« ΠροηγούμενηΣυνέχεια »
NAVIGATION is the science which teaches the mariner how to conduct his ship from any one port or place to any other. An application of its principles will enable him at any time to discover the situation of his vessel, the direction in which he ought to steer towards his intended destination or port, and his distance from such port and from the place from whence he sailed:-This science explains also the method of applying the different instruments used in its practice, and of discovering their errors and imperfections. In short, it is that branch of the seaman's education which may be learnt on shore, and apart from the ship; and furnishes a material part of all that is requisite to be known by him, on whom devolves the anxious responsibility of directing the course of a vessel through that vast and treacherous expanse of waters, which forms the greater portion of the surface of our globe, deprived of that assistance which the traveller on land receives from beaten tracks, and known landmarks, but enabled to rely with confidence on other guides, which the bounty of his Creator has provided, and which the ingenuity of man has converted to his use under such circumstances of apparent difficulty.
The term Navigation is derived from the Latin word navigo, which signifies literally, to work or manage a ship; but the art itself consists chiefly in the арplication to practice of a branch of mathematics, and a branch of the science of astronomy, which owes its name to that application.
stance, to be drawn around the angle A" CM", in fig. 5, (page 11,) and let them be so drawn, that the point C, at which the angle is formed, may be the common centre of all the circles. Then the lines A" C and M" C are said to intercept, that is, to take between them, a portion of each of the circles: in the largest and outer circle they intercept the portion A" M", which is called an arc, from its resemblance to a bow: scientific terms to the Arabians, who for as we are indebted for many of our were a people of warlike habits, they exhibit many metaphors drawn from Now the curved line the art of war.
A" M", and the corresponding intercepted portions AM, A'M', are all said to measure the angle A" C M", which is formed by the intercepting lines A" C, M" C; it matters not which of these arcs we take as a measure; for they are all the same fractional part of the respective circles to which they belong: thus, if A" M" be an eighth part of the greatest and outer circle, A M' is an eighth part of the next greatest, and AM an eighth of the smallest and innermost circle. Mathematicians conceive circles to be divided into 360 equal parts, which they call degrees; each of these degrees is again supposed to be divided into sixty equal parts, called minutes, and each of these minutes into sixty equal parts, called seconds; and, if smaller parts are required, the seconds may be divided into thirds, and the thirds into fourths, &c. each denomination being always sixty times the value of that which follows it in the order of succession; these divisions for the sake of brevity are written as follows:-thus, 57 degrees, 31 minutes, 42 seconds, 37 thirds, 48 fourths, and 32 fifths, are written
57° 31' 42" 37" 48iv 32v.
An angle is said to contain the same number of degrees, &c. which the lines containing it intercept on the circumscribed circles; if the lines A" C, M" C intercept 40° on each circle, A" C M" is an angle of 40°.
(2.) When the lines which contain an angle intercept a quarter of the circles described around it, the angle is called a right angle, and the lines containing
it are said to be perpendicular, or at right angles to another; the quarter of the circle is called a quadrant. Thus in fig. 5 the lines E" Ĉ and A" C are represented as cutting off a quarter of each of the three circles, and the angle A" CE" is a right angle, and the lines A" C and E" C are perpendicular to one another: and as the arcs, which lines containing a right angle intercept, are a quarter of the circles to which they respectively belong, which circles contain 360°, every right angle must contain 90 of those degrees.
(3.) Every triangle has six parts, the three angles and the three sides. There are various kinds of trianglesthe right angled triangle is that which has one of its angles a right angle. Thus the triangle ABC (fig. 1) is a Fig. 1. right angled triangle, having the right angle ACB as one of its angles. In this species of triangle, the sides are distinguished by mathematicians by particular names; the side AB, opposite to the right angle, is called the hypotenuse, and the two sides B C and AC are called either base or perpendicular, according to the position in which the triangle may happen to be placed: thus, in the present position of the triangle A B C, AC would be termed the base, and CB the perpendicular, CAB the angle at the base, and A B C the angle at the vertex or top.
(4.) Oblique angled triangles are those which have not any of their three angles right angles, in short, which are not right angled.
(5.) The angles of any triangle, formed by straight lines, are together equal to 180°, or two right angles; therefore in any right angled triangle A B C (fig. 1) as the angle ACB is equal to 90°, the angle at B will be equal to the angle at A subtracted from 90°, and the angle at A will be equal to the angle at B subtracted from 90°: these angles are called the complements to each other, for the one makes up what the other wants of being a right angle, or 90°.
(6.) There are mathematical instruments by which we are enabled to draw angles containing whatever number of degrees we please; but we cannot de
From the Latin word quadrans, which signifies the quarter of a small Roman coin, called an as.
From a Greek word, signifying to extend under.
pend on their accuracy to a less quantity than about the quarter of a degree or 15'; that is, were we to attempt to draw an angle containing exactly 43° 14′ 16′′, the small quantity (44") by which that angle differs from an angle of 43° 15′ would be wholly imperceptible and lost, owing to the imperfection of the means we make use of to effect our object; probably the very thickness of the lines containing our angle would be equal to nearly a quarter of a degree-it follows, therefore, that we cannot depend on an angle, thus drawn, to less than that quantity.
(7.) There is an instrument also called a plain diagonal decimal scale; this is a ruler divided into a certain number of equal parts, and these parts, by a particular contrivance, are again subdivided decimally, or into tenths and hundredths of those parts. By this instrument we are enabled, but with no great degree of accuracy, to draw lines that shall bear to each other nearly any proportions we may please to assign them.
(8.) The use of the above mentioned instruments may be illustrated as follows: suppose we know, in a particular case, that the hypotenuse of a right angled triangle is 1100 miles, and the angle at the vertex equal to 53°: then, with our plain scale, we draw a line proportionalto 1100 miles, or containing 1100 equal parts, each part being any length we choose to fix upon, and which then becomes our unit of length: let it be the line BA in fig. 1; we draw also another line BC making, with the line B A, an angle of 53°: we do not know the length the line BC ought to be when the triangle is right angled, or where we ought to stop in drawing it, but we may soon ascertain it by our ruler or scale in the following manner:-as the angle at B is 53°, if the triangle is right angled, the angle at A must be 37°, (Art. 5,) draw therefore the line A C, making an angle of 37° with the hypotenuse A B, it is obvious that the drawing this line, at this particular angle, determines the length that the line BC ought to be, when the angle at C is a right angle; and in order to ascertain the proportional lengths of the lines BC and A C, as compared with A B, we, need only measure them with our scale. This is called resolving the triangle ABC by construction. We say the proportional lengths of the lines BČ and AC, because it matters not what unit of length we make use of, in other words, what the length of each of the
1100 parts is, or how that unit is represented on our scale, provided that we apply the same unit and measure to all the three sides of the triangle. Thus it is of no consequence whether the 1100 parts of the hypotenuse be 1100 miles, or yards, or feet, &c. or whether an inch, or half an inch, &c. on our scale represent a mile, provided that in taking all the three measures the inch or half inch represents the same quantity: if half an inch, for instance, represent 100 miles when we measure AB, it must represent the same quantity of miles when we measure B C or A C.
(9.) In order, therefore, to resolve a triangle by construction, it is only necessary to have the means of constructing the data, that is, the things given, and then we measure the lines or angles that are unknown: for if the data be sufficient, the very act itself of representing them on paper enables us also to represent those lines and angles that are not given, and when those unknown quantities are represented on paper, and on a scale proportional to the known quantities, we have but to measure them and their values are then ascertained. The examples, which it will be necessary to subjoin in this Part, of this mode of resolving triangles, will sufficiently illustrate the above explanation of the phrase resolving a case by construction.
Preliminary Considerations.- On Tra
(10.) IF we often interrupted the course of useful pursuits to analyze the manner in which we perform some of the complex though common operations of life, a very small part only of that which is actually accomplished by human energies and industry would then be achieved: yet there are cases in which, in order to obtain a clear conception of matters of science or art, with which we are not familiar, it is expedient to advert to those every day occupations, which have some connexion more or less remote with the object of inquiry, and to consider the different processes, which lead to their completion, and which, generally, through the influence of habit, make no impression on our minds.
Thus the very common operation of walking from any one spot to another, not in sight at the commencement of our journey, is connected with the infinitely
more difficult one of directing the course of a ship through the ocean. The traveller on land has, in common with the mariner, to ascertain his course and distance: but the great point, in which journeying upon land and sailing on the sea differ in this respect, is, that on land the course and distance are usually determined by means of fixed marks on the surface of the earth, with the appearance of which the traveller is either acquainted, or is enabled to avail himself of their assistance by the aid of certain general considerations drawn from experience; which is indeed the source of the greater part of that more humble knowledge which all mankind possess in common. In traversing the ocean, on which vessels leave no permanent trace behind them of the path they have pursued, and the same difficulties present themselves to each succeeding voyager, the sky and the sea alone remain to direct the mariner in his course.
It is easy, however, to imagine a case in which the traveller on land may be wholly dependent on those guides, which a knowledge of some of the first principles of navigation alone can furnish. (11.) Thus (fig. 2) we will suppose Fig. 27
him at a house A situated on some extensive and barren plain, on which there is no object that can by possibility serve
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 AB 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;
(13.) Let a man (fig. 1) be supposed to travel, or a ship be supposed to sail, eleven miles, (from B to A); let the course (A B C) make an angle of 53° with the meridian line (B C); and let it 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 western course, she must sail farther than the length of the line BC before she south; we see, in fact, that she sails on makes good that distance towards the the hypotenuse of the triangle instead of the perpendicular. The course also is by the last article S 53° W.
Draw the line B A proportional to eleven miles, making an angle of 53° with another line B C drawn from the
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 AC 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 latitude, 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 calculation. The line A C, or departure, represents the amount of distance made good towards the west.
(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 42 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 42 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 purpose of 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 discovered as follows: take a long string so divided into equal parts that each part shall be the 120th part of a nautical mile, and to the end of the string attach a piece of lead. Now as a minute is of an hour, half a minute is 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 lowing the string attached to it to leaving the lead behind him, and alpass 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 between his hand and the piece of cloth at the end of the stray line will represent 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 pace at which he walks during the time the string is running through 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 1 of an hour, and during the preceding part of the hour the same rate that division, or a mile, has been perof going has been preserved, 120 times formed in 120 times half a minute, or an hour; so that if the distance between the hand and cloth at the end of the stray line is one division, the pace is
We make no mention in the text of the varia tion or deviation of the compass from the true north, this will be explained hereafter.