazzling light of the sun. These are placed at H, |