one mile per hour; if two divisions, two miles, and so forth. The line we have described resembles very nearly a seaman's log-line. Sailors call the pieces of cloth and divisions knots, so that knot and mile mean in effect the same thing: when a ship is said to be sailing eight knots an hour, the meaning is, that she runs eight divisions on the log line from the stray line in half a minute, that is, is sailing at the rate of eight miles an hour. At sea the lead is enclosed within a piece of wood, which floats on the surface of the water. On the assumption that the line is thrown down every hour, add together the distances due to each hour of the day during which the journey is prosecuted, and this will give the whole distance to be set down for each day. If our traveller were not aware of the existence of the lake, he would travel or steer NW; but we suppose him acquainted with the bearing of its eastern point; he will naturally, however, keep as near to the true direction, or course, as he can. So ships, when they are impeded by contrary winds, keep as near to their true course as circumstances will permit. Now we have seen, (Art. 11,) that the nearest course the traveller can hold at first will be due north; and the first day, or during the first 24 hours, we will suppose him to have walked 33 miles in that direction, when he arrives at D the eastern extremity of the lake; let him now alter his course, and steer for the point b, the most northern point of the lake; let him ascertain his second and third days' course to have been N 15° W, and the distance travelled, or Db, 47 miles. During the second and third days' journey our traveller has a mark to direct his steps, viz. the margin of the lake; but when he arrives at b, he will have no means of determining his direct and shortest course from thence to B, except by means of the distances travelled and courses steered during the first three days. We have seen that before B can be reached, 42 miles of northing and westing must be made; now on the first day 33 miles northing are made and no westing, therefore at the end of the first day 9 miles northing remain, and 424 westing. To ascertain the result of the second and third days' work, we have the distance Db = 47 miles, and the angle CD b = 15°; but the angle a b D = the angle b D C, for it is a property of parallel lines, that when a straight *Simson's Euclid, book i. prop, 29, line, as 6D, meets them, it makes the alternate angles equal; and the angles a b D and b D C are called the alternate angles: those angles both represent the same course, but taken in an opposite direction, that is, a b D represents a southern and eastern course, whilst CD b represents an equal northern and western course. Therefore in the right angled triangle a b D we have the angle ab D = 15° and D b = 47 miles, to find a b and a D. Draw a line proportional to 47 miles, and draw another line making an angle of 15° with the former; draw a third line at the other extremity of the first line, making an angle of 75°, the complement of 15°, with that line: then we shall have constructed the data of our question, (Art. 9,) and the triangle a b D in the figure represents such a right angled triangle as we have been describing; measure therefore ab and a D as before, and ab will be found equal to about 45 miles, and a D to about 124. We perceive, therefore, that with respect to difference of latitude, the result of the second and third days' work is 36 miles more northing than necessary; now a D = c C,* and ca= C D, for the four-sided figure a c C D is a parallelogram,† take ca= CD = 94 from a b, and c b will remain = 36 miles; take a D = c C = 124 miles from B C (= 424), and B c will remain = 30 miles, so that of departure 301 miles remain, and at the end of the third day's journey there will remain to be performed 36 miles of southing, and 30 of westing; and in the right angled triangle B cb we have bc=36 miles, and B c=30 miles given, to find the course Bbc and distance b B. Draw therefore two straight lines perpendicular to each other, the one proportional to 36 miles and the other proportional to 304 miles; let Bc and cb in the triangle B cb be those lines, measure the angle Bbc, and it will be found = 40°, and b B, or the nautical distance, will be found to be 47 miles, or = Db; so that the traveller discovers he must now for the rest of his journey steer S 40° W, and as he has 47 miles left to complete his task, we may suppose that he takes two days more to perform it. We shall put the first three days' journey into a table, which is called a traverse Simson's Euclid, book i. prop. 28 and 34. From two Greek words signifying a parallel drawing. A parallelogram is defined to be a fourthe opposite sides and angles of such a figure are sided figure having its opposite sides parallel, and proved to be equal, (15.) If there had happened to have been southing during these three days, we should have subtracted it from the northing: the same remark applies to the Departure column, in case there had been easting: for our object was to ascertain the total quantity of difference of latitude and departure made good in the first three days, for the purpose of discovering how much of each remained to be accomplished, and thus we found Bc and c b.* (16.) On account of the great use of right angled triangles in solving problems in navigation, tables have been calculated called Traverse Tables, or Difference of Latitude and Departure Tables, which give the value of the perpendicular and base of a right angled triangle to every angle of an integral number of degrees, which the hypotenuse can form with them, and to every hypotenuse from 1 to 300. Solving right angled triangles by these tables is called solving by inspection. (17.) The same principles will of course apply, whether we suppose the traverse to have been performed on land, or by a ship at sea. Thus, traverse sailing, or in other words, resolving cases of sailing in which a traverse is performed, consists simply in finding out by the solution of two or more right angled triangles, the total amount of difference of latitude and departure actually made good during the traverse; bearing it in mind, that opposite courses destroy each other, and therefore in these cases the less quantity must be subtracted from the greater for instance, suppose that a ship during a traverse of eight days, which she is compelled to make by foul winds, sails 239 miles E, 240 W, 230 N, and 230 S; it is manifest, that the two last distances, being equal and in a contrary direction, destroy each other, and that the ship after sailing 939 miles, is By calculation, AC42,43, ab= 45.4, a D= 12.17, bc=35.97, Bc = 30,26, B6 =47 and Bbc= =4094. (18.) We shall conclude the subject of traverses by observing, that when several different courses are steered during the day, the total distance on each course must appear in the traverse table for that day, and in a line with it, the difference of latitude and departure due to that distance and course. In example II. it was our object as much as possible to preserve an analogy between the two cases of travelling on land and water; but we preferred for obvious reasons illustrating the subject of traverses by an example of a journey on land instead of a voyage at sea. CHAPTER III, The Plane Chart.-Pilotage.-Sailing in Tides and Currents. (19.) THE plane chartis a representation of a small portion of the earth's surface on a large scale. In order to construct a chart of this description, we must ascertain the bearings of the objects there represented from each other; this may be done by surveying the space included within the chart; we then lay down the objects on the chart ac-. cording to their several ascertained bearings. Figure 2, for instance, may be considered as a chart of the surface of a plain, in which the position of a village, house, and lake are laid down according to their several bearings. Such a chart it will be readily seen would have been of great service to the traveller in the example; as by it he could have ascertained the course he ought to have taken on each day, without the trouble of going through the process we have described: and if there had been drawn thereon a scale on which the length of a mile in the chart had been exhibited, he could thereby have measured all the distances. A scale is therefore generally drawn on a map of this description, and also a figure of a compass, for the purpose of enabling the mariner the more easily to dete. mine the relative bearing of one spot on the chart from another. The meridian lines, or meridians, are drawn parallel to one another, (Art. 12.) (20.) These plane charts are of great 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 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 A B 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, 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. clear of all dangers, and may steer NNE for the anchorage at c." The straight lines in the figure show the marks that are to be used as guides in entering the bay, and the marks themselves are called the leading marks. (22.) Piloting vessels is however rendered in many cases very difficult, by the necessity which exists of taking into account, and allowing for the effects of tides and currents. These effects may be shown by supposing A B D C (fig. 4) Fig. 4. 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. The moon produces two tides of high water at the same time, one at places on 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 will be low water precisely at the same time, at all places which are 90° distant from the two places of high water, (Art. 35 and 36.) The action of the sun sometimes accelerates, and sometimes retards, the tides produced by the moon. The time of high water at any particular place is much affected by local circumstances; but, having ascertained the time of high water at that place, at the full and change of the moon, we are enabled to find it at any other time, supposing the longitude of the place known, by means of the Nautical Almanac, and a table calculated for the purpose. (24.) In a belt extending about 30° on each side of the equator, the wind is observed to blow all the year round from nearly the same quarter of the heavens: to the north of the equator it blows nearly from the NE quarter, and to the south of the equator from the SE quarter. These winds, from the great assistance which they afford to commerce, are called the NE and SE trade winds. When ships are bound from Europe to the West Indies, or to any part of North America, south of the parallel of about 38°, they seek the aid of these winds, but when they return, they keep away to the northward for the purpose of avoiding them. In December the NE trade is found to the south of the line, and on the other hand in June, the SE trade makes its appearance to the north of the line; but the SE trade is at other seasons of the year often found as far to the north of the line as the second degree of north latitude. The space between the second and fifth degrees of north latitude, is the inward boundary of these winds; in this space,storms, sudden squalls, and violent rains, are of frequent occurrence; but here likewise the wind often blows from the eastward, and also from the SW. There are periodical winds, which blow half the year in one direction, and half the year in the opposite direction, these are called monsoons: they are found in the Bay of Bengal, the Arabian Sea, the Mozambique Channel, on the coasts of Sumatra and Java, along the coast of China, and off the western coast of New Holland. October and April are the two months, in which the change in the direction of these winds usually takes place. It is on this account that, in books of voyages, the time of high water at different places at the full and change of the moon is so often to be met with. PART II. CHAPTER I. On Plane and Spherical Trigonometry -Logarithms. (25.) WE have seen, that the various lines, the magnitudes of which were the subjects of investigation in the first Part, constituted the sides of triangles, supposed to be drawn upon a plane surface. When we proceed to take into consideration the globular or round figure of the earth, it will not be necessary on that account to make any alteration in our hypothesis, with respect to the nature of the lines, the values of which it is necessary to ascertain; for the magnitude of the curved lines, which a vessel makes in traversing the round surface of our globe, is calculated in practice, through the medium of certain imaginary straight lines, which are proved to be either equal to such curved lines, or to have a known relation to them; such imaginary lines form the sides of plane triangles, and present no greater difficulties to the calculator, than the lines, the values of which have been hitherto obtained by the method of construction. (26.) In this our second Part, those imaginary lines will be invented, and the triangles resolved, which they constitute. It is clear, therefore, that the particular branch of mathematical science, which professes to teach the proportions and rules, derived from the relation which subsists between the sides and angles of triangles, must be of primary importance in our present inquiries. The name of that branch is Trigonometry,+ and it forms the subject of the first chapter. (27.) Every triangle having six parts, the three sides, and the three angles, (Art. 3,) one of the principal objects of the science of trigonometry is to discover rules, by which, if some of those parts be given the others can be found -Rules have accordingly been discovered, by which, if any three of those parts be given, a side being always one, the other three can be found. In right angled triangles, as the right angle is always given, if any two of the remaining parts be given, a side being one, the rest can be found. lines; and the term is used in contradistinction to a A plane triangle is one whose sides are straight spherical triangle, the sides of which are formed by portions of the circles of a sphere. This will be explained hereafter. † From two Greek words, signifying the measure of triangles |