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as a landmark to direct his steps, and that he wishes to traverse the plain to a village B at the distance of sixty nautical (Art. 44) miles from the spot on which the house is placed: let a line AC be conceived to be drawn through that spot towards the North Pole, or North point of the heavens; and let a line A B join the house and the village; and let the direction of the village from the house, or the line A B, make an angle of 45° with the line A C, that is, let the angle BAC equal 45°: draw also the line BC at right angles to A C. The angle BAC and the distance B A (=60 nautical miles) are determined, by means which will be explained hereafter, (Art. 54.)

Let us further suppose that the most eastern point of a large body of water, shown in fig. 2 by the irregular and crooked line, is situated due North of the house A, and that the situation and extent of this lake obliges our traveller, in his journey from A to B, to pursue the track marked out by the dotted line. Such a track is called by seamen a traverse; and it will be readily seen, that none of the ordinary modes which are in daily practice, of finding the way from one spot to another, will avail a person so circumstanced as in our example. (12.) Before, however, we proceed to explain all the contrivances which are to be substituted for such ordinary modes, we shall define certain terms which it will be convenient to make use of hereafter; we shall also give a more simple example than that proposed in

the last article.

Few of our readers can be ignorant, that such lines as B C and A B in fig. 2, supposed to be drawn on the round surface of the earth, cannot be straight lines; they must be curved lines. But as the earth is a globe of very large dimensions, (Art. 44,) a very small curved distance, such as sixty miles, differs very slightly from a straight line; and therefore in our investigations we may assume the line A B, and of course all the other lines in the figure smaller than A B, to be straight lines. We shall find hereafter, (Art. 52,) that no error can possibly arise from this supposition in questions similar to those solved in examples I. and II.; and such an assumption, it is evident, must materially facilitate our inquiries. Draw the right angled triangles ab D, bc B. We shall call the line A C, drawn towards the north pole, and all lines

parallel to it, as a b, meridian lines; and as those lines are drawn towards the top of our page, that top will represent the north, the right of the paper the east, the left the west, and the bottom the south. This supposition is invariably made with respect to all figures drawn to illustrate problems in navigation, so that we are enabled to discover the nature of lines by the position in which we find them, with reference to the top of the page. We perceive that the lines AB, Db, b B all make certain angles with meridian lines; these angles are called courses; the angle CAB is called the course from A to B; and as it equals 45°, the course from A to B is said to be N* 45° W, or 45° from the north, on the side of the west; and as this is just half way between the two, it is NW. The course from one place to another is also called the bearing of the latter place from the former; thus B is said to bear N 45° W from A, and A, S 45° E from B. For the present we shall call all lines drawn at right angles to meridian lines, departures. Thus B C, a D, and Bc are termed departures; also the lines which make with the meridian lines the angles called courses are called nautical distances; such are the lines A B, D b, and b B. The true nature of courses, departures, and nautical distances, will be better understood hereafter, (Art. 51 and 52.)

EXAMPLE I.

to travel, or a ship be supposed to sail, (13.) Let a man (fig. 1) be supposed eleven miles, (from B to A); let the with the meridian line (B C); and let it course (A B C) make an angle of 53° be required to find the length of the line BC, or, in other words, the distance made good in the direction B C, or from north towards the south.

It is plain, that as the ship's motion is compounded of a southern and west

ern course, she must sail farther than the length of the line BC before she makes good that distance towards the south; we see, in fact, that she sails on the hypotenuse of the triangle instead is by the last article S 53° W. of the perpendicular. The course also

Draw the line B A proportional to with another line B C drawn from the eleven miles, making an angle of 53°

It seems scarcely necessary to mention, that the letters N, S, E, and W stand for north, south, east, and west respectively. The letter b stands for by: thus, north by west is written Nb W.,

north towards the south; now the angle
BAC, the complement of the course,
(Art. 5) is 37°, draw therefore the line
AC, making an angle of 37° with the
line A B, and measure B C with the
same scale with which A B was measured
or set off; and B C will be found to be
about six miles and a half, and A C the
departure to be about eight miles and
three quarters. The line B C is called
the difference of latitude; and the four
parts or elements of the triangle ABC,
which are of importance in navigation,
are the three sides A B, B C, and A C,
and the angle A B C, which represent
the nautical distance, difference of lati-
tude, departure, and course : if any two
of these be given, the other two may be
found by construction; and, as we shall
see hereafter, by inspection or calcula-
tion. The line A C, or departure, re-
presents the amount of distance made
good towards the west.

EXAMPLE II.

(14.) We proceed to the traverse drawn in fig. 2. This case, we shall presently perceive, is, in effect, nothing

more than two solutions of the same question that was proposed in example I, and a solution of one, in which the course is the element of the triangle sought after. The bearing of the village B from the house A, (see Art. 11 and 12,) or the direct course from the latter to the former, is N 45° W; with the course N 45° W and the nautical distance A B = 60, we shall find by the mode of proceeding adopted in the last example A C = to about 424 miles, and BC= 42 miles, for in this case the departure equals the difference of latitude. So that our traveller on starting is enabled to discover by our method that he has 424 miles of northing, or difference of latitude, to make, and the same quantity of westing or departure. Having ascertained this fact before he sets off, he might now proceed, had he the means of preserving any particular direction in which he might wish to travel, and of measuring the distance he walks; therefore the mode of effecting these two objects must be next explained. He can determine his course, or the angle his track makes, with every meridian he crosses, either by the heavenly bodies, or by a compass; a description of which instrument will be given in Chapter III. of the Second Part. The compass will enable him to ascer

By calculation, BC=6.62, and AC = 8.79,

tain at any time the direction in which
he is walking ;* and, consequently, at all
times to preserve whatever direction he
may find it expedient to take; thus, for
instance, it will enable him the first day
to travel due north, for the
of
purpose
making, as the mariner terms it, the
eastern point of the lake, the bearing of
which we have supposed him to know
before he sets out. With respect to the
distances travelled, these may be dis-
covered as follows: take a long string
so divided into equal parts that each
part shall be the 120th part of a nauti-
cal mile, and to the end of the string at-
tach a piece of lead. Now as a minute
is a of an hour, half a minute is 1 of
an hour; or half a minute is the same
part of an hour, that the divisions on the
string are parts of a mile. Let pieces of
cloth be tied round the string to mark
the divisions; the first piece of cloth at
the distance of about 30 feet from the
lead; the first 30 feet of string is
called the stray line. Let the traveller
the lead on the ground and walk on,
at the end of every hour throw down
leaving the lead behind him, and al-
lowing the string attached to it to
pass through his hand as he walks; let
him mark the time by a seconds watch
when the first piece of cloth at the end
of the stray line passes, and at the end
of half a minute from that time let him
stop the string, catching it up in his
hand; and the number of divisions
and parts of divisions contained be-
tween his hand and the piece of cloth
at the end of the stray line will repre-
sent the number of miles he has
walked during the last hour, on the
supposition that he has always walked
at the same pace, (viz. the pace at which
he performed the experiment,) during
that hour: for that number represents
the time the string is running through
the pace at which he walks during
his hand; those divisions being the
same part of a mile that half a minute
is of an hour; and if part of a mile,
or one division of the string, is described
in of an hour, and during the pre-
ceding part of the hour the same rate
of going has been preserved, 120 times
that division, or a mile, has been per-
formed in 120 times half a minute, or
an hour; so that if the distance be-
tween the hand and cloth at the end of
the stray line is one division, the pace is

120

• We make no mention in the text of the raria. tion or deviation of the compass from the true north, this will be explained hereafter.

use to pilots and the masters of coasting vessels, who are not guided so much by the principles we have explained in Chapter II. as by buoys, soundings, and the bearings of the different headlands, lighthouses, and other objects on the land, in sight of which they steer their vessels. At night they are generally directed either by the bearings of the lights on the coast, or by the compass: for they are taught that, in order to avoid certain dangers, they must steer a particular course from one point to another. The bearing of any object, in sight from the deck of a ship, is ascertained immediately by looking at the compass. (Art. 45.)

(21.) As many rocks, shoals, and other dangers usually exist in the neighbourhood of land, and they are for the most part hidden below the surface of the water, even at low tide, it is the peculiar province of the pilot to be acquainted with their exact situation. The exact situation of a point may always be determined by the intersection of two straight lines, drawn from any two positions, and meeting in the required point: to illustrate this, let (fig. 3) represent an anchorage within

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dropped, this spot is c; a and b are shoals marked out by dots and the irregular lines; I, an island; A, B, C, and D are objects on shore; and E points out the situation of a hidden rock in the very middle of the channel to the harbour, which channel lies between the two shoals a and b. A, B, and E, are supposed to have such a position with respect to each other, as to lie in the same straight line. But the situation of the rock E is not determinable by the line AB E alone; for if the pilot know only, that it is situated somewhere in that line, he would still be ignorant of the exact spot which it occupied. But if the objects D and C, and the rock E, are so situated as to lie in the same straight line; then it is obvious that the point E is determined by the intersection of the two lines A BE, C D E. The pilot would then have information sufficient to enable him to avoid that rock in conducting a ship to this anchorage: for, in coming in from the south, suppose him so to place his vessel as to have the objects A and B in one line, or in one, as it is termed by seamen; now, he knows that whilst he continues to have these objects in one, he is in the exact line of the rock; but so long as D remains open to the left of C he will not strike upon it.* A safe mode of proceeding might be suggested in this case: -let him keep the object B at least a ship's length open to the right of A, and he will be in the line Fgh Bi; indeed he might keep it as far open to the right as he pleases, provided he avoids the island I, and the shoal a. When, in sailing on this line, he arrives at g, he brings D and C in one, and he knows the exact bearing of the rock E from his ship, but he cannot be said to be clear of E until he has passed the point h, which is situated due west of the rock; at that point D will be open to the right of C a certain quantity, say a ship's length; the pilot's direction may now be expressed in technical language:

"In coming from the south, keep B a ship's length open to the east of A until the most northern point of island I bears SSW, and D is open a ship's length to the east of C; you are then

Smugglers are said to make a successful application of the principle we are here illustrating, in sinking their cargoes; they sail on a line A BE, keeping two objects A and B in that one line, until they arrive at a point in which two other conspicuous objects, as D and C, become in one, and there they sink their goods: by the intersection of these two lines they can find them again at any time, when they are felieved from the fear of detection.

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to be a trough containing water; let a light body b float on the surface of the water along the line A C, whilst the trough itself is being carried in the direction A B B', or C D D': at the same instant of time at which the trough takes up the new position A'B' D'C', let the body b arrive at the point C: now the point C is transferred to C', and at that point will the body b be found at this time. This is exactly what happens in the case of a ship acted on by a current or tide; whilst the ship performs any particular course, due west for instance as in the figure, the current compels her to perform another course in the direction of its set, and a distance proportional to its velocity; this new course and distance are entered in the traverse table, and taken into account, as if the ship had made them without the assistance of the current: it amounts to the same thing in practice; for as the whole surface of water ABDC is supposed to move in the direction A B B', the same quantity of difference of latitude and departure is made good in that manner, as if the ship after its arrival at C had sailed directly from C to D, and the water had remained stationary; in both cases Ca will represent the difference of latitude made, and a D the departure. CHAPTER IV.

On Tides and Winds. (23.) THE alternate rise and fall of

The direction of a current is called its set; a

current that flows towards the NNW quarter is said to set NNW; the velocity of a current is called its drift.

the waters of the ocean, called tides, is produced by the attractive forces of the sun and moon: now, as those attractive forces vary inversely as the square of the distances of the bodies attracting from the bodies attracted, (that is, become less in the same proportion as the squares of those distances become greater,) and as the moon, though much smaller, is nearer to us than the sun, her effect in raising the waters is greater than that of the sun, which is comparatively very small. The earth by its diurnal motion round its axis from west to east (see Art. 62) causes an apparent daily revolution of the moon round the earth in a contrary direction. The waters of the sea follow the moon in this her apparent course, so far as the irregularities of the land and shores will admit of. It might be supposed, therefore, that under these circumstances it must be always high water at any place when the moon is over it: but this is not the case, for when the waters have once begun to rise they will continue to rise after the cause which influences them has produced its maximum effect, for it does not then cease to act entirely. High water, therefore, is observed to happen about three hours after the moon has passed any place, or, as it is called, passed the meridian of that place.

water at the same time, one at places on The moon produces two tides of high the earth's surface nearest to her, and another at those on the opposite side of the globe, which are the most removed from her: for she attracts the centre of the earth more than the sea on that opposite side, as being nearer to her, the effect of which is to draw that centre away from the sea, and as the sea is left behind, it appears to rise. When the sun and moon are together, as at new moon, they combine their forces in causing the tides, and make spring tides; and the same thing happens at full moon when the moon is opposite to the sun; but when the moon is in quadratures, or half-way between the change and full and full and change, the whole action of the sun in causing tides is directly opposed to that of the moon, and this produces neap tides.

We have seen that there is high water at two spots on the surface of the earth, immediately opposite to each other, at the same instant of time; and there

use to pilots and the masters of coasting vessels, who are not guided so much by the principles we have explained in Chapter II. as by buoys, soundings, and the bearings of the different headlands, lighthouses, and other objects on the land, in sight of which they steer their vessels. At night they are generally directed either by the bearings of the lights on the coast, or by the compass: for they are taught that, in order to avoid certain dangers, they must steer a particular course from one point to another. The bearing of any object, in sight from the deck of a ship, is ascertained immediately by looking at the compass. (Art. 45.)

(21.) As many rocks, shoals, and other dangers usually exist in the neighbourhood of land, and they are for the most part hidden below the surface of the water, even at low tide, it is the peculiar province of the pilot to be acquainted with their exact situation. The exact situation of a point may always be determined by the intersection of two straight lines, drawn from any two positions, and meeting in the required point: to illustrate this, let (fig. 3) represent an anchorage within

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dropped, this spot is c; a and b are shoals marked out by dots and the irregular lines; I, an island; A, B, C, and D are objects on shore; and E points out the situation of a hidden rock in the very middle of the channel to the harbour, which channel lies between the two shoals 'a and b. A, B, and E, are supposed to have such a position with respect to each other, as to lie in the same straight line. But the situation of the rock E is not determinable by the line AB E alone; for if the pilot know only, that it is situated somewhere in that line, he would still be ignorant of the exact spot which it occupied. But if the objects D and C, and the rock E, are so situated as to lie in the same straight line; then it is obvious that the point E is determined by the intersection of the two lines A B E, CD E. The pilot would then have information sufficient to enable him to avoid that rock in conducting a ship to this anchorage: for, in coming in from the south, suppose him so to place his vessel as to have the objects A and B in one line, or in one, as it is termed by seamen; now, he knows that whilst he continues to have these objects in one, he is in the exact line of the rock; but so long as D remains open to the left of C he will not strike upon it. A safe mode of proceeding might be suggested in this case: -let him keep the object B at least a ship's length open to the right of A, and he will be in the line Fgh Bi; indeed he might keep it as far open to the right as he pleases, provided he avoids the island I, and the shoala. When, in sailing on this line, he arrives at g, he brings D and C in one, and he knows the exact bearing of the rock E from his ship, but he cannot be said to be clear of E until he has passed the point h, which is situated due west of the rock; at that point D will be open to the right of C a certain quantity, say a ship's length; the pilot's direction may now be expressed in technical language:

"In coming from the south, keep B a ship's length open to the east of A until the most northern point of island I bears SSW, and D is open a ship's length to the east of C; you are then

Smugglers are said to make a successful application of the principle we are here illustrating, in sinking their cargoes; they sail on a line ABE, keeping two objects A and B in that one line, until they arrive at a point in which two other conspicuous objects, as D and C, become in one, and there they sink their goods: by the intersection of these two lines they can find them again at any time, when they are relieved from the fear of detection,

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