4.

Given log. tang. 35° 20′ = 9·850593, find log. cotang 35° 20, without using any tables at all.

5. Find the log, cosec. 68° 45' 24" from the table of natural sines only.

7.

Given log. cosine A = 9'450981, find A (1) from a table of log cosines, and (2) from a table of nat. cosines.

8. Given nat. sec. A = 2·005263, find A (1) from a table of nat. sines and cosines, and (2) secan. from a table of log. secants.

NAVIGATION.

DEFINITIONS.

NAVIGATION is a general term denoting that science which treats of the determination of the place of a ship on the sea, and which furnishes the knowledge requisite for taking a vessel from one place to another. The two fundamental problems of navigation are, therefore, the finding at sea the present position of the ship, and the determining the future

course.

The place of a ship is determined by either of two methods, which are independent of each other:-1st. By referring it to some other place, as a fixed point of land, or a previous defined place of the ship herself. 2nd. By astronomical observations.

It has been customary to employ the term NAVIGATION in a restricted sense to the first of these methods: the second is usually treated of under the head of NAUTICAL ASTRONOMY.

Navigation and Nautical Astronomy are the two great co-ordinate divisions of the "Art of Sailing on the Sea," as the old writers quaintly worded it. The first branch of the art is accomplished by means of the Mariner's Compass, which shows the direction of the ship's track; the Log, which with the help of sand-glasses for measuring small intervals of time, gives the velocity or the rate of sailing, and thence the distance run in any interval, and also a Chart of appropriate construction; in short, this branch of the art relates to the directing the ship's course under the varying forces of winds and currents, and the estimation of her change of place. The second division is that branch of practical astronomy by which the situation of the observer on the globe is ascertained by a comparison of the position of his Zenith with relation to

the heavens with the known position of the Zenith of a known place at the same moment. The principal instruments are the sextant for measuring the altitudes and taking the distances of heavenly bodies; and a chronometer to tell us the difference in time between the meridian of the ship and the first meridian; also a pre-calculated astronomical register, such as the Nautical Almanac, the Connaissance de Temps of France, &c. The solution of problems in nautical astronomy requires the use of spherical trigonometry, which is therefore characteristic of this method of navigation.

The earth is nearly a globe or sphere.

The ordinary proofs of this are of the following nature:-1st. When a vessel is scen at a considerable distance on the sea, in any part of the world, the hull is entirely or partly concealed by the water, though the masts are visible. 2nd. Ships have actually and repeatedly made the circuit of the globe; that is, by sailing from a port in a westerly direction they have returned to it in an easterly direction. 3rd. When we travel a considerable distance from north to south, a number of new stars appear, successively, in the heavens, in the quarter to which we are advancing, and many of those in the opposite quarter gradually disappear, which would not happen if the earth were a plane in that direction. 4th. In an eclipse of the moon, which is caused by the intervention of the body of the earth between the sun and moon, the shadow of the earth thrown on the moon is found in all cases, and in every position of the earth, to be a circular figure; the carth, therefore, which casts that shadow, must be a round body.

The earth, however, is not a perfect sphere, but of the figure of an oblate spheroid, being flattened in at the poles, and bulging out in a corresponding degree at the equatorial regions-the curviture being less as we recede from the equator to the poles; that is, such a figure as would be produced if a hoop were slightly flattened by pressure, and then made to revolve about the shortest diameter thus produced.

The shortest diameter (that which joins the poles) being 7899 statute miles, and that of the fullest parts (about the equator) being nearly 261

more.

We can, of course, in a work like this give no intelligible account of the refined mathematical processes by which the most probable values of the flattening in, and of the absolute dimensions have been obtained. It is suficient to say, that from a combination of the measurements of ten ares of the meridian, BESSEL has deduced the following results :

Proportion of diameters, as 299'15 to 298.15.

And from the result it follows that the polar diameter is shorter than the equatorial by about s (one three hundredth) part. This quantity is technically called the compression.

THE AXIS OF THE EARTH is one of its diameters about which it is supposed to turn round once in twenty-four hours. The direction of this rotation is from west to east, thus causing all the heavenly bodies to have an apparent motion from east to west.

POLES.

S

The extremities of the axis of the earth about which it rotates are called the poles of the earth, distinguished respectively as the North Pole and South Pole -- as N. S. (see fig.) The former being that to which we in Europe are nearest. As they are the extremities of a diameter they are 180° apart.

EQUATOR (from Latin æquare to divide into equal parts), called also by seamen the Line, is a great circle circumscribing the earth, every point of which is equally distant from both poles, as W M' E; and dividing the globe into two equal parts called hemispheres; that

By a great circle is meant a circle of the sphere having for its centre, the centre of the sphere, thus dividing the sphere into two equal parts: no greater circle can be traced upon its surface. All other circles are called small circles.

towards the north pole is called the northern hemisphere, as NW E, and the other the southern hemisphere, as SW E. (See figure, page 64).

If a plane be supposed to pass through the centre of the earth at right angles to its axis, it will intersect its surface in a great circle called the Equator.

At all places on this circle the sun rises at 6h A.M., and sets at 6h P.M., all the year round; the days and nights are therefore equal, being 12h each.

THE MERIDIAN of any place is a semi-circle passing through that place and the poles, and therefore cutting the equator at right angles, as N M'S. (See figure.) The other half of the circle is called the opposite meridian. Every point on the surface of the earth may be conceived to have a meridian passing through it; hence there may be as many meridians as there are points in the equator. Of all these innumerable meridians one is always selected as the principal or first meridian; it is a matter of arbitrary choice amongst different nations; thus the first meridian with us is that of Greenwich, whilst the French refer to Paris, &c.

Every portion of the meridian lies north and south; and places lying north and south of each other are said to be on the same meridian.

The direction of the meridian towards the north pole is called north, and marked N.; the opposite direction is called south, marked S. Directions at right angles to the meridians are called east and west; the right hand looking to the north east, the left hand west: they are marked E. and W.

Every meridian line may be said, with respect to the place through which it passes, to divide the surface of the sphere into two equal parts called the eastern and western hemispheres.

Meridians (L. Meridies, from medius dies mid-day) are so called because they mark all places which have noon at the same instant, for when any one of the meridians is exactly opposite the sun it is mid-day with all places situated on that meridian; and with the places situated on the opposite meridian it is consequently midnight. They also mark out all places which have the same longitude, and are hence called "Circles of Longitude."

LATITUDE is the distance from the equator, measured in degrees (°), minutes (), and seconds ("),* on the meridian of the place, or its angular distance from the equator measured by the arc of the meridian intercepted (cut off), between the place and the equator, or by the corresponding angle at the centre of the sphere; it is marked north

* All circles, great or small, are supposed to be divided into 360 equal parts called degrees (°). 60' (minutes) make one degree, and 60" (seconds) make one minute.

K

(N.), or south (S.), according as the place is to the north or south of the equator. Thus, the arc A' M' (fig., page 64), is the latitude of a place A', and is marked N., because A' is to the north of W M' E: whilst the latitude of B' is M' B', and is marked S., because the place B' is to the south of the equator.

As the latitude begins at the equator (lat. o°), and is reckoned thence to the poles (latitude 90°), where it terminates, therefore the greatest latitude a place can have is 90°, and all other places must have their latitude intermediate between o° and 90°.

PARALLELS OF LATITUDE are small circles of the sphere parallel to the equator, that is equidistant from it in every point, and hence all places of the same latitude being at the same distance from the equator, are said to be on the same parallel: thus (fig., page 64), O, F, and b B' are portions of parallels of latitude, and all places on O, F, and b B', &c., have the same latitude, being on the same parallel.

THE DIFFERENCE OF LATITUDE (abbreviated diff. lat.) between two places is the arc of a meridian included between their parallels of latitude, showing how far one of them is to the northward or southward of the other; thus (fig., p. 64) Ab is the difference of latitude of the two places A' B'; F S between the places FT. The difference of latitude between two places can never exceed 180°.

CO-LATITUDE is the complement of the latitude to 90°, thus the colatitude of A (fig., page 64) is A N, of B' is B S.

It is evident, that when two places are on the same side of the equator, their diff. lat. is found by subtracting the less latitude from the greater; and that when they are on opposite sides of the equator, that is, when one place is in north latitude and the other in south latitude, the sum of their latitudes is the diff. lat. Thus, the diff. of lat. of A′ B' which is A' b, is the sum of the north lat. A' M', and of the south lat. Z B', or M' B'.

or

The difference of latitude of the ship is therefore the distance made good in a north or south direction. This is also called her "northing" southing," ," these names being indicated by the initials N. and S. MERIDIONAL PARTS.-At the equator a degree of longitude is equal to a degree of latitude, but as we approach the poles, while (upon the supposition that the earth is a sphere) the degrees of latitude remain the same, the degrees of longitude become less and less. In the chart, on Mercator's projection, the degrees of longitude are made everywhere of the same length, and, therefore, to preserve the proportion that exists at different parts of the earth's surface between the degrees of latitude and the degrees of longitude, the former must be increased