with large turrets, and other accessions of building, either for attack or defence. The soldiers also annoyed their enemies with darts and slings, and, on their nearer approach, with swords and javelins; and in order that their missile weapons might be directed with greater force and certainty, the ships were equipped with several platforms, or elevations above the level of the deck. The sides of the ship were fortified with a thick fence of hides, which served to repel the darts of their adversaries, and to cover their own soldiers, who thereby annoyed the enemy with greater security.

As the invention of gun-powder has rendered useless many of the machines employed in the naval wars of the ancients, the great distance of time has also consigned many of them to oblivion: some few are, nevertheless, recorded in ancient authors, of which we shall endeavour to present a short description. And first,

The Διάπυρ was a large and massy piece of lead or iron, cast in the form of a dolphin. This machine being suspended by blocks at their mast-heads or yard-arms, ready for a proper occasion, was let down violently from thence into the adverse ships; and either penetrated through their bottom, and opened a passage for the entering waters, or by its weight immediately sunk the vessel.

The Δεξαίρας was an engine of iron crooked like a sickle, and fixed on the top of a long pole. It was employed to cut afunder the slings of the sail-yards, and, thereby letting the sails fall down, to disable the vessel from escaping, and incommodate her greatly during the action. Similar to this was another instrument, armed at the head with a broad two-edged blade of iron, wherewith they usually cut away the ropes that fastened the rudder to the vessel.

Δοξία ναιμάχης, a sort of spears or maces of an extraordinary length, sometimes exceeding 20 cubits, as appears by the 15th Iliad of Homer, by whom they are also called μαχέ.

Κιέται were certain machines used to throw large stones into the enemy's ships.

Vegetius mentions another engine, which was suspended to the main-mast, and resembled a battering-ram; for it consisted of a long beam and an head of iron, and was with great violence pushed against the sides of the enemy's galleys.

They had also a grappling-iron, which was usually thrown into the adverse ship by means of an engine: this instrument facilitated the entrance of the soldiers appointed to board, which was done by means of wooden bridges, that were generally kept ready for this purpose in the fore-part of the vessel. See the article Corvus.

The arms used by the ancients rendered the disposition of their fleets very different, according to the time, place, and circumstances. They generally considered it an advantage to be to windward, and to have the sun shining directly on the front of their enemy. The order of battle chiefly depended on their power of managing the ships, or of drawing them readily into form; and on the schemes which their officers had concerted. The fleet being composed of rowing-vessels, they lowered their sails previous to the action; they presented their prows to the enemy, and advanced against each other by the force of their oars. Before they joined

battle, the admirals went from ship to ship, and exhorted their soldiers to behave gallantly. All things being in readiness, the signal was displayed by hanging out of the admiral's galley a gilded shield, or a red garment or banner. During the elevation of this, the action continued; and by its depression, or inclination towards the right or left, the rest of the ships were directed how to attack or retreat from their enemies. To this was added the sound of trumpets; which began in the admiral's galley, and continued round the whole navy. The fight was also begun by the admiral's galley, by grappling, boarding, and endeavouring to overfet, sink, or destroy the adversary, as we have above described. Sometimes, for want of grappling-irons, they fixed their oars in such a manner as to hinder the enemy from retreating. If they could not manage their oars as dextrously as their antagonist, or fall alongside so as to board him, they penetrated his vessel with the brazen prow. The vessels approached each other as well as their circumstances would permit, and the soldiers were obliged to fight hand to hand, till the battle was decided: nor indeed could they fight otherwise with any certainty, since the shortest distance rendered their slings and arrows, and almost all their offensive weapons, ineffectual, if not useless. The squadrons were sometimes ranged in two or three right lines, parallel to each other; being seldom drawn up in one line, unless when formed into an half-moon. This order indeed appears to be the most convenient for rowing vessels that engage by advancing with their prows towards the enemy. At the battle of Ecnomus, between the Romans and the Carthaginians, the fleet of the former was ranged into a triangle, or a sort of wedge in front, and towards the middle of its depth, of two right parallel lines. That of the latter was formed into a rectangle, or two sides of a square, of which one branch extended behind, and as the opening of the other prosecuted the attack, was ready to fall upon the flank of such of the Roman galleys as should attempt to break their line. Ancient history has preserved many of these orders, of which some have been followed in later times. Thus, in a battle A. D. 1340, the English fleet was formed in two lines, the first of which contained the larger ships, the second consisted of all the smaller vessels, used as a reserve to support the former whenever necessary. In 1545, the French fleet under the command of the Marechal d'Annebault, in an engagement with the English in the Channel, was arranged in the form of a crescent. The whole of it was divided into three bodies, the centre being composed of 36 ships, and each of the wings of 30. He had also many galleys; but these fell not into the line, being designed to attack the enemy occasionally. This last disposition was continued down to the reigns of James I. and Lewis XIII.

Meanwhile the invention of gunpowder, in 1330, gradually introduced the use of fire-arms into naval war, without finally superseding the ancient method of engagement. The Spaniards were armed with cannon in a sea-fight against the English and the people of Poitou abreast of Rochelle in 1372; and this battle is the first wherein mention is made of artillery in our navies. Many years elapsed before the marine armaments were sufficiently provided with fire-arms. So great a revolution in the manner of fighting, and which

necessarily introduced a total change in the construction of ships, could not be suddenly effected. In short, the squadrons of men of war are no longer formed of rowing vessels, or composed of galleys and ships of the line; but entirely of the latter, which engage under sail, and discharge the whole force of their artillery from their sides. Accordingly, they are now disposed in no other form than that of a right line parallel to the enemy; every ship keeping close-hauled upon a wind on the same tack. Indeed, the difference between the force and manner of fighting of ships and galleys rendered their service in the same line incompatible. When we consider therefore the change introduced, both in the construction and working of ships, occasioned by the use of cannon, it necessarily follows, that squadrons of men of war must appear in the order that is now generally adopted.

The machines which owe their rise to the invention of gun-powder have now totally supplanted the others; so that there is scarce any but the sword remaining, of all the weapons used by the ancients. Our naval battles are therefore almost always decided by fire-arms, of which there are several kinds, known by the general name of artillery.

In a ship of war, fire-arms are distinguished into cannon mounted on carriages, swivel-cannon, grenadoes, and musquetry. See CANNON, &c.

Besides these machines, there are several other used in merchant-ships and privateers, as coehorns, carabines, fire-arrows, organs, flink-pots, &c. See COE-HORN, &c.

CHAP. I. Of Lines or Orders.

By orders are meant the different methods of ranging or drawing up a fleet in the several lines and forms for which it may be designed: in which two things are to be considered, 1. The position of each ship with regard to the wind: 2. The position of each ship with respect to the fleet. We cannot make any alteration in either of these circumstances, without changing the whole position of the line, which will otherwise remain complete.

The different expeditions an admiral may be ordered upon, as well as the various circumstances that occur in conducting a fleet, first gave rise to the several lines or orders into which it is formed.

When a fleet engages, it ought to be drawn up in a different form from that in which it sails. A fleet that sails in sight of an enemy must alter its position, from that which it would maintain were there none in view, or none to be expected. When a fleet sails before the wind, it has likewise its particular form of sailing; as it has also when it chases the enemy, makes a retreat, guards a freight or passage, or is obliged to force through one; or whether at anchor in a road or harbour, or going into either to insult or attack an enemy. In this variety of circumstances, proper regard must be had to the most advantageous position or form into which the fleet can be ranged before it enters upon action.

CHAP. II. Of dividing a Fleet, and of the Form of Sailing.

1. How to divide a fleet. When a fleet consists of 60 sail of the line, the admiral divides it first into three

squadrons, each of which has its divisions; and three general officers, viz. admiral, vice-admiral, and rear-admiral. Each squadron has its proper colours, and each division its proper mast: for example, the white flag is proper to the first squadron of France, the white and blue for the second, and the blue for the third. In Britain, the first admiral of the fleet carries the union-flag at the main-top-mast head; next, the admiral of the white; and then the admiral of the blue. The particular ships carry pendants of the same colour with their squadrons, and at the masts of their respective divisions; so that the last ship in the division of the blue squadron wears a blue pendant at the mizen-top-mast head.

The general officers, or commanders of divisions, place themselves in the centre of their divisions. We must except the three commanding admirals, who in a sailing position lead their respective squadrons.

2. The sailing form of a Fleet. The following is judged the best, and is that which is put in practice upon most occasions, whether upon expeditions, looking out for an enemy, &c. It consists in dividing the fleet into three columns, or parallel lines, either upon a wind or large, as the admiral may think most expedient. Thus will the course and distance of the columns, as well as each ship's station, be determined and regulated; observing at the same time that they keep abreast of each other as near as possible.

By this manner of sailing, the fleet is closed as much as possible, can better observe the proper signals, and is ready to be ranged or formed into any position or line that the admiral shall judge proper. In sailing, care must be taken to preserve the just distance between the columns; in order to which, it will be best for the ships in general to regulate themselves by some of the centre ships of the column to windward, rather than the sternmost, as they are often too far in their rear to follow their motions.

The most natural course in this order of sailing is to go nearest upon a wind on either tack, or to go away large three or four points: however, the fleet may steer away more or less from the wind, or even right afore it, as may be judged most expedient.

In all forms of sailing, the transports, tenders, &c. are ordered to keep to windward, for the following reasons. 1. They are by such a situation more out of danger from the enemy, as they can the readier bear down into the body of the fleet to avoid them. 2. They can more expeditiously execute the admiral's orders, when necessary. 3. They do not delay the fleet by waiting for them, being fitter to make sail before than upon a wind.

This kind of ships ought not to keep farther than half a league, for the same reasons that they are ordered to keep to windward; but we must observe, that when the fleet is not drawn up in a line or three columns, the fire-ships, &c. ought not to be farther distant from the men of war than they are from one another.

The same reasons which prevail in placing the fire-ships, &c. to windward of a fleet in a sailing posture, will equally hold good to place them to leeward when obliged to make a retreat. 1. They are there in less danger from the pursuit of an enemy, because they are

Of Chacing. surrounded by the fleet to guard them, in the form of a half-moon, from any attack that may be made upon them. 2. The fleet going large in this form, these small ships may shorten sail to wait for orders. 3. If the fleet should be obliged to resume the line of battle again to engage the enemy, they will still be in the best position to sail in.

In the order of retreat, the store-ships, &c. keep themselves at a greater distance than in any other form, 1. That they may not retard the progress of the fleet. 2. When the fleet forms again into a line of battle, they may keep better the proper distance required.

The fire-ships, &c. of the fleet to leeward ought to keep themselves a little a-head of those ships whose orders they are to follow; to the end that they may be the readier to join them upon occasion.

CHAP. III. Of Chacing.

Plate CXCVI. fig. 1. 1. To chace with the greatest advantage. If the ship that chaces is a great way to leeward of the chace, she should continue on the same board, till she can tack upon her: that is to say, arriving at the point E, she will find the chace at the point F; so that the angle FED will make four points, or 45°.

The above method is thought to be the best by the most experienced seamen; because by working in this manner you keep nigher the chace, and by making two tacks you will fetch her wake.

2. The ship D may continue on the same tack, till she entirely cuts off the chace C; but then she runs the risk of losing sight of her, by continuing too long upon the same tack: a fog, a shift of wind, a headland, night coming on, and many other incidents that frequently happen at sea, may give the chace an opportunity of escaping, by altering her course, &c. therefore we should never put this method in practice, but when very near her, or when we chace a friend in order to join him.

Fig. 2. 3. If the ship A chacing the ship B to windward, is at a very great distance, she must continue on the same tack till she gets upon her beam; then the ship A will tack, and stand on again the same with the chace, till she brings the ship B abreast of her. She must continue on in the same manner, tacking every time she gets abreast of the chace, till she is no longer apprehensive of losing sight of her. Note. This is to be understood when the ship A is at a very great distance from the chace, because that then she would run too great a length from her, were she to continue on the same tack till she could fetch the ship B: but then again, when the ship A is but at a small distance from the ship B, she would lose too much time if she were to tack always when she got a-breast of the chace.

Fig. 3. 4. We have already observed, that if the ship B is to leeward of the ship A which she chaces, and under no apprehension of losing sight of her; or because she might not be at any great distance from her, or chaces her large, in proper settled weather, when the days are long; or that she might be a friend, whom you would willingly join: then the ship B ought to continue on the same tack with the ship A, till she can cut her off upon the other tack.

Now we must know how to determine if the ship B in tacking will fetch the ship A. 1. It is evident, that if the ship B, which is supposed to be a better sailer

than the chace A, continues still on the same board Of Chacing. till she gets as far to windward as the ship A, she may easily intercept her by tacking upon her: this appears, at first sight, as if we should reject this method, because it makes the ship B continue on one tack longer than is necessary: but still good judges think it should not be entirely rejected, because the ship B may thoroughly make up for her loss of time in continuing on so long a board; for she may then bear down upon the ship A as much as she thinks proper, at the same time obliging her not to alter her course, as was remarked above. 2. The ship B will exactly know when she ought to tack upon the chace A, if she calculates the time from her being abreast of her till she thinks proper to tack upon her again: for continuing on the same length of time, she may be certain she cannot fail of intercepting her.

5. When the admiral would have the whole fleet to chace, or a particular squadron only, he will make known the same accordingly by hoisting the usual signal for making sail, whether his intention be to chace or join some ships that appear in sight, or stretch out a-head, and make the land. In the first case, the ships should prepare themselves immediately, that no time may be lost in wearing or tacking, if occasion for either. In the second, the headmost ships should bring to, to found, if the coast is not thoroughly known to the fleet.

6. The squadron that chaces, or the cruisers detached from the fleet, should be very careful not to engage too far in the chace, for fear of being overpowered; however, not to omit, at the same time, thoroughly satisfying themselves with regard to the object of their chace, if possible. They must pay great attention to the admiral's signals at all times, to prevent separation; in order to which, they should collect themselves before night, especially if there be any appearance of thick or foggy weather coming on; and endeavour to join the fleet again.

7. If the admiral would have the whole fleet to chace, without observing any particular form, he will, to avoid confusion, prepare them accordingly for it by signal to look out and watch his motions; to which he will join the general signal to chace at large: then the ships are immediately to get ready to make sail, as soon as the admiral shall think proper to signify his orders to chace in any particular quarter, or for any other movement he would have them execute: thus he will be able to inform himself (as the signal might perhaps be for that purpose only) which are the best sailing ships, and which the most experienced and skilful officers in his fleet; all which, with their several methods of working and sailing, will give him an opportunity of knowing them more thoroughly, that he may employ them accordingly, whenever the service should require the exertion of their respective abilities and experience.

When the admiral would have only a particular squadron to chace at pleasure, he will make a preparatory signal for that squadron to look out and watch his motions with its distinguishing flag, and that of chacing at large; but the ships are not to begin the chace before the signal for the execution of the particular motion is hoisted at the mast-head which denotes the said squadron.

The ships are diligently to observe when the admiral makes the signal to give over chace, that each, regarding the admiral's ship as a fixed point, is to work back, or make sail into her station, to form the order or line again, as expeditiously as the nature of the chace and distance will permit.

8. After the ship has signified to the admiral that she expects to come up with the chace, and that, if an enemy, she can attack her to advantage, she must be then very attentive to the admiral's signs in return, whether of approbation or disapprobation, which no doubt he will make upon the occasion, lest she should unwarily engage too far, and against the admiral's orders. On the signal of disapprobation, she must absolutely quit the chace, and return again into the fleet at all events.

9. The same signal that the admiral makes to give over the engagement, will serve at the same time for the ships to rally or return again to their respective stations; the commanders of squadrons will repeat it, that the ships may work properly and with expedition, to form their line as before; each commanding officer with respect to the commander in chief, and the rest of the ships with regard to the chiefs of their divisions or commanders of squadrons.

If the action continue till night, the admiral will make the general signal for rallying, when each commander of a squadron is to make the same for his particular ships.

Sometimes the signal for discontinuing the action might regard only a particular part of the fleet, which will be signified accordingly by the proper distinguishing flag of that body or division.

10. To avoid the chace. If the ship that is chased be to windward, she must keep on the tack on which she finds she gains most on the enemy, to keep him at the greatest distance; if to leeward, she will go right before the wind, or more or less large, according as she finds either most to her advantage, and more agreeable to her particular properties in sailing or working.

CHAP. IV. Of Anchoring.

WHEN a fleet comes to an anchor, there are five circumstances to be considered: 1. That the ground be good and holding. 2. That the place be well sheltered against the reigning winds that blow on the coast where you anchor. 3. That you may easily get under sail with the same wind that may serve an enemy, and at the same time be able to dispute the advantage of the wind with them. 4. That you can readily form the line as soon as you get under sail. 5. That the ships may have room to keep clear of each other in getting under way: in order to which we should give the ships a good berth when we come to anchor, making one or many lines, about three cables length afunder, and 120 fathom between each ship.

EXAMPLE. It was no doubt by such wise precautions that the duke of York saved his fleet in Solebay in the year 1672; it was composed of 60 English and 30 French ships. His royal highness kept the sea a long time to draw out the Dutch to a decisive action: but seeing they still persisted to secure themselves amidst the banks and shoals of their own coast, and could not by any means force them to a battle, he

took the resolution of returning to Solebay, to refresh and recruit his men with proper necessaries. Admiral de Ruyter, who commanded the Dutch fleet, thought proper not to let slip so happy a conjuncture, as he imagined, of surprising the English as they lay at an anchor in the road: he accordingly set sail with all his fleet, which was equal to the duke of York's, on the 6th of June, and stretched over on the English coast, with the wind at N. E. for Solebay, where he did not doubt but he should meet with the enemy in some disorder and confusion. But the duke, like an experienced officer, ordered the count d'Etrées, vice-admiral, and afterwards maréchal of France, who commanded the van, to anchor out in the offing; placing himself, with the rest of his fleet, in such a manner as to enable him to receive the Dutch admiral in a proper position, whenever he should be informed of his coming. Upon his appearance, the count d'Etrées formed the line with incredible alertness, kept close to the wind, and having stretched out the length of the squadron of Zealand, commanded by vice-admiral Barker, begun the action the 7th of June at eight in the morning, and fought the enemy with such bravery, that several of their ships were disabled: he had even made the proper disposition to re-tack, and charge thro' Barker's squadron, if the calm which came on had not prevented his glorious design. The duke of York was engaged at the same time with de Ruyter, whilst the earl of Sandwich in the rear attacked the Dutch rear-admiral Van Ghent: but the clouds of smoke being dispersed, and the ships no longer under command in the calm, the two fleets found themselves so intermixed and embarrassed with each other, as greatly heightened the horror of the action, and made it the bloodiest that ever was fought. The gallant earl of Sandwich perished with his ship, that was set on fire by a Dutch fire-ship: soon after which, his death was revenged by that of the admiral of the Amsterdam squadron, and by the loss of two Dutch line-of-battle ships, one of which was taken, and the other sunk. The duke of York shifted his flag twice. In fine, the battle lasted with incredible obstinacy on both sides till night, which favoured the retreat of the Dutch: the duke pursued them next day home to their very banks; where, having sheltered themselves, they escaped a total defeat from the hands of a victorious enemy.

We see by the preceding example, how important and necessary it is to be always in readiness to get under sail to receive an enemy; and we may learn by the following example how dangerous it is to wait for an enemy at anchor.

The maréchal duke de Vivonne, viceroy of Sicily for the king of France, having intelligence that the enemy, after the engagement off Agulla, had retired to the port of Palermo, resolved to go and attack them in the road. He accordingly embarked on board the Sceptre, commanded by M. de Tourville as commodore, who hoisted an admiral's flag, and arrived the 2d of June 1676 in sight of Palermo, having 27 ships of the line and 25 galleys. He sent his cruisers to reconnoitre the position of the enemy, who brought him intelligence that they consisted of 27 ships and 29 galleys; that they lay at anchor in a line, fronting the fort of Castel-del-Mar, and under its cannon; and were defended on the right by the grand tower and the

the artillery of the ramparts of the city, and on the left by the batteries of the mole. The marquis de Priuli, chef d'escadre (commodore), was detached with nine ships and five fireships, and the chevaliers de Breteuil and Bethormas with seven galleys, to bear down upon the enemy to the left; all which was executed with so much bravery and resolution, that the vanguard of the enemy were obliged to cut and slip, and run ashore under the batteries of the town, where the fireships burnt three of their ships to the water's-edge: the whole was destroyed by the French with very little loss.

CHAP. V. Line of Battle.

I. THE ancients, as already observed, ranged their ships or galleys so as to present them in front to their enemy; because the machines they then made use of were fixed in the heads or prows of their vessels: the same reason now prevails with regard to the galleys, (see the article GALLEY), which are drawn up in the form of a crescent or half-moon, whose ends or horns are opposed to the enemy; in the middle of which is the admiral, from whence he the more distinctly observes the motions of his fleet throughout. The two fleets being thus drawn up, approach each other to a convenient distance; when the engagement beginning at the ends of the half-moon, they extend themselves insensibly till the whole fleet is engaged, and each partakes of the danger and glory of the action. See fig. 4.

EXAMPLE. The famous battle of Lepanto is the most remarkable action of this kind that ever happened. It was fought between the Turks and Christians in the gulph of Lepanto, the 7th of October 1571. The Christian fleet of galleys consisted of 205, large and small; the Turks had near 260: both formed two long lines, each inclining towards the end, where they began the engagement. Don John of Austria, generalissimo of the Christian forces, had placed himself in the centre of his fleet, and gave the command of his right wing (van) to the famous admiral Doria, and his left (rear) to Michael Barbarigo. The bashaw Pertau, general of the Turks, had likewise placed himself, together with the bashaw Ali, in the centre of his fleet: and gave the command of his right wing to the bashaws of Alexandria, Mehmet and Siroco, and his left to Uluchiali, governor of Algier. The action began at two o'clock in the afternoon; which was first brought on by rowing towards each other with all their might, accompanied all the time with the most alarming shouts and outcries. The left wing of the Christians performed wonders: Barbarigo attacked the Turks with such incredible fury, that the barbarians could no longer resist the incessant fire of the Christians, but precipitately ran themselves ashore on the neighbouring coast, some plunging into the sea, others leaving their galleys to the bravery and mercy of the conquerors. The defeat was so general, that the Turks escaped with only 30 galleys. There perished in this day's bloody action 25,000 Turks; 3500 prisoners were taken, and 130 galleys; the Christians lost on their side 10,000 men, and 15 galleys: they might have then destroyed the whole Ottoman power, had they known how to have made the greatest advantage of so glorious a victory.

II. In an engagement of men of war, the fleets are drawn up in a line of battle on two parallel lines upon a wind. The ships keep close to the wind on the line they are formed in, and are commonly at a cable's length distant one from the other, the fireships, transports, tenders, &c. keeping at half a league's distance on the opposite side of the enemy. Thus the fleets AB, CD, (fig. 5.) that are engaged, are ranged under an easy sail with their larboard tacks on board; and the fireships EE of the fleet AB are to windward, the fireships FF are to leeward of their fleet CD.

EXAMPLE. This form was observed, for the first time, in the famous battle of the Texel, where the duke of York defeated the Dutch on the 13th of June 1665.

We, as well as the French, owe the entire perfection of this order to his royal highness. The English fleet consisted of 100 ships of the line; that of Holland was more numerous, though not in three-deck ships: the two fleets found themselves nigh each other early in the morning, the wind being at S. W. they ranging themselves in two lines at S. S. E. each extending itself about five leagues in length, the English having the advantage of the wind. The duke of York, commander in chief of the English fleet, had placed himself in the centre, and gave the command of the van-guard to prince Rupert, and the rear to lord Sandwich. The Dutch admiral Opdam had opposed himself in the centre of his fleet to the duke of York, and vice-admiral Tromp against prince Rupert. They cannonaded each other from 3 o'clock in the morning till 11, with great fury and intrepidity, the victory still declaring for neither side. The Dutch took one English ship, which too rashly attempted to force through their line: but they falling off to S. E. found the English fire greatly annoyed them. About 11 o'clock the duke of York bore down with his whole line upon the enemy, he himself bearing down at the same time upon Opdam: this disposition and resolution of his royal highness elevated the courage and spirit of both parties to an almost invincible obstinacy. The terrible roaring of the cannon, wrecks of ships, fall of masts, together with a thick smoke intermixed with flashes of fire from the ships that blew up, heightened the horror of this action beyond the power of imagination. It is related of admiral Opdam, that, amidst all this scene of carnage and destruction, he sat with the greatest composure on his poop, viewing, and giving orders to repair as much as possible the damage and disorder he sustained from the duke of York; animating his men all the time both by his words and actions. At two o'clock in the afternoon, his royal highness made the signal for the whole line to bear down together upon the enemy; which obliged the Dutch to alter their disposition of keeping close to the wind any longer. Opdam only, with one of his ships, called the Prince of Orange, of three decks, still kept his station; but soon after, Opdam having received a whole broadside from the duke of York, his ship blew up, without its being ever known by what accident, though five of the men were saved. The Dutch, having already lost many of their ships, and seeing their admiral blow up, put before the wind for the Texel; the duke of York pursuing them with great resolution and bravery to the

A diagram illustrating a naval tactical formation. It shows a three-sided triangular arrangement of three three-masted sailing ships. Ship A is on the left, ship B is on the right, and ship C is at the top. Ship D is positioned on the shore in front of ship B. Ship E is located in the water between ships A and C. Dashed lines connect the ships to form the triangle and indicate the formation's geometry. The background features a calm sea, distant hills, and stylized clouds in the upper left corner.

A diagram illustrating a naval tactical formation. It shows three three-masted sailing ships on the water. Ship A is in the center, ship B is on the left, and ship C is on the right. The ships are arranged in a loose triangular formation. The background shows a calm sea, distant hills, and stylized clouds in the upper left corner.

A diagram illustrating a naval tactical formation. It shows three three-masted sailing ships on the water. Ship A is on the left, ship B is on the right, and ship C is at the top. Ship A is positioned in front of ship B. Dashed lines connect the ships to form a triangle and indicate the formation's geometry. The background features a calm sea, distant hills, and stylized clouds in the upper left corner.

A diagram illustrating a naval tactical formation. It shows a large number of small sailing ships arranged in a dense, rectangular formation on the water. The ships are positioned in two rows, with the front row slightly staggered. The background shows a calm sea, distant hills, and stylized clouds in the upper left corner.

This engraving, labeled 'Fig. 1.', depicts a large-scale naval battle formation. The scene is set in a wide bay with rocky shorelines on the left and right. In the center, a large fleet of sailing ships is arranged in a complex, multi-layered formation. The ships are organized into several distinct groups, with some groups labeled with letters: 'C' and 'E' on the left side, 'F' and 'G' in the upper left, and 'D' and 'B' on the right side. The ships are shown in various stages of maneuvering, with some sails partially set. Above the fleet, a decorative emblem featuring a winged figure is centered in the sky. The water is rendered with fine lines to suggest movement and texture.

Fig. 2.

This engraving, labeled 'Fig. 2.', depicts a smaller-scale naval battle formation. The scene is set in a bay with a rocky cliff on the left and a distant shoreline on the right. The fleet is arranged in a more compact formation than in Fig. 1. The ships are organized into several groups, with some groups labeled with letters: 'A' and 'B' on the left side, and 'C' and 'D' on the right side. The ships are shown in various stages of maneuvering, with some sails partially set. The water is rendered with fine lines to suggest movement and texture. The overall composition is more focused and detailed than the larger formation in Fig. 1.

A Fleet to Windward. the very entrance of their port: he took, burnt, and destroyed, 22 ships of the line, 20 of which were from 50 to 80 guns; and gained over them the most glorious victory that was ever obtained at sea. The whole action cost him but one man of war, with the loss of 300 or 400 men.

III. Before we enter into action, or form the line of battle, we must consider first the advantages or disadvantages of being to windward or to leeward.

Advantages of being to WINDWARD.

1. THE fleet to windward can edge down to the enemy, when and as near as it shall think convenient: consequently regulates the time and distance most advantageous to come to action.

2. IF the fleet to windward is more in number, it may easily detach some ships to send after the rear of the enemy, which must undoubtedly throw them into confusion. Thus, the fleet AB being more in number than the fleet CD, may easily detach the ships EF to double upon the rear D, which cannot long resist such superior fire; therefore must give them an opportunity (with the rest of the ships that will of course join them) to range along the whole length of the enemy's line. This is an advantage the fleet to leeward cannot have, let it be ever so numerous; for the rear of its line will be in a manner useless.

EXAMPLE. The advantage of the wind could never be more favourable than it was in an action off Agusta, the 22d of April 1676, when the combined fleets of Spain and Holland avoided a total defeat by having the advantage of the wind. The French fleet, commanded by Monf. Duquesne, consisted of 27 ships of the line. The marquis of Almira, lieutenant-general, commanded the van-guard; and Monf. Gabbare, chef d'escadre (commodore), the rear. The enemy's fleet consisted of about the same number of ships, but had besides 9 galleys; De Ruyter commanded the van-guard, the Spanish admiral was in the centre, and the Dutch vice-admiral commanded the rear. The two fleets met off Agusta early in the morning; but the enemy kept their wind till 4 o'clock in the afternoon, when de Ruyter bore down upon the French rear in good order, where they received him in the same manner with equal intrepidity: there were many ships disabled on both sides. The marquis d'Almira was carried off by a cannon-ball, and de Ruyter was mortally wounded. These two accidents caused some disorder in the van of the fleet; but the chevalier de Vallballe, chef d'escadre, supplying the place of the marquis d'Almira, behaved so remarkably gallant, that he was just upon the point of taking and destroying a part of the enemy's fleet, had not their galleys most opportunely taken their disabled ships in tow, and saved them from falling into his hands. The action began later in the centre, and had scarcely reached the rear guard: the enemy having the wind, had availed themselves so well of that advantage, that they continued the engagement no longer than was necessary to save their honour; waiting for the approaching night to retire from the pursuit of the victors.

3. IF any of the ships of the fleet to leeward should be disabled, whether in the van or rear, or even in the centre, the fleet to windward may with

the greater ease send down their fire-ships upon them, or send a detachment after any part of the flying enemy.

4. We must likewise attribute, amidst other advantages of being to windward, that of being sooner freed from the inconvenience of the smoke of the enemy as well as of our own. 1. The wind repelling back again the smoke of the cannon into the ship, so greatly inconveniences those quartered at the guns, as totally to deprive them for some time of the sight of the enemy. 2. The same smoke must likewise much embarrass the sailors in working the ship; as it is often found by experience, that the sails and rigging are set on fire by the combustible matter and fiery particles incorporated with the smoke; besides many other fatal accidents incident to ships in that unhappy situation.

Advantages of being to LEEWARD.

It must be acknowledged, that a fleet to leeward has likewise great advantages; and there are who maintain, that the advantage of being to leeward is at least equal to that of being to windward. But when they consider all circumstances more attentively, they will find the advantage of being to windward the greatest a fleet can possibly have, whether superior or inferior to the enemy: though we must allow at the same time, that, on some extraordinary occasions, it may be more advisable to get to leeward if we can, that is, when it blows hard, and the sea runs so high, that the weather fleet cannot open its lower tier, when obliged to engage a greater number of ships, or in an action between two single ships. But still, in an engagement between two fleets, in moderate, proper weather for engaging, that which has the weather-gage has greatly the advantage.

1. THE fleet to leeward fire to windward, and consequently the ships may make use of their lower tier, without being under any apprehensions that a sudden squall of wind should overpower them, by the water rushing in irresistibly between decks: an advantage (in some measure) the English fleet under Sir Edward Hawke, had over the French fleet commanded by Monf. Consans, in that ever-glorious and memorable action off Belleisle, the 20th of November 1759, where they fatally experienced the difference of our superior skill, undaunted resolution, and seamanship. This circumstance is certainly the greatest advantage a fleet to leeward can have, especially when it blows hard, with a great sea. One can hardly conceive the confusion and disorder sudden gusts of wind occasion between the men between decks, when the waves come pouring in, and lay a ship upon her broad-side, so as often to endanger her oversetting, or going to the bottom before the ports can be secured.

2. THE fleet to leeward can easier cover any of their ships that should be disabled in action, which must greatly embarrass the fleet to windward to effect, without running the risk of being destroyed by the enemy in attempting it; however, these are disasters which both are equally subject to.

3. THE fleet to leeward may easier make a retreat if beaten; whereas the fleet to windward cannot so well escape, without being reduced to the necessity of forcing its way through the enemy's line, which must be attended with the most fatal consequences.

It must be acknowledged, that a fleet which puts

before the wind runs a great risk, if the enemy is in a condition to pursue it: but then there are some circumstances wherein the fleet to leeward may boldly venture to crowd away before the wind; that is to say, when night approaches, the wind freshens, the sea rises, or the enemy is embarrassed with a convoy, which may prevent their pursuing them.

EXAMPLE. The allies availed themselves of these advantages, in the year 1689, off Bantry. The count de Chateau-Renaud commanded the French fleet of 24 ships of the line, and convoyed 3000 men to Ireland, with a great quantity of stores, provisions, and ammunition. My lord Herbert, who commanded a squadron of much the same force, having intelligence that the French landed their troops in Bantry-bay, resolved to go and attack them there, not doubting but he should find them in some disorder: but the count had taken his measures so prudently, and was so well prepared, that he advanced in good order to receive the English, and attacked them with so much bravery, that he soon obliged them to crowd away all the sail they could make before the wind; pursuing them till night put an end to the chase: and the count, having thus happily landed his troops, returned to Brest, where he received the just applause due to his successful expedition; having in eleven days carried succours into Ireland, beat the enemy, took a considerable convoy, and reconducted back again his fleet to Brest in good order and condition.

IV. But to return to the explanation of the line of battle: We have already observed, that fleets in action ought to be ranged on two parallel lines; for, if formed otherwise, by inclining in the van and rear, the headmost and sternmost ships will be engaged, whilst the ships in the centre will be out of the reach of each other's guns: a consequence too obvious to need any demonstration.

The ships ought to keep at a cable's length from each other, or closer, if judged convenient or necessary: otherwise, if too far asunder, one ship of such line will be exposed to the fire of two ships at a time, from the closer and more regular line of the enemy.

The size of the ships is likewise too important a point not to be properly considered in a line of battle, as it contributes more to its strength than the number of the fleet; for two reasons. 1. A large ship carries more guns, and heavier metal: so that a fleet consisting of such ships is of greater force than a more numerous fleet of smaller ships, though drawn up in a closer line; because they engage the enemy with more, as well as heavier artillery, in the same space. 2. The great ships are stronger timbered, and consequently better able to resist the shot of the enemy; therefore of greater service in action than a fleet of smaller ships, notwithstanding the advantage of a closer line; because that each ship of the former is attacked only by a less number of guns of the latter that can do her any damage.

V. To form a line of battle from the order of retreat. Suppose the fleet AGF in a retreating form, to change it into a line of battle, the headmost ship A must haul up upon a wind, and the rest of the fleet,

running large four points on the same tack, will form itself in her wake, or the line IH. To dispose
the Wind.

This evolution is so regular, so simple, and short, that it makes this order of retreat preferable to any other; for a fleet that retreats may be often obliged to come to action: it would be then greatly embarrassed, if it could not immediately resume the line again by so easy and regular a method as this. In effect, by way of illustration, let us suppose, that the enemy LLM pres the fleet so close in its pursuit, as to force it to come to an engagement; then the ships that were sailing large, haul upon a wind all together, as soon as possible, on the larboard tack, the head A hauling up at the same time. This manœuvre (method of working) can cause no confusion in the fleet: on the contrary, it then presents its sides with greater advantage to the enemy; and the ships that range themselves upon a wind in the wake of the ship A, will force between two fires the enemy's ships M.

We suppose that the enemy attack only one side; for they will find it difficult in effect to attack both, without running the risk at the same time of being separated: but admit that the enemy did attack on both sides, you may still perform the evolution equally the same; and the ships GF would present their sides to the enemy, as well as when they were sailing large or before the wind.

CHAP. VI. Some necessary Manœuvres before an Engagement.

§ 1. To dispose the Wind with the Enemy.

1. THE fleet to leeward should avoid extending itself the length of the enemy's line, in order to oblige them to edge down upon theirs, if they intend to attack them; which will be a means, if they still persist in doing so, of losing the advantage of the wind.

It is impossible for a fleet to leeward to gain to windward so long as the enemy keep their wind, without a change happens in their favour. Therefore all that a fleet to leeward can do, must be to wait with patience for such a happy change, which they will undoubtedly avail themselves of, as well as any mistake or inadvertency the enemy may commit in the mean time. And as long as the fleet to leeward does not extend its line the length of the enemy's, it will be impossible for the latter to bring them to action, without running the hazard, by bearing down, of losing the advantage of the wind, which both fleets will be so desirous of preserving.

2. In fine, that an admiral may benefit by the shifts of wind that frequently happen, he must in a manner foresee them; which will not appear so extraordinary to officers of any experience, who know what winds reign most on the coast, or off the head-lands where they may expect an enemy: and though an admiral may be sometimes out in his conjecture, he also as often succeeds so happily as to gain the advantage of his enemy.

Mont. du Queues, the French admiral, by his superior skill in these particulars, gained a considerable advantage over the Dutch fleet, when he engaged them off Strombolo in the year 1676. He waited till the next day for a shift of wind; which happened in his favour as he foresaw, and gave him an opportunity

To avoid or force an Action.

ty of tacking to windward of the enemy, and bearing down upon them in good order: an advantage they neglected the day before; which fatal oversight they could never afterwards recover.

3. The disposition or projecting of head-lands, of currents in the Mediterranean, and tides in the ocean, contribute greatly towards gaining the wind of the enemy: for sometimes you may only range a coast along, or keep out in the offing, to gain a few leagues upon a tack; and we may say, we think with justice, that the knowledge of these advantages is as essential to an admiral at sea, as the geography of the several countries, with their woods, roads, course of rivers, &c. he is obliged to march through, is to a general of an army on shore.

4. The fleet to windward ought to keep the enemy as much as possible always a breast of it; because, by doing so, they will preserve the advantage they may have, unless the wind greatly changes against them. They should force them likewise to keep their wind, unless they think it more prudent not to engage; but when that is the case, they should keep entirely out of sight of the enemy.

§ 2. To avoid an Action.

1. THE fleet to windward can never be forced to engage; because it can always continue on that tack, which keeps the enemy at the greatest distance from it, by stretching out upon one tack whilst they continue upon the other.

If the wind was not so subject to change, it would be very easy for the fleet to windward to keep in sight of the enemy, without being under any apprehensions of being forced to come to action; but the inconstancy of the wind obliges the most experienced admirals to avoid meeting the enemy, when they think it improper to engage them. The reason of this maxim is founded upon the impossibility of an inferior fleet's avoiding an action, when in presence for any time of a superior fleet.

2. If the fleet that endeavours to avoid coming to action be to leeward, they will edge away the same as the enemy; but, at the same time, they should not go away right afore the wind, without making their retreat in a half-moon, if in sight of the enemy. So that the fleet to leeward, which is not for engaging, seeing the enemy still persist in chasing them, will bear away as they do, in order to keep them at the same distance.

There are some circumstances in which the fleet to leeward may put afore the wind, without ranging it into the order of a retreat; that is, when it only designs to prolong the engagement, or is resolved to engage the enemy, if they still continue to pursue them to bring them to action. But, except on such extraordinary occasions, the form of a retreat puts the fleet into the best posture of defence, and with the least hazard and danger.

§ 3. To force the Enemy to Action.

AXIOM I. We may look upon it as a general maxim: "When two fleets of equal force remain long in sight, they may alternately force each other to bring on an action." The following reasons support this maxim.

If the fleet that wants to bring on an engagement is to leeward, they must endeavour to keep on that tack which forereaches most upon the enemy, that they may keep them better in view, till the wind may happen to change in their favour.

The least experience at sea will serve to convince us, that it is almost impossible for a fleet that once discovers itself to the enemy, ever to retire or escape, unless it secures itself in some port or harbour; for fleets are generally at sea at a season of the year when the nights are very short, and the days long; so that any stratagems or false courses they may use, will avail them but little to escape the pursuit of a watchful enemy: besides, a fleet would not run the hazard of crowding too much sail by night, for fear of being separated, which may be attended with fatal consequences. A recent example of such conduct happened with Monf. de la Clue in 1759, who, by crowding away too much sail at night, to push through the gut of Gibraltar with a strong easterly wind, before morning lost sight of half his fleet, and subjected himself of course, by such imprudence, to fall much the easier victim to admiral Boscawen, who was in close pursuit of him with his whole squadron, and engaged him the next day with a superior force; which obliged the French admiral to make a running fight, tho' it availed him but little, as five out of his squadron were burnt or taken on the coast of Portugal.

AXIOM II. It is scarcely possible for a much inferior fleet to remain long in presence of an enemy, without being forced to an action. 1. A fleet that is superior in number may send a detachment of its best cruisers after the flying squadron, and soon bring it to action. 2. It may divide itself into three squadrons, leaving a considerable interval between each; then, whatever course the enemy may take to escape, one or other will be always ready to intercept it.

The only resource an inferior squadron can have in such circumstances is, to bear away in the form of a half-moon: though even then, it can have no great hopes of avoiding an engagement, if the enemy persists in chasing it to bring it to action, unless they steer for some harbour or friendly asylum to secure themselves in.

COROLLARY. We may from all this draw the following conclusive inference, that it is almost impossible for an inferior fleet, under any pretext whatever, to continue long in the presence of one greatly superior to it, without being forced to action.

§ 4. To double an Enemy.

To facilitate this, the superior fleet must endeavour to stretch out the length of the enemy's line, and at the same time, leaving ships a-stern, to close and double upon that of the enemy's, and force them between two fires.

1. If the superior fleet is to windward, it may so much the easier double its rear upon that of the enemy's, and force it between two fires; and even if it should be to leeward, it should likewise leave some ships a-stern of it, because of the wind's often changing during the action; besides, the fleet to leeward may insensibly edge away in the heat of the engagement, to give its rear an opportunity of doubling upon the enemy, by immediately luffing up close to the wind

2. There are who maintain, that the enemy's line should be doubled a-head rather than a-stern: because, say they, if the enemy's van is once put into disorder, it will of course fall a-stern upon the rest of the line, and throw it into confusion. But, on the contrary, it seems plain, that the ships will be less exposed, and find it safer to double upon the enemy's line a-stern: for if a ship should be disabled a-head, it does not appear how she can recover her own line again; whereas if a ship should be disabled in attempting the same in the enemy's rear, she cannot be attacked by any of their line, without exposing themselves at the same time to the fire of two ships; therefore may remain a-stern out of danger, till she has repaired her damages again.

3. It seems equally clear, that if the ships E, M, of the fleet CD, had doubled the head A, they would run a great hazard of being destroyed: for if the ship E should be disabled, how can she recover her own line again? how easy might the enemy destroy her? On the contrary, if the ships LN, of the fleet FG, have doubled the rear I of the enemy, and the ship L should be disabled, she may remain a-stern, without being under any apprehension from the rear I, which is already hard pressed by the ships G, N.

Nothing can illustrate this method of working a fleet better than the famous engagement off la Hogue in the year 1692, between the count de Tourville and admiral Russel. The French, having the wind, bore down in good order upon the English: but, being at the same time so much inferior in number, it was impossible for them to extend their line the length of the enemy's; therefore could not prevent the English from extending their rear a great way a-stern of the French, which made their line so much the longer in attempting it, and consequently the ships wider asunder, (a great disadvantage against a close line): The wind, which was at first S. W. changing to the N. W. gave the rear of the English an opportunity of still closing its line more, and doubling upon the French; so that the count de Tourville with his division soon found himself surrounded by his enemies on all sides; in which unlucky situation he distinguished himself with the greatest bravery and resolution imaginable, tho' overpowered by numbers, whose great superiority of force could be no longer resisted.

To prevent any of the enemy's line from doubling upon your's, you must not suffer them to extend any of their ships beyond your rear; in order to which, there are several methods to be taken when your fleet is inferior in number.

1. If you are to windward, you need not extend your line the length of the enemy's van, but attack your second division F with your van A; by which means their first division FG will be in a manner useless; and if they should stretch out a-head to tack upon you, they will lose too much time, and run the risk of being separated by the calm which generally happens in the course of a sea-engagement, occasioned by the continual discharge of cannon on both sides; you may even leave a great opening in the centre E, provided you take the necessary precautions to prevent

your van-guard from being cut off: and thus, however inferior you may be in number, you will have it in your power to interrupt the enemy's line from extending itself beyond, or a-stern of, your rear.

EXAMPLE. Admiral Herbert's method of ranging his fleet, when he engaged the French off Beachy-head, in the year 1690, was generally approved of. He had some few ships less than the enemy, and was resolved to use his utmost efforts against their rear; to effect which, he ordered the first division of the Dutch to bear down upon the second division of the French, at the same time opening his fleet in the centre, leaving a great space a-breast of the main body of the enemy. He then closes his rear, which he opposes to theirs, keeping himself with his division at some distance a-breast of the centre: then closing his ships as much as possible, he opposes them to the enemy's rear, at the same time reserving his own division to attack the French, if they should attempt to push through the opening in the middle, in order to double upon the Dutch. By this method (which shewed great foresight and experience) he rendered the enemy's first division almost useless, because of its being obliged to stretch out a long way a-head to tack upon his van: and the calm which afterwards came on had in a great measure deprived it of partaking of the danger and glory of the action.

2. If the inferior fleet is to leeward, you might leave a greater interval in the centre and less in the van; but then you should have a small corps de reserve of capital ships and fire-ships, that the enemy may not take the advantage of the intervals in your fleet to cut off your line.

3. There are some again for giving it as a general rule, that the commanding officers of the inferior fleet should oppose themselves to the respective general officers of the enemy; by this means several of their ships will remain useless in the intervals, and will be rendered incapable of doubling upon you.

This method has its inconveniences, because the van and rear of each division is exposed to the fire of two ships at a time, and does not secure the last division from the hazard of being doubled by the enemy's rear; but, to remedy this, you may place the larger ships in the van and rear of each division, and order it so that the last division may not have it in its power to extend its rear a-stern of yours.

4. Again, others will have it, that the three squadrons of the inferior fleet should attack each a squadron of the superior fleet; observing at the same time, that each squadron extends its line far enough to prevent the opposite line from leaving any ships a-stern of it, but rather a-head.

5. In fine, there are who rather choose, that the inferior fleet should stretch its line so long, as to leave a great distance between the ships, that it may extend its line the length of the enemy's. But this seems to be the worst method that can be taken; because it gives the enemy's fleet all the advantage it can desire of exerting its whole force upon the inferior line: tho' it must be allowed, upon certain occasions, this method would be very proper to follow; such as, when the enemy's ships, though more in number, are not of such force and weight of metal as the ships of the inferior fleet.

To force
the enemy's
Line.

§ 6. To receive a Fleet that bears down upon you.

THE fleet to leeward seeing the enemy bear down upon it, will of course range itself, as expeditiously as possible, into a line of battle, by edging away a little, to gain as much time as may be necessary to form the line without confusion. They should not omit at the same time leaving some small intervals between the divisions, that the fleet may be the better able to distinguish, and have more room for action: then each commander will exert his utmost to keep his ship a-breadth of any ship of the enemy's that happens to fall to his lot; either by making more or less sail, or even tacking, (if absolutely necessary), to preserve his station with regard to the enemy.

§ 7. To force through the Enemy's Line.

1. WE find in the several relations of the wars between the English and Dutch, that they had often alternately traversed and charged through each other's fleets; that is to say, the fleet to leeward CHD, having stretched out a little a-head, tacked upon the enemy AB, and forced through their line at E, then re-tacking upon them immediately at C, on the other side, gained the wind of them; but then again, the others, in their turn, regained the same advantage of them, and cut them off from their line. Thus they mutually traversed each other, cutting off and destroying one another's ship's, with an invincible obstinacy and bravery not to be described.

This manner of fighting and working a fleet is equally daring and hazardous, and requires the most consummate ability as well as experience to succeed in it so happily as the count d'Estrées did in an engagement with the Dutch in the Texel in the year 1673. He traversed and charged through the squadron of Zealand, gained the wind of them, and threw them into such disorder, that the victory, which was before doubtful, now manifestly declared in his favour.

2. It should seem easy for the fleet to windward to hinder the enemy to leeward from forcing through their line; which, whenever they attempted, the other fleet may tack at the same time all together, and thereby effectually prevent them from succeeding.

3. It does not therefore appear why we should be under any apprehension from a fleet that attempts to force through our line. It even seems that it should never be put in practice but in the following circumstances: Such as being sometimes obliged to it, to avoid a greater evil; if the enemy should leave a great opening in the centre, and render part of your fleet useless; or if a number of ships should be disabled, &c.

4. You are sometimes obliged to traverse the enemy's line to disengage some of your own ships which may happen to be cut off; in that case, you must boldly risk something, at the same time not forget the necessary precautions. 1. To close your ships as much as possible. 2. To make all the sail you can, without waiting to attack the enemy as you force through their line. 3. As soon as you have got through, you should tack again without loss of time, lest the enemy should stand on upon the same tack with the ships that had broke through their line.

Admiral de Ruyter put this sort of traverses in practice to the greatest advantage, when he beat the Eng-

lish fleet under general Monk in the year 1666, three days successively, on the north of England.

CHAP. VII. Of a general Action between two Fleets.

§ 1. General Observations.

THE engagement will not begin before the admiral makes the signal, unless an action is insensibly brought on by some unavoidable circumstances in the line or position of the van or rear of both fleets, in forming or approaching each other: the admiral, in such case, will make the proper signal for the van or rear, by the distinguishing flag of either of these divisions; which will undoubtedly regulate the necessary manœuvres of the rest of the fleet through the whole line.

The admiral in action carries but little sail: in which, however, he must conduct himself by the motions of the enemy; the ships always observing to keep close in the line; and wherever they do not, the ships which immediately follow, should pay no regard to those that precede them, if they should unguardedly leave too great an opening from the rest, unless ordered so to do by signal from the admiral.

The ships ought to be particularly careful not to fire till they find themselves high enough for the line to do effectual execution; otherwise it will be but expending a quantity of ammunition to very little purpose. They ought principally to level well their guns, without that hurry or confusion too often practised in firing broadsides; and from which so little advantage in general is derived, to answer the end proposed, that of defeating the enemy, which may be much sooner accomplished by a more regular and steady fire, constantly kept up without intermission, the better to embarrass their line, and divert their attention, more than broad-side and broad-side, with some intervals between (as must naturally happen), will ever effect. We ought to be convinced of this general truth, That of all actions, a sea-fight (except in the article of boarding) should be conducted with the least hurry or precipitation in order to succeed.

A captain must not quit his post in the line upon any pretence whatever, unless his ship should be so greatly incommoded as to render her incapable of continuing the action: the little sail a fleet is under at such time, in general, may give the ships, though damaged in their rigging, &c. time enough to repair their defects, without causing an unnecessary interruption in the line, by withdrawing out of action, when their service might perhaps be of the utmost importance to the rest of the fleet.

A captain, through too impetuous a desire of distinguishing himself, ought never to break the order of the line, however inviting the advantage of an attack might then appear to him to secure success: he must wait with patience the signal from the admiral or commanding officer of his division, because it is always more essential to preserve and support a close line in action, as it constitutes the principal strength of a fleet in general, than to attend to a particular attack between two ships, which commonly decides but little with regard to the whole, however glorious in appearance, unless with a view at the same time of taking or destroying a flag-ship of the enemy's, and where success alone, even then, can justify the attempt.

Tho.

The two immediate seconds to the admiral ought to direct part of their fire against the enemy's flag-ship, or any other that may attack their admiral; so that their chief attention should be employed more in his defence than in that of their own proper ships, as they must sacrifice every other consideration to the honour of their flag.

The same attention must likewise be paid to any other ship that may find herself engaged with one of the enemy's flag-ships; the next to her a-head and a-stern should serve in that respect as seconds, by dividing part of their fire against such flag-officer, in order to make him strike the sooner.

If any flag-officer stand in need of being succoured, he will of course make a signal for the corps-de-reserve; or if there should be none, he will signify the same to his division; on which his two seconds, with those nearest him, will close in to cover him, and continue on the action: the frigates of his squadron will likewise be ready to give him the necessary assistance; and, if he should still continue the attack, he will be in a particular manner supported by his whole division.

Those ships that happen to be most exposed to danger, will naturally make the ordinary signals upon the occasion, if they should receive any hurt or damage, in order to be supported by such in the line as are within reach of them.

§ 2. To detach from the Line a Corps-de-reserve.

WHEN a fleet is so far superior in number as to be able to extend itself both a-head and a-stern considerably beyond the enemy's line, if then the admiral should think it expedient, that such ships as may not be a-breast of the enemy, should detach themselves from the line, and form to windward or to leeward into a body of reserve; those of the second division of the van-guard, or those of the second in the rear, will immediately detach themselves from the body of the fleet, after the repetition of the signal from the commanding officers of their divisions, and place themselves in a line with the frigates nearest a-breast of the centre of the fleet, if to windward; or if to leeward, somewhat a-head of the same; being careful at the same time to keep within reach of observing distinctly all the signals and motions of the fleet, to be the readier at hand to re-place such of the ships as may happen to be dismasted or drove out of the line, where all intervals must be properly strengthened and carefully filled up again without loss of time.

The oldest captain, after the senior officer who commands the reserve, ought to relieve the first, or close that part of the line where the disabled ship has been obliged to quit; and so on successively of the rest.

The commanding officer of the body of reserve will not be detached with the whole corps, unless on some pressing occasion to fortify the line, where such reinforcement is immediately necessary; if to defend one of the flag-officers of the three squadrons, he will be followed by the next senior officer of the reserve, who was not before detached, in order to place themselves as seconds, the first a-head and the other a-stern of the flag they are to support, without any diminution of the honour of his own proper seconds at the same time, as they are only called in thro' necessity on that emergency, being not engaged before, and conse-

quently better able to assist and support the admiral; their duty being likewise to exert their utmost efforts in attacking or boarding (if possible) the enemy's flag-ship to force him to yield, except they are particularly ordered off to some other quarter or part of the line.

The admiral will sometimes order the whole body of reserve to reinforce one of the three squadrons of the fleet, as he may see occasion; which when he does, the corps must make all the sail it can, that each ship may place herself, successively, the first in the first interval, the second in the second, and so on throughout; but if the admiral should want only part of that body, he will make the signal accordingly.

If the admiral, commanding an equal or superior number of ships to the enemy, should judge it necessary to have a small reserve of one or two ships for each of the three squadrons of the fleet, the ships for that purpose in each of the three bodies are made known by signal for the reserve; they will immediately draw out of the line upon hoisting the same, and form themselves on the line with the frigates, at a convenient distance from their commanding officer: that is to say, the first a-breast, and under cover of the headmost second; and the other a-breast, and under cover of the admiral, to be in readiness to run in between him and one of his seconds, to enable him the better to continue on the action with fresh vigour, and press the enemy with unremitting ardour to strike as soon as possible.

The corps-de-reserve is generally formed at the same time with the line, to prevent any irregularity that may happen on leaving any intervals or openings; tho' the admiral may, if he thinks proper, draw ships out of the line during the action, to form a body of reserve, according to the time or circumstances of his situation, &c.

WHEN the admiral finds he has no further occasion for his body of reserve, he will make proper signals for such ships to resume their respective posts in the line again; the corps-de-reserve will always repeat the signals which regard themselves particularly.

§ 3. Of Boarding.

WHEN the admiral shall judge it necessary or convenient to prepare the fleet for action, he will make the signal proper for the occasion, and the fleet will at the same time make the necessary disposition for boarding. If the admiral design to board any of the enemy's ships, he will undoubtedly make the proper signal for the whole fleet, a particular squadron, or, in fine, for a particular ship, by the different position of the signal, and the distinguishing mark of such squadron or division, or particular pennant of such ship.

If any captain in the fleet think he can board with success one of the enemy's ships, he will signify the same to the admiral by hoisting the boarding flag, together with his particular pendant, to be more plainly distinguished; the admiral, in return, will make the proper signal of approbation, or otherwise, if he disapprove the attempt, by letting fly at the same time that ship's particular pennant, that she may observe the signal the better.

WHEN a captain seems to express an ardent desire of distinguishing himself by boarding one of the enemy's ships, he ought to consider well the ill consequences that

Economy of a naval Engagement. that might perhaps attend such enterprise, if he fail of success; for the breaking the order or disposition of the line, by quitting his post, may be of much greater disadvantage to the whole, than any advantage arising from his victory, except that over a flag-ship.

§ 4. The Fire-ships to prepare.

WHEN the admiral makes the signal for his fleet to prepare for action, the fire-ships will at the same time get ready their grappling-irons, fire-engines, &c. for boarding, and will likewise dispose all their combustibles into their proper channels of communication, &c. as soon as possible after the action begins: all which, when ready, they will take care to make known by signal to the particular division or squadron they belong to, and they of course will repeat the same to the admirals.

The fire-ships must be particularly careful in placing themselves out of the reach of enemy's guns, which they may do a-breadth, and under shelter of their own ships in the line, and not in the openings between the ships, unless to prevent any of the enemy's ships that should attempt to force through their line; when they must, in such case, use their utmost efforts to prevent them. They ought always to be very attentive to the admiral's signals, as well as those of the commanding officer of the particular squadrons they belong to, that they may lose no time when the signal is made for them to act, which they must quickly answer by a signal in return.

Although no ship in the line might be particularly appointed to lead down or protect the fire-ships, besides the frigates already ordered by especial appointment to attend that service; yet notwithstanding, the ship a-head of which the fire-ship passes in her way to the enemy is to escort her, whatever division she may belong to, and must assist her with a boat well-manned and armed, as well as any other succour she may stand in need of: the two next ships to her must likewise give her all necessary assistance. The captain of a fire-ship is to consider, in short, that he is answerable for the event, in proportion as he expects to be honourably rewarded, if he succeed in so daring and hazardous an enterprise.

§ 5. Particular Description of the Economy of a naval Engagement.

SINCE a general engagement of fleets or squadrons of men of war is nothing else than a variety of particular actions of single ships with each other, in a line of battle; it may not be improper to begin by describing the latter, and then proceed to represent the usual manner of conducting the former.

I. The whole economy of a naval engagement may be arranged under the following heads, viz. the preparation, the action, and the repair, or refitting for the purposes of navigation.

1. The preparation is begun by issuing the order to clear the ship for action, which is repeated by the boatswain and his mates at all the hatchways, or stair-cases, leading to the different batteries. As the management of the artillery in a vessel of war requires a considerable number of men, it is evident that the officers and sailors must be restrained to a narrow space in their usual habitations, in order to preserve the in-

ternal regularity of the ship. Hence the hammocks, or hanging-beds, of the latter are crowded together as close as possible between the decks, each of them being limited to the breadth of fourteen inches. They are hung parallel to each other, in rows stretching from one side of the ship to the other, nearly throughout her whole length, so as to admit no passage but by stooping under them. As the cannon therefore cannot be worked while the hammocks are suspended in this situation, it becomes necessary to remove them as quick as possible. By this circumstance a double advantage is obtained: the batteries of cannon are immediately cleared of an encumbrance, and the hammocks are converted into a sort of parapet, to prevent the execution of small-shot on the quarter-deck, tops, and fore-castle. At the summons of the boatswain, Up all hammocks! every sailor repairs to his own, and, having stowed his bedding properly, he cords it up firmly with a lashing or line provided for that purpose. He then carries it to the quarter-deck, poop, or fore-castle, or wherever it may be necessary. As each side of the quarter-deck and poop is furnished with a double net-work, supported by iron cranes fixed immediately above the gunnel or top of the ship's side, the hammocks thus corded are firmly stowed by the quarter-master between the two parts of the netting, so as to form an excellent barrier. The tops, waists, and fore-castle, are then fenced in the same manner.

Whilst these offices are performed below, the boatswain and his mates are employed in securing the sail-yards, to prevent them from tumbling down when the ship is cannonaded, as she might thereby be disabled, and rendered incapable of attack, retreat, or pursuit. The yards are now likewise secured by strong chains or ropes, additional to those by which they are usually suspended. The boatswain also provides the necessary materials to repair the rigging, wherever it may be damaged by the shot of the enemy, and to supply whatever parts of it may be entirely destroyed. The carpenter and his crew in the mean while prepare his shot-plugs and mauls, to close up any dangerous breaches that may be made near the surface of the water; and provide the iron-work necessary to rest the chain-pumps, in case their machinery should be wounded in the engagement. The gunner with his mates and quarter-gunners is busied in examining the cannon of the different batteries, to see that their charges are thoroughly dry and fit for execution; to have every thing ready for furnishing the great guns and small arms with powder, as soon as the action begins; and to keep a sufficient number of cartridges continually filled, to supply the place of those expended in battle. The master and his mates are attentive to have the sails properly trimmed, according to the situation of the ship; and to reduce or multiply them, as occasion requires, with all possible expedition. The lieutenants visit the different decks, to see that they are effectually cleared of all encumbrance, so that nothing may retard the execution of the artillery; and to enjoin the other officers to diligence and alertness, in making the necessary dispositions for the expected engagement, so that every thing may be in readiness at a moment's warning.

When the hostile ships have approached each other to a competent distance, the drums beat to arms. The

The boatswain and his mates pipe, All hands to quarters! at every hatchway. All the persons appointed to manage the great guns immediately repair to their respective stations. The crowbars, handspikes, rammers, spunges, powder-horns, matches, and train-tackles, are placed in order by the side of every cannon. The hatches are immediately laid, to prevent any one from deserting his post by escaping into the lower apartments. The marines are drawn up in rank and file on the quarter-deck, poop, and forecastle. The lashings of the great guns are cast loose, and the tompions withdrawn. The whole artillery, above and below, is run out at the ports, and levelled to the point-blank range ready for firing.

2. The necessary preparations being completed, and the officers and crew ready at their respective stations to obey the order, the commencement of the action is determined by the mutual distance and situation of the adverse ships, or by the signal from the commander in chief of the fleet or squadron. The cannon being levelled in parallel rows, projecting from the ship's side, the most natural order of battle is evidently to range the ships abreast of each other, especially if the engagement is general. The most convenient distance is properly within the point-blank range of a musket, so that all the artillery may do effectual execution.

The combat usually begins by a vigorous cannonade, accompanied with the whole efforts of the swivel-guns and the small arms. The method of firing in platoons, or volleys of cannon at once, appears inconvenient in the sea-service, and perhaps should never be attempted unless in the battering of a fortification. The sides and decks of the ship, although sufficiently strong for all the purposes of war, would be too much shaken by so violent an explosion and recoil. The general rule observed on this occasion throughout the ship, is to load, fire, and sponge, the guns with all possible expedition, yet without confusion or precipitation. The captain of each gun is particularly enjoined to fire only when the piece is properly directed to its object, that the shot may not be fruitlessly expended. The lieutenants, who command the different batteries, traverse the deck to see that the battle is prosecuted with vivacity; and to exhort and animate the men to their duty. The midshipmen second these injunctions, and give the necessary assistance wherever it may be required, at the guns committed to their charge. The gunner should be particularly attentive that all the artillery is sufficiently supplied with powder, and that the cartridges are carefully conveyed along the decks in covered boxes. The havoc produced by a continuation of this mutual assault may be readily conjectured by the reader's imagination: battering, penetrating, and splintering the sides and decks; shattering or dismounting the cannon; mangling and destroying the rigging; cutting afunder or carrying away the masts and yards; piercing and tearing the sails so as to render them useless; and wounding, disabling, or killing the ship's company! The comparative vigour and resolution of the assailants to effect these pernicious consequences in each other, generally determine their success or defeat: we say generally, because the fate of the combat may sometimes be decided by an unforeseen incident, equally fortunate for the one and fatal to the other. The defeated ship having acknowledged

the victory, by striking her colours, is immediately taken possession of by the conqueror, who secures her officers and crew as prisoners in his own ship; and invests his principal officer with the command of the prize until a captain is appointed by the commandee in chief.

3. The engagement being concluded, they begin to repair: the cannon are secured by their breechings and tackle, with all convenient expedition. Whatever sails have been rendered unserviceable are unbent; and the wounded masts and yards struck upon deck, and fished, or replaced by others. The standing rigging is knotted, and the running-rigging spliced wherever necessary. Proper sails are bent in the room of those which have been displaced as useless. The carpenter and his crew are employed in repairing the breaches made in the ship's hull, by shot-plugs, pieces of plank, and sheet-lead. The gunner and his assistants are busied in replenishing the allotted number of charged cartridges, to supply the place of those which have been expended, and in refitting whatever furniture of the cannon may have been damaged by the late action.

Such is the usual process and consequences of an engagement between two ships of war, which may be considered as an epitome of a general battle between fleets or squadrons. The latter, however, involves a greater variety of incidents, and necessarily requires more comprehensive skill and judgment in the commanding officer.

II. When the admiral, or commander in chief, of a naval armament has discovered an enemy's fleet, his principal concern is usually to approach it, and endeavour to come to action as soon as possible. Every inferior consideration must be sacrificed to this important object; and every rule of action should tend to hasten and prepare for so material an event. The state of the wind, and the situation of his adversary, will in some measure dictate the conduct necessary to be pursued with regard to the disposition of his ships on this occasion. To facilitate the execution of the admiral's orders, the whole fleet is ranged into three squadrons, each of which is classed into three divisions, under the command of different officers. Before the action begins, the adverse fleets are drawn up in two lines, as above described. As soon as the admiral displays the signal for the line of battle, the several divisions separate from the columns, in which they were disposed in the usual order of sailing, and every ship crowds into its station in the wake of the next ahead; and a proper distance from each other, which is generally about 50 fathoms, is regularly observed from the van to the rear. The admiral, however, will occasionally contract or extend his line, so as to conform to the length of that of his adversary, whose neglect or inferior skill on this occasion he will naturally convert to his own advantage, as well as to prevent his own line from being doubled, a circumstance which might throw his van and rear into confusion.

When the adverse fleets approach each other, the courses are commonly hauled up in the brails, and the top-gallant sails and stay-sails furled. The movement of each ship is chiefly regulated by the main and fore-top sails, and the jib; the mizen-top sail being referred

Economy of a naval Engagement. ved to hasten or retard the course of the ship, and, in fine, by filling or backing, hoisting or lowering it, to determine her velocity.

The frigates, tenders, and fire-ships, being also hauled upon a wind, lie at some distance, ready to execute the admiral's orders or those of his seconds, leaving the line of battle between them and the enemy. If there are any transports and storeships attendant on the fleet, these are disposed still further distant from the action. If the fleet is superior in number to that of the enemy, the admiral usually selects a body of reserve from the different squadrons, which will always be of use to cover the fire-ships, bomb-vessels, &c. and may fall into the line in any case of necessity: these also are stationed at a convenient distance from the line, and should evidently be opposite to the weakest parts thereof.

And here it may not be improper to observe, with an ingenious French author (M. de Morogues), that order and discipline give additional strength and activity to a fleet. If thus a double advantage is acquired by every fleet, it is certainly more favourable to the inferior, which may thereby change its disposition with greater facility and dispatch than one more numerous, yet without being separated. When courage is equal to both, good order is then the only resource of the smaller number. Hence we may infer, that a smaller squadron of men of war, whose officers are perfectly disciplined in working their ships, may, by its superior dexterity, vanquish a more powerful one, even at the commencement of the fight; because the latter, being less expert in the order of battle, will, by its separation, suffer many of the ships to remain useless, or not sufficiently near, to protect each other.

The signal for a general engagement is usually displayed when the opposite fleets are sufficiently within the range of point-blank shot, so that they may level the artillery with certainty of execution, which is near enough for a line of battle. The action is begun and carried on throughout the fleet in the manner we have already described between single ships; at which time the admiral carries little sail, observing, however, to regulate his own motions by those of the enemy. The ships of the line meanwhile keep close in their stations, none of which should hesitate to advance in their order, although interrupted by the situation of some ship a-head, which has negligently fallen a-shore of her station.

The various exigencies of the combat call forth the skill and resources of the admiral, to keep his line as complete as possible when it has been unequally attacked; by ordering ships from those in reserve to supply the place of others which have suffered greatly by the action; by directing his fire-ships at a convenient time to fall aboard the enemy; by detaching ships from one part of the line or wing which is stronger, to another which is greatly pressed by superior force, and requires assistance. His vigilance is ever necessary to review the situation of the enemy from van to rear; every motion of whom he should, if possible, anticipate and frustrate. He should seize the favourable moments of occasion, which are rapid in their progress, and never return. Far from being disconcerted by any unforeseen incident, he should endeavour, if possible, to make it subservient to his de-

sign. His experience and reflection will naturally furnish him with every method of intelligence to discover the state of his different squadrons and divisions. Signals of inquiry and answers, of request and assent, of command and obedience, will be displayed and repeated on this occasion. Tenders and boats will also continually be detached between the admiral and the commanders of the several squadrons or divisions.

As the danger presses on him, he ought to be fortified by resolution and presence of mind; because the whole fleet is committed to his charge, and the conduct of his officers may, in a great degree, be influenced by his intrepidity and perseverance. In short, his renown or infamy may depend on the fate of the day.

If he conquers in battle, he ought to prosecute his victory as much as possible, by seizing, burning, or destroying the enemy's ships. If he is defeated, he should endeavour, by every resource his experience can suggest, to save as many of his fleet as possible, by employing his tenders, &c. to take out the wounded and put fresh men in their places; by towing the disabled ships to a competent distance; and by preventing the execution of the enemy's fire-ships.

By what we have observed, the real force or superiority of a fleet consists less in the number of vessels, and the vivacity of the action, than in good order, dexterity in working the ships, presence of mind, and skilful conduct in the captains.

CHAP. VIII. Of Retreat and Chace.

I. WHEN a fleet is obliged to retreat in sight of an enemy, the best way to effect it securely will be by sailing in a kind of half-moon, the admiral making the obtuse angle A, and to windward in the form Plate CC. BAC; one part of his fleet to sail on the starboard, whilst the other goes away on the larboard tack: keeping the fire-ships, transports, &c. in the middle.

This manner of ranging a fleet seems the most advisable, because the enemy can never approach those that endeavour to escape, without exposing themselves at the same time to the fire of the ships to windward: thus the enemy's ships D can never approach the ships E without exposing themselves at the same time to the fire of the admiral A, as likewise to that of his seconds. If the admiral thinks this form gives too great an extent to his fleet, he may easily close his wings or quarters, and make the half-moon more complete; in the midst of which he may place his convoy in safety.

EXAMPLE. This order of retreat was exactly followed by the Dutch admiral Van Tromp, in his engagement with the English off Portland, in the year 1653. The English fleet consisted of 70 sail, under the command of admiral Blake; and the Dutch were as strong, convoying about 200 rich merchant-ships. The two fleets met off Portland, where the English used their utmost efforts to bring on an action. The Dutch had the advantage of the wind, and endeavoured to avoid an engagement to preserve their convoy; but Van Tromp, considering rightly, that if the wind should happen to change, he must be under the necessity of fighting with less advantage, determined upon bearing down to the enemy, making his signal

at the same time for the convoy to keep to windward: he then divided his fleet into three squadrons, and attacked the enemy with great bravery: they received him with equal resolution; which made the action very desperate on both sides, several ships being disabled, sunk, burnt, or destroyed; and nothing but the darkness of the approaching night could separate two such obstinate enemies: during which, each prepared to renew the action, that still remained undecided, with greater fury the next day. Van Tromp found himself much embarrassed how to act; but, after many deliberations, he resolved at last upon that of a retreat: he therefore drew up his fleet into a half-moon, placing his convoy in the middle; that is to say, his own ship to windward formed the obtuse angle of the half-moon, the rest ranging themselves on two lines upon a wind on the same tack, in order to form the faces of the half-moon to cover their convoy. He then made what sail he could, and went away large, firing to the right and left of him at all those that attempted to insult his wings or quarters; and would have entirely saved his convoy, if some of his ships had not basely deserted him. The English ships immediately took the advantage of the intervals these ships left in the face of their floating half-moon, and carried off several of their merchant-ships; which obliged admiral Tromp to replace himself in the line of battle as before, and continue the engagement till night should give him an opportunity of returning his order of retreat. He was chaced the next day by the enemy; but, after sustaining a few broadsides, he got safe into harbour, having acquired by his great valour and skill a rich convoy to his country, that nar-

The most natural course in this form of sailing is to steer away before the wind: but, if necessary, the ships may bear away large upon either tack, or even may keep close upon a wind.

In flying, or retreating, the uncertainty of the weather is to be considered: it may become calm, or the wind may shift favourably. The admiral's schemes may be assisted by the approach of night, or the proximity of the land; since he ought rather to run the ships ashore, if practicable, than suffer them to be taken afloat, and thereby transfer additional strength to the enemy. In short, nothing should be neglected that may contribute to the preservation of his fleet, or prevent any part of it from falling into the hands of the conqueror.

II. When you chace a fleet that endeavours to escape, you detach your best cruisers after them, in order to pick up the stragglers, or force them to action; the body of the victorious fleet should keep the same order or line with the enemy, as nigh as possible, to be ready for action, if necessary. This is only to be understood when the fleet that is chaced may not be so inferior to the other but that it may hazard an action; for if the one bears no proportion to the other, they must bear down upon them in the same manner as a conquering army ashore carries all before it when it has forced an enemy's camp: otherwise, were the conquerors to wait to draw up in form, the enemy would undoubtedly take the advantage of such an opportunity to make their escape.

Navarre,
Naude.

NAVARRE, a province of Spain, part of the ancient kingdom of Navarre, erected soon after the invasion of the Moors; and is otherwise called Upper Navarre, to distinguish it from Lower Navarre, belonging to the French. It is bounded on the south and east by Arragon, on the north by the Pyrenees, and on the west by Old Castile and Biscay; extending from south to north about 80 miles, and from east to west about 75. It abounds in sheep and cattle; game of all kinds, as boars, stags, and roebucks; and in wild-fowl, horses, and honey; yielding also some grain, wine, oil, and a variety of minerals, medicinal waters, and hot baths. Some of the ancient chiefs of this country were called Sobrarbores, from the custom, as it is supposed, which prevailed among some of those free nations of choosing and swearing their princes under some particular tree. The name of the province is supposed to be a contraction of Nava Errea, signifying, in the language of the Vascones, its ancient inhabitants, "a land of valleys."—For the particulars of its history, see the article SPAIN.

NAUDE (Gabriel), a critic and physician in the 17th century, was born at Paris; and became librarian to the cardinals Bagni and Antonio Barberini at Rome, and afterwards to cardinal Mazarin, who made

him canon of Verdun and prior of Lartige in Limosin. Christina queen of Sweden at length invited him into her dominions, and bestowed many marks of her favour and esteem upon him. He returned from thence, and died at Abbeville in 1653. His principal works are, 1. Syntagma de studio liberali. 2. Syntagma de studio militari. 3. An apology for the great men who have been accused of magic. 4. Instructions concerning the chimerical society of the Rosicrucians. 5. Advice on collecting a library. 6. An appendix to the life of Lewis XI. 7. The science of princes, or political considerations on the body of a state, &c. Naude's works abound with many curious and interesting particulars.

NAVE, in architecture, the body of a church, where the people are disposed, reaching from the belfry, or rail of the door, to the chief choir. Some derive the word from the Greek ναός, "a temple;" and others from ναυς, "a ship," by reason the vault or roof of a church bears some resemblance to a ship.

NAVEL, in anatomy, the centre of the lower part of the abdomen; being that part where the umbilical vessels passed out of the placenta of the mother. See ANATOMY, p. 366. note (G).

NAVEL-Word, in botany. See COTYLEDON.

NAVEW, in botany. See BRASSICA.

NAVIGATION, is the art of conducting or carrying a ship from one port to another.

THE poets refer the invention of the art of navigation to Neptune, some to Bacchus, others to Hercules, others to Jason, and others to Janus, who is said to have made the first ship. Historians ascribe it to the Æginætes, the Phœnicians, Tyrians, and the ancient inhabitants of Britain. Some will have it, the first hint was taken from the flight of the kite; others, as Oppian, (De piscibus, lib. i.) from the fish called nautilus: others ascribe it to accident.—Scripture refers the origin of so useful an invention to God himself, who gave the first specimen thereof in the ark built by Noah under his direction. For the raillery the good man underwent on account of his enterprise shews evidently enough the world was then ignorant of anything like navigation, and that they even thought it impossible.

However, history represents the Phœnicians, especially those of their capital Tyre, as the first navigators; being urged to seek a foreign commerce by the narrowness and poverty of a slip of ground they possessed along the coasts; as well as by the convenience of two or three good ports, and by their natural genius to traffic. Accordingly, Lebanon, and the other neighbouring mountains, furnishing them with excellent wood for ship-building, in a short time they were masters of a numerous fleet, which constantly hazarding new navigations, and settling new trades, they soon arrived at an incredible pitch of opulence and populousness: infomuch as to be in a condition to send out colonies, the principal of which was that of Carthage; which, keeping up their Phœnician spirit of commerce, in time not only equalled Tyre itself, but vastly surpassed it; sending its merchant-fleets through Hercules's pillars, now the straits of Gibraltar, along the western coasts of Africa and Europe; and even, if we believe some authors, to America itself.

Tyre, whose immense riches and power are represented in such lofty terms both in sacred and profane authors, being destroyed by Alexander the Great, its navigation and commerce were transferred by the conqueror to Alexandria, a new city, admirably situated for those purposes; proposed for the capital of the empire of Asia, which Alexander then meditated. And thus arose the navigation of the Egyptians; which was afterwards so cultivated by the Ptolemies, that Tyre and Carthage were quite forgot.

Egypt being reduced into a Roman province after the battle of Actium, its trade and navigation fell into the hands of Augustus; in whose time Alexandria was only inferior to Rome: and the magazines of the capital of the world, were wholly supplied with merchandizes from the capital of Egypt.

At length, Alexandria itself underwent the fate of Tyre and Carthage; being surprised by the Saracens, who, in spite of the emperor Heraclius, overspread

the northern coasts of Africa, &c. whence the merchants being driven, Alexandria has ever since been in a languishing state, though still it has a considerable part of the commerce of the Christian merchants trading to the Levant.

The fall of Rome and its empire drew along with it not only that of learning and the polite arts, but that of navigation; the barbarians, into whose hands it fell, contenting themselves with the spoils of the industry of their predecessors.

But no sooner were the more brave among those nations well settled in their new provinces; some in Gaul, as the Franks; others in Spain, as the Goths; and others in Italy, as the Lombards; but they began to learn the advantages of navigation and commerce, and the methods of managing them, from the people they subdued; and this with so much success, that in a little time some of them became able to give new lessons, and set on foot new institutions for its advantage. Thus it is to the Lombards we usually ascribe the invention and use of banks, book-keeping, exchanges, recharges, &c.

It does not appear which of the European people, after the settlement of their new masters, first betook themselves to navigation and commerce.—Some think it began with the French; though the Italians seem to have the justest title to it, and are accordingly ordinarily looked on as the restorers hereof, as well as of the polite arts, which had been banished together from the time the empire was torn asunder. It is the people of Italy then, and particularly those of Venice and Genoa, who have the glory of this restoration; and it is to their advantageous situation for navigation they in great measure owe their glory. In the bottom of the Adriatic were a great number of marshy islands, only separated by narrow channels, but those well screened, and almost inaccessible, the residence of some fishermen, who here supported themselves by a little trade of fish and salt, which they found in some of these islands. Thither the Veneti, a people inhabiting that part of Italy along the coasts of the gulph, retired, when Alaric king of the Goths, and afterwards Attila king of the Huns, ravaged Italy.

These new islanders, little imagining that this was to be their fixed residence, did not think of composing any body politic; but each of the 72 islands of this little Archipelago continued a long time under its several masters, and each made a distinct commonwealth. When their commerce was become considerable enough to give jealousy to their neighbours, they began to think of uniting into a body. And it was this union, first begun in the sixth century, but not completed till the eighth, that laid the sure foundation of the future grandeur of the state of Venice. From the time of this union, their fleets of merchantmen were sent to all the parts of the Mediterranean; and at last to those of Egypt, particularly Cairo, a new city, built by the Saracen princes on the eastern banks of the Nile, where they traded for their spices and other products of the Indies. Thus they flourished, increased their

commerce, their navigation, and their conquests on the terra firma, till the league of Cambrai in 1508, when a number of jealous princes conspired to their ruin; which was the more easily effected by the diminution of their East-India commerce, of which the Portuguese had got one part, and the French another. Genoa, which had applied itself to navigation at the same time with Venice, and that with equal success, was a long time its dangerous rival, disputed with it the empire of the sea, and shared with it the trade of Egypt and other parts both of the east and west.

Jealousy soon began to break out; and the two republics coming to blows, there was almost continual war for three centuries ere the superiority was ascertained; when, towards the end of the 14th century, the battle of Chioza ended the strife: the Genoese, who till then had usually the advantage, having now lost all; and the Venetians, almost become desperate, at one happy blow, beyond all expectation, secured to themselves the empire of the sea, and superiority in commerce.

About the same time that navigation was retrieved in the southern parts of Europe, a new society of merchants was formed in the north, which not only carried commerce to the greatest perfection it was capable of till the discovery of the East and West Indies, but also formed a new scheme of laws for the regulation thereof, which still obtain under the names of Uses and Customs of the Sea. This society is that famous league of the Hanse-towns, commonly supposed to have begun about the year 1164. See HANSE Towns.

For the modern state of navigation in England, Holland, France, Spain, Portugal, &c. See COMMERCE, COMPANY, &c.

We shall only add, that, in examining the reasons of commerce's passing successively from the Venetians, Genoese, and Hanse-towns, to the Portuguese and Spaniards, and from these again to the English and Dutch; it may be established as a maxim, that the relation between commerce and navigation, or, if we may be allowed to say it, their union is so intimate, that the fall of the one inevitably draws after it the other; and that they will always either flourish or dwindle together. Hence so many laws, ordinances, statutes, &c. for its regulation; and hence particularly that celebrated act of navigation, which an eminent foreign author calls the palladium or tutelary deity of the commerce of England; which is the standing rule, not only of the British among themselves, but also of other nations with whom they traffic.

The art of navigation hath been exceedingly improved in modern times, both with regard to the form of the vessels themselves, and with regard to the methods of working them. The use of rowers is now entirely superseded by the improvements made in the formation of the sails, rigging, &c. by which means the ships can not only sail much faster than formerly, but can tack in any direction with the greatest facility. It is also very probable that the ancients were neither so well skilled in finding the latitudes, nor in steering their vessels in places of difficult navigation, as the moderns. But the greatest advantage which the mo-

derans have over the ancients is from the mariner's compass, by which they are enabled to find their way with as great facility in the midst of an immeasurable ocean, as the ancients could have done by creeping along the coast, and never going out of sight of land. Some people indeed contend, that this is no new invention, but that the ancients were acquainted with it. They say, that it was impossible for Solomon to have sent ships to Ophir, Tarshish, and Parvaim, which last they will have to be Pera, without this useful instrument. They insist, that it was impossible for the ancients to be acquainted with the attractive virtue of the magnet, and to be ignorant of its polarity. Nay, they affirm, that this property of the magnet is plainly mentioned in the book of Job, where the loadstone is mentioned by the name of topaz, or the stone that turns itself. But it is certain, that the Romans, who conquered Judæa, were ignorant of this instrument; and it is very improbable, that such an useful invention, if once it had been commonly known to any nation, would have been forgot, or perfectly concealed from such a prudent people as the Romans, who were so much interested in the discovery of it.

Among those who do agree that the mariner's compass is a modern invention, it hath been much disputed who was the inventor. Some give the honour of it to Flavio Gioia of Amalfi in Campania*, who lived about the beginning of the 14th century; while others say that it came from the east, and was earlier known in Europe. But, at whatever time it was invented, it is certain, that the mariner's compass was not commonly used in navigation before the year 1420. In that year the science was considerably improved under the auspices of Henry duke of Viseo, brother to the king of Portugal. In the year 1485, Roderic and Joseph, physicians to king John II. of Portugal, together with one Martin de Bohemia, a Portuguese native of the island of Fayal, and scholar to Regiomontanus, calculated tables of the sun's declination for the use of sailors, and recommended the astrolabe for taking observations at sea. Of the instructions of Martin, the celebrated Christopher Columbus is said to have availed himself, and to have improved the Spaniards in the knowledge of the art; for the farther progress of which a lecture was afterwards founded at Seville by the emperor Charles V.

The discovery of the variation is claimed by Columbus, and by Sebastian Cabot. The former certainly did observe this variation without having heard of it from any other person, on the 14th of September 1492, and it is very probable that Cabot might do the same. At that time it was found that there was no variation at the Azores, where some geographers have thought proper to place the first meridian; though it hath since been observed that the variation alters in time.—The use of the cross-staff now began to be introduced among sailors. This ancient instrument is described by John Werner of Nuremberg, in his annotations on the first book of Ptolemy's Geography, printed in 1514. He recommends it for observing the distance between the moon and some star, in order thence to determine the longitude.

At this time the art of navigation was very imperfect on account of the inaccuracies of the plane chart, which

which was the only one then known, and which, by its gross errors, must have greatly misled the mariner, especially in voyages far distant from the equator. Its precepts were probably at first only set down on the sea-charts, as is the custom at this day: but at length there were two Spanish treatises published in 1545; one by Pedro de Medina; the other by Martin Cortes, which contained a complete system of the art, as far as it was then known. These seem to have been the oldest writers who fully handled the art; for Medina, in his dedication to Philip prince of Spain, laments that multitudes of ships daily perished at sea, because there were neither teachers of the art, nor books by which it might be learned; and Cortes, in his dedication, boasts to the emperor, that he was the first who had reduced navigation into a compendium, valuing himself much on what he had performed. Medina defended the plane chart; but he was opposed by Cortes, who shewed its errors, and endeavoured to account for the variation of the compass, by supposing the needle to be influenced by a magnetic pole (which he called the point attractive) different from that of the world: which notion hath been farther prosecuted by others. Medina's book was soon translated into Italian, French, and Flemish, and served for a long time as a guide to foreign navigators. However Cortes was the favourite author of the English nation, and was translated in 1561; while Medina's work was entirely neglected, though translated also within a short time of the other. At that time the system of navigation consisted of the following particulars, and others similar: An account of the Ptolemaic hypothesis, and the circles of the sphere; of the roundness of the earth, the longitudes, latitudes, climates, &c. and eclipses of the luminaries; a calendar; the method of finding the prime, epoch, moon's age, and tides; a description of the compass, an account of its variation, for the discovering of which Cortes said an instrument might easily be contrived; tables of the sun's declination for four years, in order to find the latitude from his meridian altitude; directions to find the same by certain stars: of the course of the sun and moon; the length of the days; of time and its divisions; the method of finding the hour of the day and night; and lastly, a description of the sea-chart, on which to discover where the ship is, they made use of a small table, that shewed, upon an alteration of one degree of the latitude, how many leagues were run in each rhumb, together with the departure from the meridian. Besides, some instruments were described, especially by Cortes; such as one to find the place and declination of the sun, with the days, and place of the moon; certain dials, the astrolabe, and cross-staff; with a complex machine to discover the hour and latitude at once.

About the same time were made proposals for finding the longitude by observations on the moon. In 1530, Gemma Frisius advised the keeping of the time by means of small clocks or watches, then, as he says, newly invented. He also contrived a new sort of cross-staff and an instrument called the nautical quadrant; which last was much praised by William Cuningham, in his Astronomical Glasi, printed in the year 1559.

In 1537 Pedro Nunez, or Nonius, published a book

in the Portuguese language, to explain a difficulty in navigation proposed to him by the commander Don Martin Alphonso de Sufa. In this he exposes the errors of the plane chart, and likewise gives the solution of several curious astronomical problems; amongst which is that of determining the latitude from two observations of the sun's altitude and intermediate azimuth being given. He observed, that though the rhumbs are spiral lines, yet the direct course of a ship will always be in the arch of a great circle, whereby the angle with the meridians will continually change: all that the steersman can here do for the preserving of the original rhumb, is to correct these deviations as soon as they appear sensible. But thus the ship will in reality describe a course without the rhumb-line intended; and therefore his calculations for assigning the latitude, where any rhumb-line crosses the several meridians, will be in some measure erroneous. He invented a method of dividing a quadrant by means of concentric circles, which, after being much improved by Dr Halley, is used at present, and is called a nonius.

In 1577, Mr William Bourne published a treatise, in which, by considering the irregularities in the moon's motion, he shews the errors of the sailors in finding her age by the epoch, and also in determining the hour from observing on what point of the compass the sun and moon appeared. He advises, in sailing towards the high latitudes, to keep the reckoning by the globe, as there the plane chart is most erroneous. He despairs of our ever being able to find the longitude, unless the variation of the compass should be occasioned by some such attractive point, as Cortes had imagined; of which, however, he doubts: but as he had shewn how to find the variation at all times, he advises to keep an account of the observations, as useful for finding the place of the ship; which advice was prosecuted at large by Simon Stevin in a treatise published at Leyden in 1599; the subject of which was the same year printed at London in English by Mr Edward Wright, intitled the Haven-finding Art. In this ancient tract also is described the way by which our sailors estimate the rate of a ship in her course, by an instrument called the log. This was so named from the piece of wood or log that floats in the water while the time is reckoned during which the line that is fastened to it is veering out. The author of this contrivance is not known; neither was it taken notice of till 1607, in an East-India voyage published by Purchas: but from this time it became famous, and was much taken notice of by almost all writers on navigation in every country; and it still continues to be used as at first, though many attempts have been made to improve it, and contrivances proposed to supply its place; many of which have succeeded in quiet water, but proved useless in a stormy sea.

In 1581 Michael Coignet, a native of Antwerp, published a treatise in which he animadverted on Medina. In this he shewed, that as the rhumbs are spirals, making endless revolutions about the poles, numerous errors must arise from their being represented by straight lines on the sea-charts; but though he hoped to find a remedy for these errors, he was of opinion that the proposals of Nonius were scarcely practicable, and therefore in a great measure useless. In treating

treating of the sun's declination, he took notice of the gradual decrease in the obliquity of the ecliptic; he also described the cross-staff with three transverse pieces, as it is at present made, and which he owned to have been then in common use among the sailors. He likewise gave some instruments of his own invention; but all of them are now laid aside, excepting perhaps his nocturnal. He constructed a sea-table to be used by such as sailed beyond the 60th degree of latitude; and at the end of the book is delivered a method of sailing on a parallel of latitude by means of a ring dial and a 24 hour-glass. The same year the discovery of the dipping needle was made by Mr Robert Forman*. In his publication on that art he maintains, in opposition to Cortes, that the variation of the compass was caused by some point on the surface of the earth, and not in the heavens: he also made considerable improvements in the construction of compasses themselves; shewing especially the danger of not fixing, on account of the variation, the wire directly under the flower-de-luce; as compasses made in different countries have placed it differently. To this performance of Forman's is always prefixed a discourse on the variation of the magnetic needle, by Mr William Burrough, in which he shews how to determine the variation in many different ways. He also points out many errors in the practice of navigation at that time, and speaks in very severe terms concerning those who had published upon it. All this time the Spaniards had continued to publish treatises on the art. In 1585 an excellent compendium was published by Roderico Zamorano; which contributed greatly towards the improvement of the art, particularly in the sea-charts. Globes of an improved kind, and of a much larger size than those formerly used, were now constructed, and many improvements were made in other instruments; however, the plane chart continued still to be followed, though its errors were frequently complained of. Methods of removing these errors had indeed been sought after; and Gerard Mercator seems to have been the first who found the true method of doing this so as to answer the purposes of seamen. His method was to represent the parallels both of latitude and longitude by parallel straight lines, but gradually to augment the former as they approached the pole. Thus the rhumbs, which otherwise ought to have been curves, were now also extended into straight lines; and thus a straight line drawn between any two places marked upon the chart would make an angle with the meridians, expressing the rhumb leading from the one to the other. But though, in 1569, Mercator published an universal map constructed in this manner, it doth not appear that he was acquainted with the principles on which this proceeded; and it is now generally believed, that the true principles on which the construction of what is called Mercator's chart depends, were first discovered by an Englishman, Mr Edward Wright.

Mr Wright supposes, but, according to the general opinion, without sufficient grounds, that this enlargement of the degrees of latitude was known and mentioned by Ptolemy, and that the same thing had also been spoken of by Cortes. The expressions of Ptolemy alluded to, relate indeed to the proportion between the distances of the parallels and meridians; but instead of proposing any gradual enlargement of the

parallels of latitude, in a general chart, he speaks only of particular maps; and advises not to confine a system of such maps to one and the same scale, but to plan them out by a different measure, as occasion might require: only with this precaution, that the degrees of longitude in each should bear some proportion to those of latitude; and this proportion is to be deduced from that which the magnitude of the respective parallels bear to a great circle of the sphere. He adds, that in particular maps, if this proportion be observed with regard to the middle parallel, the inconvenience will not be great tho' the meridians should be straight parallels to each other. Here he is said only to mean, that the maps should in some measure represent the figures of the countries for which they are drawn. In this sense Mercator, who drew maps for Ptolemy's tables, understood him: thinking it, however, an improvement not to regulate the meridians by one parallel, but by two; one distant from the northern, the other from the southern extremity of the map by a fourth part of the whole depth; by which means, in his maps, though the meridians are straight lines, yet they are generally drawn inclining to each other towards the poles. With regard to Cortes, he speaks only of the number of degrees of latitude, and not of the extent of them; nay, he gives express directions that they should all be laid down by equal measurement on a scale of leagues adapted to the map.

For some time after the appearance of Mercator's map, it was not rightly understood, and it was even thought to be entirely useless, if not detrimental. However, about the year 1592, its utility began to be perceived; and seven years after, Mr Wright printed his famous treatise entitled, The Correction of certain Errors in Navigation; where he fully explained the reason of extending the length of the parallels of latitude, and the uses of it to navigators. In 1610, a second edition of Mr Wright's book was published with improvements. An excellent method was proposed of determining the magnitude of the earth; at the same time it was judiciously proposed to make our common measures in some proportion to a degree on its surface, that they might not depend on the uncertain length of a barley-corn. Some of his other improvements were, "The table of latitudes for dividing the meridian computed to minutes;" whereas it had only been divided to every tenth minute. He also published a description of an instrument which he calls the sea-rings; and by which the variation of the compass, altitude of the sun, and time of the day, may be determined readily at once in any place, provided the latitude is known. He shewed also how to correct the errors arising from the excentricity of the eye in observing by the cross-staff. He made a total amendment in the tables of the declinations and places of the sun and stars from his own observations made with a six-foot quadrant in the years 1594, 95, 96, and 97. A sea-quadrant to take altitudes by a forward or backward observation; and likewise with a contrivance for the ready finding the latitude by the height of the pole-star, when not upon the meridian. To this edition was subjoined a translation of Zamorano's Compendium above-mentioned; in which he corrected some mistakes in the original; adding a large table of the variation of the compass observed in very different parts

* See Dipping-needle.

parts of the world, to shew that it was not occasioned by any magnetical pole.

These improvements soon became known abroad. In 1608, a treatise intitled, Hyponnemata Mathematica, were published by Simon Stevin, for the use of Prince Maurice. In that part relating to navigation, the author having treated of sailing on a great circle, and shewn how to draw the rhumbs on a globe mechanically, sets down Wright's two tables of latitude and of rhumbs, in order to describe these lines more accurately, pretending even to have discovered an error in Wright's table. But all Stevin's objections were fully answered by the author himself, who shewed that they arose from the gross way of calculating made use of by the former.

In 1624, the learned Wellebrordus Snellius, professor of mathematics at Leyden, published a treatise of navigation on Wright's plan, but somewhat obscurely; and as he did not particularly mention all the discoveries of Wright, the latter was thought by some to have taken the hint of all his discoveries from Snellius. But this supposition is long ago refuted; and Wright enjoys the honour of those discoveries which is justly his due.

Mr Wright having shewn how to find the place of the ship on his chart, observed that the same might be performed more accurately by calculation: but considering, as he says, that the latitudes, and especially the courses at sea, could not be determined so precisely, he forbore setting down particular examples; as the mariner may be allowed to save himself this trouble, and only mark out upon his chart the ship's way after the manner then usually practised. However, in 1614, Mr Raphe Handson, among his nautical questions subjoined to a translation of Pitiscus's trigonometry, solved very distinctly every case of navigation, by applying arithmetical calculations to Wright's table of latitudes, or of meridional parts, as it hath since been called. Though the method discovered by Wright for finding the change of longitude by a ship sailing on a rhumb is the proper way of performing it, Handson also proposes two ways of approximation to it without the assistance of Wright's division of the meridian line. The first was computed by the arithmetical mean between the co-sines of both latitudes; the other by the same mean between the secants as an alternative, when Wright's book was not at hand; tho' this latter is wider from the truth than the first. By the same calculations also he shewed how much each of these compendiums deviates from the truth, and also how widely the computations on the erroneous principles of the plane chart differ from them all. The method, however, commonly used by our sailors is commonly called the middle latitude; which, though it errs more than that by the arithmetical mean between the two co-sines, is preferred on account of its being less operative: yet in high latitudes it is more eligible to use that of the arithmetical mean between the logarithmic co-sines, equivalent to the geometrical mean between the co-sines themselves; a method since proposed by Mr John Bassat. The computation by the middle latitude will always fall short of the true change of longitude; that by the geometrical mean will always exceed; but that by the arithmetical mean falls short in latitudes above 45 degrees, and exceeds in les-

ser latitudes. However, none of these methods will differ much from the truth when the change of latitude is sufficiently small.

About this time logarithms were invented by John Napier, baron of Merchiston in Scotland, and proved of the utmost service to the art of navigation. They were first applied by Mr Edward Gunter in 1620. He constructed a table of artificial sines and tangents to every minute of the quadrant. These were applied according to Wright's table of meridional parts, and have been found extremely useful in other branches of the mathematics. He contrived also a most excellent ruler, commonly known by the name of Gunter's scale, on which were inscribed the logarithmic lines for numbers, and for sines and tangents of arches*. He also greatly improved the sector for the same purposes. He shewed also how to take a back-observation by the cross-staff, whereby the error arising from the eccentricity of the eye is avoided. He described likewise another instrument of his own invention, called the cross-beam, for taking altitudes of the sun or stars, with some contrivances for the more ready collecting the latitude from the observation. The discoveries concerning the logarithms were carried to France in 1624 by Mr Edmund Wingate, who published two small tracts in that year at Paris. In one of these he taught the use of Gunter's scale; and in the other, of the tables of artificial sines and tangents, as modelled according to Napier's last form, erroneously attributed by Wingate to Briggs.

Gunter's ruler was projected into a circular arch by the reverend Mr William Oughtred in 1633, and its uses fully shown in a pamphlet intitled, The circles of proportion; where, in an appendix, are well handled several important points in navigation. It has also been made in the form of a sliding ruler.

The logarithmic tables were first applied to the different cases of sailing by Mr Thomas Addison, in his treatise intitled, Arithmetical navigation, printed in 1625. He also gives two traverse tables, with their uses; the one to quarter points of the compass, the other to degrees. Mr Henry Gellibrand published his discovery of the changes of the variation of the compass, in a small quarto pamphlet, intitled, A discourse mathematical on the variation of the magnetical needle, printed in 1635. This extraordinary phenomenon he found out by comparing the observations made at different times near the same place by Mr Burrough, Mr Gunter, and himself, all persons of great skill and experience in these matters. This discovery was likewise soon known abroad; for Father Athanasius Kircher, in his treatise intitled, Magnes, first printed at Rome in 1641, informs us, that he had been told it by Mr John Greaves; and then gives a letter of the famous Marinus Mersennus, containing a very distinct account of the same.

As altitudes of the sun are taken on shipboard by observing his elevation above the visible horizon, to collect from thence the sun's true altitude with correctness, Wright observes it to be necessary that the dip of the horizon below the observer's eye should be brought into the account, which cannot be calculated without knowing the magnitude of the earth. Hence he was induced to propose different methods for finding this; but complains that the most effectual was out

of his power to execute; and therefore contented himself with a rude attempt, in some measure sufficient for his purpose: and the dimensions of the earth deduced by him corresponded so well with the usual divisions of the log-line, that as he writ not an express treatise on navigation, but only for correcting such errors as prevailed in general practice, the log-line did not fall under his notice. Mr Richard Norwood, however, put in execution the method recommended by Mr Wright as the most perfect for measuring the dimensions of the earth, with the true length of the degrees of a great circle upon it; and, in 1635, he actually measured the distance between London and York; from whence, and the summer solstitial altitudes of the sun observed on the meridian at both places, he found a degree on a great circle of the earth to contain 367,196 English feet, equal to 57,300 French fathoms or toises: which is very exact, as appears from many measures that have been made since that time. Of all this Mr Norwood gave a full account in his treatise called The seaman's practice, published in 1637. He there shows the reason why Snellius had failed in his attempt; he points out also various uses of his discovery, particularly for correcting the gross errors hitherto committed in the divisions of the log-line. These necessary amendments, however, were little attended to by the sailors, whose obstinacy in adhering to established errors has been complained of by the best writers on navigation; but at length they found their way into practice, and few navigators of reputation now make use of the old measure of 42 feet to a knot. In that treatise also Mr Norwood describes his own excellent method of setting down and perfecting a sea-reckoning, by using a traverse table; which method he had followed and taught for many years. He shows also how to rectify the course by the variation of the compass being considered; as also how to discover currents, and to make proper allowance on their account. This treatise, and another on trigonometry, were continually reprinted, as the principal books for learning scientifically the art of navigation. What he had delivered, especially in the latter of them, concerning this subject, was contracted as a manual for sailors, in a very small piece called his Epitome; which useful performance has gone through a great number of editions. No alterations were ever made in the Seaman's Practice till the 12th edition in 1676, when the following paragraph was inserted in a smaller character: "About the year 1672, Monsieur Picart has published an account in French, concerning the measure of the earth, a breviary whereof may be seen in the Philosophical Transactions, n^o 112; wherein he concludes one degree to contain 365,184 English feet, nearly agreeing with Mr Norwood's experiment;" and this advertisement is continued through the subsequent editions as late as the year 1732. About the year 1645, Mr Bond published in Norwood's epitome a very great improvement in Wright's method by a property in his meridian line, whereby its divisions are more scientifically assigned than the author himself was able to effect; which was from this theorem, that these divisions are analogous to the excesses of the logarithmic tangents of half the respective latitudes augmented by 45 degrees above the logarithm of the radius. This he afterwards explained more fully in

the third edition of Gunter's works, printed in 1653; where, after observing that the logarithmic tangents from 45° upwards increase in the same manner that the secants added together do; if every half degree be accounted as a whole degree of Mercator's meridional line. His rule for computing the meridional parts belonging to any two latitudes, supposed on the same side of the equator, is to the following effect. "Take the logarithmic tangent, rejecting the radius, of half each latitude, augmented by 54 degrees; divide the difference of those numbers by the logarithmic tangent of 50° 30', the radius being likewise rejected; and the quotient will be the meridional parts required, expressed in degrees." This rule is the immediate consequence from the general theorem. That the degrees of latitude bear to one degree, (or 60 minutes, which in Wright's table stands for the meridional parts of one degree), the same proportion as the logarithmic tangent of half any latitude augmented by 45 degrees, and the radius neglected, to the like tangent of half a degree augmented by 45 degrees, with the radius likewise rejected. But here was farther wanting the demonstration of this general theorem, which was at length supplied by Mr James Gregory of Aberdeen, in his Exercitationes Geometricæ, printed at London in 1668; and afterwards more concisely demonstrated, together with a scientific determination of the divisor, by Dr Halley in the Philosophical Transactions for 1695, n^o 219. from the consideration of the spirals into which the rhumbs are transformed in the stereographic projection of the sphere upon the plane of the equinoctial; and which is rendered still more simple by Mr Roger Cotes, in his Logometria, first published in the Philosophical Transactions for 1714, n^o 388. It is moreover added in Gunter's book, that if $\backslash frac\{1\}\{2\}$ of this division, which does not sensibly differ from the logarithmic tangent of 45° 1' 30" (with the radius subtracted from it), be used, the quotient will exhibit the meridional parts expressed in leagues: and this is the divisor set down in Norwood's Epitome. After the same manner the meridional parts will be found in minutes, if the like logarithmic tangent of 45° 1' 30", diminished by the radius, be taken; that is, the number used by others being 12633, when the logarithmic tables consist of eight places of figures besides the index. In an edition of the seaman's calendar, Mr Bond declared, that he had discovered the longitude by having found out the true theory of the magnetic variation; and to gain credit to his assertion, he foretold, that at London, in 1657, there would be no variation of the compass, and from that time it would gradually increase the other way; which happened accordingly. Again, in the Philosophical Transactions for 1668, n^o 40. he published a table of the variation for 49 years to come. Thus he acquired such reputation, that his treatise, intitled, The longitude found, was, in 1676, published by the special command of Charles II. and approved by many celebrated mathematicians. It was not long, however, before it met with opposition; and, in 1678, another treatise, intitled, The longitude not found, made its appearance; and as Mr Bond's hypothesis did not in any manner answer its author's sanguine expectations, the affair was undertaken by Dr Halley. The result of his speculations was, that the magnetic needle is influenced

THEORY. influenced by four poles; but this wonderful phenomenon seems hitherto to have eluded all our researches. In 1700, however, Dr Halley published a general map, with curve lines expressing the paths where the magnetic needle had the same variation; which was received with universal applause. But as the positions of these curves vary from time to time, they should frequently be corrected by skilful persons; as was done in 1744 and 1756, by Mr William Mountain, and Mr James Dodson, F. R. S. In the Philosophical Transactions for 1690, Dr Halley also gave a dissertation on the monsoons; containing many very useful observations for such as sail to places subject to these winds.

After the true principles of the art were settled by Wright, Bond, and Norwood, the authors on navigation became so numerous, that it would be impossible to enumerate them; and every thing relative to it was settled with an accuracy not only unknown to former ages, but which would have been

reckoned utterly impossible. The earth being found THEORY. to be a spheroid, and not a perfect sphere, with the shortest diameter passing through the poles, a tract was published in 1741 by the Rev. Dr Patrick Murdoch, wherein he accommodated Wright's sailing to such a figure; and Mr Colin Maclaurin, the same year, in the Philosophical Transactions, n^o 461, gave a rule for determining the meridional parts of a spheroid; which speculation is farther treated of in his book of Fluxions, printed at Edinburgh in 1742.

Among foreign nations also many treatises were now published; but excepting the remarkable discovery of the longitude by Mr Harrison, no considerable improvement hath been made any-where. Indeed, the subject hath been so much canvassed and studied by men of learning and ingenuity in all nations, that there seems to be little room for farther improvements; and the art of navigation seems to be nearly brought to as much perfection as it is capable of.

PART I. THEORY OF NAVIGATION.

THE motion of a ship in the water is well known to depend on the action of the wind upon its sails, regulated by the direction of the helm. As the water is a resisting medium, and the bulk of the ship very considerable, it thence follows, that there is always a great resistance on her fore-part; and when this resistance becomes sufficient to balance the moving force of the wind upon the sails, the ship attains her utmost degree of velocity, and her motion is no longer accelerated. This velocity is different according to the different strength of the wind: but the stronger the wind, the greater resistance is made to the ship's passage through the water; and hence, though the wind should blow ever so fiercely, there is also a limit to the velocity of the ship: for the sails and ropes can bear but a certain force of air; and when the resistance on the fore-part becomes more than equivalent to their strength, the velocity can be no longer increased, and the tackle gives way.

The direction of a ship's motion depends on the situation of her sails with regard to the wind. The most natural and easy position is, when she runs directly before it; but this is not often the case, on account of the variable nature of the winds, and the situations of the different ports to which the ship may be bound. When the wind therefore happens not to be favourable, the rudder and sails must be managed in such a manner that the ship may make an angle with the direction of the current of air, as represented Plate CCI. fig. 4. Thus, supposing a ship at D, bound for the port B. Supposing DG the length of the keel, it must be kept by the rudder in such a position as to make the acute angle EDB with the direction of the wind. If, when she arrives at B, it is required to sail to another port A, the keel must be kept in the position BF; and thus, by continually making the angle EBA with the direction of the wind, she will arrive at the desired port: and in this manner may a ship be steered to any other port, suppose to C or H.

The reason of the ship's motion in these cases is, Vol. VII.

that the water resists the side more than the fore-part, and that in the same proportion that her length exceeds her breadth. This proportion is so considerable, that the ship continually flies off where the resistance is least, and that sometimes with great swiftness. In this way of sailing, however, there is a great limitation: for if the angle made by the keel with the direction of the wind be too acute, the ship cannot be kept in that position; neither is it possible for a large ship to make a more acute angle with the wind than 67½ degrees; though small sloops, it is said, may make an angle of 56 or 57 degrees with it. In all these cases, however, the velocity of the ship is greatly retarded; and not only on account of the obliquity of her motion, but by reason of what is called her lee-way. This is occasioned by the yielding of the water on the lee-side of the ship, by which means the vessel acquires a compound motion, partly in the direction of the wind, and partly in that which is necessary for attaining the desired haven. Thus, supposing a ship to set out from B, in the direction BA, the force of the wind will have such an impression upon her, that, instead of keeping the straight path BFFF, she will follow that of Babc, &c. and thus will fall short of her intended port by some considerable space, as Af.

It is perhaps impossible to lay down any mathematical principles on which the lee-way of a ship could be properly calculated; only we may see in general that it depends on the strength of the wind, the roughness of the sea, and the velocity of the ship. When the wind is not very strong, the resistance of the water on the lee-side bears a very great proportion to that of the current of air; and therefore it will yield but very little: however, supposing the ship to remain in the same place, it is evident, that the water having once begun to yield will continue to do so for some time, even though no additional force was applied to it; but as the wind continually applies the same force as at first, the lee-way of the ship must go on constantly increasing till the resistance of the water

THEORY. on the lee side balances the force applied on the other, when it will become uniform, as doth the motion of a ship sailing before the wind. If the ship changes her place with any degree of velocity, then every time she moves her own length, a new quantity of water is to be put in motion, which hath not yet received any momentum, and which of consequence will make a greater resistance than it can do when the ship remains in the same place. In proportion to the swiftness of the ship, then, the lee-way will be the less: but if the wind is very strong, the velocity of the ship bears but a small proportion to that of the current of air; and the same effects must follow as though the ship moved slowly, and the wind was gentle; that is, the ship must make a great deal of lee-way.—The same thing happens when the sea rises high, whether the wind is strong or not; for then the whole water of the ocean, as far as the swell reaches, hath acquired a motion in a certain direction, and that to a very considerable depth. The mountainous waves will not fail to carry the ship very much out of her course; and this deviation will certainly be according to their velocity and magnitude. In all cases of a rough sea, therefore, a great deal of lee-way is made.—Another circumstance also makes a variation in the quantity of the lee-way; namely, the lightness or heaviness of the ship; it being evident, that when the ship sinks deep in the water, a much greater quantity of that element is to be put in motion before she can make any lee-way, than when she swims on the surface. As therefore it is impossible to calculate all these things with mathematical exactness, it is plain that the real course of a ship is exceedingly difficult to be found, and frequent errors must be made, which only can be corrected by celestial observations.

In many places of the ocean there are currents, or places where the water, instead of remaining at rest, runs with a very considerable velocity for a great way in some particular direction, and which will certainly carry the ship greatly out of her course. This occasions an error of the same nature with the lee-way; and therefore, whenever a current is perceived, its velocity ought to be determined, and the proper allowances made.

Another source of error in reckoning the course of a ship proceeds from the variation of the compass. There are few parts of the world where the needle points exactly north; and in those where the variation is known, it is subject to very considerable alterations. By these means the course of the ship is mistaken; for as the sailors have no other standard to direct them than the compass, if the needle, instead of pointing due north, should point north-east, a prodigious error would be occasioned during the course of the voyage, and the ship would not come near the port to which she was bound. To avoid errors of this kind the only method is, to observe the azimuths as frequently as possible, by which the difference of variation will be perceived, and the proper allowances can then be made for errors in the course which this may have occasioned.

Errors will arise in the reckoning of a ship, especi-

THEORY. ally when she sails in high latitudes, from the spheroidal figure of the earth; for as the polar diameter of our globe is found to be considerably shorter than the equatorial one, it thence follows, that the farther we remove from the equator the longer are the degrees of latitude. Of consequence, if a navigator assigns any certain number of miles for the length of a degree of latitude near the equator, he must vary that measure as he approaches towards the poles, otherwise he will imagine that he hath not sailed so far as he actually hath done. It would therefore be necessary to have a table containing the length of a degree of latitude in every different parallel from the equator to either pole; as without this a troublesome calculation must be made at every time the navigator makes a reckoning of his course. Such a table, however, hath not yet appeared; neither indeed seems it to be easy to make it, on account of the difficulty of measuring the length even of one or two degrees of latitude in different parts of the world. Sir Isaac Newton first discovered this spheroidal shape of the earth; and shewed, from experiments on pendulums, that the polar diameter was to the equatorial one as 229 to 230. This proportion, however, hath not been admitted by succeeding calculators. The French mathematicians who measured a degree on the meridian in Lapland, made the proportion between the equatorial and polar diameters to be as 1 to 0.9891. Those who measured a degree at Quito in Peru, made the proportion 1 to 0.99624, or 266 to 265. M. Bouguer makes the proportion to be as 179 to 178; and M. Buffon, in one part of his theory of the earth, makes the equatorial diameter exceed the polar one by $\backslash frac\{1\}\{250\}$ of the whole. From these variations it appears that the point is not exactly determined, and consequently that any corrections which can be made with regard to the spheroidal figure of the earth must be very uncertain.

It is of consequence to navigators in a long voyage to take the nearest way to their port; but this can seldom be done without considerable difficulty. The shortest distance between any two points of a sphere is measured by an arch of a great circle intercepted between them; and therefore, excepting where both places lie under the same parallel of latitude, it is advisable to direct the ship along a great circle of the earth's surface. But this is a matter of considerable difficulty, because there are no fixed marks by which it can be readily known whether the ship sails in the direction of a great circle or not. For this reason the sailors commonly choose to direct their course by the rhumbs, or the bearing of the place by the compass. These bearings do not point out the shortest distance between places; because, on a globe, the rhumbs are spirals, and not arches of great circles. However, when the places lie directly under the equator, or exactly under the same meridian, the rhumb then coincides with the arch of a great circle, and of consequence shews the nearest way. The sailing on the arch of a great circle is called great circle sailing; and the cases of it depend all on the solution of problems in spheric trigonometry.

Fig. 1. NEPERS RODS

Fig. 3. OLIVE PRESS

Fig. 10.

Fig. 11. PERAMBULATOR

A Bell Sculp.

THE main end of all practical navigation is to conduct the ship in safety to her destined port; and for this purpose it is of the utmost consequence to know in what particular part of the surface of the globe she is at any particular time. This can only be done by having an accurate map of the sea-coasts of all the countries of the world, and, by tracing out the ship's progress along the map, to know at what time she approaches the desired haven, or how she is to direct her course in order to reach it. It is therefore a matter of great importance for navigators to be furnished with maps, or charts, as they are called, not only very accurate in themselves, but such as are capable of having the ship's course easily traced upon them, without the trouble of laborious calculations, which are ready to create mistakes.—The names of the two great divisions of navigation are taken merely from the kind of charts made use of. Plane sailing is that in which the plane chart is made use of; and Mercator's sailing, or globular sailing, is that in which Mercator's chart is used. In both these methods, it is easy to find the ship's place with as great exactness as the chart will allow, either by the solution of a case in plane trigonometry, or by geometrical construction.

As a necessary preliminary to our understanding this method of navigation, we shall here give the construction of the plane chart.

1. This chart supposes the earth to be a plane, and the meridians parallel to one another; and likewise the parallels of latitude at equal distance from one another, as they really are upon the globe. Tho' this method be in itself evidently false; yet, in a short run, and especially near the equator, an account of the ship's way may be kept by it tolerably well.

Having determined the limits of the chart, that is, how many degrees of latitude and longitude, or meridional distance (they being in this chart the same), it is to contain: suppose from the lat. of $20^\{\backslash circ\}$ N. to the lat. of $71^\{\backslash circ\}$ N.; and from the longitude of London in $0^\{\backslash circ\}$ deg. to the lon. of $50^\{\backslash circ\}$ W.; then choose a scale of equal parts, by which the chart may be contained within the size of the sheet of paper on which it is intended to be drawn. In the chart annexed, the scale is such, that each degree of latitude and longitude is $\backslash frac\{1\}\{4\}$ part of an inch.

Plate CCI. fig. 5. Make a parallelogram ABCD, the length of which AB from north to south shall contain $51^\{\backslash circ\}$ degrees, the difference of latitude between the limits of $20^\{\backslash circ\}$ and $71^\{\backslash circ\}$; and the breadth AD from east to west shall contain the proposed $50^\{\backslash circ\}$ degrees of longitude, the degrees being taken from the said scale of 8 degrees to an inch; and this parallelogram will be the boundaries of the chart.

About the boundaries of the chart make scales containing the degrees, halves and quarters of degrees (if the scale is large enough); drawing lines across the chart thro' every 5 or 10 degrees; let the degrees of latitude and longitude have their respective numbers

annexed, and the sheet is then fitted to receive the places intended to be delineated thereon.

On a strait slip of pasteboard, or stiff paper, let the scale of the degrees and parts of degrees of longitude, in the line AD, be laid close to the edge; and the divisions numbered from the right hand towards the left, being all west longitude.

Seek in a geographical table for the latitudes and longitudes of the places contained within the proposed limits; and let them be written out in the order in which they increase in latitude.

Then, to lay down any place, lay the edge of the pasteboard scale to the divisions on each side the chart, shewing the latitude of the place; so that the beginning of its divisions fall on the right-hand border AB; and against the division shewing the longitude of the given place make a point, and this gives the position of the place proposed; and in like manner are all the other places to be laid down.

Draw wavy lines from one point to the other, where the coast is contiguous, and thus the representation of the lands within the proposed limits will be delineated.

Write the names to the respective parts, and in some convenient place insert a compass, and the chart will be completed.

2. The angle formed by the meridian and rhumb that a ship sails upon, is called the ship's course. Thus if a ship sails on the NNE rhumb, then her course will be $22^\{\backslash circ\}\; 30\text{'}$; and so of others.

3. The distance between two places lying on the same parallel counted in miles of the equator, or the distance of one place from the meridian of another counted as above on the parallel passing over that place, is called meridional distance; which, in plane sailing, goes under the name of departure.

4. Let A ($n^\{\backslash circ\}\; 3.$) denote a certain point on the Plate CCI. earth's surface, AC its meridian, and AD the parallel of latitude passing through it; and suppose a ship to sail from A on the NNE rhumb till she arrive at B; and through B draw the meridian BD, (which, according to the principles of plane sailing, must be parallel to CA,) and the parallel of latitude BC: then the length of AB, viz. how far the ship has sailed upon the NNE rhumb, is called her distance; AC or BD will be her difference of latitude, or northing; CB will be her departure, or easting; and the angle CAB will be the course. Hence it is plain, that the distance sailed will always be greater than either the difference of latitude or departure; it being the hypotenuse of a right-angled triangle, whereof the other two are the legs; except the ship sails either on a meridian or a parallel of latitude: for if the ship sails on a meridian, then it is plain, that her distance will be just equal to her difference of latitude, and she will have no departure; but if she sail on a parallel, then her distance will be the same with her departure, and she will have no difference of latitude. It is evident also from the figure, that if the course be less than 4 points, or 45 degrees, its complement, viz. the other oblique angle, will be greater than 45 degrees, and

PRACTICE and so the difference of latitude will be greater than the departure; but if the course be greater than 4 points, then the difference of latitude will be less than the departure; and lastly, if the course be just 4 points, the difference of latitude will be equal to the departure.

5. Since the distance, difference of latitude, and departure, form a right-angled triangle, in which the oblique angle opposite to the departure is the course, and the other its complement; therefore, having any two of these given, we can (by plain trigonometry) find the rest; and hence arise the cases of plane-sailing, which are as follow.

CASE I. Course and distance given, to find difference of latitude and departure.

EXAMPLE. Suppose a ship sails from the latitude of $30^\{\backslash circ\}\; 25\text{'}$ north, NNE, 32 miles, (n^o 4.): Required the difference of latitude and departure, and the latitude come to. Then (by right-angle trigonometry,) we have the following analogy, for finding the departure, viz.

so the ship has made 12.25 miles of departure easterly, or has got so far to the eastward of her meridian. Then for the difference of latitude or northing the ship has made, we have (by rectangular trigonometry) the following analogy, viz.

so the ship has differed her latitude, or made of northing, 29.57 minutes.

And since her former latitude was north, and her difference of latitude also north; therefore, To the latitude sailed from $30^\{\backslash circ\},\; 25\text{'}\; N$ add the difference of latitude $00^\{\backslash circ\},\; 29.57$

and the sum is the latitude come to $30^\{\backslash circ\},\; 54.57\; N$

By this case are calculated the tables of difference of latitude, and departure, to every degree, point, and quarter-point of the compass.

CASE II. Course and difference of latitude given, to find distance and departure.

EXAMPLE. Suppose a ship in the latitude of $45^\{\backslash circ\}\; 25\text{'}$ north, sails NE$\backslash frac\{1\}\{4\}$NE$\backslash frac\{1\}\{4\}$ easterly (n^o 5.) till she come to the latitude of $46^\{\backslash circ\}\; 55\text{'}$ north: Required the distance and departure made good upon that course.

Since both latitudes are northerly, and the course also northerly; therefore, From the latitude come to $46^\{\backslash circ\},\; 55\text{'}$ subtract the latitude sailed from $45^\{\backslash circ\},\; 25\text{'}$ and there remains $01^\{\backslash circ\},\; 30\text{'}$

the difference of latitude, equal to 90 miles.

And (by rectangular trigonometry) we have the following analogy, for finding the departure BD, viz.

so the ship has got 73.84 miles to the eastward of her

former meridian.

Again, for the distance AD, we have (by rectangular trigonometry) the following proportion, viz.

CASE III. Difference of latitude and distance given, to find course and departure.

EXAMPLE. Suppose a ship sails from the latitude of $56^\{\backslash circ\}\; 50\text{'}$ north, on a rhumb between south and west, 126 miles, and she is then found by observation to be in the latitude of $55^\{\backslash circ\}\; 40\text{'}$ north: Required the course she sailed on, and her departure from the meridian. N^o 6.

Since the latitudes are both north, and the ship sailing towards the equator; therefore,

and the remainder $01^\{\backslash circ\},\; 40\text{'}$ equal to 70 miles, is the difference of latitude.

By rectangular trigonometry we have the following proportion for finding the angle of the course F, viz.

which, because she sails between south and west, will be south $56^\{\backslash circ\}\; 15\text{'}$ west, or SW$\backslash frac\{1\}\{4\}$W. Then, for the departure, we have (by rectangular trigonometry) the following proportion, viz.

consequently she has made 104.8 miles of departure westerly.

CASE IV. Difference of latitude and departure given, to find course and distance.

EXAMPLE. Suppose a ship sails from the latitude of $44^\{\backslash circ\}\; 50\text{'}$ north, between south and east, till she has made 64 miles of easting, and is then found by observation to be in the latitude of $42^\{\backslash circ\}\; 56\text{'}$ north: Required the course and distance made good.

Since the latitudes are both north, and the ship sailing towards the equator; therefore,

and there remains $01^\{\backslash circ\},\; 54\text{'}$ equal to 114 miles, the difference of latitude or southing.

In this case (by rectangular trigonometry) we have the following proportion to find the course KGL (N^o 7.) viz.

which, because the ship is sailing between south and east, will be south $29^\{\backslash circ\}\; 19\text{'}$ east, or SSE $\backslash frac\{1\}\{4\}$ east nearly.

Then for the distance, we shall have (by rectangular trigonometry) the following analogy, viz.

CASE V. Distance and departure given, to find course and difference of latitude.

EXAMPLE. Suppose a ship at sea fails from the latitude of 34° 24' north, between north and west 124 miles, and is found to have made of westing 86 miles: Required the course steered, and the difference of latitude or northing made good.

In this case (by rectangular trigonometry) we have the following proportion for finding the course ADB, (N° 8.) viz.

Then for the difference of latitude, we have (by rectangular trigonometry) the following analogy, viz.

Hence, to find the latitude the ship is in, since both latitudes are north, and the ship sailing from the equator; therefore,

the sum is - - - - - 35°, 53
the latitude the ship is in north.

CASE VI. Course and departure given, to find distance and difference of latitude.

EXAMPLE. Suppose a ship at sea, in the latitude of 24° 30' south, sails SE $\backslash frac\{1\}\{4\}$ S, till she has made of easting 96 miles: Required the distance and difference of latitude made good on that course.

In this case (by rectangular trigonometry and by case 2.) we have the following proportion for finding the distance, (N° 9.) viz.

Then, for the difference of latitude, we have (by rectangular trigonometry) the following analogy, viz.

to the difference of latitude GH - 143.7 - 2.15738 equal to 2°, 24' nearly. Consequently, since the latitude the ship failed from was south, and she sailing still towards the south,

and the sum - - - - - 26°, 54
is the latitude she is come to south.

6. When a ship fails on several courses in 24 hours,

the reducing all these into one, and thereby finding the course and distance made good upon the whole, is commonly called the reducing of a traverse.

7. At sea they commonly begin each day's reckoning from the noon of that day, and from that time they set down all the different courses and distances sailed by the ship till noon next day upon the log-board; then from these several courses and distances, they compute the difference of latitude and departure for each course (by Case 1. of Plane Sailing;) and these, together with the courses and distances, are set down in a table, called the traverse table, which consists of five columns: in the first of which are placed the courses and distances; in the two next, the differences of latitude belonging to these courses, according as they are north or south; and in the two last are placed the departures belonging to these courses, according as they are east or west. Then they sum up all the northings and all the southings; and taking the difference of these, they know the difference of latitude made good by the ship in the last 24 hours, which will be north or south, according as the sum of the northings or southings is greatest: the same way, by taking the sum of all the eastings, and likewise of all the westings, and subtracting the lesser of these from the greater, the difference will be the departure made good by the ship last 24 hours, which will be east or west according as the sum of the eastings is greater or less than the sum of the westings; then from the difference of latitude and departure made good by the ship last 24 hours, found as above, they find the true course and distance made good upon the whole (by Case 4. of Plane Sailing), as also the course and distance to the intended port.

EXAMPLE. Suppose a ship at sea, in the latitude of 48° 24' north at noon any day, is bound to a port in the latitude of 43° 40' north, whose departure from the ship is 144 miles east; consequently the direct course and distance of the ship is SSE $\backslash frac\{1\}\{4\}$ east 315 miles; but by reason of the shifting of the winds she is obliged to steer the following courses till noon next day, viz. SE $\backslash frac\{1\}\{4\}$ S 56 miles, SSE 64 miles, NW $\backslash frac\{1\}\{4\}$ W 48 miles, SW $\backslash frac\{1\}\{4\}$ W 54 miles, and SE $\backslash frac\{1\}\{4\}$ S 74 miles: Required the course and distance made good the last 24 hours, and the bearing and distance of the ship from the intended port.

The solution of this traverse depends entirely on the 1st and 4th Cases of Plane Sailing; and first we must (by Case 1.) find the difference of latitude and departure for each course. Thus,

1 Course SE $\backslash frac\{1\}\{4\}$ S distance 56 miles.

For departure.

For difference of latitude.

2. Course SSE and distance 64 miles.

For departure.

Now these several courses and distances, together with the differences of latitude and departures deduced from them, being set down in their proper columns in the traverse table, will stand as follows.

From the above table it is plain, since the sum of the northings is 26.67, and of the southings 214.58, the difference between these, viz. 187.91, will be the southing made good by the ship the last 24 hours; also the sum of the eastings being 102.55, and of the

westings 55.58, the difference 46.97 will be the easting or departure made good by the ship's last 24 hours; consequently, to find the true course and distance made good by the ship in that time, it will be (by Case 4. of Plane Sailing.)

which is S & E $\backslash frac\{1\}\{2\}$ east nearly. Then for the distance, it will be,
As radius - - - - - 10.00000
is to the difference of latitude - 187.91 2.27393
so is the secant of the course - 14°, 03' 10.01319
to the distance - 193.7 - 2.28712
consequently the ship has made good the last 24 hours, on a S & E $\backslash frac\{1\}\{2\}$ east course, 193.7 miles: and since the ship is sailing towards the equator; therefore,

From the latitude failed from 48°, 24' N
take the diff. of latitude made good - 3, 08 S

there remains - - - - - 45, 16 N
the latitude the ship is in north. And because the port the ship is bound for lies in the latitude of 43° 40' N. and consequently south of the ship; therefore,
From the latitude the ship is in - 45°, 16' N
take the latitude she is bound for - 43, 40 N

and there remains - - - - - 1, 36
or 96 miles, the difference of latitude or southing the ship has to make. Again, the whole easting the ship had to make being 144 miles, and she having already made 46.97 or 47 miles of easting; therefore the departure or easting she still has to make will be 97 miles: consequently, to find the direct course and distance between the ship and the intended port, it will be (by Case 4. of Plane Sailing.)

And
As radius - - - - - 10.00000
is to the difference of latitude 96 - 1.98227
so is the secant of the course 45°, 19' 10.15293
to the distance - 136.5 - 2.13620
whence the true bearing and distance of the intended port is SE, 136.5 miles.

1. SINCE the parallels of latitude do always decrease Plate CCII. the nearer they approach the pole, it is plain a degree on any of them must be less than a degree upon the equator. Now in order to know the length of a degree on any of them, let PB (n° 10.) represent half the earth's axis, PA a quadrant of a meridian, and consequently A a point on the equator, C a point on the meridian, and CD a perpendicular from that point upon the axis, which plainly will be the sine of CP the distance of that point from the pole, or the co-sine of CA its distance from the equator; and CD will be to AB, as the sine of CP, or co-sine of CA, is to the radius. Again, if the quadrant PAB is turned round upon the axis PB, it is plain the point A will describe the circumference of the equator whose radius is AB, and

and any other point C upon the meridian will describe the circumference of a parallel whose radius is CD.

COR. I. Hence (because the circumference of circles are as their radii) it follows, that the circumference of any parallel is to the circumference of the equator, as the co-sine of its latitude is to radius.

COR. II. And since the wholes are as their similar parts, it will be, As the length of a degree on any parallel is to the length of a degree upon the equator, so is the co-sine of the latitude of that parallel to radius.

COR. III. Hence, as radius is to the co-sine of any latitude, so are the minutes of difference of longitude between two meridians, or their distance in miles upon the equator, to the distance of these two meridians on the parallel in miles.

COR. IV. And as the co-sine of any parallel is to radius, so is the length of any arch on that parallel (intercepted between two meridians) in miles, to the length of a similar arch on the equator, or minutes of difference of longitude.

COR. V. Also, as the co-sine of any one parallel is to the co-sine of any other parallel, so is the length of any arch on the first, in miles, to the length of the same arch on the other in miles.

2. From what has been said, arises the solution of the several cases of parallel sailing, which are as follow.

CASE I. Given the difference of longitude between two places, both lying on the same parallel; to find the distance between those places.

EXAMPLE I. Suppose a ship in the latitude of $54^\{\backslash circ\}\; 20\text{'}$ north, sails directly west on that parallel till she has differed her longitude $12^\{\backslash circ\}\; 45\text{'}$; required the distance sailed on that parallel.

First, The difference of longitude reduced into minutes, or nautical miles, is 765', which is the distance between the meridian sailed from, and the meridian come to, upon the equator; then to find the distance between these meridians on the parallel of $54^\{\backslash circ\}\; 20\text{'}$, or the distance sailed, it will be, by Cor. 3. of the last article,

As radius - - - - - 10.00000
is to the co-sine of the lat. - $54^\{\backslash circ\}\; 20\text{'}$ - 9.76572
so are the minutes of diff. long. 765 - 2.88366
to the distance on the parallel 446.1 - 2.64938

EXAMPLE II. A degree on the equator being 60 minutes or nautical miles; required the length of a degree on the parallel of $51^\{\backslash circ\}\; 32\text{'}$.

By Cor. 3. of the last article, it will be
As radius - - - - - 10.00000
is to the co-sine of the latitude - $51^\{\backslash circ\}\; 32\text{'}$ 9.79383
so are the minutes in 1 degree on the equator. 60 1.77815
to - - - - - 37.32 1.57198
the miles answering to a degree on the parallel of $51^\{\backslash circ\}\; 32\text{'}$

By this problem the following table is constructed, shewing the geographic miles answering to a degree on any parallel of latitude; in which you may observe, that the columns marked at the top with D. L. contain the degrees of latitude belonging to each parallel; and the adjacent columns marked at the top, Miles, contain the geographic miles answering to a degree upon these parallels.

A TABLE shewing how many Miles answer to a Degree of Longitude, at every Degree of Latitude.

Though this table does only shew the miles answering to a degree of any parallel, whose latitude consists of a whole number of degrees; yet it may be made to serve for any parallel whose latitude is some number of degrees and minutes, by making the following proportion, viz.

As 1 degree, or 60 minutes, is to the difference between the miles answering to a degree in the next greater and next less tabular latitude than that proposed; so is the excess of the proposed latitude above the next tabular latitude, to a proportional part; which, subtracted from the miles answering to a degree of longitude in the next less tabular latitude, will give the miles answering to a degree in the proposed latitude.

EXAMPLE. Required to find the miles answering to a degree on the parallel of $56^\{\backslash circ\}\; 44\text{'}$.

First, The next less parallel of latitude in the table than that proposed, is that of $56^\{\backslash circ\}$, a degree of which (by the table) is equal to 33.55 miles; and the next greater parallel of latitude in the table, than that proposed, is that of $57^\{\backslash circ\}$, a degree of which is (by the table) equal to 32.68 miles; the difference of these is 87, and the distance between these parallels is 1 degree, or 60 minutes; also the distance between the parallel of $56^\{\backslash circ\}$, and the proposed parallel of $56^\{\backslash circ\}\; 44\text{'}$, is 44 minutes: then by the preceding proportion it will be, as 60 is to 87, so is 44 to 638, the difference between a degree on the parallel of $56^\{\backslash circ\}$ and a degree on the parallel of $56^\{\backslash circ\}\; 44\text{'}$; which therefore, taken from 33.55, the miles answering to a degree on the parallel of $56^\{\backslash circ\}$, leaves 32.912, the miles answering to a degree on the parallel of $56^\{\backslash circ\}\; 44\text{'}$, as was required.

CASE II. The distance sailed in any parallel of latitude, or the distance between any two places on that parallel, being given; to find the difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $55^\{\backslash circ\}\; 36\text{'}$ north,

PRACTICE north fails directly east 685.6 miles: Required how much she has differed her longitude.

By Cor. 4. Art. 1. of this section, it will be
As the co-sine of the lat. - $55^\{\backslash circ\}\; 36\text{'}$ - 9.75202
is to radius - - - - - 10.00000
so is the distance failed - 685.6 - 2.83607
to min. of diff. of long. - 1213 - 3.08405
which reduced into degrees, by dividing by 60,
makes $20^\{\backslash circ\}\; 13\text{'}$, the difference of longitude the ship has
made.

This also may be solved by help of the foregoing table, viz. by finding from it the miles answering to a degree on the proposed parallel, and dividing with this the given number of miles, the quotient will be the degrees and minutes of difference of longitude required.

Thus in the last example, we find, from the foregoing table, that a degree on the parallel of $55^\{\backslash circ\}\; 36\text{'}$ is equal to 33.89 miles; by this we divide the proposed number of miles 685.6, and the quotient is 20.13 degrees, i. e. $20^\{\backslash circ\}\; 13\text{'}$, the difference of longitude required.

CASE III. The difference of longitude between two places on the same parallel, and the distance between them, being given; to find the latitude of that parallel.

EXAMPLE. Suppose a ship fails on a certain parallel directly west 624 miles, and then has differed her longitude $18^\{\backslash circ\}\; 46\text{'}$, or 1126 miles: Required the latitude of the parallel she failed upon.

By Cor. 3. Art. 1. of this section, it will be,
As the min. of diff. long. - 126 - 3.05154
is to the distance failed - 624 - 2.79518
so is radius - - - - - 10.00000
to the co-sine of the lat. - $56^\{\backslash circ\}\; 21\text{'}$ - 9.74364
consequently the latitude of the ship or parallel she failed upon was $56^\{\backslash circ\}\; 21\text{'}$.

From what has been said, may be solved the following problems.

PROB. I. Suppose two ships in the latitude of $46^\{\backslash circ\}\; 30\text{'}$ north, distant afunder 654 miles, sail both directly north 256 miles, and consequently are come to the latitude of $50^\{\backslash circ\}\; 46\text{'}$ north: Required their distance on that parallel.

By Cor. 6. Art. 1. of this section, it will be,
As the co-sine of - $46^\{\backslash circ\}\; 30\text{'}$ - 9.83781
is to the co-sine of - $50^\{\backslash circ\}\; 46\text{'}$ - 9.80105
so is - 654 - 2.81558
to - 601 - 2.77882
the distance between the ships when on the parallel of $50^\{\backslash circ\}\; 46\text{'}$.

PROB. II. Suppose two ships in the latitude of $45^\{\backslash circ\}\; 48\text{'}$ north, distant 846 miles, sail directly north till the distance between them is 624 miles: Required the latitude come to, and the distance failed.

By Cor. 5. Art. 1. of this section, it will be,
As their first distance - 846 - 2.92737
is to their second distance - 624 - 2.79518
so is the co-sine of - $45^\{\backslash circ\}\; 48\text{'}$ - 9.84334
to the cosine - $59^\{\backslash circ\}\; 04\text{'}$ - 9.71115
the latitude of the parallel the ships are come to.

Consequently to find their distance failed,
From the latitude come to - $59^\{\backslash circ\}\; 04\text{'}$
subtract the latitude failed from - $45^\{\backslash circ\}\; 48\text{'}$
and there remains - - - - - 13, 16

equal to 796 miles, the difference of latitude or distance failed.

§ 3. Of Middle-latitude Sailing.

I. WHEN two places lie both on the same parallel, we shewed in the last section, how, from the difference of longitude given, to find the miles of sailing or westing between them, et cetera. But when two places lie not on the same parallel, then their difference of longitude cannot be reduced to miles of sailing or westing on the parallel of either place: for if counted on the parallel of that place that has the greatest latitude, it would be too small; and if on the parallel of that place having the least latitude, it would be too great. Hence the common way of reducing the difference of longitude between two places, lying on different parallels, to miles of sailing or westing, et cetera, is by counting it on the middle parallel between the places, which is found by adding the latitudes of the two places together, and taking half the sum, which will be the latitude of the middle parallel required. And hence arises the solution of the following cases.

CASE I. The latitudes of two places, and their difference of longitude, given; to find the direct course and distance.

EXAMPLE. Required the direct course and distance between the Lizard in the latitude of $50^\{\backslash circ\}\; 00\text{'}$ north, and longitude of $5^\{\backslash circ\}\; 14\text{'}$ west, and St Vincent in the latitude of $17^\{\backslash circ\}\; 10\text{'}$ N. and longitude of $24^\{\backslash circ\}\; 20\text{'}$ W.

First, To the latitude of the Lizard $50^\{\backslash circ\}\; 00\text{'}$ N.
add the latitude of St Vincent - $17^\{\backslash circ\}\; 10\text{'}$
The sum is - - - - - 67, 10
Half the sum or latitude of the middle parallel is } - 33, 35 N.
Also the difference of latitude is } - 33, 50
equal to 1970 miles of southing. Again,
From the longitude of St Vincent 24, 20 W.
take the longitude of the Lizard - 5, 14
there remains - - - - - 16, 06
equal to 1146 min. of diff. of long. west.

Then for the miles of westing, or departure, it will be, (by Case 1. of Parallel Sailing)

As radius - - - - - 10.00000
is to the co-sine of the } middle parallel } - $33^\{\backslash circ\}\; 35\text{'}$ - 9.92069
so is min. diff. of long. - 1146 - 3.05918
to the miles of westing - 954.7 - 2.97987
And for the course it will be, (by Case 4. of Plane Sailing)

As the diff. of lat. - 1970 - 3.29447
is to radius - - - - - 10.00000
so is the departure - 954.7 - 2.97987
to the tang. of the course $25^\{\backslash circ\}\; 51\text{'}$ - 9.68540
which, because it is between south and west, it will be SSW $\backslash frac\{1\}\{2\}$ west nearly.

For the distance, it will be, by the same case,
As radius - - - - - 10.00000
is to the diff. of lat. - 1970 - 3.29447
so is the secant of the course $25^\{\backslash circ\}\; 51\text{'}$ - 10.04579
to the distance - 2189 - 3.34026
whence the direct course and distance from the Lizard to St Vincent is SSW $\backslash frac\{1\}\{2\}$ 2189 W miles.

CASE II. One latitude, course, and distance failed being

PRACTICE being given; to find the other latitude and difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $50^\{\backslash circ\}\; 00\text{'}$ north, sails south $50^\{\backslash circ\}\; 06\text{'}$ west, 150 miles: Required the latitude the ship has come to, and how much she has differed her longitude.

First, For the difference of latitude, it will be, (by Case 1. of Plane Sailing.)

and there remains - - - 48, 24 the latitude she has come to north. Consequently the latitude of the middle parallel will be $49^\{\backslash circ\}\; 12\text{'}$.

Then for departure or westing it will be, by the same Case,

As for the difference of longitude, it will be, (by Case 2. of Plane Sailing.)

CASE III. Course and difference of latitude given; to find the distance failed, and difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $53^\{\backslash circ\}\; 34\text{'}$ north, sails SE&S, till by observation she is found to be in the latitude of $51^\{\backslash circ\}\; 12\text{'}$, and consequently has differed her latitude $2^\{\backslash circ\}\; 22\text{'}$, or 142 miles: Required the distance failed, and the difference of longitude.

First, for the departure, it will be, (by Case 2. of Plane Sailing.)

Then, since the latitude failed from was $53^\{\backslash circ\}\; 34\text{'}$ north, and the latitude come to $51^\{\backslash circ\}\; 12\text{'}$ north; therefore the middle parallel will be $52^\{\backslash circ\}\; 23\text{'}$; and consequently, for the difference of longitude, it will be (by Case 2. of Parallel Sailing.)

CASE IV. Difference of latitude and distance failed.

ed, given; to find the course and difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $43^\{\backslash circ\}\; 26\text{'}$ north, sails between south and east, 246 miles, and then is found by observation to be in the latitude of $41^\{\backslash circ\}\; 06\text{'}$ north: Required the direct course and difference of longitude.

First, For the course, it will be, (by Case 3. of Plane Sailing.)

Then for departure, it will be, by the same Case,

Lastly, For the difference of longitude, it will be, (by Case 2. of Parallel Sailing.)

CASE V. Course and departure given, to find difference of latitude, difference of longitude, and distance failed.

EXAMPLE. Suppose a ship in the latitude of $48^\{\backslash circ\}\; 23\text{'}$ north, sails SW&S, till she has made of westing 123 miles: Required the latitude come to, the difference of longitude, and the distance failed.

First, For the distance, it will be, (by Case 6. of Plane Sailing.)

And for the difference of latitude, it will be, by the same Case,

Then to find the difference of longitude, it will be, (Case 2. of Parallel Sailing.)

CASE VI. Difference of latitude and departure given, to find course, distance, and difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $46^\{\backslash circ\}\; 37\text{'}$ north, sails between south and east, till she has made of easting 146 miles, and is then found by observation to be in the latitude of $43^\{\backslash circ\}\; 24\text{'}$ north:

PRACTICE Required the course, distance, and difference of longitude.

First, By Case 4. of Plane Sailing, it will be for the course,

As the diff. of latitude - 193 - 2.28556
is to departure - 146 - 2.16137
so is radius - - - 10.00000
to the tang. of the course $36^\{\backslash circ\}\; 55\text{'}$ - 9.87581
which, because the ship is sailing between south and east, will be south $36^\{\backslash circ\}\; 55\text{'}$ east, or SE $\backslash frac\{1\}\{2\}$ S $\backslash frac\{1\}\{2\}$ E nearly.

For the distance, it will be, by the same Case,
As radius - - - 10.00000
is to the diff. of latitude 193 - 2.28556
so is the secant of the course $36^\{\backslash circ\}\; 55\text{'}$ - 10.09718
to the distance - 241.4 - 2.38274

Then for the difference of longitude, it will be, by Case 2. of Parallel Sailing,
As the co-sine of the mid. par. $45^\{\backslash circ\}\; 00\text{'}$ 9.84949
is to the departure - 146 - 2.16137
so is radius - - - 10.00000
to min. of diff. of longitude 205 - 2.31188
equal to $3^\{\backslash circ\}\; 25\text{'}$, the difference of longitude easterly.

CASE VII. Distance and departure given, to find difference of latitude, course, and difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $33^\{\backslash circ\}\; 40\text{'}$ north, sails between south and east 165 miles, and has then made of easting 112.5 miles: Required the difference of latitude, course, and difference of longitude.

First, for the course, it will be, by Case 5. of Plane sailing,

As the distance - 165 - 2.21748
is to radius - - - 10.00000
so is the departure 102.5 - 2.05115
to the sine of the course $42^\{\backslash circ\}\; 59\text{'}$ - 9.83367
which, because the ship sails between south and east, will be south $42^\{\backslash circ\}\; 59\text{'}$ east, or SE $\backslash frac\{1\}\{2\}$ E $\backslash frac\{1\}\{2\}$ S nearly.

And for the difference of latitude, it will be, by the same Case,

As radius - - - 10.00000
is to the distance - 165 - 2.21748
so is the co-sine of the course $42^\{\backslash circ\}\; 59\text{'}$ - 9.86436
to the difference of lat. - 120.7 - 2.08184
equal to $2^\{\backslash circ\}\; 00\text{'}$; consequently the latitude come to will be $31^\{\backslash circ\}\; 40\text{'}$ north, and the latitude of the middle parallel will be $32^\{\backslash circ\}\; 40\text{'}$. Hence, to find the difference of longitude, it will be, by Case 2. of Parallel Sailing,
As the co-sine of the mid. par. $32^\{\backslash circ\}\; 40\text{'}$ 9.92522
is to the departure - 112.5 - 2.05115
so is radius - - - 10.00000
to min. of diff. of long. - 133.6 - 2.12593
equal to $2^\{\backslash circ\}\; 13\text{'}$ nearly, the difference of longitude easterly.

CASE VIII. Difference of longitude and departure given; to find difference of latitude, course, and distance failed.

EXAMPLE. Suppose a ship in the latitude of $50^\{\backslash circ\}\; 46\text{'}$ north, sails between south and west, till her difference of longitude is $3^\{\backslash circ\}\; 12\text{'}$, and is then found to have departed from her former meridian 126 miles; required the difference of latitude, course, and distance failed.

First, For the latitude she has come to, it will be, by Case 3. of Parallel Sailing,

As min. of diff. of long. - 192 - 2.28330
is to departure - - - 126 - 2.10037
so is radius - - - 10.00000
to the co-sine of the mid. par. $48^\{\backslash circ\}\; 59\text{'}$ 9.81797

Now since the middle latitude is equal to half the sum of the two latitudes (by art. 1. of this sect.) and so the sum of the two latitudes equal to double the middle latitude; it follows, that if from double the middle latitude we subtract any one of the latitudes, the remainder will be the other. Hence from twice $48^\{\backslash circ\}\; 59\text{'}$, viz. $97^\{\backslash circ\}\; 58\text{'}$, taking $50^\{\backslash circ\}\; 46\text{'}$ the latitude failed from, there remains $47^\{\backslash circ\}\; 12\text{'}$ the latitude come to; consequently the difference of latitude is $3^\{\backslash circ\}\; 34\text{'}$, or 214 minutes.

Then for the course, it will be, by Case 4. of Plane Sailing,

As diff. of lat. - 214 - 2.33041
is to radius - - - 10.00000
so is the departure - 126 - 2.10037
to the tang. of the course $30^\{\backslash circ\}\; 29\text{'}$ - 9.76996
which, because it is between south and west, will be south $30^\{\backslash circ\}\; 29\text{'}$ west, or SSW $\backslash frac\{1\}\{2\}$ W nearly.

And for the distance, it will be, by the same Case,

As radius - - - 10.00000
is to the diff. of lat. - 214 - 2.33041
so is the secant of the course $30^\{\backslash circ\}\; 29\text{'}$ - 10.06461
to the distance - 248.4 - 2.39502

2. From what has been said, it will be easy to solve a traverse, by the rules of Middle-latitude Sailing.

EXAMPLE. Suppose a ship in the latitude of $43^\{\backslash circ\}\; 25\text{'}$ north, sails upon the following courses, viz. SW $\backslash frac\{1\}\{2\}$ S 63 miles, SSW $\backslash frac\{1\}\{2\}$ W 45 miles, S $\backslash frac\{1\}\{2\}$ E 54 miles, and SW $\backslash frac\{1\}\{2\}$ W 74 miles: Required the latitude the ship has come to, and how far she has differed her longitude.

First, By Case 2. of this Sect. find the difference of latitude and difference of longitude belonging to each course and distance, and they will stand as in the following table.

Hence it is plain the ship has differed her latitude 186.2 minutes, or $3^\{\backslash circ\}\; 6\text{'}$, and so has come to the latitude of $40^\{\backslash circ\}\; 19\text{'}$ north, and has made of difference of longitude 143.8 minutes, or $2^\{\backslash circ\}\; 23\text{'}\; 48\text{'}\text{'}$ westerly.

3. This method of sailing, though it be not strictly true, yet it comes very near the truth, as will be evident, by comparing an example wrought by this method with the same wrought by the method delivered

PRACTICE vered in the next section, which is strictly true; and it serves, without any considerable error, in runnings of 450 miles between the equator and parallel of 30 degrees, of 300 miles between that and the parallel of 60 degrees, and of 150 miles as far as there is any occasion, and consequently must be sufficiently exact for 24 hours run.

§. 4. Of Mercator's sailing.

1. THOUGH the meridians do all meet at the pole, and the parallels to the equator do continually decrease, and that in proportion to the co-sines of their latitudes; yet in old sea-charts the meridians were drawn parallel to one another, and consequently the parallels of latitude made equal to the equator, and so a degree of longitude on any parallel as large as a degree on the equator: also in these charts the degrees of latitude were still represented (as they are in themselves) equal to each other, and to those of the equator. By these means the degrees of longitude being increased beyond their just proportion, and the more so the nearer they approach the pole, the degrees of latitude at the same time remaining the same, it is evident places must be very erroneously marked down upon these charts with respect to their latitude and longitude, and consequently their bearing from one another very false.

2. To remedy this inconvenience, so as still to keep the meridians parallel, is plain we must protract, or lengthen, the degrees of latitude in the same proportion as those of longitude are, that so the proportion in easting and westing may be the same with that of southing and northing, and consequently the bearings of places from one another be the same upon the chart as upon the globe itself.

Plate CCII. Let ABD (No 11.) be a quadrant of a meridian, A the pole, D a point on the equator, AC half the axis, B any point upon the meridian, from which draw BF perpendicular to AC, and BG perpendicular to CD; then BG will be the sine, and BF or CG the co-sine of BD the latitude of the point B; draw D the tangent and CE the secant of the arch CD. It has been demonstrated in Sect. 2. that any arch of a parallel is to the like arch of the equator as the co-sine of the latitude of that parallel is to radius. Thus any arch as a minute on the parallel described by the point B, will be to a minute on the equator as BF or CG is to CD; but since the triangles CGB CDE are similar, therefore CG will be to CD as CB is to CE, i. e. the co-sine of any parallel is to radius as radius is to the secant of the latitude of that parallel. But it has been just now shown, that the co-sine of any parallel is to radius, as the length of any arch as a minute on that parallel is to the length of the like arch on the equator: therefore the length of any arch as a minute on any parallel, is to the length of the like arch on the equator, as radius is to the secant of the latitude of that parallel; and so the length of any arch, as a minute on the equator, is longer than the like arch of any parallel in the same proportion as the secant of the latitude of that parallel is to radius. But since in this projection the meridians are parallel, and consequently each parallel of latitude equal to the equator, it is plain the length of any arch as a minute on any parallel, is increased beyond its

just proportion, at such rate as the secant of the latitude of that parallel is greater than radius; and therefore, to keep up the proportion of northing and southing to that of easting and westing, upon this chart, as it is upon the globe itself, the length of a minute upon the meridian at any parallel must also be increased beyond its just proportion at the same rate, i. e. as the secant of the latitude of that parallel is greater than radius. Thus to find the length of a minute upon the meridian at the latitude of 75 degrees, since a minute of a meridian is every-where equal on the globe, and also equal to a minute upon the equator, let it be represented by unity; then making it as radius is to the secant of 75 degrees, so is unity to a fourth number, which is 3.864 nearly; and consequently, by whatever line you represent one minute on the equator of this chart, the length of one minute on the enlarged meridian at the latitude of 75 degrees, or the distance between the parallel of 75° 00' and the parallel of 75° 01', will be equal to 3 of these lines, and $\backslash frac\{3.864\}\{100\}$ of one of them. By making the same proportion, it will be found, that the length of a minute on the meridian of this chart at the parallel of 60°, or the distance between the parallel of 60° 00' and that of 60° 01', is equal to two of these lines. After the same manner, the length of a minute on the enlarged meridian may be found at any latitude; and consequently, beginning at the equator, and computing the length of every intermediate minute between that and any parallel, the sum of all these shall be the length of a meridian intercepted between the equator and that parallel; and the distance of each degree and minute of latitude from the equator upon the meridian of this chart, computed in minutes of the equator, forms what is commonly called a table of meridional parts.

If the arch BD (No 11.) represent the latitude of any point B, then (CD being radius) CE will be the secant of that latitude: but it has been shown above, that radius is to secant of any latitude, as the length of a minute upon the equator is to the length of a minute on the meridian of this chart at that latitude; therefore CD is to CE, as the length of a minute on the equator is to the length of a minute upon the meridian, at the latitude of the point B. Consequently, if the radius CD be taken equal to the length of a minute upon the equator, CE, or the secant of the latitude, will be equal to the length of a minute upon the meridian at that latitude. Therefore, in general, if the length of a minute upon the equator be made radius, the length of a minute upon the enlarged meridian will be every-where equal to the secant of the arch contained between it and the equator.

COR. 1. Hence it follows, since the length of every intermediate minute between the equator and any parallel, is equal to the secant of the latitude, (the radius being equal to a minute upon the equator), the sum of all these lengths, or the distance of that parallel on the enlarged meridian from the equator, will be equal to the sum of all the secants, to every minute contained between it and the equator.

COR. 2. Consequently the distance between any two parallels on the same side of the equator is equal to the difference of the sums of all the secants contained be-

PRACTICE between the equator and each parallel, and the distance between any two parallels on contrary sides of the equator is equal to the sum of the sums of all the secants contained between the equator and each parallel.

3. By the tables of meridional parts given by all the writers on this subject, may be constructed the nautical chart, commonly called Mercator's chart. Thus, for example, let it be required to make a chart that shall commence at the equator, and reach to the parallel of 60 degrees, and shall contain 80 degrees of longitude.

Draw the line EQ representing the equator, (see N^o 12.): then take, from any convenient line of equal parts, 4800, (the number of minutes containing in 80 degrees), which set off from E to Q, and this will determine the breadth of the chart.

Divide the line EQ into eight equal parts, in the points 10, 20, 30, &c, each containing 10 degrees; and each of these divided into 10 equal parts, will give the single degrees upon the equator: then through the points E, 10, 12, &c. drawing lines perpendicular to EQ, these shall be meridians.

From the scale of equal parts take 4527.4, (the meridional parts answering to 60 degrees), and set that off from E to A and from Q to B, and join AB; then this line will represent the parallel of 60, and will determine the length of the chart.

Again, from the scale of equal parts take 603.1, (the meridional parts answering to 10 degrees), and set that off from E to 10 on the line EA; and through the point 10 draw 10, 10, parallel to EQ; and this will be the parallel of 10 degrees. The same way, setting off from E on the line EA, the meridional parts answering to each degree, &c. of latitude, and through the several points drawing lines parallel to EQ, we shall have the several parallels of latitude.

If the chart does not commence from the equator, but is only to serve for a certain distance on the meridian between two given parallels on the same side of the equator; then the meridians are to be drawn as in the last example: and for the parallels of latitude you are to proceed thus, viz. From the meridional parts answering to each point of latitude in your chart subtract the meridional parts answering to the least latitude, and set off the differences severally, from the parallel of the least latitude, upon the two extreme meridians; and the lines joining these points of the meridians shall represent the several parallels upon your chart.

Thus let it be required to draw a chart that shall serve from the latitude of 20 degrees north to 60 degrees north, and that shall contain 80 degrees of longitude.

Having drawn the line DC to represent the parallel of 20 degrees (see N^o 12.) and the meridians to it, as in the foregoing example; set off 663.3 (the difference between the meridional parts answering to 30 degrees, and those of 20 degrees) from D to 30, and from C to 30; then join the points 30 and 30 with a right line, and that shall be the parallel of 30. Also set off 1397.6 (the difference between the meridional parts answering to 40 degrees, and those of 20 degrees) from D to 40, and from C to 40; and joining the points 40 and 40 with a right line, that shall be the parallel

of 40. And proceeding after the same way, we may PRACTICE draw as many of the intermediate parallels as we have occasion for.

But if the two parallels of latitude that bound the chart, are on the contrary sides of the equator; then draw a line representing the equator and meridians to it, as in the first example; and from the equator set off on each side of it the several parallels contained between it and the given parallels as above, and your chart is finished.

If Mercator's chart, constructed as above, hath its equator extended on each side of the point E 180 degrees, and if the several places on the surface of the earth be there laid down according to their latitudes and longitudes, we shall have what is commonly called Mercator's map of the earth. This map is not to be considered as a similar and just representation of the earth's surface; for in it the figures of countries are distorted, especially near the poles: but since the degrees of latitude are every where increased in the same proportion as those of longitude are, the bearings between the places will be the same in this chart as on the globe; and the proportions between the latitudes, longitudes, and nautical distances, will also be the same on this chart, as on the globe itself; by which means the several cases of navigation are solved after a most easy manner, and adapted to the meanest capacities.

N. B. Here you must take notice, that in all charts the upper part is the north side, and the lower part or bottom is the south side; also that part of it towards the right-hand is the east, and that towards the left-hand the west side of the chart.

4. Since, according to this projection, the meridians are parallel right lines; it is plain, that the rhumbs which form always equal angles with the meridians, will be straight lines; which property renders this projection of the earth's surface much more easy and proper for the use than any other.

5. This method of projecting the earth's surface upon a plane, was first invented by Mr Edward Wright, but first published by Mercator; and hence the sailing by the chart was called Mercator's sailing.

6. In N^o 13. let A and E represent two places upon Mercator's chart, AC the meridian of A, and CE the parallel of latitude passing through E; draw AE, and set off upon AC the length AB equal to the number of minutes contained in the difference of latitude between the two places, and taken from the same scale of equal parts the chart was made by, or from the equator, or any graduated parallel of the chart, and through B draw BD parallel to CE meeting AE in D. Then AC will be the enlarged difference of latitude, AB the proper difference of latitude, CE the difference of longitude, BD the departure, AE the enlarged distance, and AD the proper distance, between the two places A and E; also the angle BAD will be the course, and AE the rhumb-line between them.

7. Now since in the triangle ACE, BD is parallel to one of its sides CE; it is plain the triangles ACE, ABD, will be similar, and consequently the sides proportional. Hence arise the solutions of the several cases in this sailing, which are as follow.

CASE I. The latitudes of two places given, to find the meridional or enlarged difference of latitude between

NAVIGATION.

Plate CCH.

A circular compass rose with 32 points. The outer ring is marked in degrees from 0 to 360. The inner ring shows the 16 cardinal and intercardinal directions: N, NNE, NE, ENE, E, ESE, S, SSW, SW, WSW, W, NNW, NW, NNE. The center has a stylized needle pointing North.

N.2.

A diagram showing a point P at the top. A curve passes through points A, B, C, D, and Q. Lines are drawn from P to each of these points. A vertical line segment PH is also shown, where H is on the curve.

N.3.

A right-angled triangle with vertices C, B, and D. The vertical side is CD, labeled 'Diff. Lat. N'. The horizontal side is BD, labeled 'Def. E'. The hypotenuse is CB, labeled 'Center Distance NNE'.

A right-angled triangle with vertices B, C, and A. The vertical side is BC, labeled 'Diff. Lat. N'. The horizontal side is AC, labeled 'Dist. NNE 32 Miles'. The hypotenuse is AB, labeled 'Def. E'.

N.5.

A right-angled triangle with vertices B, D, and A. The vertical side is BD, labeled 'Diff. Lat. N 00'. The horizontal side is AD, labeled 'Dist. NE 1 E 10 1'. The hypotenuse is AB, labeled 'Def. E'.

N.6.

A right-angled triangle with vertices F, E, and D. The vertical side is FE, labeled '70'. The horizontal side is DE, labeled '104.8'. The hypotenuse is FD, labeled 'Dist. 112 MS'.

N.7.

A right-angled triangle with vertices G, K, and L. The vertical side is GK, labeled '70'. The horizontal side is KL, labeled 'East 04'. The hypotenuse is GL, labeled 'Dist. 112 MS'.

N.8.

A right-angled triangle with vertices A, B, and D. The vertical side is BD, labeled 'Westing 86'. The horizontal side is AD, labeled '121.12'. The hypotenuse is AB, labeled '45.36'.

N.9.

A right-angled triangle with vertices G, H, and M. The vertical side is GH, labeled 'SEBs 172.8'. The horizontal side is HM, labeled 'Easting 90'. The hypotenuse is GM, labeled '182.8'.

N.12.

A large rectangular grid with a coordinate system. The horizontal axis is labeled with values from 70 to 10. The vertical axis is labeled with values from 0 to 60. Points are labeled with letters: A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. Lines connect various points, forming a complex geometric shape.

N.10.

A diagram showing a curve segment from point A to point P. Points B, C, and D are marked along the curve. Horizontal lines are drawn from points B, C, and D to the vertical line AP.

N.13.

A right-angled triangle with vertices A, B, and C. Point D is on the hypotenuse AC, and point E is on the vertical side AB. A horizontal line is drawn from D to the vertical side AB.

A diagram showing a curve segment from point A to point D. Points B, C, and E are marked along the curve. A vertical line is drawn from point B to the horizontal line AD, and a horizontal line is drawn from point C to the vertical line AD.

N.14.

A right-angled triangle with vertices A, B, and C. Point D is on the hypotenuse AC, and point E is on the vertical side BC. A horizontal line is drawn from D to the vertical side BC.

Of this case there are three varieties, viz. either one of the places lies on the equator, or both on the same side of it, or lastly, on different sides.

1. If one of the proposed places lies on the equator, then the meridional difference of latitude is the same with the latitude of the other place, taken from the table of meridional parts.

EXAMPLE. Required the meridional difference of latitude between St Thomas, lying on the equator, and St Antonio in the latitude of $17^\{\backslash circ\}\; 20\text{'}$ north. We look in the tables for the meridional parts answering to $17^\{\backslash circ\}\; 20\text{'}$, and find it to be 1056.2, the enlarged difference of latitude required.

2. If the two proposed places be on the same side of the equator, then the meridional difference of latitude is found by subtracting the meridional parts answering to the least latitude from those answering to the greatest, and the difference is that required.

EXAMPLE. Required the meridional difference of latitude between the Lizard in the latitude of $50^\{\backslash circ\}\; 00\text{'}$ north, and Antigua in the latitude of $17^\{\backslash circ\}\; 30\text{'}$ north. From the meridional parts of $50^\{\backslash circ\}\; 00\text{'}$ - 3474.5 subtract the meridional parts of $17^\{\backslash circ\}\; 30\text{'}$ - 1066.7

there remains - - - 2407.8
the meridional difference of latitude required.

3. If the places lie on different sides of the equator, then the meridional difference of latitude is found by adding together the meridional parts answering to each latitude, and the sum is that required.

EXAMPLE. Required the meridional difference of latitude between Antigua in the latitude of $17^\{\backslash circ\}\; 30\text{'}$ north, and Lima in Peru in the latitude of $12^\{\backslash circ\}\; 30\text{'}$ south.

To the merid. parts answering to $17^\{\backslash circ\}\; 30\text{'}$ 1066.7
add these answering to $12^\{\backslash circ\}\; 30\text{'}$ 756.1

the sum is - - - 1822.8
the meridional difference of latitude required.

CASE II. The latitudes and longitudes of two places given, to find the direct course and distance between them.

EXAMPLE. Required to find the direct course and distance between the Lizard in the latitude of $50^\{\backslash circ\}\; 00\text{'}$ north, and Port-Royal in Jamaica in the latitude of $17^\{\backslash circ\}\; 40\text{'}$; differing in longitude $70^\{\backslash circ\}\; 46\text{'}$, Port-Royal lying so far to the westward of the Lizard.

From the latitude of the Lizard - $50^\{\backslash circ\}\; 00\text{'}$
subtract the latitude of Port-Royal - $17^\{\backslash circ\}\; 40\text{'}$

and there remains - - - 32, 20
equal to 1940 minutes, the proper difference of latitude.

Then from the meridional parts of $50^\{\backslash circ\}\; 00\text{'}$ 3474.5
subtract those of $17^\{\backslash circ\}\; 40\text{'}$ 1077.2

and there remains - - - 2397.3
the meridional or enlarged difference of longitude.

GEOMETRICALLY. Draw the line AC ($N^\{\backslash circ\}\; 14.$) representing the meridian of the Lizard at A, and set off from A, upon that line, AE equal to 1940 (from any scale of equal parts) the proper difference of latitude, also AC equal to 2397.3 (from the same scale) the

meridional or enlarged difference of latitude. Upon the PRACTICE point C raise CB perpendicular to AC, and make CB equal to 4246, the minutes of difference of longitude.

Join AB, and through E draw ED parallel to BC: so the case is constructed; and AD applied to the same scale of equal parts the other legs were taken from, will give the direct distance, and the angle DAE measured by the line of chords will give the course.

For the angle of the course EAD, it will be, (by rectangular trigonometry,)

AC : CB :: R : T, BAC, i. e.

As the meridional diff. of lat. 2397.3 - 3.37970
is to the difference of long. 4246.0 - 3.62798
so is radius - - - 10.00000
to the tang. of the direct course $60^\{\backslash circ\}\; 33\text{'}$ 10.34828
which, because Port-Royal is southward of the Lizard, and the difference of longitude westerly, will be south $60^\{\backslash circ\}\; 33\text{'}$ west, or SWW $\backslash frac\{1\}\{4\}$ west nearly.

Then for the distance AD, it will be, (by rectangular trigonometry),

R : AE :: Sec. A : AD, i. e.

As the radius - - - 10.00000
is the proper diff. of lat. - 1940 - 3.28780
so is the secant of the course - $60^\{\backslash circ\}\; 33\text{'}$ 10.30833
to the distance - 3945.6 - 3.59613
consequently the direct course and distance between the Lizard and Port-Royal in Jamaica, is south $60^\{\backslash circ\}\; 33\text{'}$, 3945.6 miles.

CASE III. Course and distance failed given, to find difference of latitude and difference of longitude.

EXAMPLE. Suppose a ship from the Lizard in the latitude of $50^\{\backslash circ\}\; 00\text{'}$ north, sails south $35^\{\backslash circ\}\; 40\text{'}$ west 156 miles: Required the latitude come to, and how much she has altered her longitude.

GEOMETRICALLY. 1. Draw the line BK ($N^\{\backslash circ\}\; 15.$) Plate CCIII representing the meridian of the Lizard at B; from B draw the line BM, making with BK an angle equal to $35^\{\backslash circ\}\; 40\text{'}$, and upon this line set off BM equal to 156 the given distance, and from M let fall the perpendicular MK upon BK.

Then for BK the proper difference of latitude, it will be, (by rectangular trigonometry,)

R : MB :: S, BMK : BK,

i. e. As radius - - - 10.00000
is to the distance - 156 - 2.19312
so is the co-sine of the course $35^\{\backslash circ\}\; 40\text{'}$ - 9.90978
to the proper difference of lat. 127 - 2.10290
equal to $2^\{\backslash circ\}\; 07\text{'}$; and since the ship is sailing from a north latitude towards the south, therefore the latitude come to will be $47^\{\backslash circ\}\; 53\text{'}$ north. Hence the meridional difference of latitude will be 193.4.

2. Produce BK to D, till BD be equal to 193.4; through D draw DL parallel to MK, meeting DM produced in L; then DL will be the difference of longitude: to find which by calculation, it will be, (by rectangular trigonometry,)

R : BD :: T, LBD : DL,

i. e. As radius - - - 10.00000
is to the meridional diff. of lat. 193.4 - 2.28646
so is the tangent of the course $35^\{\backslash circ\}\; 40\text{'}$ 9.85594
to minutes of diff. of long. - 138.8 - 2.14240
equal to $2^\{\backslash circ\}\; 18\text{'}\; 48\text{'}\text{'}$, the difference of longitude the ship has made westerly.

CASE IV. Given course and both latitudes, viz. the la-

PRACTICE latitude failed from, and the latitude come to; to find the distance failed, and the difference of longitude.

EXAMPLE. Suppose a ship in the latitude of $50^\{\backslash circ\}\; 20\text{'}$ north, fails south $33^\{\backslash circ\}\; 45\text{'}$ east, until by observation she is found to be in the latitude of $51^\{\backslash circ\}\; 45\text{'}$ north: Required the distance failed, and the difference of longitude.

GEOMETRICALLY. Draw AB ($N^\{\backslash circ\}\; 16.$) to represent the meridian of the ship in the first latitude, and set off from A to B 155 the minutes of the proper difference of latitude, also AG equal to 257.9 the minutes of the enlarged difference of latitude. Through B and G, draw the lines BC and GK perpendicular to AG; also draw AK, making with AG an angle of $33^\{\backslash circ\}\; 45\text{'}$, which will meet the two former lines in the points C and K; so the case is constructed, and AC and GK may be found from the line of equal parts: To find which,

By CALCULATION;

First, For the difference of longitude, it will be, (by rectangular trigonometry,)

$R\; :\; AG\; ::\; T,\; GAK\; :\; GK,$
i. e. As radius - - - 10.00000
is to the enlarged diff. of lat. 257.9 - 2.41145
so is the tang. of the course $33^\{\backslash circ\}\; 45\text{'}$ - 9.82489
to min. of diff. of longitude 172.3 - 2.23634
equal to $2^\{\backslash circ\}\; 52\text{'}\; 18\text{'}\text{'}$, the difference of longitude the ship has made easterly.

This might also have been found, by first finding the departure BC, (by Case 2. of Plane Sailing), and then it would be

$AB\; :\; BC\; ::\; AG\; :\; GK$, the difference of longitude required.

Then for the direct distance AC, it will be, (by rectangular trigonometry,)

$R\; :\; AB\; ::\; \backslash text\{Sec.\; A\}\; :\; AC,$
i. e. As radius - - - 10.00000
is to the proper diff. of lat. - 155 - 2.19033
so is the secant of the course $33^\{\backslash circ\}\; 45\text{'}$ - 10.08015
to the direct distance - 186.4 - 2.27048
consequently the ship has failed south $33^\{\backslash circ\}\; 45\text{'}$ east 186.4 miles, and has differed her longitude $2^\{\backslash circ\}\; 52\text{'}\; 18\text{'}\text{'}$ easterly.

CASE V. both latitudes, and distance failed, given; to find the direct course, and difference of longitude.

EXAMPLE. Suppose a ship from the latitude of $45^\{\backslash circ\}\; 26\text{'}$ north, fails between north and east 195 miles, and then by observation she is found to be in the latitude of $48^\{\backslash circ\}\; 6\text{'}$ north: Required the direct course and difference of longitude.

GEOMETRICALLY. Draw AB ($N^\{\backslash circ\}\; 17.$) equal to 160 the proper difference of latitude, and from the point B raise the perpendicular BD; then take 195 in your compasses, and setting one foot of them in A, with the other cross the line BD in D. Produce AB, till AC be equal to 233.6 the enlarged difference of latitude. Through C draw CK parallel to BD, meeting AD produced in K: so the case is constructed; and the angle A may be measured by the line of chords, and CK by the line of equal parts: To find which,

By CALCULATION:

First, For the angle of the course BAD, it will be (by rectangular trigonometry,)

$AB\; :\; R\; ::\; AD\; :\; \backslash text\{Sec.\; A.\; \}\; i.\; e.$
As the proper diff. of lat. - 160 - 2.20412

is to radius - - - 10.00000 PRACTICE
so is the distance - 195 - 2.29003
to the secant of the course $34^\{\backslash circ\}\; 52\text{'}$ - 10.08591
which, because the ship is failing between north and east, will be north $34^\{\backslash circ\}\; 52\text{'}$ east, or NEEN $1^\{\backslash circ\}\; 7\text{'}$ easterly.

Then for the difference of longitude, it will be, (by rectangular trigonometry,)

$R\; :\; AC\; ::\; T,\; A\; :\; CK,$
i. e. As radius - - - 10.00000
is to the merid. diff. of lat. - 233.6 - 2.36847
so is the tang. of the course $34^\{\backslash circ\}\; 52\text{'}$ - 9.84307
to min. of diff. of longitude 162.8 - 2.21154
equal to $2^\{\backslash circ\}\; 42\text{'}\; 48\text{'}\text{'}$, the difference of longitude easterly.

CASE VI. One latitude, course, and difference of longitude, given; to find the other latitude, and distance failed.

EXAMPLE. Suppose a ship from the latitude of $48^\{\backslash circ\}\; 50\text{'}$ north, fails south $34^\{\backslash circ\}\; 40\text{'}$ west, till her difference of longitude is $2^\{\backslash circ\}\; 42\text{'}$: Required the latitude come to, and the distance failed.

GEOMETRICALLY. 1. Draw AE ($N^\{\backslash circ\}\; 18.$) to represent the meridian of the ship in the first latitude, and make the angle EAC equal to $34^\{\backslash circ\}\; 40\text{'}$, the angle of the course; then draw FC parallel to AE, at the distance of 164 the minutes of difference of longitude, which will meet AC in the point C. From C let fall upon AE the perpendicular CE; then AE will be the enlarged difference of latitude. To find which by calculation, it will be, (by rectangular trigonometry,)

$T,\; A\; :\; R\; ::\; CE\; :\; AE,$
i. e. As the tang. of the course $34^\{\backslash circ\}\; 40\text{'}$ - 9.83984
is to the radius - - - 10.00000
so is min. of diff. longitude - 164 - 2.21484
to the enlarged diff. of latitude 237.2 - 2.37500
and because the ship is failing from a north latitude southerly, therefore

From the merid. parts of }
the latitude failed from } - $48^\{\backslash circ\}\; 50\text{'}$ 3366.9
take the merid. difference of latitude - 237.2
and there remains - - - 3129.7
the meridional parts of the latitude come to, viz. $46^\{\backslash circ\}\; 09\text{'}$.

Hence for the proper difference of latitude,
From the latitude failed from - $48^\{\backslash circ\}\; 50\text{'}\; N$
take the latitude come to - $46^\{\backslash circ\}\; 09\text{'}\; N$

and there remains - - - 2, 41
equal to 161, the minutes of difference of latitude.

2. Set off upon AE the length AD equal to 161 the proper difference of latitude, and thro' D draw DB parallel to CE: then AB will be the direct distance. To find which by calculation, it will be, (by rectangular trigonometry,)

$R\; :\; AD\; ::\; \backslash text\{Sec.\; A\}\; :\; AB,$
i. e. As radius - - - 10.00000
is to the proper diff. of latitude - 161 - 2.20683
so is the secant of the course $34^\{\backslash circ\}\; 40\text{'}$ - 10.08488
to the direct distance - 195.8 - 2.29171

CASE VII. One latitude, course, and departure given; to find the other latitude, distance failed, and difference of longitude.

EXAMPLE. Suppose a ship fails from the latitude of $54^\{\backslash circ\}$

PRACTICE 54° 36' north, south 42° 33' east, until she has made of departure 116 miles: Required the latitude she is in, her direct distance failed, and how much she has altered her longitude.

GEOMETRICALLY. 1. Having drawn the meridian AB, (n° 19.) make the angle BAD equal to 42° 33'. Draw FD parallel to AB at the distance of 116, which will meet AD in D. Let fall upon AB the perpendicular DB. Then AB will be the proper difference of latitude, and AD the direct distance: to find which by calculation, first, for the distance AD it will be (by rectangular trigonometry.)

Then for the proper difference of latitude, it will be, (by rectangular trigonometry.)

2. Produce AB to E, till AE be equal to 212.2; and through E draw EC parallel to BD, meeting AD produced in C; then EC will be the difference of longitude; to find which by calculation, it will be, (by rectangular trigonometry.)

This might have been found otherwise, thus: because the triangles ACE, ADB, are similar; therefore it will be,

CASE VIII. Both latitudes and departure given, to find course, distance, and difference of longitude.

EXAMPLE. Suppose a ship from the latitude of 46° 20' N. fails between south and west, till she has made of departure 126.4 miles; and is then found by observation to be in the latitude of 43° 35' north: Required the course and distance failed, and difference of longitude.

GEOMETRICALLY. Draw AK (n° 20.) to represent the meridian of the ship in her first latitude; set off upon it AC, equal to 165, the proper difference of latitude. Draw BC perpendicular to AC, equal to 126.4 the departure, and join AB. Set off from A, AK equal to 233.3, the enlarged difference of latitude; and through K draw KD parallel to BC, meeting AB produced in D; so the case is constructed, and DK will be the difference of longitude, AB the distance, and the angle A the course; to find

which,

By CALCULATION:

First, For DC the difference of longitude, it will be,

Then for the course it will be, (by rectangular trigonometry.)

Lastly, For the distance AB, it will be, (by rectangular trigonometry.)

CASE IX. One latitude, distance failed, and departure given; to find the other latitude, difference of longitude, and course.

EXAMPLE. Suppose a ship in the latitude of 48° 33' north, fails between south and east 138 miles, and has then made of departure 112.6: Required the latitude come to, the direct course, and difference of longitude.

GEOMETRICALLY. 1. Draw BD (n° 21.) for the meridian of the ship at B; and parallel to it draw FE, at the distance of 112.6, the departure. Take 138, the distance, in your compasses, and fixing one point of them in B, with the other cross the line FE in the point E; then join B and E, and from E let fall upon BD the perpendicular ED; so BD will be the proper difference of latitude, and the angle B will be the course; to find which, by calculation,

First, For the course it will be, (by rectangular trigonometry.)

Then for the difference of latitude, it will be, (by rectangular trigonometry.)

2dly, Produce B to A, till BA be equal to 117.7; and

PRACTICE and through A draw AC parallel to DE, meeting BE produced in C; then AC will be the difference of longitude; to find which by calculation, it will be,

i. e. As the proper diff. of latitude 79.8 1.90180 is to the departure 112.6 2.05154 so is the enlarged diff. of latitude 117.7 2.07078 to the diff. of longitude 166.1 2.22044 equal to $2^\{\backslash circ\}\; 46\text{'}\; 06\text{'}\text{'}$, the difference of longitude easterly.

10. From what has been said, it will be easy to solve a traverse according to the rules of Mercator's sailing.

EXAMPLE. Suppose a ship at the Lizard in the latitude $50^\{\backslash circ\}\; 00\text{'}$ north, is bound to the Madeira in the latitude of $32^\{\backslash circ\}\; 20\text{'}$ north, the difference of longitude between them being $11^\{\backslash circ\}\; 40\text{'}$, the west end of the Madeira lying so much to the westward of the Lizard, and consequently the direct course and distance (by Case 2. of this Sect.), is south $26^\{\backslash circ\}\; 15\text{'}$ west 1181.9 miles; but by reason of the winds she is forced to sail on the following courses, (allowance being made for lee-way and variation, &c.), viz. SSW 44 miles, SW $\backslash frac\{1\}\{2\}$ W 36 miles, SW & S 56 miles, and S & E 28 miles: Required the latitude the ship is in, her bearing and distance from the Lizard, and her direct course and distance from the Madeira, at the end of these courses.

The geometrical construction of this traverse is performed by laying down the two ports according to construction of Case 2. of this Section, and the several courses and distances according to Case 3. by which we have the following solution by calculation.

1. Course SSW, distance 44 miles.

For difference of latitude.

As radius - - - - 10.00000
is to the distance - 44 - 1.64345
so is the co-sine of the course $22^\{\backslash circ\}\; 30\text{'}$ 9.96562
to the difference of latitude 40.65 1.60907
and since the course is southerly, therefore the latitude come to will be $49^\{\backslash circ\}\; 20\text{'}$ north, and consequently the meridional difference of latitude will be 61.8.
Then,

For difference of longitude.

As radius - - - - 10.00000
is to the enlarged diff. of lat. 61.8 - 1.79099
so is the tang. of the course $22^\{\backslash circ\}\; 30\text{'}$ 9.61722
to min. of diff. of longitude 25.6 - 1.40821
2. Course SW $\backslash frac\{1\}\{2\}$ W, distance 36 miles.

For difference of latitude.

As radius - - - - 10.00000
is to the distance - 36 - 1.55630
so is the co-sine of the course $16^\{\backslash circ\}\; 52\text{'}$ 9.98090
to the difference of latitude 34.46 - 1.53720
and since the course is southerly, therefore the latitude come to will be $48^\{\backslash circ\}\; 45\text{'}$. Hence the meridional difference of latitude will be 53.4. Then,

For difference of longitude.

As radius - - - - 10.00000
is to the enlarged diff. of lat. 53.4 - 1.72754
so is the tang. of the course $16^\{\backslash circ\}\; 52\text{'}$ 9.48171
to the difference of longitude 16.19 1.20925

3. Course SW & S, distance 56 miles.

For difference of latitude.

As radius - - - - 10.00000
is to the distance - 56 - 1.74819

so is the co-sine of the course $33^\{\backslash circ\}\; 45\text{'}$ 9.91985
to the difference of latitude 46.56 - 1.66804
consequently the latitude come to is $47^\{\backslash circ\}\; 59\text{'}$; and therefore the enlarged difference of latitude will be 69.2.

Then,

For difference of longitude.

As radius - - - - 10.00000
is to the enlarged diff. of lat. 69.2 - 1.84011
so is the tang. of the course $33^\{\backslash circ\}\; 45\text{'}$ 9.82489
to the difference of longitude 46.24 1.66500

4. Course S & E, distance 28 miles.

For difference of latitude.

As radius - - - - 10.00000
is to the distance - 28 - 1.44716
so is the co-sine of the course $11^\{\backslash circ\}\; 15\text{'}$ 9.99157
to the difference of latitude 27.46 1.43873
consequently the latitude come to will be $47^\{\backslash circ\}\; 31\text{'}$; and hence the meridional difference of latitude will be 43.2.
Then,

For difference of longitude.

As radius - - - - 10.00000
is to the enlarged diff. of lat. 43.2 - 1.63548
so is the tang. of the course $11^\{\backslash circ\}\; 15\text{'}$ 9.29866
to the diff. of longitude 8.59 - 0.93414

Now these several courses and distances, together with the difference of latitude and longitude belonging to each of them, being set down in their proper columns in the Traverse Table, will stand as follow.

Hence it is plain that the ship has made of southing 149.13 minutes, and consequently has come to the latitude of $47^\{\backslash circ\}\; 31\text{'}$ north, and so the meridional difference of latitude between that and her first latitude will be 226.1: And since she has made of difference of longitude 79.44 minutes westerly; therefore, for the direct course and distance between the Lizard and the ship, it will be, (by Case 2. of this Section).

For the direct course.

As the merid. diff. of latitude 226.1 - 2.35430
is to radius - - - - 10.00000
so is the difference of longitude 79.44 - 1.90004
to the tang. of the course $19^\{\backslash circ\}\; 22\text{'}$ 9.54593
which, because the difference of latitude is southerly, and the difference of longitude westerly, will be south $19^\{\backslash circ\}\; 22\text{'}$ west, or SW $8^\{\backslash circ\}\; 7\text{'}$ westerly. Then,

For the direct distance.

As radius - - - - 10.00000
is to the proper diff. of lat. 149.13 - 2.17349
so is the secant of the course $19^\{\backslash circ\}\; 22\text{'}$ 10.02530

For the direct course:

For the direct distance:

11. It is very common in working a day's reckoning at sea, to find the difference of latitude and departure to each course and distance; and adding all the departures together, and all the differences of latitudes for the whole departure, and difference of latitude made good that day, from thence (by Case 8. of this Section) to find the difference of longitude, &c. made good that day. Now that this method is false, will evidently appear, if we consider that the same departure, reckoned on two different parallels, will give unequal differences of longitude; and consequently, when several departures are compounded together and reckoned on the same parallel, the difference of longitude resulting from that cannot be the same with the sum of the differences of longitude resulting from the several departures on different parallels; and therefore we have chosen, in the last example of a traverse, to find the difference of longitude answering to each particular course and distance, the sum of which must be the true difference of longitude made good by the ship on these several courses and distances.

12. We shewed, at Art. 3. of this Section, how to construct a Mercator's chart; and now we shall proceed to its several uses, contained in the following problems.

PROB. I. Let it be required to lay down a place upon the chart, its latitude, and the difference of longitude between it and some known place upon the chart being given.

EXAMPLE. Let the known place be the Lizard by VOL. VII.

ing on the parallel of 50° 00' north, and the place to be laid down St Katharine's on the east coast of America, differing in longitude from the Lizard 42° 36', lying so much to the westward of it.

Let L represent the Lizard on the chart, (see N° 12.) lying on the parallel of 50° 00' north, its meridian. Set off AE from E upon the equator EQ 42° 36', towards Q, which will reach from E to F. Through F draw the meridian FG, and this will be the meridian of St Katharine's; then set off from Q to H upon the graduated meridian QB, 28 degrees; and thro' H draw the parallel of latitude HM, which will meet the former meridian in K, the place upon the chart required.

PROB. II. Given two places upon the chart, to find their difference of latitude and difference of longitude.

Through the two places draw parallels of latitude; then the distance between these parallels, numbered in degrees and minutes upon the graduated meridian, will be the difference of latitude required; and thro' the two places drawing meridians, the distance between these, counted in degrees and minutes on the equator or any graduated parallel, will be the difference of longitude required.

PROB. III. To find the bearing of one place from another upon the chart.

EXAMPLE. Required the bearing of St Katharine's at K (see N° 12.) from the Lizard at L.

Draw the meridian of the Lizard AE, and join K and L with the right line KL; then by the line of chords measuring the angle KLE, and with that entering the tables, we shall have the thing required.

This may also be done, by having compasses drawn on the chart, (suppose at two of its corners); then lay the edge of a ruler over the two places, and let fall a perpendicular, or take the nearest distance from the centre of the compass next the first place, to the ruler's edge; then with this distance in your compasses, slide them along by the ruler's edge, keeping one foot of them close to the ruler, and the other as near as you can judge perpendicular to it, which will describe the rhumb required.

PROB. IV. To find the distance between two given places upon the chart.

This problem admits of four cases, according to the situation of the two places with respect to one another.

CASE I. When the given places lie both upon the equator.

In this case their distance is found by converting the degrees of difference of longitude intercepted between them into minutes.

CASE II. When the two places lie both on the same meridian.

Draw the parallels of those places; and the degrees upon the graduated meridian, intercepted between those parallels, reduced to minutes, give the distance required.

CASE III. When the two places lie on the same parallel.

EXAMPLE. Required to find the distance between the points K and N (see N° 12.) both lying on the parallel of 28° 00' north. Take from your scale the chord

PRACTICE. chord of $60^\backslash circ$ or radius in your compasses, and with that extent on KN as a base make the isosceles triangle KPN; then take from the line of sines the co-sine of the latitude, or sine of $72^\backslash circ$, and set that off from P to S and T. Join S and T with the right line ST, and that applied to the graduated equator will give the degrees and minutes upon it equal to the distance; which, converted into minutes, will be the distance required.

The reason of this is evident from the section of Parallel Sailing: for it has been there demonstrated, that radius is to the co-sine of any parallel, as the length of any arch on the equator, to the length of the same arch on that parallel. Now in this chart KN is the distance of the meridians of the two places K and N upon the equator; and since, in the triangle PKN, ST is the parallel to KN, therefore $PN\; :\; PT\; ::\; NK\; :\; TS$. Consequently TS will be the distance of the two places K and N upon the parallel of $28^\backslash circ$.

If the parallel the two places lie on be not far from the equator, and they not far asunder; then their distance may be found thus: Take the distance between them in your compasses, and apply that to the graduated meridian, so as the one foot may be as many minutes above as the other is below the given parallel; and the degrees and minutes intercepted, reduced to minutes, will give the distance.

Or it may also be found thus: Take the length of a degree on the meridian at the given parallel, and turn that over on the parallel from the one place to the other, as oft as you can; then as oft as that extent is contained between the places, so many times 60 miles will be contained in the distance between them.

CASE IV. When the places differ both in longitude and latitude.

EXAMPLE. Suppose it were required to find the distance between the two places a and e upon the chart. By

Prob. II. Find the difference of latitude between them; and take that in your compasses from the graduated equator, which set off on the meridian of a, from a to b; then thro' b draw bc parallel to de; and taking ac in your compasses, apply it to the graduated equator, and it will shew the degrees and minutes contained in the distance required, which multiplied by 60 will give the miles of distance.

The reason of this is evident from Art. 6. of this Section: for it is plain ad is the enlarged difference of latitude, and ab the proper; consequently ae the enlarged distance, and ac the proper.

PROB. V. To lay down a place upon the chart, its latitude and bearing from some known place upon the chart being known, or (which is the same) having the course and difference of latitude that a ship has made, to lay down the running of the ship, and find her place upon the chart.

EXAMPLE. A ship from the Lizard in the latitude of $50^\backslash circ\; 00\text{'}$ north, sails SSW till she has differed her latitude $36^\backslash circ\; 40\text{'}$. Required her place upon the chart.

Count from the Lizard at L, on the graduated meridian downwards (because the course is southerly) $36^\backslash circ\; 40\text{'}$ to e; through which draw a parallel of latitude, which will be the parallel the ship is in; then from L draw a SSW line Lf, cutting the former pa-

rallel in f, and this will be the ship's place upon the PRACTICE chart.

PROB. VI. One latitude, course, and distance, sailed, given; to lay down the running of the ship, and find her place upon the chart.

EXAMPLE. Suppose a ship at a in the latitude of $20^\backslash circ\; 00\text{'}$ north, sails north $37^\backslash circ\; 20\text{'}$, east 191 miles: Required the ship's place upon the chart.

Having drawn the meridian and parallel of the place a, set off the rhumb-line ae, making with ab an angle of $37^\backslash circ\; 20\text{'}$; and upon it set off 191 from a to c; thro' c draw the parallel cb; and taking ab in your compasses, apply it to the graduated equator, and observe the number of degrees it contains; then count the same number of degrees on the graduated meridian from C to b, and through b draw the parallel be, which will cut ac produced in the point e, the ship's place required.

PROB. VII. Both latitudes and distance sailed, given; to find the ship's place upon the chart.

EXAMPLE. Suppose a ship sails from a, in the latitude of $20^\backslash circ\; 00\text{'}$ north, between north and east 191 miles, and is then in the latitude of $45^\backslash circ\; 00\text{'}$ north: Required the ship's place upon the chart.

Draw de the parallel of $45^\backslash circ$, and set off upon the meridian of a upwards, ab equal to the proper difference of latitude taken from the equator or graduated parallel. Through b draw bc parallel to de; then with 191 in your compasses, fixing one foot of them in a, with the other cross bc in c. Join a to c with the right line ac; which produced will meet de in e, the ship's place required.

PROB. VIII. One latitude, course, and difference of longitude, given; to find the ship's place upon the chart.

EXAMPLE. Suppose a ship from the Lizard in the latitude of $50^\backslash circ\; 00\text{'}$ north, sails SW&W, till her difference of longitude is $42^\backslash circ\; 36\text{'}$: Required the ship's place upon the chart.

Having drawn AE the meridian of the Lizard at L, count from E to F upon the equator $42^\backslash circ\; 36\text{'}$; and through F draw the meridian EG; then from L draw the SW&W line LK, and where this meets FG, as at K, will be the ship's place required.

PROB. IX. One latitude, course, and departure, given; to find the ship's place upon the chart.

EXAMPLE. Suppose a ship at a in the latitude of $20^\backslash circ\; 00\text{'}$ north, sails north $37^\backslash circ\; 20\text{'}$ east, till she has made of departure 116 miles: Required the ship's place upon the chart.

Having drawn the meridian of a, at the distance of 116, draw parallel to it the meridian kl. Draw the rhumb-line ac, which will meet kl in some point e; then through c draw the parallel cb, and ab will be the proper difference of latitude, and bc the departure. Take ab in your compasses, and apply it to the equator or graduated parallel; then observe the number of degrees it contains, and count so many on the graduated meridian from C upwards to b. Through b draw the parallel be, which will meet ac produced in some point as e, which is the ship's place upon the chart.

PROB. X. One latitude, distance, and departure, given; to find the ship's place upon the chart.

EXAMPLE. Suppose a ship at a in the latitude of $20^\backslash circ\; 00\text{'}$ north, sails 191 miles between north and east, and then

PRACTICE then is found to have made of departure 116 miles: Required the ship's place upon the chart.

Having drawn the meridian and parallel of the place $a$, set off upon the parallel $am$ equal to 116, and thro' $m$ draw the meridian $kl$. Take the given distance 191 in your compasses; setting one foot of them in $a$, with the other cross $kl$ in $c$. Join $ac$, and through $c$ draw the parallel $cb$; so $cb$ will be the departure, and $ab$ the proper difference of latitude; then proceeding with this, as in the foregoing problem, you will find the ship's place to be $e$.

PROB. XI. The latitude sailed from, difference of latitude, and departure, given; to find the ship's place upon the chart.

EXAMPLE. Suppose a ship from $a$ in the latitude of $20^\{\backslash circ\}\; 00\text{'}$ north, sails between north and east, till she be in the latitude of $45^\{\backslash circ\}\; 00\text{'}$ north, and is then found to have made of departure 116 miles: Required the ship's place upon the chart.

Having drawn the meridian of $a$, set off upon it, from $a$ to $b$, 25 degrees, (taken from the equator or graduated parallel), the proper difference of latitude; then through $b$ draw the parallel $bc$, and make $bc$ equal to 116 the departure, and join $ac$. Count from the parallel of $a$ on the graduated meridian upwards to $b$ 25 degrees, and through $b$ draw the parallel $be$, which will meet $ac$ produced in some point $e$, and this will be the place of the ship required.

13. In the section of Plane Sailing, it is plain, that the terms meridional distance, departure, and difference of longitude, were synonymous, constantly signifying the same thing; which evidently followed from the supposition of the earth's surface being projected on a plane in which the meridians were made parallel, and the degrees of latitude equal to one another and to those of the equator. But since it has been demonstrated (in this section) that if, in the projection of the earth's surface upon a plane, the meridians be made parallel, the degrees of latitude must be unequal, still increasing the nearer they come to the pole; it follows, that these terms must denote lines really different from one another.

§ 6. Of Oblique Sailing.

THE questions that may be proposed on this head being innumerable, we shall only give a few of the most useful.

PROB. I. Coasting along the shore, I saw a cape bear from me NNE; then I stood away NW&W 20 miles, and I observed the same cape to bear from me NE&E: Required the distance of the ship from the cape at each station.

Plate CCIII. GEOMETRICALLY. Draw the circle NWSE ($N^\{\backslash circ\}\; 22.$) to represent the compass, NS the meridian, and WE the east and west line, and let C be the place of the ship in her first station; then from C set off upon the NW&W line, CA 20 miles, and A will be the place of the ship in her second station.

From C draw the NNE line CB, and from A draw AB parallel to the NE&E line CD, which will meet CB in B the place of the cape, and CB will be the distance of it from the ship in its first station, and AB the distance in the second: to find which,

By CALCULATION;

In the triangle ABC are given AC, equal to 20

miles; the angle ACB, equal to $78^\{\backslash circ\}\; 45\text{'}$, the distance between the NNE and NW&W lines; also the angle ABC, equal to BCD, equal to $33^\{\backslash circ\}\; 45\text{'}$, the distance between the NNE and NE&E lines; and consequently the angle A, equal to $67^\{\backslash circ\}\; 30\text{'}$.

Hence for CB, the distance of the cape from the ship in her first station, it will be (by oblique trigonometry)

i. e. As the sine of the angle B $33^\{\backslash circ\}\; 45\text{'}$ - 9.74473 is to the distance run AC 20 - 1.30103 so is the sine of BAC 67. 30 - 9.96562 to CB 33.26 - 1.52191 the distance of the cape from the ship at the first station. Then for AB, it will be, by oblique trigonometry,

i. e. as the sine of B $33^\{\backslash circ\}\; 45\text{'}$ - 9.74473 is to AC 20 - 1.30103 so is the sine of C 78. 45 - 9.99157 to AB 35.31 - 1.54786 the distance of the ship from the cape at her second station.

PROB. II. Coasting along the shore, I saw two headlands; the first bore from me NE&E 17 miles, the other SSW miles: Required the bearing and distance of these headlands from one another.

GEOMETRICALLY. Having drawn the compass NWSE ($N^\{\backslash circ\}\; 23.$) let C represent the place of the ship; set off upon the NE&E line CA 17 miles from C to A, and upon the SSW line CB 20 miles from C to B, and join AB: then A will be the first headland, and B the second; also AB will be their distance, and the angle A will be the bearing from the NE&E line: to find which,

By CALCULATION;

In the triangle ACB are given, AC 17, CB 20, and the angle ACB equal to $101^\{\backslash circ\}\; 15\text{'}$, the distance between the NE&E and SSW lines. Hence (by oblique angular trigonometry) it will be

As the sum of the sides AC and CB - 37 1.56820 is to their difference - 3 0.47712 so is the tang. of $\backslash frac\{1\}\{2\}$ the sum } of the angles A and B } 39. 22. 9.91417 to the tang. of half their diff. - 3, 49 8.82309 consequently the angle A will be $43^\{\backslash circ\}\; 11\text{'}$, and the angle B $35^\{\backslash circ\}\; 34\text{'}$; also the bearing of B from A will be S&W $1^\{\backslash circ\}\; 49\text{'}$ westerly, and the bearing of A from B will be N&E $1^\{\backslash circ\}\; 49\text{'}$ easterly.

Then for the distance AB, it will be, (by oblique angular trigonometry),

i. e. As the sine of A $43^\{\backslash circ\}\; 11\text{'}$ - 9.83527 is to CB 20 - 1.30103 so is the sine of C $101,\; 15$ - 9.99157 to AB 28.67 - 1.45733 the distance between the two headlands.

PROB. III. Coasting along the shore, I saw two headlands; the first bore from me NW&N, and the second NNE; then standing away E&N $\backslash frac\{1\}\{2\}$ northerly 20 miles, I found the first bore from me WNW $\backslash frac\{1\}\{2\}$ westerly, and the second NW $\backslash frac\{1\}\{2\}$ westerly: Required the bearing and distance of these two headlands.

GEOMETRICALLY. Having drawn the compass NWSE ($N^\{\backslash circ\}\; 24.$) let C represent the first place of the ship;

PRACTICE ship: from which draw the NW&N line CB, and the NNE line CD, also the E&N $\backslash frac\{1\}\{2\}$ N line CA, which make equal to 20. From A draw AB parallel to the WNW $\backslash frac\{1\}\{2\}$ W line, and AD parallel to the N&W $\backslash frac\{1\}\{2\}$ W meeting the two first lines in the points B and D; then B will be the first and D the second headlands. Join the points B and D, and BD will be the distance between them, and the angle CDB the bearing from the NNE line: to find which,

1. In the triangle ABC are given the angle BCA, equal to $104^\{\backslash circ\}\; 04\text{'}$, the distance between the NW&N line, and the ENE $\backslash frac\{1\}\{2\}$ E line; the angle BAC, equal to $36^\{\backslash circ\}\; 34\text{'}$, the distance between the WSW $\backslash frac\{1\}\{2\}$ W line and the WNW $\backslash frac\{1\}\{2\}$ W line; the angle ABC equal to $39^\{\backslash circ\}\; 22\text{'}$, the distance between the ESE $\backslash frac\{1\}\{2\}$ E line; and the SWS line, also the side CA equal to 20 miles: whence for CB, it will be (by oblique trigonometry)

the distance between the first headland and the ship in her first station.

2. In the triangle ACD, are given the angle ACD, equal to $47^\{\backslash circ\}\; 49\text{'}$, the distance between the ENE $\backslash frac\{1\}\{2\}$ E line, and the NNE line; the angle CAD, equal to $92^\{\backslash circ\}\; 49\text{'}$, the distance between the WSW $\backslash frac\{1\}\{2\}$ W line; and the N&W $\backslash frac\{1\}\{2\}$ W line, the angle CDA equal to $39^\{\backslash circ\}\; 22\text{'}$, the distance between the SSW line and the S&E $\backslash frac\{1\}\{2\}$ E line; also the leg CA equal to 20.

Hence for CD, it will be (by oblique trigonometry)

the distance between the second headland and the ship in her first station.

3. In the triangle BCD are given BC 18.79, CD 31.5, and the angle BCD equal to $56^\{\backslash circ\}\; 15\text{'}$, the distance between the NW&N line and the NNE line.

Hence for the angle CDB, it will be (by oblique trigonometry)

consequently the angle CBD is $87^\{\backslash circ\}\; 10\text{'}$, and the angle CDB $36^\{\backslash circ\}\; 35\text{'}$. Hence the bearing of the first headland from the second will be S $59^\{\backslash circ\}\; 8\text{'}$, W or SWS $\backslash frac\{1\}\{2\}$ W nearly; and for the distance between them, it will be,

the distance between the two headlands.

This, and the first problem, are of great use in drawing the plot of any harbour, or laying down any sea-coast.

Suppose a ship that makes her way good within 64 points of the wind, at north, is bound to a port bearing east 86 miles distance from her: Required the

course and distance upon each tack, to gain the intended port. PRACTICE

GEOMETRICALLY. Having drawn the compass NE SW, ($N^\{\backslash circ\}\; 25.$) let C represent the ship's place, and set off upon the east line CA 86 miles, so A will be the intended port. Draw CD and CB on each side of the north line at 64 points distance from it, and through A draw AB parallel to CD meeting CB in B; then the ENE $\backslash frac\{1\}\{2\}$ E line CB, will be the course of the ship upon the starboard tack, and CB its distance on that tack; also the ESE $\backslash frac\{1\}\{2\}$ E line Ab, will be the course on the larboard tack, and BA the distance on that tack: to find which,

In the triangle ABC are given the angle ACB, equal to $16^\{\backslash circ\}\; 53\text{'}$, the distance between the east and ENE $\backslash frac\{1\}\{2\}$ E line; the angle CBA, equal to $146^\{\backslash circ\}\; 14\text{'}$, the distance between the ENE $\backslash frac\{1\}\{2\}$ E and the WNW $\backslash frac\{1\}\{2\}$ W lines; the angle BAC equal to $16^\{\backslash circ\}\; 53\text{'}$, the distance between the east and ESE $\backslash frac\{1\}\{2\}$ E lines; also AC 86 miles.

Hence, since the angle at A and C are equal, the legs CB and BA will likewise be equal; to find either of which (suppose CB) it will be (by oblique angled trigonometry)

the distance the ship must sail on each tack.

There is a great variety of useful questions of this nature that may be proposed; but the nature of them being better understood by practice at sea, we shall leave them, and go on to Great Circle Sailing.

A GREAT many cases might be proposed in this kind of sailing; but as they serve rather for exercises in the solution of spheric triangles than for any real use towards the navigating of a ship, we shall only give the solution of one problem, as being the most generally useful.

PROB. Given the latitudes and longitudes of two places on the earth: Required the nearest distance on the surface, together with the angles of position (or that which a great circle, passing over both places, makes with the meridian of one of them) from either place to the other.

CASE I. When both places lie under the same meridian, their difference of latitude shews their nearest distance.

CASE II. When the two places lie under the equator, their distance is equal to the difference of longitude between them.

CASE III. When the places lie under the same parallel of latitude.

EXAMPLE. What is the least distance between St Mary's in Lat. $37^\{\backslash circ\}\; 00\text{'}$ N. Long. $25^\{\backslash circ\}\; 0\text{'}$ W. and Cape Henry, in Lat. $37^\{\backslash circ\}\; 00\text{'}$ Long. $76^\{\backslash circ\}\; 23\text{'}$ west?

Describe a circle PESQ representing the meridian of one of the places; suppose of the eastern one, as St Mary's; draw the line EQ representing the equator, and at right angles to it draw the line PS, for the axis of the earth, the extremity of which, P, is the north pole, and S the south pole; and on this circle lay off from P to A the complement of the lati-

PRACTICE latitude of St Mary's, the eastern place. On the equator, from Q to C, lay off the difference of longitude between the two places; and through the points P, C, S, describe a circle, which will be the meridian of the other place Cape Henry; on which lay from P to B the co-latitude of this place, which is done by describing the arc $Aa$ about the pole P according to the rules of projection, at the distance of the co-latitude. Through the points ABD describe a great circle; then will A represent St Mary's, and B Cape Henry; PA and PB are their co-latitudes; the angle APB, which is measured by the arc QC, is the difference of longitude; the arc AB is the nearest distance of these places; the angle PAB is the angle of position from A to B; and the angle PBA is the angle of position from B to A. The arc AB, and the angle PAB or PBA, may be measured according to the rules laid down under the article PROJECTION. Now, the places having the same latitude, PA is equal to PB, and the angles PAB and PBA are likewise equal. Therefore if the arc PI be described, making the angle API = $25^\{\backslash circ\}\; 41\backslash frac\{1\}\{2\}\text{'}$, the half of the difference of longitude; PI will be perpendicular to AB, and bisect it. And in the triangle AIP, right-angled at I, there will be given the hypotenuse AP = $53^\{\backslash circ\}\; 00\text{'}$ the angle API = $25^\{\backslash circ\}\; 41\backslash frac\{1\}\{2\}\text{'}$; to find the leg AI = half the distance sought, and the angle PAI = the angle of position. Then, for the distance: As radius is to the sine of the hypotenuse PA, so is the sine of the given angle API to the sine of the leg AI. Or,

which doubled, gives $40^\{\backslash circ\}\; 31\text{'}$ for the distance; and this distance, reduced to nautical miles, is 2431; less by 31 than that given by parallel sailing.—For the angle of position, As radius is to the co-sine of the hypotenuse PA, so is the tangent of the given angle API to the co-tangent of the angle A. Or,

Hence it appears, that to sail from A to B, or from B to A, the ship must first steer N. $73^\{\backslash circ\}\; 51\text{'}$ W. or E. and then gradually increase her course till she comes to I, where it will be due west or east; and from thence the course is to be gradually diminished again till she comes to the other port, where it will be $73^\{\backslash circ\}\; 51\text{'}$, the same as she set out with.

CASE IV. When one place has latitude, and the other has none.

EXAMPLE. What is the nearest distance between the island of St Thomas, in lat. $0^\{\backslash circ\}\; 00\text{'}$, and long. $1^\{\backslash circ\}\; 00\text{'}$? The co-latitude of St Julian is $41^\{\backslash circ\}\; 09\text{'}$; and the difference of longitude between the two places is $66^\{\backslash circ\}\; 10\text{'}$.—Let the point A (plate CCL. fig. 7.) be St Thomas, and P and S the north and south poles. Make AC, the measure of the angle ASC, equal to $66^\{\backslash circ\}\; 10\text{'}$ the difference of longitude. Then, as Port St Julian is in south latitude, about S the south pole at the distance of Julian's co-latitude, describe the arc $aa$, cutting SCP, the meridian of Julian in B, through

the points A, B, E, a great circle being described, the arc AB is the distance sought. The distances and angles may now be measured according to the rules of projection, or it falls under a case in spheric trigonometry: for, in the quadrantal triangle ASB, there are given the co-latitude of St Thomas or AS = $90^\{\backslash circ\}\; 00\text{'}$; the co-latitude of St Julian, or SB = $41^\{\backslash circ\}\; 09\text{'}$; the difference of longitude, or the angle ASB = $66^\{\backslash circ\}\; 10\text{'}$, from whence all the rest may be found. Or, in the supplemental triangle, ACB, right-angled at C, there is given the latitude of St Julian's, or the leg CB = $48^\{\backslash circ\}\; 51\text{'}$; the difference of longitude, or the leg CA = $66^\{\backslash circ\}\; 10\text{'}$, whence the rest may easily be found; and hence it will appear, that a ship sailing from the island of St Thomas must first shape her course south $51^\{\backslash circ\}\; 22\text{'}$ W; and then, by constantly altering her course towards the west, so as to arrive at Port St Julian on a course S. $71^\{\backslash circ\}\; 36\text{'}$ W, she will have sailed the shortest distance between these places.

CASE V. When the latitudes of the given places are both north, or both south.

EXAMPLE. What is the nearest distance between the Lizard and the island of Bermudas, and also the angles of position?—The difference of longitude of the two places is $58^\{\backslash circ\}\; 11\text{'}$.

Make PA (Plate CCL. fig. 8.) = $57^\{\backslash circ\}\; 25\text{'}$, the co-latitude of Bermudas; Pa = $40^\{\backslash circ\}\; 03\text{'}$, the co-latitude of the Lizard; and with the tangent of Pa describe the arc $aa$. With the secant of $58^\{\backslash circ\}\; 11\text{'}$, the difference of longitude, describe arcs from P and S, which gives the centre of the circle PCS the meridian of the Lizard; its intersection with $aa$ gives B, the place of the Lizard. The arcs of the circle and angles may be measured by spheric trigonometry as before. Had the eastern place, the Lizard, been put upon the primitive circle, the great circle AB would have been difficult to describe; and therefore the western place was put upon it, it being a matter of indifference which of the places are so taken.

CASE VI. When one of the given places has north latitude, and the other has south latitude.

EXAMPLE. What is the nearest distance from the island of St Helena to the island of Bermudas, and also the angles of position at each place; the difference of longitude between the two being $57^\{\backslash circ\}\; 43\text{'}$?

Make QA (fig. 9.) = $15^\{\backslash circ\}\; 55\text{'}$, the lat. of St Helena; describe the arc $aa$ about P, with the tangent of P a = $57^\{\backslash circ\}\; 25\text{'}$, the co-latitude of Bermudas. Arcs described from P, S, with the secant of $57^\{\backslash circ\}\; 43\text{'}$, the difference of longitude, will give the centre of the circle PCS, the meridian of Bermudas; and its intersection B with $aa$, is the place of Bermudas. Describe a great circle through A, B, D; the intercepted arc AB is the distance sought; and the angles PAB, ABS, are the positions required, which must be measured according to the rules of spheric trigonometry. From the solutions of these triangles it will appear, that when a ship sails from St Helena to Bermudas on the arc of a great circle, she must first shape her course N. $48^\{\backslash circ\}\; 00\text{'}$ W. and gradually alter it from the north towards the west, so as to arrive at Bermudas on a course N. $50^\{\backslash circ\}\; 01\text{'}$ west, after having run $73^\{\backslash circ\}\; 26\text{'}$, or 4406 miles. The course found by Mercator's sailing is N. $48^\{\backslash circ\}\; 45\text{'}$ W. and the distance is 4414 miles.—By this it appears

PRACTICE appears, that when the places are one in N. latitude, and the other in S. latitude; neither of them being very far from the equator, there is but a small difference between the results found by Mercator's and great circle sailing: for, near the equator, the rhumb-lines do not differ much from great circles.

From the solution of the foregoing cases, it is plain, 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, founded upon this principle, that, in small arcs, the difference between the arc and its tangent is so little, that they may be taken one for the other in any nautical operations. Upon this principle, the great circles of the earth are supposed to be made up of short right lines, each of which is a segment of a rhumb-line. And on this supposition the solution of the following problem is founded.

Having given the latitudes and longitudes of the places sailed from, and bound to; to find 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 six cases.

2. Find the greatest latitude the great circle runs through; that is, 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 rad. : tan. greatest lat. :: cos. 1 polar ang. : tan. 1 lat.
:: cos. 2 polar ang. : tan. 2 lat.
Ec. Ec.

In the triangle PIB.

Given PB = 53° 00'

the angle PBI = 73 09

To find PI.

PRACTICE 4. Having the several latitudes passed thro', and the difference of longitude between each, find, by Mercator's sailing, 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 the truth; but it is sufficient to compute to every five degrees of difference of longitude, the length of an arc of five degrees differing from its chord or tangent only by 0.0002.

EXAMPLE. A ship being bound from a place in lat. 37° 00' N. lon. 22° 56' W. to a place in the same lat. and in lon. 76° 23' W. it is proposed she shall sail as near the arc of a great circle as she can, by altering her course at every five degrees difference of longitude: Required the latitude at each time of altering the course, and also the courses and distances between those several latitudes.

Place sailed from lat. is 37° 00' N. the long. 22° 56' W.
Place bound to lat. is 37 00 N. the long. 76 23 W.

The diff. of long. 53 27

The figure being described, and the computation made, the distance BA is found to be 42° 06', and the angle A or the angle B = 73° 09', the angle of position.

Now the triangle APB, being isosceles, the perpendicular PI falls in the middle of AB; and the latitudes, courses, and distances, being known in running the half BI, those in the half IA will also be known.

Let the points a, b, c, d, &c. be the places arrived at on each alteration of five degrees of longitude: then will the arcs Pa, Pb, Pc, Pd, &c. be the respective co-latitudes of those places, and are the hypothenuses of the right-angled spheric triangles PIa, PIb, PIc, PId, &c.

Now the angle IPB = $(\backslash frac\{53^\{\backslash circ\}\; 27\text{'}\}\{2\})$ 26° 43½'; the angle IPa = 21° 43½'; the angle IPb = 16° 43½'; the angle IPc = 11° 43½'; the angle IPd = 6° 43½', are the several polar angles.

The degrees and min. set over each column, are the polar angles used in that proportion, and the corresponding latitudes stand at bottom.

The first term of these proportions being radius, and the second term constant, the operations may be very

expediently performed thus.

On a slip of paper let the log. of the second or constant term be written of the same size with the printed figures; apply this log. co-tang. successively to the log. co-sines of the polar angles: Then the sum of the

PRACTICE the two log^s. being written down each time, will give the log. tangents of the several latitudes arrived at.

By this method, each proportion will be worked by writing down only one line.

Hence it appears, the ship must first sail from the lat. 37° 00' N. to lat. 38° 05' N.; thence to lat. 38° 56' N.; thence to lat. 39° 33'; thence to lat. 39° 57' N.; thence to lat. 40° 09' N., which is the greatest latitude she must go to; and from thence she must proceed through the latitudes 39° 57', 39° 33', 38°

56', 38° 05', and so to 37° 00', the parallel she set out from, and in which she is to find the place she is bound to.

Now between these several latitudes, with the respective differences of longitude, find by Mercator's sailing the courses and distances.

If the results of the several operations, in the questions of great circle sailing be entered in such a table as the following, it will be found of some convenience to the operator.

In the first column are the angles at the pole contained between the perpendicular and the several meridians differing by 5° of longitude.

In the second column, the departed longitude 22° 56' being increased by the differences of longitude, make the successive longitudes come to.

In the third column are the successive latitudes passed thro' in sailing from the place set out from to the greatest latitude.

In the fourth and fifth columns are the differences between the longitudes and latitudes in the second and third columns.

In the sixth column are the meridional parts to the successive latitudes; and in the seventh column are the meridional diff. of latitudes.

The eighth and ninth columns contain the courses and distances between the places answering to the second and third columns.

The numbers in the third, eighth, and ninth columns, are found by working the logarithmic proportions on a waste paper; but the work is here omitted, as it is so easily supplied.

Now the column of distances being summed up amounts to 1261.9; which being doubled, gives 2523.8 miles for the distance between the two places.

And the courses the ship must steer are, 1st, N. 74° 43' W.; 2d, N. 77° 44' W.; 3d, N. 80° 57' W.; 4th, N. 84° 04' W.; 5th, N. 87° 46' W.; 6th, S. 87° 46' W.; 7th, S. 84° 04' W.; 8th, S. 80° 57' W.; 9th, S. 77° 44' W.; 10th, S. 74° 43' W.; and on these courses she must run the respective distances standing against them.

Having now shown the method of solving the different cases of navigation mathematically, and supposing the course of the ship and distance run to be always exactly known, we shall now proceed to give an account of those mechanical methods by which the ship's course is observed, and the frequent variations and errors in it corrected at convenient times.

§ 7. Of the Log-line and Compass.

1. THE method commonly made use of for mea-

suring a ship's way at sea, or how far she runs in a given space of time, is by the LOG-LINE, and Half-minute Glass.

2. The log, fig. 3. is generally about a quarter of Plate an inch thick, and five or six inches from the angular CLXXI. point a to the circumference b. It is balanced by a thin plate of lead, nailed upon the arch, so as to swim perpendicularly in the water, with about $\backslash frac\{1\}\{2\}$ impressed under the surface. The line is fastened to the log by means of two legs a and b, fig. 2. one of which passes thro' a hole a at the corner, and is knotted on the opposite side; whilst the other leg is attached to the arch by a pin b, fixed in another hole, so as to draw out occasionally. By these legs the log is hung in equilibrium, and the line, which is united to it, is divided into certain spaces, which are in proportion to an equal number of geographical miles, as a half minute or quarter minute is to an hour of time.

3. These spaces are called knots, because at the end of each of them there is a piece of twine with knots in it, increased between the strands of the line, which shews how many of these spaces or knots are run out during the half minute. They commonly begin to be counted at the distance of about 10 fathom or 60 feet from the log; that so the log, when it is hove overboard, may be out of the eddy of the ship's wake before they begin to count; and for the more ready discovery of this point of commencement, there is commonly fastened at it a piece of red rag.

4. The log, being thus prepared, and hove overboard from the poop, and the line veered out (by the help of a reel (fig. 4.) that turns easily, and about which it is wound) as fast as the log will carry it away, or rather as the ship sails from it, will shew, according to the time of veering, how far the ship has run in a given time, and consequently her rate of sailing.

5. A degree of a meridian, according to the exactest measures, contains about 69 545 English miles; and each mile by the statute being 5280 feet, therefore a degree of a meridian will be about 367,200 feet; whence the $\backslash frac\{1\}\{60\}$ of that, viz. a minute, or nautical mile, must contain 6120 standard feet; consequently, since $\backslash frac\{1\}\{2\}$ minute

PRACTICE $\backslash frac\{1\}\{2\}$ minute is the $\backslash frac\{1\}\{120\}$ part of an hour, and each knot being the same part of a nautical mile, it follows, that each knot will contain the $\backslash frac\{1\}\{120\}$ of 6120 feet, viz. 51 feet.

6. Hence it is evident, that whatever number of knots the ship runs in half a minute, the same number of miles she will run in one hour, supposing her to run with the same degree of velocity during that time; and therefore it is the general way to heave the log every hour to know her rate of sailing: but if the force or direction of the wind vary, and not continue the same during the whole hour; or if there has been more sail set, or any sail handed, that so the ship has run swifter or slower in any part of the hour than she did at the time of heaving the log; then there must be an allowance made accordingly for it, and this must be according to the discretion of the artist.

7. Sometimes, when the ship is before the wind, and there is a great sea setting after her, it will bring home the log, and consequently the ship will sail faster than is given by the log. In this case it is usual, if there be a very great sea, to allow one mile in ten, and less in proportion, if the sea be not so great. But for the generality, the ship's way is really greater than that given by the log; and therefore, in order to have the reckoning rather before than behind the ship, (which is the safest way), it will be proper to make the space on the log-line between knot and knot to consist of 50 feet instead of 51.

8. If the space between knot and knot on the log-line should happen to be too great in proportion to the half-minute glass, viz. greater than 50 feet, then the distance given by the log will be too short; and if that space be too small, then the distance run (given by the log) will be too great: therefore, to find the true distance run in either case, having measured the distance between knot and knot, we have the following proportion, viz.

As the true distance, 50 feet, is to the measured distance; so are the miles of distance given by the log, to the true distance in miles that the ship has run.

EXAMPLE I. Suppose a ship runs at the rate of 6$\backslash frac\{1\}\{2\}$ knots in half a minute; but measuring the space between knot and knot, I find it to be 56 feet: Required the true distance in miles.

Making it, As 50 feet is to 56 feet, so is 6.25 knots to 7 knots; I find that the true rate of sailing is 7 miles in the hour.

EXAMPLE II. Suppose a ship runs at the rate of 6$\backslash frac\{1\}\{2\}$ knots in half a minute; but measuring the space between knot and knot, I find it to be only 44 feet: Required the true rate of sailing.

Making it, As 50 feet is to 44 feet, so is 6.5 knots to 5.72 knots; I find that the true rate of sailing is 5.72 knots in the hour.

9. Again, supposing the distance between knot and knot on the log-line to be exactly 50 feet, but that the glass is not 30 seconds; then, if the glass require longer time to run than 30 seconds, the distance given will be too great, if estimated by allowing one mile for every knot run in the time the glass runs; and, on the contrary, if the glass requires less time to run than 30 seconds, it will give the distance failed too small. Consequently, to find the true distance in either case, we must measure the time the glass requires to run out

(by the method in the following article); then we have PRACTICE the following proportion, viz.

As the number of seconds the glass runs, is to half a minute, or 30 seconds; so is the distance given by the log, to the true distance.

EXAMPLE I. Suppose a ship runs at the rate of 7$\backslash frac\{1\}\{2\}$ knots in the time the glass runs; but measuring the glass, I find it runs 34 seconds: Required the true distance failed.

Making it, As 34 seconds is to 30 seconds, so is 7.5 to 6.6; I find that the ship sails at the rate of 6.6 miles an hour.

EXAMPLE II. Suppose a ship runs at the rate of 6$\backslash frac\{1\}\{2\}$ knots; but measuring the glass, I find it runs only 25 seconds: Required the true rate of sailing.

Making it, As 25 seconds is to 30 seconds, so is 6.5 knots to 7.8 knots; I find that the true rate of sailing is 7.8 miles an hour.

10. In order to know how many seconds the glass runs, you may try it by a watch or clock that vibrates seconds; but if neither of these be at hand, then take a line, and to the one end fastening a plummet, hang the other upon a nail or peg, so as the distance from the peg to the centre of the plummet be 39$\backslash frac\{1\}\{2\}$ inches: then this put into motion will vibrate seconds; i. e. every time it passes the perpendicular, you are to count one second; consequently, by observing the number of vibrations that it makes during the time the glass is running, we know how many seconds the glass runs.

11. If there be an error both in the log-line and half-minute glass, viz. if the distance between knot and knot and the log-line be either greater or less than 50 feet, and the glass runs either more or less than 30 seconds; then the finding out the ship's true distance will be somewhat more complicate, and admit of three cases, viz.

CASE I. If the glass runs more than 30 seconds, and the distance between knot and knot be less than 50 feet, then the distance given by the log-line, viz. by allowing 1 mile for each knot the ship sails while the glass is running, will always be greater than the true distance, since either of these errors gives the distance too great. Consequently, to find the true rate of sailing in this case, we must first find (by Art. 8.) the distance, on the supposition that the log-line is only wrong, and then with this (by Art. 9.) we shall find the true distance.

EXAMPLE. Suppose a ship is found to run at the rate of 6 knots; but examining the glass, I find it runs 35 seconds; and measuring the log-line, I find the distance between knot and knot to be but 46 feet: Required the true distance run.

First, (by Art. 8.) We have the following proportion, viz. As 50 feet : 46 feet :: 6 knots : 5.52 knots. Then (by Art. 9.) As 35 seconds : 30 seconds :: 5.52 knots : 4.73 knots. Consequently the true rate of sailing is 4.73 miles an hour.

CASE II. If the glass be less than 30 seconds, and the place between knot and knot be more than 50 feet; then the distance given by the log will always be less than the true distance, since either of these errors lessens the true distance.

EXAMPLE. Suppose a ship is found to run at the rate of 7 knots; but examining the glass, I find it runs only 25 seconds; and measuring the space between

PRACTICE between knot and knot on the log-line, I find it is 54 feet: Required the true rate of sailing.

First, (by Art. 9.) As 25 seconds: 30 seconds :: 7 knots: 8.4 knots. Then (by Art. 8.) As 50 feet: 54 feet :: 8.4 knots: 9.072 knots. Consequently the true rate of sailing is 9.072 miles an hour.

CASE III. If the glass runs more than 30 seconds, and the space between knot and knot be greater than 50 feet; or if the glass runs less than 30 seconds, and the space between knot and knot be less than 50 feet: then, since in either of these two cases the effects of the errors are contrary, it is plain the distance will sometimes be too great, and sometimes too little, according as the greater quantity of the error lies; as will be evident from the following examples.

EXAMPLE I. Suppose a ship is found to run at the rate of 9$\backslash frac\{1\}\{2\}$ knots per glass; but examining the glass, it is found to run 36 seconds; and by measuring the space between knot and knot, it is found to be 58 feet: Required the true rate of sailing.

First, (by Art. 8.) As 50 feet: 58 feet :: 9.5 knots: 11.02 knots. Then (by Art. 9.) As 38 seconds: 30 seconds :: 11.02 knots: 8.7 knots. Consequently the ship's true rate of sailing is 8.7 miles an hour.

EXAMPLE II. Suppose a ship runs at the rate of 6 knots per glass; but examining the glass, it is found to run only 20 seconds; and by measuring the log-line, the distance between knot and knot is found to be but 38 feet: Required the true rate of sailing.

First, (by Art. 8.) As 50 feet: 38 feet :: 6 knots: 4.56 knots. Then (by Art. 9.) As 20 seconds: 30 seconds :: 4.56 knots: 6.84 knots. Consequently the true rate of sailing is 6.84 miles an hour.

But if in this case it happen, that the time the glass takes to run be to the distance between knot and knot, as 30, the seconds in half a minute, is to 50, the true distance between knot and knot; then it is plain, that whatever number of seconds the glass consists of, and whatever number of feet is contained between knot and knot, yet the distance given by the log-line will be the true distance in miles.

12. Though the method of measuring the ship's way by the log-line, described in the foregoing articles, be that which is now commonly made use of; yet it is subject to several errors, and these very considerable. For, first, the half-minute or quarter-minute glasses (by which and the log the ship's way is determined) are seldom or never true, because dry and wet weather have a great influence on them; so that at one time they may run more, and at another time fewer, than 30 seconds; and it is evident that a small error in the glass will cause a sensible one in the ship's way. Again, the chief property of the log is to have it swim upright, or perpendicular to the horizon: but this is too often wanting in logs, because few seamen examine whether it is so or not, and generally take it upon trust, being satisfied if it weigh a little more at the stern than the head. And from this there flows an error in the reckoning; for if the log does not swim upright, it will not hold water, nor remain steady in the place where it is heaved, since the least check in the hand in veering the line will make it come up several feet: this repeated will make the errors become fathoms, and perhaps knots, which, how insignificant soever they appear, are miles and parts of miles, and amount to a good deal in a long voyage. Another

inconvenience attending the log-line is its stretching and shrinking; for when a new line is first used, let it be ever so well stretched upon the deck, and measured as true as possible, yet after wetting it shrinks considerably; and consequently to be the better assured of the ship's way by the log-line, we ought to measure and alter the knots on it every time before we use it; but this is seldom done oftener than once a week, and sometimes not above once or twice in a whole voyage: also when the line is measured to its greatest degree of shrinking, it is generally left there; and when, by much use, it comes to stretch again, it is seldom or never mended, though it will stretch beyond what it first shrunk. These and many other errors, too well known, attending that method of measuring the ship's way by the log-line, plainly accounts for a great many errors committed in reckonings. So it is to be wished, that either this method were improved or amended, or that some other method less subject to error were found out.

13. The meridian and prime vertical of any place cuts the horizon in 4 points, at 90 degrees distance from one another, viz. North, South, East and West; that part of the meridian which extends itself from the place to the north point of the horizon is called the north line; that which tends to the south point of the horizon is called the south line; and that part of the prime vertical which extends towards the right hand of the observer, when his face is turned to the north, is called the east line; and lastly, that part of the prime vertical which tends towards the left hand is called the west line; the four points in which these lines meet the horizon are called the cardinal points.

14. In order to determine the course of the winds, and to discover their various alterations or shiftings, each quadrant of the horizon, intercepted between the meridian and prime vertical, is usually divided into eight equal parts, and consequently the whole horizon into thirty-two; and the lines drawn from the place on which the observer standeth, to the points of division in his horizon, are called rhumb-lines; the four principal of which are those described in the preceding article, each of them having its name from the cardinal point in the horizon towards which it tends: the rest of the rhumb-lines have their names compounded of the principal lines on each side of them, as in the figure (Plate CII. N^o 1.); and over whichever of these lines the course of the wind is directed, that wind takes its name accordingly.

15. The instrument commonly used at sea for directing the ship's way is called the MARINER'S COMPASS; which consists of a card and two boxes. The card is a circle made to represent the horizon, whose circumference is quartered and divided into degrees, and also into thirty-two equal parts, by lines drawn from the centre to the several points of division, called points of the compass. On the back-side of the card, and just below the south and north line, is fixed a steel needle with a brass cupola, or hollow centre in the middle, which is placed upon the end of a fine pin, upon which the card may easily turn about: the needle is touched with a loadstone, by which a certain virtue is infused into it, that makes it (and consequently the south and north line on the card above it) hang nearly in the plane of the meridian; by which means the south and

PRACTICE north lines on the card produced would meet the horizon in the south and north points; and consequently all the other lines on the card produced would meet the horizon in the respective points.

Plate CCII. 16. The card is represented in No. 1, in which you may observe, that the capital letters N, S, E, W, denote the four cardinal points, viz. N the North, S the South, &c. and the small letter b signifies the word by. The rhumbs in the middle between any two of the cardinals are expressed by the letters denoting these cardinals, that which denotes the point lying in the meridian having the precedence; thus the rhumb in the middle between the north and east is expressed N. E. which is to be read North-east; also S. W. denotes the South-west rhumb, &c.: the other rhumbs are expressed according to their situation with respect to these middle rhumbs and the nearest cardinals, as is plain from the foresaid figure.

17. The card is put into a round box, made for it, having a pin erected in the middle, upon which the hollow centre of the needle is fixed, so as the card may lie horizontal, and easily vibrate according to the motion of the needle: the box is covered over with a smooth glass, and is hung in a brass hoop upon two cylindrical pins, diametrically opposite to one another; and this hoop is hung within another brass circle, upon two pins at right angles with the former. These two circles, and the box, are placed in another square wooden box, so that the innermost box, and consequently the card, may keep horizontal which way ever the ship heels.

18. Since the meridians do all meet at the poles, and there form certain angles with one another; and since, if we move ever so little towards the east or west, from one place to another, we thereby change our meridian, and in every place the east and west line being perpendicular to the meridian; it follows, that the east and west line in the first place will not coincide with the east and west line in the second, but be inclined to it at a certain angle: and consequently all the other rhumb-lines at each place will be inclined to each other, they always forming the same angles with the meridian. Hence it follows, that all rhumbs, except the four cardinals, must be curves or helispherical lines, always tending towards the pole, and approaching it by infinite gyrations or turnings, but never falling into it. Thus let P (No. 2.) be the pole, EQ an arch of the equator, PE, PA, &c. meridians, and EFGHKL any rhumb: then because the angles PEF, PFG, &c. are by the nature of the rhumb-line equal, it is evident that it will form a curve-line on the surface of the globe, always approaching the pole P, but never falling into it; for if it were possible for it to fall into the pole, then it would follow, that the same line could cut an infinite number of other lines at equal angles, in the same point; which is absurd.

19. Because there are 32 rhumbs (or points in the compass) equally distant from one another, therefore the angle contained between any two of them adjacent will be $11^\{\backslash circ\}\; 15\text{'}$, viz. $\backslash frac\{1\}\{32\}$ part of $360^\{\backslash circ\}$; and so the angle contained between the meridian and the N.E., will be $11^\{\backslash circ\}\; 15\text{'}$, and between the meridian and the N.N.E. will be $22^\{\backslash circ\}\; 30\text{'}$; and so of the rest, as in the following table.

A TABLE of the Angles which every $\backslash frac\{1\}\{32\}$ Point of the Compass makes with the Meridian.

§ 8. Concerning Currents, and how to make proper allowances.

1. CURRENTS are certain settings of the stream, by which all bodies (as ships, &c.) moving therein, are compelled to alter their course or velocity, or both; and submit to the motion impressed upon them by the current.

CASE I. If the current sets just with the course of the ship, i. e. moves on the same rhumb with it; then the motion of the ship is increased, by as much as is the drift or velocity of the current.

EXAMPLE. Suppose a ship sails S.E.S. at the rate of 6 miles an hour, in a current that sets S.E.S. 2 miles an hour: Required her true rate of sailing.

Here it is evident that the ship's true rate of sailing will be 8 miles an hour.

CASE II. If the current sets directly against the ship's course, then the motion of the ship is lessened by

PRACTICE by as much as is the velocity of the current.

EXAMPLE. Suppose a ship sails SSW at the rate of 10 miles an hour, in a current that sets NNE 6 miles an hour: Required the ship's true rate of sailing.

Here it is evident that the ship's true rate of sailing will be 4 miles an hour. Hence it is plain.

COR. I. If the velocity of the current be less than the velocity of the ship, then the ship will get so much a-head as is the difference of these velocities.

COR. II. If the velocity of the current be greater than that of the ship, then the ship will fall so much a-stern as is the difference of these velocities.

COR. III. Lastly, If the velocity of the current be equal to that of the ship, then the ship will stand still; the one velocity destroying the other.

CASE III. If the current thwarts the course of the ship, then it not only lessens or augments her velocity, but gives her a new direction compounded of the course she steers, and the setting of the current, as is manifest from the following

LEMMA. If a body at A (N^o 26.) be impelled by two forces at the same time, the one in the direction AB capable to carry that body from A to B in a certain space of time, and the other in the direction AD capable to carry it from A to D in the same time; complete the parallelogram ABCD, and draw the diagonal AC; then the body at A, agitated by these two forces together, will move along the line BC, and will be in the point C at the end of the time in which it would have moved along AD or AB with the forces separately applied.

Hence the solution of the following examples will be evident.

EXAMPLE I. Suppose a ship sails (by the compass) directly south 96 miles in 24 hours, in a current that sets east 45 miles in the same time: Required the ship's true course and distance.

GEOMETRICALLY. Draw AD (see N^o 26.) to represent the south and north line of the ship at A, which make equal to 96; from D draw DC perpendicular to AD, equal to 45; and join AC. Then C will be the ship's true place, AC her true distance, and the angle CAD the true course. To find which,

By CALCULATION;

First, For the true course DAC, it will be, (by rectangular trigonometry),
As the apparent distance AD - 96 - 1.98227
is to the current's motion DC - 45 - 1.65321
so is radius - - - - - 10.00000
to the tangent of the true course DAC - 25°, 07' 9.67094
consequently the ship's true course is S 25° 07' E, or SSE 2° 37' easterly.

Then for the true distance AC, it will be, (by rectangular trigonometry),
As the sine of the course A - 25°, 07' - 9.62784
is to the departure DC - 45 - 1.65321
so is radius - - - - - 10.00000
to the true distance AC - 106 - 2.02537

EXAMPLE. Suppose a ship sails SE 120 miles in 20 hours, in a current that sets W&N at the rate of 2 miles an hour: Required the ship's true course and distance sailed in that time.

GEOMETRICALLY. Having drawn the compass

NESW (N^o 27.) let C represent the place the ship sailed from; draw the SE line CA, which make equal to 120; then will A be the place the ship capped at.

From A draw AB parallel to the W&N line CD, equal to 40, the motion of the current in 20 hours, and join CB; then B will be the ship's true place at the end of 20 hours, CB her true distance, and the angle SCB her true course. To find which,

By CALCULATION;

In the triangle ABC, are given CA 120, AB 40, and the angle CAB equal to 34° 45', the distance between the E&S and SE lines, to find the angles B and C, and the side CB.

First, For the angles C and B, it will be, (by oblique trigonometry),

As the sum of the sides CA and AB 160 - 2.20412
is to their difference - 80 - 1.90309
so is the tang. of half the sum }
of the angles B and C - 73°, 07' 10.51783
to the tang. of half their diff. - 59°, 45' 10.21680
consequently the angle B will be 131°, 52', and the angle ACB 14° 23'. Hence the true course is S 30° 37' E, or SSE 2° 07' easterly.

Then for the true distance CB, it will be, (by oblique trigonometry),

As the sine of B - 131°, 52' - 9.87198
is to AC - 120 - 2.07918
so is the sine of A - 33°, 45' - 9.74474
to the true distance CB 89.53 - 1.95194

EXAMPLE III. Suppose a ship coming out from sea in the night, has sight of Scilly light, bearing NE&N distance 4 leagues, it being then flood tide setting ENE 2 miles an hour, and the ship running after the rate of 5 miles an hour: Required upon what course and how far she must sail to hit the Lizard, which bears from Scilly E^½S distance 17 leagues.

GEOMETRICALLY. Having drawn the compass NESW (N^o 28.) let A represent the ship's place at sea, and draw the NE&N line AS, which make equal to 12 miles; so S will represent Scilly.

From S draw SL equal to 51 miles, and parallel to the E^½S line; then L with represent the Lizard.

From L draw LC parallel to the ENE line, equal to 2 miles, and from C draw CD equal to 5 miles meeting AL in D; then from A draw AB parallel to CD meeting LC produced in B; and AB will be the required distance, and SAB the true course. To find which,

By CALCULATION;

In the triangle ASL are given the side AS equal to 12 miles, the side SL equal to 51, and the angle ASL equal to 118° 07', the distance between the NE&N and W&N lines; to find the angles SAL and SLA. Consequently, (by oblique trigonometry), it will be,

As the sum of the sides AS and SL - 63 - 1.79934
is to their difference - 39 - 1.59166
so is the tang. of half the sum }
of the angles SAL and SLA } 30°, 56' 9.77763
to the tang. of half their diff. - 20°, 21' 9.56935
consequently the angle SAL, will be 51° 17'; and to the direct bearing of the Lizard from the ship will be N 85° 02' E, or E&N 6° 17' E; and for the distance AL, it will be, (by oblique trigonometry),

the distance between the ship and the Lizard.

Again, in the triangle DLC, are given the angle L equal to 17° 32', the distance between the ENE and N 85° 02' E lines; the side LC, equal to 2 miles, the current's drift in an hour; and the side CD, equal to 5 miles, the ship's run in the same time. Hence for the angle D, it will be, (by oblique trigonometry), As the ship's run in 1 hour DC - 5 - 0.69897 is to the fine of L - 17°, 32' - 9.47894 so is the current's drift LC - 2 - 0.30103 to the fine of D - 6°, 55' - 9.08100 consequently, since by construction the angle LAB is equal to the angle LDC, the course the ship must steer is S 88°, 03' E.

Then for the distance AB, it will be, (by oblique trigonometry), As the fine of B - 155°, 33' - 9.61689 is to AL - 57.65 - 1.76080 so is the fine of L - 17.32 - 9.47894 to AB - 41.96 - 1.62285 consequently, since the ship is sailing at the rate of 5 miles an hour, it follows, that in sailing 8^h 24^m S 88° 03' E, she will arrive at the Lizard.

EXAMPLE IV. A ship from a certain headland in the latitude of 34° 00' north, sails SE 8 S 12 miles in 3 hours, in a current that sets between north and east; and then the same headland is found to bear WNW, and the ship to be in the latitude of 33° 52' north: Required the setting and drift of the current.

GEOMETRICALLY. Having drawn the compass NESW (N° 29.) let A represent the place of the ship, and draw the SE 8 S line AB, equal to 12 miles, also the ESE line AC.

Set off from A upon the meridian AD, equal to 8 miles, the difference of latitude, and through D draw DC parallel to the east and west line WE, meeting AC in C. Join C and B with the right line BC; then C will be the ship's place, the angle ABC the setting of the current from the SE 8 S line, and the line BC will be the drift of the current in 3 hours. To find which,

By CALCULATION;

In the triangle ABC, right-angled at D, are given the difference of latitude AD equal to 8 miles, the angle DAC equal to 67° 30'. Whence for AC, the distance the ship has sailed, it will be

Again, in the triangle ABC, are given AB equal to 12 miles, AC equal to 20.9, and the angle BAC equal to 30° 45', the distance between the SE 8 S and ESE lines. Whence, for the angle at B, it will be, As the sum of the sides AC and AB 32.9 1.51720 is to their difference - 8.9 - 0.94930 so is the tang. of half the } 73°, 07' - 10.51806 sum of the angles B and C } to tang. of $\backslash frac\{1\}\{2\}$ their diff. - 41°, 43' $\backslash frac\{1\}\{2\}$ - 9.95025 consequently the angle B is 114° 51', and so the set-

ting of the current will be N 81° 06' E, or E 8° 21' E. Then for BC, the current's drift in 3 hours, it will be, As the fine of B - 114°, 51' - 9.92700 is to the distance run AC 20.9 - 1.32025 so is the fine of A - 33°, 45' - 9.74474 to BC - 12.8 - 1.10719 the current's drift in 3 hours; and consequently the current sets E 8° 21' E 4.266 miles an hour.

§. 9. Concerning the VARIATION of the COMPASS, and how to find it from the true and observed AMPLITUDES or AZIMUTHS of the sun.

1. The variation of the compass is how far the north or south point of the needle stands from the true south or north point of the horizon towards the east or west; or it is an arch of the horizon intercepted between the meridian of the place of observation and the magnetic meridian.

2. It is absolutely necessary to know the variation of the compass at sea, in order to correct the ship's course; for since the ship's course is directed by the compass, it is evident that if the compass be wrong, the true course will differ from the observed, and consequently the whole reckoning differ from the truth.

3. The sun's true amplitude is an arch of the horizon comprehended between the true east or west point thereof, and the centre of the sun at rising or setting; or it is the number of degrees, &c. that the centre of the sun is distant from the true east or west point of the horizon, towards the south or north.

4. The sun's magnetic amplitude is the number of degrees that the centre of the sun is from the east or west point of the compass, towards the south or north point of the same at rising or setting.

5. Having the declination of the sun, together with the latitude of the place of observation, we may from thence find the sun's true amplitude, by the following astronomical proposition, viz.

As the co-line of the latitude is to the radius

So is the fine of the sun's declination to the fine of the sun's true amplitude which will be north or south according as the sun's declination is north or south.

EXAMPLE. Required the sun's true amplitude in the latitude of 41° 50' north, on the 23d day of April 1731.

First, I find (from the tables of the sun's declination) that the sun's declination the 23d of April is 15° 54' north; then for the true amplitude, it will be, by the former analogy,

which is north, because the declination is north at that time; and consequently, in the latitude of 40° 50' north, the sun rises on the 23d of April 21° 35' from the east part of the horizon towards the north, and sets so much from the west the same way.

6. The sun's true azimuth is the arch of the horizon intercepted between the meridian and the vertical circle passing through the centre of the sun at the time of observation.

PRACTICE 7. The sun's magnetic azimuth is the arch of the horizon, intercepted between the magnetic meridian and the vertical, passing through the sun.

8. Having the latitude of the place of observation, together with the sun's declination and altitude at the time of observation, we may find his true azimuth after the following method, viz.

Make it,
As the tangent of half the complement of the latitude is to the tangent of half the sum of the distance of the sun from the pole and complement of the altitude

So is the tangent of half the difference between the distance of the sun from the pole and complement of the altitude

To the tangent of a fourth arch
which fourth arch added to half the complement of the latitude will give a fifth arch, and this fifth arch lessened by the complement of the latitude will give a sixth arch.

Then make it,

As the radius is to the tangent of the altitude
so is the tangent of the sixth arch to the co-sine of the sun's azimuth
which is to be counted from the south or north, to the east or west, according as the sun is situated with respect to the place of observation.

If the latitude of the place and declination of the sun be both north or both south, then the declination taken from $90^\backslash circ$ will give the sun's distance from the pole; but if the latitude and declination be on contrary sides of the equator, then the declination added to $90^\backslash circ$ will give the sun's distance from the nearest pole to the place of observation.

EXAMPLE. In the latitude of $51^\backslash circ\; 32\text{'}$ north, the sun having $19^\backslash circ\; 39\text{'}$ north declination, his altitude was found by observation to be $38^\backslash circ\; 18\text{'}$: Required the azimuth.

By the first of the foregoing analogies, it will be
As the tangent of $\backslash frac\{1\}\{2\}$ the complement of the latitude } $19^\backslash circ,\; 14\text{'}\; 9.54269$
is to the tangent of $\backslash frac\{1\}\{2\}$ the sum }
of the distance of the sun } $61^\backslash circ,\; 01\text{'}\; 10.25655$
from the pole and complement of the altitude }
so is the tangent of half their } $9^\backslash circ,\; 19\text{'}\; 7.21499$
difference }
to the tang. of a 4th arch } $40^\backslash circ,\; 20\text{'}\; 9.92885$
which fourth arch $40^\backslash circ\; 20\text{'}$, added to $19^\backslash circ\; 14\text{'}$, half the complement of the latitude, gives a fifth arch $59^\backslash circ\; 34\text{'}$; and this fifth arch lessened by $38^\backslash circ\; 18\text{'}$, the complement of the latitude, gives the sixth arch $21^\backslash circ\; 06\text{'}$; then for the azimuth, it will be, by the second of the preceding analogies,

As radius - - - - - 10.00000
is to the tang. of the altitude $38^\backslash circ,\; 18\text{'}$ 9.89749
so is the tang. of the sixth arch $21^\backslash circ,\; 06\text{'}$ - 9.58644
to the co-sine of the azimuth $72^\backslash circ,\; 15\text{'}$ - 9.48393
which, because the latitude is north and the sun south of the place of observation, must be counted from the south towards the east or west; and consequently, if the altitude of the sun was taken in the morning, the azimuth will be $S\; 72^\backslash circ\; 15\text{'}\; E$, or $ESE\; 4^\backslash circ\; 45\text{'}\; E$; but if the altitude was taken in the afternoon, the

azimuth will be $S\; 72^\backslash circ\; 15\text{'}\; W$, or $WSW\; 4^\backslash circ\; 45\text{'}$ PRACTICE westerly.

9. Having found the sun's true amplitude or azimuth by the preceding analogies, and his magnetic amplitude or azimuth by observation, it is evident, if they agree, there is no variation; but if they disagree, then if the true and observed amplitudes at the rising or setting of the sun be both of the same name, i. e. either both north, or both south, their difference is the variation; but if they be of different names, i. e. one north and the other south, their sum is the variation. Again, if the true and observed azimuth be both of the same name, i. e. either both east or both west, their difference is the variation: but if they be of different names, their sum is the variation: And to know whether the variation is easterly, observe this general rule, viz.

Let the observer's face be turned to the sun: then if the true amplitude or azimuth be to the right-hand of the observed, the variation is easterly; but if be to the left, westerly.

To explain which, Let NESW ($N^\backslash circ\; 30.$) represent Plate CCIII a compass, and suppose the sun is really EBS at the time of observation, but the observer sees him off the east point of the compass, and so the true amplitude or azimuth of the sun is to the right of the magnetic or observed; here it is evident that the EBS point of the compass ought to lie where the east point is, and so the north where the NBS is; consequently the north point of the compass is a point too far east, i. e. the variation in this case is easterly. The same will hold when the amplitude or azimuth is taken on the west side of the meridian.

Again, let the true amplitude or azimuth be to the left-hand of the observed. Thus, suppose the sun is really EBN at the time of observation, but the observer sees him off the east point of the compass, and so the true amplitude or azimuth to the left of the observed: Here it is evident that the EBN point of the compass ought to stand where the east point is, and so the north where the NBE point is; consequently the north point of the compass lies a point too far westerly; so in this case the variation is west. The same will hold when the sun is observed on the west side of the meridian.

EXAMPLE I. Suppose the sun's true amplitude at rising is found to be $E\; 14^\backslash circ\; 20\text{'}\; N$, but by the compass it is found to be $E\; 26^\backslash circ\; 12\text{'}$: Required the variation, and which way it is.

Since they are both the same way, therefore
From the magnetic amplitude - $E\; 26^\backslash circ,\; 12\text{'}\; N$.
take the true amplitude - $E\; 14^\backslash circ,\; 20\text{'}\; N$.

and there remains the variation - $11^\backslash circ,\; 52\text{'}\; E$.
which is easterly, because in this case the true amplitude is the right of the observed.

EXAMPLE II. Suppose the sun's true amplitude at setting is $W\; 34^\backslash circ\; 26\text{'}\; S$, and his magnetic amplitude $W\; 23^\backslash circ\; 13\text{'}\; S$: Required the variation, and which way it is.

Since they lie both the same way, therefore
From the sun's true amplitude - $W\; 43^\backslash circ,\; 26\text{'}\; S$.
take his magnetic amplitude - $W\; 23^\backslash circ,\; 13\text{'}\; S$.
there remains the variation - $11^\backslash circ,\; 13\text{'}\; W$.
which

PRACTICE which is westerly, because the true amplitude, in this case, is to the left-hand of the observed.

EXAMPLE III. Suppose the sun's true altitude at rising is found to be $13^\{\backslash circ\}\; 24\text{'}\; N$, and his magnetic $E\; 12^\{\backslash circ\}\; 32\text{'}\; S$: Required the variation, and which way it lies.

Since the true and observed amplitudes lie different ways, therefore

which is westerly, because the true amplitude is, in this case, to the left of the observed.

EXAMPLE IV. Suppose the sun's true altitude at setting is found to be $W\; 8^\{\backslash circ\}\; 24\text{'}\; N$, but his magnetic amplitude is $W\; 10^\{\backslash circ\}\; 13\text{'}\; S$: Required the variation.

which is easterly, because the true amplitude is to the right of the observed.

EXAMPLE V. Suppose the sun's true azimuth at the time of observation is found to be $N\; 86^\{\backslash circ\}\; 40\text{'}\; E$, but by the compass it is $N\; 73^\{\backslash circ\}\; 24\text{'}\; E$: Required the variation, and which way it lies.

which is easterly, because the true azimuth is to the right of the observed.

EXAMPLE VI. Suppose the sun's true azimuth is $S\; 3^\{\backslash circ\}\; 24\text{'}\; E$, and the magnetical $S\; 4^\{\backslash circ\}\; 36\text{'}\; W$: Required the variation, and which way it lies.

which is westerly, because the true azimuth is, in this case, to the left of the observed.

10. The variation of the compass was first observed at London, in the year 1580, to be $11^\{\backslash circ\}\; 15\text{'}$ easterly, and in the year 1622 it was $6^\{\backslash circ\}\; 0\text{'}$ E.; also in the year 1634, it was $4^\{\backslash circ\}\; 05\text{'}$ E. still decreasing, and the needle approaching the true meridian, till it coincided with it, and then there was no variation; after which, the variation began to be westerly; and in the year 1672, it was observed to be $2^\{\backslash circ\}\; 30\text{'}\; W$; also in the year 1683, it was $4^\{\backslash circ\}\; 30\text{'}\; W$; and since that time the variation still continues at London to increase westerly; but how far it will go that way, time and observations will probably be the only means to discover.

Again, at Paris, in the year 1640, the variation was $3^\{\backslash circ\}\; 00\text{'}\; E$; and in the year 1666, there was no variation; but in the year 1681, it was $2^\{\backslash circ\}\; 30\text{'}\; W$, and still continues to go westerly.

In short, from observations made in different parts of the world, it appears, that in different places the variation differs both as to its quantity and denomination, it being east in one place, and west in another; the true cause and theory of which, for want of a suf-

ficient number of observations, has not as yet been fully explained.

§ 10. The Method of keeping a Journal at sea; and how to correct it, by making proper allowance for the Lee-way, Variation, &c.

1. LEEWAY is the angle that the rhumb-line, upon which the ship endeavours to sail, makes with the rhumb she really sails upon. This is occasioned by the force of the wind or surge of the sea, when she lies to the windward, or is close-hauled; which causes her to fall off and glide sideways from the point of the compass she capes at. Thus let NESW ($N^\{\backslash circ\}\; 31^\{\backslash circ\}$) represent the compass; and suppose a ship at C capes at, or endeavours to sail upon, the rhumb Ca; but by the force of the wind, and surge of the sea, she is obliged to fall off, and make her way good upon the rhumb Cb: then the angle aCb is the lee-way; and if that angle be equal to one point, the ship is said to make one point lee-way; and if equal to two points, the ship is said to make two points lee-way, &c.

The quantity of this angle is very uncertain, because some ships, with the same quantity of sail, and with the same gale, will make more lee-way than others; it depending much upon the mould and trim of the ship, and the quantity of water that she draws. The common allowances that are generally made for the lee-way, are as follow.

1. If a ship be close hauled, has all her sails set, the water smooth, and a moderate gale of wind, she is then supposed to make little or no lee-way.

2. If it blow so fresh as to cause the small sails to be handed, it is usual to allow one point.

3. If it blow so hard that the top-sails must be close reefed, then the common allowance is two points for lee-way.

4. If one top-sail must be handed, then the ship is supposed to make between two and three points lee-way.

5. When both top-sails must be handed, then the allowance is about four points for lee-way.

6. If it blows so hard as to occasion the fore-course to be handed, the allowance is between $5\backslash frac\{1\}\{2\}$ and 6 points.

7. When both main and fore-courses must be handed, then 6 or $6\backslash frac\{1\}\{2\}$ points are commonly allowed for lee-way.

8. When the mizen is handed, and the ship is trying a-hull, she is then commonly allowed about 7 points for lee-way.

3. Though these rules are such as are generally made use of, yet since the lee-way depends much upon the mould and trim of the ship, it is evident that they cannot exactly serve to every ship; and therefore the best way is to find it by observation. Thus, let the ship's wake be set by a compass in the poop, and the opposite rhumb is the true course made good by the ship; then the difference between this and the course given by the compass in the binacle, is the lee-way required. If the ship be within sight of land; then the lee-way may be exactly found by observing a point on the land which continues to bear the same way, and the distance between the point of the compass

PRACTICE passes it lies upon, and the point the ship capes at, will be the lee-way. Thus, suppose a ship at C, is lying up NW, towards A; but instead of keeping that course, she is carried on the NNE line CB, and consequently the point B continues to bear the same way from the ship: Here it is evident, that the angle ACB, or the distance between the NW line that the ship capes at, and the NNE line that the ship really sails upon, will be the lee-way.

4. Having the course steered, and the lee-way, given; we may from thence find the true course by the following method, viz. Let your face be turned directly to the windward; and if the ship have her larboard tacks on board, count the lee-way from the course steered towards the right hand; but if the starboard tacks be on board, then count it from the course steered towards the left hand. Thus, suppose the wind at north, and the ship lies up within 6 points of the wind, with her larboard tacks on board, making one point lee-way; here it is plain, that the course steered is ENE, and the true course E$\backslash frac\{1\}\{2\}$N: also suppose the wind is at NNW, and the ship lies up within 6$\backslash frac\{1\}\{2\}$ points of the wind, with her starboard tack on board, making 1$\backslash frac\{1\}\{2\}$ point lee-way; it is evident that the true course, in this case, is WSW.

5. We have shewed, in the last section, how to find the variation of the compass; and from what has been said there, we have this general rule for finding the ship's true course, having the course steered and the variation given, viz. Let your face be turned towards the point of the compass upon which the ship is steered; and if the variation be easterly, count the quantity of it from the course steered towards the right hand; but if westerly, towards the left hand; and the course thus found is the true course steered. Thus, suppose the course steered is NNE, and the variation one point easterly; then the true course steered will be NNE. Also suppose, the course steered is NE$\backslash frac\{1\}\{2\}$E, and the variation one point westerly; then, in this case, the true course will be NE; and so of others.

Hence, by knowing the lee-way variation, and course steered, we may from thence find the ship's true course; but if there be a current under foot, then that must be tried, and proper allowances made for it, as has been shown in the section concerning Currents, from thence to find the true course.

6. After making all the proper allowances for finding the ship's true course, and making as just an estimate of the distance as we can; yet by reason of the many accidents that attend a ship in a day's running, such as different rates of sailing between the times of heaving the log; the want of due care at the helm by not keeping her steady, but suffering her to yaw and fall off; sudden storms, when no account can be kept, &c.; the latitude by account frequently differs from the latitude by observation; and when that happens, it is evident there must be some error in the reckoning; to discover which, and where it lies, and also how to correct the reckoning, you may observe the following rules.

1^st. If the ship sail near the meridian, or within 2 or 2$\backslash frac\{1\}\{2\}$ points thereof; then if the latitude by account disagrees with the latitude by observation, it is most likely that the error lies in the distance run; for it is plain, that in this case it will require a very sensible

error in the course to make any considerable error in the difference of latitude, which cannot well happen if due care be taken at the helm, and proper allowances be made for the lee-way, variation, and currents. Consequently, if the course be pretty near the truth, and the error in the distance run regularly through the whole, we may, from the latitude obtained by observation, correct the distance and departure by account, by the following analogies, viz.

As the difference of latitude by account

is to the true difference of latitude,

So is the departure by account

to the true departure,

And so is the direct distance by account

to the true direct distance.

The reason of this is plain: for let AB (N$^\{\backslash circ\}$ 33$^\{\backslash circ\}$) denote the meridian of the ship at A, and suppose the ship sails upon the rhumb AE near the meridian, till by account she is found in C, and consequently her difference of latitude by account is AB; but by observation she is found in the parallel ED, and so her true difference of latitude is AD, her true distance AE, and her true departure DE; then, since the triangles ABC ADE are similar, it will be AB: AD :: BC: DE, and AB: AD :: AC: AE.

EXAMPLE. Suppose a ship from the latitude of 45$^\{\backslash circ\}$ 20' north, after having sailed upon several courses near the meridian for 24 hours, her difference of latitude is computed to be upon the whole 95 miles southerly, and her departure 34 miles easterly; but by observation she is found to be in the latitude of 43$^\{\backslash circ\}$ 10' north, and consequently her true difference of latitude is 130 miles southerly: then for the true departure, it will be, As the difference of latitude by account 95 is to the true difference of latitude 130; so is the departure by account 34 to the true departure 46.52, and so is the distance by account 100.9 to the true distance 138.

2^dy. If the courses are for the most part near the parallel of east and west, and the direct course be within 5$\backslash frac\{1\}\{2\}$ or 6 points of the meridian; then if the latitude by account differs from the observed latitude, it is most probable that the error lies in the course or distance, or perhaps both; for in this case it is evident, the departure by account will be very nearly true; and thence by the help of this, and the true difference of latitude, may the true course and direct distance be readily found by Case 4. of Plane Sailing.

EXAMPLE. Suppose a ship from the latitude of 43$^\{\backslash circ\}$ 50' north, after having sailed upon several courses near the parallel of east and west, for the space of 24 hours, is found by dead reckoning to be in the latitude of 42$^\{\backslash circ\}$ 45' north, and to have made 160 miles of westing; but by a good observation the ship is found to be in the latitude of 42$^\{\backslash circ\}$ 35' north: Required the true course, and direct distance sailed.

With the true difference of latitude 75 miles, and departure 160 miles, we shall find (by Case 4. of Plane Sailing) the true course to be S. 64$^\{\backslash circ\}$ 53' W. and the direct distance 176.7 miles.

3^dy. If the courses are for the most part near the middle of the quadrant, and the direct course within 2 or 6 points of the meridian; then the error may be either in the course or in the distance, or in both, which

PRACTICE which will cause an error both in the difference of latitude and departure; to correct which, having found the true difference of latitude by observation, with this, and the direct distance by dead reckoning, find a new departure (by Case 3. of Plane Sailing;) then half the sum of this departure, and that by dead reckoning, will be nearly equal to the true departure; and consequently with this, and the true difference of latitude, we may (by Case 4. of Plane Sailing) find the true course and distance.

EXAMPLE. Suppose a ship from the latitude of $44^\{\backslash circ\}\; 38\text{'}$ north, sails between south and east upon several courses, near the middle of the quadrant, for the space of 24 hours, and is then found by dead reckonings to be in the latitude of $42^\{\backslash circ\}\; 15\text{'}$ north, and to have made of easting 136 miles; but by observation she is found to be in the latitude of $42^\{\backslash circ\}\; 04\text{'}$ north: Required her true course and distance.

With the true distance of latitude 154 miles, and the direct distance by dead reckoning 197.4, you will find (by Case 3. of Plane Sailing) the new departure to be 123.4, and half the sum of this and the departure by dead reckoning will be 123.7 the true departure; then with this, and the true difference of la-

titude, you will find (by Case 4. of Plane Sailing), PRACTICE the true course to be S. $39^\{\backslash circ\}\; 00\text{'}$ E. and the direct distance 198.2 miles.

7. In keeping a ship's reckoning at sea, the common method is to take from the log-board the several courses and distances sailed by the ship last 24 hours, and to transfer these together with the most remarkable occurrences into the log-book, into which also are inserted the courses corrected, and the difference of latitude and difference of longitude made good upon each; then the whole day's work being finished in the log-book, if the latitude by account agree with the latitude by observation, the ship's place will be truly determined; if not, then the reckoning must be corrected according to the preceding rules, and placed in the journal.

The form of the Log-book and Journal, together with an example of 2 days work, you have here subjoined.

Note, to express the days of the week, they commonly use the characters by which the sun and planets are expressed, viz. $\backslash odot$ denotes Sunday, $\backslash text\{M\}$ Monday, $\backslash text\{T\}$ Tuesday, $\backslash text\{W\}$ Wednesday, $\backslash text\{Th\}$ Thursday, $\backslash text\{F\}$ Friday, and $\backslash text\{S\}$ Saturday.

§ 11. The Form of the Log-Book, with the Manner of working Days Works at Sea.

Hence the ship, by account, has come to the latitude of $47^\{\backslash circ\}\; 46\text{'}$ north, and has differed her longitude $2^\{\backslash circ\}\; 5\text{'}$ westerly; so this day I have made my way good S. $31^\{\backslash circ\}\; 31\text{'}$ W. distance 157.4 miles.

At noon the Lizard bore from me N. $31^\{\backslash circ\}\; 31\text{'}$ E. distance 157.4 miles; and having observed the latitude, I found it agreed with the latitude by account.

Hence the ship, by account, has come to the latitude of 47° 17' north, and has differed her longitude 49' easterly; consequently she has got 1° 16' to the westward of the Lizard, and has made her way good the last 24 hours 849° 08' E, distance 44.3 miles.

At noon the Lizard bore from me north 17° 7' east, distance 170.6 miles.

This day I had an observation, and found the latitude by account to disagree with the latitude by observation by 11 minutes, I being so much further to the southward than by dead reckoning, which by the third of the preceding rules I correct as in the Journal.

A JOURNAL from the Lizard towards Jamaica in the ship Neptune, J. M. commander.