SAILING
, in a general sense, denotes the movement by which a vessel is wafted along the surface of the water, by the action of the wind upon her sails.
Sailing is also used for the art or act of navigating; or of determining all the cases of a ship's motion, by means of sea charts &c. These charts are constructed either on the supposition that the earth is a large extended flat surface, whence we obtain those that are called plane charts; or on the supposition that the earth is a sphere, whence are derived globular charts. Accordingly, Sailing may be distinguished into two general kinds, viz, plane Sailing, and globular Sailing. Sometimes indeed a third sort is added, viz, spheroidical Sailing, which proceeds upon the supposition of the spheroidical figure of the earth.
Plane Sailing is that which is performed by means of a plane chart; in which case the meridians are considered as parallel lines, the parallels of latitude are at right angles to the meridians, the lengths of the degrees on the meridians, equator, and parallels of latitude, are every where equal.
In Plane Sailing, the principal terms and circumstances made use of, are, course, distance, departure, difference of latitude, rhumb, &c; for as to longitude, that has no place in plane Sailing, but belongs properly to globular or spherical sailing. For the explanation of all which terms, see the respective articles.
If a ship sails either due north or south, she sails on a meridian, her distance and difference of latitude are the same, and she makes no departure; but where the ship sails either due east or west, she runs on a parallel of latitude, making no difference of latitude, and her departure and distance are the same. It may farther be observed, that the departure and difference of latitude always make the legs of a right-angled triangle, whose hypotenuse is the distance the ship has sailed; and the angles are the course, its complement, and the right angle; therefore among these four things, course, distance, difference of latitude, and departure, any two of them being given, the rest may be found by plane trigonometry.
Thus, in the annexed figure, suppose the circle FHFH to represent the horizon of the place A, from whence a ship sails; AC the rhumb she sails upon, and C the place arrived at: then HH represents the parallel of latitude she sailed from, and CC the parallel of the latitude arrived in: so that AD becomes the difference of latitude. DC the departure, AC the distance sailed, [angle]DAC is the course, and [angle]DCA the comp. of the course. And all these particulars will be alike represented, whether the ship sails in the NE, or NW, or SE, or SW quarter of the horizon.
From the same figure, in which AE or AF or AH represents the rad. of the tables, EB the sine of the course, AB the cosine of the course, we may easily deduce all the proportions or canons, as they are usually called by mariners, that can arise in Plane Sailing; because the triangles ADC and ABE and AFG are evidently similar These proportions are exhibited in the following Table, which consists of 6 cases, according to the varieties of the two parts that can be given. |
| Case. | Given. | Required. | Solutions. |
| 1 | [angle]A and AC, i. e. course and distance. | AD and DC, i. e. difference of latitude and departure. | AE : AB :: AC : AD, i. e. rad. : s. course :: dist. : dif. lat. AE : EB :: AC : DC, i. e. rad. : cos. course :: dist. : depart. |
| 2 | [angle]A and AD, i. e. course and difference of latitude. | AC and DC, i. e. distance and departure. | AB: AE :: AD : AC, i. e. cos. cour. : rad. :: dif. lat. : dist. AB : BE :: AD : DC, i. e. cos.cour.:s.cour.::dif.lat.:dep. |
| 3 | [angle]A and DC, i. e. course and departure. | AC and AD, i. e. distance and difference of latitude. | BE : AE :: DC : AC, i. e. s. cour. : rad. :: depart. : dist. BE : AB :: DC : AD, i. e. s.cour.:cos.cour.::dep.:dif.lat. |
| 4 | AC and AD, i. e. distance and difference of latitude. | [angle]A and DC, i. e. course and departure. | AC : AD :: AE : AB, i. e. dist. : dif. lat. :: rad. : cos. course. AE : EB :: AC : DC, i. e. rad. : s. course :: dist. : depart. |
| 5 | AC and DC, i. e. distance and departure. | [angle]A and AD, i. e. course and difference of latitude. | AC : DC :: AE : EB, i. e. dist. : dep. :: rad. : s. course. AE : AB :: AC : AD, i. e. rad. : cos. cour. :: dist. : dif. lat. |
| 6 | AD and DC, i. e. difference of latitude and departure. | [angle]A and AC, i. e. course and distance. | AD : DC :: AF : FG, i. e. dif. lat. : dep. :: rad. : tang. course. BE : AE :: DC : AC, i. e. s. cour. : rad. :: dep. : dist. |
For the ready working of any single course, there is a table, called a Traverse Table, usually annexed to treatises of navigation; which is so contrived, that by finding the given course in it, and a distance not exceeding 100 or 120 miles, the usual extent of the table; then the difference of latitude and the departure are had by inspection. And the same table will serve for greater distances, by doubling, or trebling, or quadrupling, &c, or taking proportional parts. See Traverse Table.
An ex. to the first case may suffice to shew the method. Thus, A ship from the latitude 47° 30′ N, has sailed SW by S 98 miles; required the departure made, and the latitude arrived in.
1. By the Traverse Table. In the column of the course, viz 3 points, against the distance 98, stands the number 54.45 miles for the departure, and 81.5 miles for the diff. of lat.; which is 1° 21′ 1/2; and this being taken from the given lat. 47° 30′, leaves 46° 8′1/2 for the lat. come to.
2. By Construction. Draw the me- ridian AD; and drawing an arc, with the chord of 60, make PQ or angle A equal to 3 points; through Q draw the distance AQE = 98 miles, and through E the departure ED perp. to AD. Then, by measuring, the diff. of lat. AD measures about 81 1/2 miles, and the departure DE about 54 1/2 miles.
3. By Computation.
| First, as radius | 10.00000 |
| to sin. course 33° 45′ | 9.74474 |
| so dist. 98 | 1.99123 |
| to depart. 54.45 | 1.73597 |
| Again, as radius | 10.00000 |
| to cos. course | 9.91985 |
| so dist. 98 | 1.99123 |
| to diff. of lat. 81.48 | 1.91108 |
4. By Gunter's Scale. The extent from radius, or 8 points, to 3 points, on the line of sine rhumbs, applied to the line of numbers, will reach from 98 to 54 1/2 the departure. And the extent from 8 points to 5 points, of the rhumbs, reaches from 98 to 81 1/2 on the line of numbers, for the difference of latitude.
And in like manner for other cases.
Traverse Sailing, or Compound Courses, is the uniting of several cases of plane sailing together into one; as when a ship sails in a zigzag manner, certain distances upon several different courses, to find the whole difference of latitude and departure made good on all of them. This is done by working all the cases separately, by means of the traverse table, and constructing the figure as in this example. |
Ex. A ship sailing from a place in latitude 24° 32′ N, has run five different courses and distances, as set down in the 1st and 2d columns of the following traverse table; required her present latitude, with the departure, and the direct course and distance, between the place sailed from, and the place come to.
| Traverse Table. | |||||
| Courses. | Dist. | N | S | E | W |
| SW b S | 45 | 25.0 | 37.4 | ||
| ESE | 50 | 19.1 | 46.2 | ||
| SW | 30 | 21.2 | 21.2 | ||
| SE b E | 60 | 33.3 | 49.9 | ||
| SW b S 1/4 W | 63 | 50.6 | 37.5 | ||
| 149.2 | 96.1 | 96.1 |
| that is | 2° | 29′, | |
| which taken from | 24 | 32, | the latitude dep. from, |
| leaves | 22 | 3 N, | the latitude come to. |
To Construct this Traverse. With the chord of 60 degrees describe the circle N 135 S &c, and quarter it by the two perpendicular diameters; then from S set upon it the several courses, to the points marked 1, 2, 3, 4, 5, through which points draw lines from the centre A, or conceive them to be drawn; lastly, upon the first line lay off the first distance 45 from A to B, also draw BC = 50 and parallel to A 2, and CD = 30 parallel to A 3, and DE = 60 parallel to A 4, and EF = 63, parallel to A 5; then it is found that the point F falls exactly upon the meridian NAF produced, thereby shewing that there is no departure; and by measuring AF, it gives 149 miles for the difference of latitude.
Oblique Sailing, is the resolution of certain cases and problems in Sailing by oblique triangles, or in which oblique triangles are concerned.
In this kind of Sailing, it may be observed, that to set an object, means to observe what rhumb or point of the nautical compass is directed to it. And the bearing of an object is the rhumb on which it is seen; also the bearing of one place from another, is reckoned by the name of the rhumb passing through those two places.
In every figure relating to any case of plane Sailing, the bearing of a line, not running from the centre of the circle or horizon, is found by drawing a line parallel to it, from the centre, and towards the same quarter.
Ex. A ship sailing at sea, observed a point of land to bear E by S; and then after sailing NE 12 miles, its bearing was found to be SE by E. Required the place of that point, and its distance from the ship at the last observation.
Construction. Draw the meridian line NAS, and, assuming A for the first place of the ship, draw AC the E by S rhumb, and AB the NE one, upon which lay off 12 miles from A to B; then draw the meridian BT parallel to NS, from which set off the SE by E point BC, and the point C will be the place of the land required; then the distance BC measures 26 miles.
By Computation. Here are given the side AB, and the two angles A and B, viz, the [angle]A = 5 points or 56° 15′, and the [angle]B = 9 points or 101° 15′; consequently the [angle]C = 2 points or 22° 30′. Then, by plane trigonometry,
| As sin. [angle]C 22° 30′ | 9.58284 |
| To sin. [angle]B 56 15 | 9.91985 |
| So is AB 12 miles | 1.07918 |
| To BC 26.073 miles | 1.41619 |
Sailing to Windward, is working the ship towards that quarter of the compass from whence the wind blows.
For rightly understanding this part of navigation, it will be necessary to explain the terms that occur in it, though most of them may be seen in their proper places in this work.
When the wind is directly, or partly, against a ship's direct course for the place she is bound to, she reaches her port by a kind of zigzag or z like course; which is made by sailing with the wind first on one side of the ship, and then on the other side.
In a ship, when you look towards the head, Starboard denotes the right hand side. | Larboard the left hand side. Forwards, or afore, is towards the head. Aft, or abaft, is towards the stern.
The beam signifies athwart or across the middle of the ship.
When a ship sails the same way that the wind blows, she is said to sail or run before the wind; and the wind is said to be right aft, or right astern; and her course is then 16 points, or the farthest possible, from the wind, that is from the point the wind blows from.—When the ship sails with the wind blowing directly across her, she is said to have the wind on the beam; and her course is 8 points from the wind.—When the wind blows obliquely across the ship, the wind is said to be abaft the beam when it pursues her, or blows more on the hinder part, but before the beam when it meets or opposes her course, her course being more than 8 points from the wind in the former case, but less than 8 points in the latter case.—When a ship endeavours to sail towards that point of the compass from which the wind blows, she is said to sail on a wind, or to ply to windward.—And a vessel sailing as near as she can to the point from which the wind blows, she is said to be close hauled. Most ships will lie within about 6 points of the wind; but sloops, and some other vessels, will lie much nearer. To know how near the wind a ship will lie; observe the course she goes on each tack, when she is close hauled; then half the number of points between the two courses, will shew how near the wind the ship will lie.
The windward, or weather side, is that side of the ship on which the wind blows; and the other side is called the leeward, or lee side.—Tacks and sheets are large ropes fastened to the lower corners of the fore and main sails; by which either of these corners is hauled fore or aft.—When a ship sails on a wind, the windward tacks are always hauled forwards, and the leeward sheets aft.—The starboard tacks are aboard, when the starboard side is to windward, and the larboard side to leeward. And the larboard tacks are aboard, when the larboard side is to windward, and the starboard to leeward.
The most common cases in turning to windward may be constructed by the following precepts. Having drawn a circle with the chord of 60°, for the compass, or the horizon of the place, quarter it by drawing the meridian and parallel of latitude perpendicular to each other, and both through the centre; mark the place of the wind in the circumference; draw the rhumb passing through the place bound to, and lay on it, from the centre, the distance of that place. On each side of the wind lay off, in the circumference, the points or degrees shewing how near the wind the ship can lie; and draw these rhumbs.—Now the first course will be on one of these rhumbs, according to the tack the ship leads with. Draw a line through the place bound to, parallel to the other rhumb, and meeting the first; and this will shew the course and distance on the other tack.
Ex. The wind being at north, and a ship bound to a port 25 miles directly to windward; beginning with the starboard tacks, what must be the course and distance on each of two tacks to reach the port?
Construction. Having drawn the circle &c, as above described, where A is the port, AP and AQ the two rhumbs, each within 6 points of AN; in NA produced take AB = 25 miles, then B is the place of the ship; draw BC parallel to AP, and meeting QA produced in C; so shall BC and CA be the distances on the two tacks; the former being WNW, and the latter ENE.
Computation.
| Here | [angle]B = NAP = 6 points, |
| and | [angle]A = NAQ = 6 points, |
| theref. | [angle]C = 4 points. |
Sailing in Currents, is the method of determining the true course and distance of a ship when her own motion is affected and combined with that of a current.
A current or tide is a progressive motion of the water, causing all floating bodies to move that way towards which the stream is directed.—The setting of a tide, or current, is that point of the compass towards which the waters run; and the drift of the current is the rate at which it runs per hour.
The drift and setting of the most remarkable tides and currents, are pretty well known; but for unknown currents, the usual way to find the drift and setting, is thus: Let three or four men take a boat a little way from the ship; and by a rope, fastened to the boat's stem, let down a heavy iron pot, or loaded kettle, into the sea, to the depth of 80 or 100 fathoms, when it can be done: by which means the boat will ride almost as steady as at anchor. Then heave the log, and the number of knots run out in half a minute will give the current's rate, or the miles which it runs per hour; and the bearing of the log shews the setting of the current.
A body moving in a current, may be considered in three cases: viz,
1. Moving with the current, or the same way it sets.
2. Moving against it, or the contrary way it sets.
3. Moving obliquely to the current's motion.
In the 1st case, or when a ship sails with a current, its velocity will be equal to the sum of its proper motion, and the current's drift. But in the 2d case, or when a ship sails against a current, its velocity will be equal to the difference of her own motion and the drift of the current: so that if the current drives stronger than the wind, the ship will drive astern, or lose way. In the 3d case, when the current sets oblique to the course of the ship, her real course, or that made good, will be somewhere between that in which the ship endeavours to go, and the track in which the current tries to drive her; and indeed it will always be along the diagonal of a parallelogram, of which one side represents the ship's course set, and the other adjoining side is the current's drift. |
Thus, if ABDC be a parallelo- gram. Now if the wind alone would drive the ship from A to B in the same time as the current alone would drive her from A to C: then, as the wind neither helps nor hinders the ship from coming towards the line CD, the current will bring her there in the same time as if the wind did not act. And as the current neither helps nor hinders the ship from coming towards the line BD, the wind will bring her there in the same time as if the current did not act. Therefore the ship must, at the end of that time, be sound in both those lines, that is, in their meeting D. Consequently the ship must have passed from A to D in the diagonal AD.
Hence, drawing the rhumbs for the proper course of the ship and of the current, and setting the distances off upon them, according to the quantity run by each in the given time; then forming a parallelogram of these two, and drawing its diagonal, this will be the real course and distance made good by the ship.
Ex. 1. A ship sails E. 5 miles an hour, in a tide setting the same way 4 miles an hour: required the ship's course, and the distance made good.
| The ship's motion is | 5m. E. |
| The current's motion is | 4m. E. |
| Theref. the ship's run is | 9m. E. |
Ex. 2. A ship sails SSW. with a brisk gale, at the rate of 9 miles an hour, in a current setting NNE. 2 miles an hour: required the ship's course, and the distance made good.
| The ship's motion is | SSW. 9m. |
| The current's motion is | NNE. 2m. |
| Theref. ship's true run is | SSW. 7m. |
Ex. 3. A ship running south at the rate of 5 miles an hour, in 10 hours crosses a current, which all that time was setting east at the rate of 3 miles an hour: required the ship's true course and distance sailed.
Here the ship is first supposed to be at A, her imaginary course is along the line AB, which is drawn south, and equal to 50 miles, the run in 10 hours; then draw BC east, and equal to 30 miles, the run of the ourrent in 10 hours. Then the ship is found at C, and her true path is in the line AC = 58.31 her distance, and her course is the angle at A = 30° 58′ from the south towards the east.
Globular Sailing is the estimating the ship's motion and run upon principles derived from the globular figure of the earth, viz, her course, distance, and difference of latitude and longitude.
The principles of this method are explained under the artioles Rhumb-line, Mercator's Chart, and MERIDIONAL Parts; which see.
Globular Sailing, in the extensive sense here applied to the term, comprehends Parallel Sailing, Middle-latitude Sailing, and Mercator's Sailing; to which may be added Circular Sailing, or Great-circle Sailing. Of each of which it may be proper here to give a brief account.
Parallel Sailing is the art of finding what distance a ship should run due east or west, in sailing from the meridian of one place to that of another place, in any parallel of latitude.
The computations in parallel sailing depend on the following rule: As radius, To cosine of the lat. of any parallel; So are the miles of long. between any two meridians, To the dist. of these meridians in that parallel. Also, for any two latitudes, As the cosine of one latitude, Is to the cosine of another latitude; So is a given meridional dist. in the 1st parallel, To the like meridional dist. in the 2d parallel.
Hence, counting 60 nautical miles to each degree of longitude, or on the equator; then, by the first rule the number of miles in each degree on the other parallels, will come out as in the following table.
| Table of Meridional Distances. | |||||
| Lat. | Miles. | Lat. | Miles. | Lat. | Miles. |
| 1 | 59.99 | 31 | 51.43 | 61 | 29.09 |
| 2 | 59.96 | 32 | 50.88 | 62 | 28.17 |
| 3 | 59.92 | 33 | 50.32 | 63 | 27.24 |
| 4 | 59.85 | 34 | 49.74 | 64 | 26.30 |
| 5 | 59.77 | 35 | 49.15 | 65 | 25.36 |
| 6 | 59.67 | 36 | 48.54 | 66 | 24.41 |
| 7 | 59.56 | 37 | 47.92 | 67 | 23.44 |
| 8 | 59.42 | 38 | 47.28 | 68 | 22.48 |
| 9 | 59.26 | 39 | 46.63 | 69 | 21.50 |
| 10 | 59.09 | 40 | 45.96 | 70 | 20.52 |
| 11 | 58.89 | 41 | 45.28 | 71 | 19.53 |
| 12 | 58.69 | 42 | 44.59 | 72 | 18.54 |
| 13 | 58.46 | 43 | 43.<*>8 | 73 | 17.54 |
| 14 | 58.22 | 44 | 43.16 | 74 | 16.54 |
| 15 | 57.95 | 45 | 42.43 | 75 | 15.53 |
| 16 | 57.67 | 46 | 41.68 | 76 | 14.51 |
| 17 | 57.38 | 47 | 40.92 | 77 | 13.50 |
| 18 | 57.06 | 48 | 40.15 | 78 | 12.48 |
| 19 | 56.73 | 49 | 39.36 | 79 | 11.45 |
| 20 | 56.38 | 50 | 38.57 | 80 | 10.42 |
| 21 | 56.01 | 51 | 37.76 | 81 | 9.38 |
| 22 | 55.63 | 52 | 36.94 | 82 | 8.35 |
| 23 | 55.23 | 53 | 36.11 | 83 | 7.32 |
| 24 | 54.81 | 54 | 35.27 | 84 | 6.28 |
| 25 | 54.38 | 55 | 34.41 | 85 | 5.23 |
| 26 | 53.93 | 56 | 33.55 | 86 | 4.18 |
| 27 | 53.46 | 57 | 32.68 | 87 | 3.14 |
| 28 | 52.97 | 58 | 31.79 | 88 | 2.09 |
| 29 | 52.47 | 59 | 30.90 | 89 | 1.05 |
| 30 | 51.96 | 60 | 30.00 | 90 | 0.00 |
See another table of this kind, allowing 69 1/<*> English miles to one degree, under the article DEGREE.
To sind the meridional distance to any number of minutes between any of the whole degrees in the table, as for instance in the parallel of 48° 26′; take out the tabular distances for the two whole degrees between which the parallel or the odd minutes lie, as for 48° and 49°; subtract the one from the other, and take the proportional part of the remainder for the odd minutes, by multiplying it by those minutes, and dividing by 60; and lastly, subtract this proportional part from the greater tabular number. Thus,
| Lat. 48° | 40.15 | |
| Lat. 49. | 39.36 | |
| As 60′ : 26′ :: | 0.79 | rem. : 0.34 |
| 26 | ||
| 474 | ||
| 158 | ||
| 60) 20.54 | ||
| 0.34 | pro. part | |
| Taken from | 40.15 | for lat. 48° |
| Leaves merid. dist. | 39.81 | for lat. 48° 26′ |
And, in like manner, by the counter operation, to find what latitude answers to a given meridional distance. As, for ex. in what latitude 46.08 miles answer to a degree of longitude.
| From | 46.63 | for 39° | from 46.63 | for 39° |
| Take | 45.96 | for 40° | take 46.08 | given number. |
| Then as | 0.67 | : 60′ :: | 0.55 | : 49′ |
| 67) 3300 | ||||
| 49′ | pro. part. |
Ex. 3. Given the latitude and meridional distance; to find the corresponding difference of longitude. As, if a ship, in latitude 53° 36′, and longitude 10° 18 east, sail due west 236 miles; required her present longitude.
Here, by the first rule,
| As cos. lat. | 53° 36′ | comp. | 0.22664 |
| To radius | 90 00 | 10.00000 | |
| So merid. dist. | 236 | m. | 2.37291 |
| To diff. long. 397.7 | 2.59955 |
| Its 60th gives | 6° 38′ | W. diff. long. |
| Taken from | 10 18 | E. long. from |
| Leaves | 3 40 | E. long. come to. |
By the table; the length of a degree on the parallel of 53° 36′ is 35.6. Then as 35.6 : 60 :: 236 : 397.7, the diff. of long. the same as before.
Middle-latitude Sailing, is a method of resolving the cases of globular Sailing by means of the middle latitude between the latitude departed from, and that come to. This method is not quite accurate, being only an approximation to the truth, and it makes use of the principles of plane Sailing and parallel Sailing conjointly.
The method is founded on the supposition that the departure is reckoned as a meridional distance in that latitude which is a middle parallel between the latitude sailed from, and the latitude come to. And the method is not quite accurate, because the arithmetical mean, or half sum of the cosines of two distant latitudes, is not exactly the cosine of the middle latitude, or half the sum of those latitudes; nor is the departure between two places, on an oblique rhumb, equal to the meridional distance in the middle latitude; as is presumed in this method. Yet when the parallels are near the equator, or near to each other, in any latitude, the error is not considerable.
This method seems to have been invented on account of the easy manner in which the several cases may be resolved by the traverse table, and when a table of meridional parts is wanting. The computations depend on the following rules:
1. Take half the sum, or the arithmetical mean, of the two given latitudes, for the middle latitude. Then,
2. As cosine of middle latitude, Is to the radius; So is the departure, To the diff. of longitude. And,
3. As cosine of middle latitude, Is to the tangent of the course; So is the difference of latitude, To the difference of longitude.
Mercator's Sailing, is the art of resolving the several cases of globular Sailing, by plane trigonometry, with the assistance of a table of meridional parts, or of logarithmic tangents. And the computations are performed by the following rules:
1. As meridional diff. lat. To diff. of longitude; So is the radius, To tangent of the course.
2. As the proper diff. lat. To the departure; So is merid. diff. lat. To diff. of longitude.
3. As diff. log. tang. half colatitudes, To tang. of 51° 38′ 09″; So is a given diff. longitude, To tangent of the course.
The manner of working with the meridional parts and logarithmic tangents, will appear from the two following cases.
1. Given the latitudes of two places; to sind their meridional difference of latitude.
By the Merid. Parts. When the places are both on | the same side of the equator, take the difference of the meridional parts answering to each latitude; but when the places are on opposite sides of the equator, take the sum of the same parts, for the meridional difference of latitude sought.
By the Log. Tangents. In the former case, take the difference of the long. tangents of the half colatitudes; but in the latter case, take the sum of the same; then the said difference or sum divided by 12 63, will give the meridional difference of latitude sought.
2. Given the latitude of one place, and the meridional difference of latitude between that and another place; to find the latitude of this latter place.
By the Merid. Parts. When the places have like names, take the sum of the merid. parts of the given lat. and the given diff.; but take the difference between the same when they have unlike names; then the result, being found in the table of meridional parts, will give the latitude sought.
By the Log. Tangents. Multiply the given meridional diff. of lat. by 12.63; then in the former case subtract the product from the log. tangent of the given half colatitude, but in the latter case add them; then seek the degrees and minutes answering to the result among the log. tangents, and these degrees, &c. doubled will be the colatitude sought.
Circular Sailing, or Great-circle Sailing, is the art of finding what places a ship must go through, and what courses to steer, that her track may be in the arc of a great circle on the globe, or nearly so, passing through the place sailed from and the place bound to.
This method of Sailing has been proposed, because the shortest distance between two places on the sphere, is an arc of a great circle intercepted between them, and not the spiral rhumb passing through them, unless when that rhumb coincides with a great circle, which can only be on a meridian, or on the equator.
As the solutions of the cases in Mercator's Sailing are performed by plane triangles, in this method of Sailing they are resolved by means of spherical triangles. A great variety of cases might be here proposed, but those that are the most useful, and most commonly occur, pertain to the following problem.
Problem I. Given the latitudes and longitudes of two places on the earth; to find their nearest distance on the surface, together with the angles of position from either place to the other.
This problem comprehends 6 cases.
Case 1. When the two places lie under the same meridian; then their difference of latitude will give their distance, and the position of one from the other will be directly north and south.
Case 2. When the two places lie under the equator; their distance is equal to their difference of longitude, and the angle of position is a right angle, or the course from one to the other is due east or west.
Case 3. When both places are in the same parallel of latitude. Ex. gr. The places both in 37° north, but the longitude of the one 25° west, and of the other 76° 23′ west.
Let P denote the north pole, and A and B the two places on the same parallel BDA, also BIA their distance asunder, or the arc of a great circle passing through them. Then is the angle A or B that of position, and the angle BPA = 51° 23′ the difference of longitude, and the side PA or PB = 53° the colatitude.
Draw PI perp. to AB, or bisecting the angle at P. Then in the triangle API, right-angled at I, are given the hypotenuse AP = 53°, and the angle API = 25° 41′ 30″; to find the angle of position A or B = 73° 51′; and the half distance AI = 20° 15′1/2; this doubled gives 40° 31′ for the whole distance AB, or 2431 nautical miles, which is 31 miles less than the distance along ADB, or by parallel Sailing.
Case 4. When one place has latitude, and the other has none, or is under the equator. For example, suppose the Island of St. Thomas, lat. 0°, and long. 1° 0′ east, and Port St. Julian, in lat. 48° 51′ south, and long. 65° 10′ west.
| Port St. Julian, lat. | 48° 51′ S. | long. | 65° 10′W. |
| Isle St. Thomas | 0 00 | " | 1 00 E |
| Julian's colat. | 41 09 Diff. | long. | 66 10 |
Hence, if S denote the south pole, A the Isle St. Thomas at the equator, and B St. Julian; then in the triangle are given SA a quadrant or 90°, BS = 41° 9′ the colat. of St. Julian, and the [angle]S = 66° 10′ the dif. of longitude; to find AB = 74° 35′ = 4475 miles, which is less by 57 miles than the distance found by Mercator's Sailing; also the angle of position at A = 51°22′, and the angle of position B = 108° 24′.
Case 5. When the two given places are both on the same side of the equator; for example the Lizard, and the island of Bermudas.
| The Lizard, lat. | 49° 57′N. | long. 5° | 21′W. |
| Bermudas, | 32 35 N. | 63 | 32 W. |
| 58 | 11 |
Here, if P be the north pole, L the Lizard, and B Bermudas; there are given, PL = 40° 03′ colat. of the Lizard, PB = 57 25 colat. of Bermudas, [angle]P = 58 11 diff. of longitude; to sind BL = 45° 44 = 2744 miles the distance, and [angle] of position B = 49° 27′, also [angle] of position L = 90° 31′.
Case 6. When the given places lie on different sides of the equator; as suppose St. Helena and Bermudas. Here |
PB = 57° 25′ polar dist. Bermudas,
PH = 105 55 polar dist. St. Helena,
[angle]P = 57 43 diff. long.
To find BH = 73° 26′ = 4406 miles, the distance, also the angle of position H = 48° 0′, and the angle of position B = 121° 59′.
From the solutions of the foregoing cases it appears, that to sail on the arc of a great circle, the ship must continually alter her course; but as this is a difficulty too great to be admitted into the practice of navigation, it has been thought sufficiently exact to effect this business by a kind of approximation, that is, by a method which nearly approaches to the sailing on a great circle: namely, upon this principle, that in small arcs, the difference between the arc and its chord or tangent is so small, that they may be taken for one another in any nautical operations: and accordingly it is supposed that the great circles on the earth are made up of short right lines, each of which is a segment of a rhumb line. On this supposition the solution of the following problem is deduced.
Problem II. Having given the latitudes and longitudes of the places sailed from and bound to; to sind the successive latitudes on the arc of a great circle in those places where the alteration in longitude shall be a given quantity; together with the courses and distances between those places.
1. Find the angle of position at each place, and their distance, by one of the preceding cases.
2. Find the greatest latitude the great circle runs through, i. e. find the perpendicular from the pole to that circle; and also find the several angles at the pole, made by the given alterations of longitude between this perpendicular and the successive meridians come to.
3. With this perpendicular and the polar angles severally, find as many corresponding latitudes, by saying, as radius : tang. greatest lat. :: cos. 1st polar angle : tang. 1st lat. :: cos. 2d polar angle : tang. of 2d lat. &c.
4. Having now the several latitudes passed through, and the difference of longitude between each, then by Mercator's Sailing find the courses and distances between those latitudes. And these are the several courses and distances the ship must run, to keep nearly on the arc of a great circle.
The smaller the alterations in longitude are taken, the nearer will this method approach to the truth; but it is sufficient to compute to every 5 degrees of difference of longitude; as the length of an arc of 5 degrees differs from its chord, or tangent, only by 0.002.
The track of a ship, when thus directed nearly in the arc of a great circle, may be delineated on the Mercator's chart, by marking on it, by help of the latitudes and longitudes, the successive places where the ship is to alter her course; then those places or points, being joined by right lines, will shew the path along which the ship is to sail, under the proposed circumstances.
On the subject of these articles, see Robertson's Elements of Navigation, vol. 2.
Spheroidical Sailing, is computing the cases of navigation on the supposition or principles of the spheroidical figure of the earth. See Robertson's Navigation, vol. 2, b. 8. sect. 8.
Sailing, in a more confined sense, is the art of conducting a ship from place to place, by the working or handling of her sails and rudder.
To bring Sailing 10 certain rules, M. Renau computes the force of the water, against the ship's rudder, stem, and side; and the force of the wind against her sails. In order to this, he first considers all fluid bodies, as the air, water, &c, as composed of little particles, which when they act upon, or move against any surface, do all move parallel to one another, or strike against the surface after the same manner. Secondly, that the motion of any body, with regard to the surface it strikes, must be either perpendicular, parallel, or oblique.
From these principles he computes, that the force of the air or water, striking perpendicularly upon a sail or rudder, is to the force of the same striking obliquely, in the duplicate ratio of radius to the sine of the angle of incidence: and consequently that all oblique forces of the wind against the sails, or of the water against the rudder, will be to one another in the duplicate ratio of the sines of the angles of incidence.
Such are the conclusions from theory; but it is very different in real practice, or experiments, as appears from the tables of experiments inserted at the article Resistance.
Farther, when the different degrees of velocity are considered, it is also found that the forces are as the squares of the velocities of the moving air or water nearly; that is, a wind that blows twice as swift, as another, will have 4 times the force upon the sail; and when 3 times as swift, 9 times the force, &c. And it being also indifferent, whether we consider the motion of a solid in a fluid at rest, or of the fluid against the solid at rest; the reciprocal impressions being always the same; if a solid be moved with different velocities in the same fluid matter, as water, the different resistances which it will receive from that water, will be in the same proportion as the squares of the velocities of the moving body.
He then applies these principles to the motions of a ship, both forwards and sideways, through the water, when the wind, with certain velocities, strikes the sails in various positions. After this, the author proceeds to demonstrate, that the best position or situation of a ship, so as she may make the least lee-way, or side motion, but go to windward as much as possible, is this: that, let the sail have what situation it will, the ship be always in a line bisecting the complement of the wind's angle of incidence upon the sail. That is, supposing the sail in the position BC, and the wind blow- ing from A to B, and consequently the angle of the wind's incidence on the sail is ABC, the complement of which is CBE: then must the ship be put in the position BK, or move in the line BL, bisecting the [angle] CBE. |
He shews farther, that the angle which the sail ought to make with the wind, i. e. the angle ABC, ought to be but 24 degrees; that being the most advantageous situation to go to windward the most possible.
To this might be added many curious particulars from Borelli de Vi Percussionis, concerning the different directions given to a vessel by the rudder, when sailing with a wind, or floating without sails in a current: in the former case, the head of the ship always coming to the rudder, and in the latter always flying off from it; as also from Euler, Bouguer, and Juan, who have all written learnedly on this subject.
