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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.

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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.


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 DCMACB; 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 ray AC; and DHM is the angle between the mirrors.

By the preceding article, with Euc. 29. 1, and 15. 1,



But by Euc. 32. 1 and 15. 1,

Therefore CPM-2DHM.

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,

between the index mirror and the fore horizon glass. They may be taken out when necessary, and placed at N between the index mirror and the back horizon glass.

94. This instrument which is in form an octant, is called a quadrant, because the graduation extends to 90 degrees, although the arc on which these degrees are marked is only the eighth part of a circle. The light coming from the object is first reflected by the index glass C, (Fig. 17,) and thrown upon the horizon glass E, by which it is reflected to the eye at G. If the index be brought to 0, so as to make the index glass and the horizon glass parallel; the object will appear in its true situation. (Art. 92. Cor. 2.) But if the index glass be turned, so as to make with the horizon glass an angle of a certain number of degrees; the apparent direction of the object will be changed twice as many degrees.

Now the graduation is adapted to the apparent change in the situation of the object, and not to the motion of the index. If the index move over 45 degrees, it will alter the apparent place of the object 90 degrees. The arc is commonly graduated a short distance on the other side of 0 towards P. This part is called the arc of excess.

95. The quadrant is used at sea, to measure the angular distances of the heavenly bodies from each other, and their elevations above the horizon. One of the objects is seen in its true situation, by looking through the transparent part of the horizon glass. The other is seen by reflection, by looking on the silvered part of the same glass. By turning the index, the apparent place of the latter may be changed, till it is brought in contact with the other. The motion of the index which is necessary to produce this change, determines the distance of the two objects.*

96. To find the distance of the moon from a star.-Hold the quadrant so that its plane shall pass through the two objects. Look at the star through the transparent part of the horizon glass, and then turn the index till the nearest edge of the image of the moon is brought in contact with the star. This will measure the distance between the star and one edge of the moon. By adding the semi-diameter of the moon, we shall have the distance of its center from the star.

For the adjustments of the quadrant, see Vince's Practical Astronomy, Mackay's Navigation, or Bowditch's Practical Navigator.

The distance of the sun from the moon, or the distance of two stars from each other, may be measured in a similar


97. To measure the altitude of the sun above the horizon. -Hold the instrument so that its plane shall pass through the sun, and be perpendicular to the horizon. Then move the index till the lower edge of the image of the sun is brought in contact with the horizon, as seen through the transparent part of the glass.

The altitude of any other heavenly body may be taken in the same manner.

98. To measure altitudes by the back observation.—When the index stands at 0, the index glass is at right angles with the back horizon glass. (Art. 93.) The apparent place of the object as seen by reflection from this glass, must therefore be changed 180 degrees; (Art. 92. Cor. 1.) that is, it must appear in the opposite point of the heavens. In taking altitudes by the back observation, if the object is in the east, the observer faces the west; or if it be in the south, he faces the north; and moves the index, till the image formed by reflection is brought down to the horizon.

This method is resorted to, when the view of the horizon in the direction of the object is obstructed by fog, hills, &c.

99. Dip or Depression of the Horizon. In taking the altitude of a heavenly body at sea, with Hadley's Quadrant, the reflected image of the object is made to coincide with the most distant visible part of the surface of the ocean. A plane passing through the eye of the observer, and thus touching the ocean, is called the marine horizon of the place of observation. If BAB' (Fig. 13.) be the surface of the ocean, and the observation be made at T, the marine horizon is TA. But this is different from the true horizon at T, becausė the eye is elevated above the surface. Considering the earth as a sphere, of which C is the center, the true horizon is TH perpendicular to TC. The marine horizon TA falls below this. The angle ATH is called the dip or depression of the horizon. This varies with the height of the eye above the surface. Allowance must be made for it, in observations for determining the altitude of a heavenly body above the true horizon.

In the right angled triangle ATC, the angle ACT is equal to the angle of depression ATH; for each is the complement

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