By this word we express that noble art, or, more purely, the qualifications which enable a man to exercise the noble art of working a ship. A seaman, in the language of the profession, is not merely a mariner or labourer on board a ship, but a man who understands the structure of this wonderful machine, and every subordinate part of its mechanism, so as to enable him to employ it to the best advantage for pushing her forward in a particular direction, and for avoiding the numberless dangers to which she is exposed by the violence of the winds and waves. He also knows what courses can be held by the ship, according to the wind that blows, and what cannot, and which of these is most conducive to her progress in her intended voyage; and he must be able to perform every part of the necessary operation with his own hands. As the seamen express it, he must be able "to hand, reef, and steer."

We are justified in calling it a noble art, not only by its importance, which it is quite needless to amplify or embellish, but by its immense extent and difficulty, and the prodigious number and variety of principles on which it is founded—all of which must be possessed in such a manner that they shall offer themselves without reflection in an instant, otherwise the pretended seaman is but a lubber, and cannot be trusted on his watch.

The art is practised by persons without what we call education, and in the humbler walks of life, and therefore it suffers in the estimation of the careless spectator. It is thought little of, because little attention is paid to it. But if multiplicity, variety, and intricacy of principles, and a systematic knowledge of these principles, entitle any art to the appellation of scientific and liberal, seamanship claims these epithets in an eminent degree. We are amused with the pedantry of the seaman, which appears in his whole language. Indeed it is the only pedantry that amuses. A scholar, a soldier, a lawyer, nay, even the elegant courtier, would disgust us, were he to make the thousandth part of the allusions to his profession that is well received from the jolly seaman; and we do the seaman no more than justice. His profession may engross his whole mind; otherwise he can never learn it. He possesses a prodigious deal of knowledge; but the honest tar cannot tell what he knows, or rather what he feels, for his science is really at his fingers' ends. We can say with confidence, that if a person of education, versed in mechanics, and acquainted with the structure of a ship, were to observe with attention the movements which are made on board a first or second rate ship of war during a shifting storm, under the direction of an intelligent officer, he would be rapt in admiration.

What a pity it is that an art so important, so difficult, and so intimately connected with the invariable laws of mechanical nature, should be so held by its possessors, that it cannot improve, but must die with each individual. Having no advantages of previous education,

tion, they cannot arrange their thoughts; they can hardly be said to think. They can far less express or communicate to others the intuitive knowledge which they possess; and their art, acquired by habit alone, is little different from an instinct. We are as little entitled to expect improvement here as in the architecture of the bee or the beaver. The species (pardon the allusion, ye generous hearts of oak) cannot improve. Yet a ship is a machine. We know the forces which act on it, and we know the results of its construction—all these are as fixed as the laws of motion. What hinders this to be reduced to a set of practical maxims, as well founded and as logically deduced as the working of a steam engine or a cotton mill. The fitter or the spinner acts only with his hands, and may "whistle as he works, for want of thought;" but the mechanist, the engineer, thinks for him, improves his machine, and directs him to a better practice. May not the rough seaman look for the same assistance; and may not the ingenious speculator in his closet unravel the intricate thread of mechanism which connects all the manual operations with the unchangeable laws of nature, and both furnish the seaman with a better machine and direct him to a more dexterous use of it?

We cannot help thinking that much may be done; may, we may say that much has been done. We think highly of the progressive labours of Renard, Pitot, Bouguer, Du Hamel, Groignard, Bernoulli, Euler, Romme, and others; and are both surprised and sorry that Britain has contributed so little in their attempts. Gordon is the only one of our countrymen who has given a professedly scientific treatise on a small branch of the subject. The government of France has always been strongly impressed with the notion of great improvements being attainable by systematic study of this art; and we are indebted to the endeavours of that ingenious nation for any thing of practical importance that has been obtained. M. Bouguer was professor of hydrology at one of the marine academies of France, and was enjoined, as part of his duty, to compose dissertations both on the construction and the working of ships. His Traité du Navire, and his Manœuvre des Voiles, are undoubtedly very valuable performances: So are those of Euler and Bernoulli, considered as mathematical dissertations, and they are wonderful works of genius, considered as the productions of persons who hardly ever saw a ship, and were totally unacquainted with the profession of a seaman. In this respect Bouguer had great superiority, having always lived at a sea port, and having made many very long voyages. His treaties therefore are infinitely better accommodated to the demands of the seaman, and more directly instructive; but still the author is more a mathematician than an artist, and his performance is intelligible only to mathematicians. It is true, the academical education of the young gentlemen of the French navy is such, that a great number of them may acquire the preparatory knowledge that is necessary; and we are well informed that, in this respect, the officers of the British navy are greatly inferior to them.

But this very circumstance has furnished to many persons an argument against the utility of those performances. It is said that, notwithstanding this superior mathematical education, and the possession of those boasted performances of M. Bouguer, the French are greatly inferior, in point of seamanship, to our countrymen, who have not a page in their language to instruct them, and who could not peruse it if they had it." Nay, so little do the French themselves seem sensible of the advantage of these publications, that no person among them has attempted to make a familiar abridgement of them, written in a way fitted to attract attention; and they still remain neglected in their original abstruse and unintelligible form.

We wish that we could give a satisfactory answer to this observation. It is just, and it is important. These very ingenious and learned dissertations are by no means so useful as we should expect. They are large books, and appear to contain much; and as their plan is logical, it seems to occupy the whole subject, and therefore to have done almost all that can be done. But, alas! they have only opened the subject, and the study is yet in its infancy. The whole science of the art must proceed on the knowledge of the impulsion of the wind and water. These are the forces which act on the machine; and its motions, which are the ultimatum of our research, whether as an end to be obtained or as a thing to be prevented, must depend on these forces. Now it is with respect to this fundamental point that we are as yet almost totally in the dark. And, in the performances of M. Bouguer, as also in those of the other authors we have named, the theory of these forces, by which their quantity and the direction of their action fundamentally affected, is altogether erroneous; and its results will prudently deviate so enormously from what is observed in the examples; of a ship, that the person who should direct the operations on shipboard, in conformity to the maxims deducible from M. Bouguer's propositions, would be baffled in most of his attempts, and be in danger of losing the ship. The whole proceeds on the supposed truth of that theory which states the impulse of a fluid to be in the proportion of the square of the sine of the angle of incidence; and that its action on any small portion, such as a square foot of the sails or hull, is the same as if that portion were detached from the rest, and were exposed, single and alone, to the wind or water in the same angle. But we have shown, in the article RESISTANCE OF FLUIDS, both from theory and experience, that both of these principles are erroneous, and this to a very great degree, in cases which occur most frequently in practice, that is, in the small angles of inclination. When the wind falls nearly perpendicular on the sails, theory is not very erroneous; but in these cases, the circumstances of the ship's situation are generally such that the practice is easy, occurring almost without thought; and in this case, too, even considerable deviations from the very best practice are of no great moment. The interesting cases, where the intended movement requires or depends upon very oblique actions of the wind on the sails, and its practicability or impracticability depends on a very small variation of this obliquity; a mistake of the force, either as to intensity or direction, produces a mighty effect on the resulting motion. This is the case in tacking to windward; the most important of all the general problems of seamanship. The trim of the sails, and the course of the ship, so as to gain most on the wind, are very nice things; that is, they are confined within very narrow limits, and a small mistake produces a very considerable effect. The same thing obtains in many of the nice problems. The error in the second assertion of the theory is still greater, and the action on one part of the sail or hull is so greatly modified by its action on another adjoining part, that a stay-sail is often seen hanging like a loose rag, although there is nothing between it and the wind; and this merely because a great sail in its neighbourhood tends off a lateral stream of wind, which completely hinders the wind from getting at it. Till the theory of the action of fluids be established, therefore, we cannot tell what are the forces which are acting on every point of the sail and hull: Therefore we cannot tell either the mean intensity or direction of the whole force which acts on any particular sail, nor the intensity and mean direction of the resistance to the hull; circumstances absolutely necessary for enabling us to say what will be their energy in producing a rotation round any particular axis. In like manner, we cannot, by such a computation, find the spontaneous axis of conversion (see Rotation), or the velocity of such conversion. In short, we cannot pronounce with tolerable confidence a priori what will be the motions in any case, or what dispositions of the sails will produce the movement we wish to perform. The experienced seaman learns by habit the general effects of every disposition of the sails; and though his knowledge is far from being accurate, it seldom leads him into any very blundering operation. Perhaps he seldom makes the best adjustment possible, but seldom still does he deviate very far from it; and in the most general and important problems, such as working to windward, the result of much experience and many corrections has settled a trim of the sails, which is certainly not far from the truth, but (it must be acknowledged) deviates widely and uniformly from the theories of the mathematician's closet. The honest tar, therefore, must be indulged in his joke on the useless labours of the mathematician, who can neither hand, reef, nor steer.

After this account of the theoretical performances in the art of seamanship, and what we have said in another place on the small hopes we entertain of seeing a perfect theory of the impulse of fluids, it will not be expected that we enter very minutely on the subject in this place; nor is it our intention. But let it be observed, that the theory is defective in one point only; and although this is a most important point, and the errors in it destroy the conclusions of the chief propositions, the reasonings remain in full force, and the modus operandi is precisely such as is stated in the theory. The principles of the art are therefore to be found in these treatises; but false inferences have been drawn, by computing from erroneous quantities. The rules and the practice of the computation, however, are still beyond controversy: Nay, since the process of investigation is legitimate, we may make use of it in order to discover the very circumstance in which we are at present mistaken: for by converting the proposition, instead of finding the motions by means of the supposed forces, combined with the known mechanism, we may discover the forces by means of this mechanism and the observed motions.

We shall therefore in this place give a very general view of the movements of a ship under sail, showing how they are produced and modified by the action of the wind on her sails, the water on her rudder and on her bows. We shall not attempt a precise determination of any of these movements; but we shall say enough to enable the curious landsman to understand how this mighty machine is managed amidst the fury of the winds and waves: and, what is more to our wish, we hope to enable the uninstructed but thinking seaman to generalize that knowledge which he possesses; to clear his ideas, and give them a sort of rational system; and even to improve his practice, by making him sensible of the immediate operation of every thing he does, and in what manner it contributes to produce the movement which he has in view.

A ship may be considered at present as a mass of inert matter in free space, at liberty to move in every direction, according to the forces which impel or resist her; and when she is in actual motion, in the direction of her course, we may still consider her as at rest in absolute space, but exposed to the impulse of a current of water opposite moving equally fast in the opposite direction: for in both cases the prelude of the water on her bows is the same; and we know that it is possible, and frequently happens in currents, that the impulse of the wind on her sails, and that of the water on her bows, balance each other so precisely, that she not only does not stir from the place, but also remains steadily in the same position, with her head directed to the same point of the compass. This state of things is easily conceived by any person accustomed to consider mechanical subjects, and every seaman of experience has observed it. It is of importance to consider it in this point of view, because it gives us the most familiar notion of the manner in which these forces of the wind and water are set in opposition, and made to balance or not to balance each other by the intervention of the ship, in the same manner as the goods and the weights balance each other in the scales by the intervention of a beam or teelyard.

When a ship proceeds steadily in her course, without changing her rate of sailing, or varying the direction of the wind her head, we must in the first place conceive the accumulated impulses of the wind on all her sails as precisely opposite to those of the water on her bows. In the next place, because the ship the bow does not change the direction of her keel, she resembles the balanced teelyard, in which the energies of the two weights, which tend to produce rotations in opposite directions, and thus to change the position of the beam, mutually balance each other round the fulcrum; so the energies of the actions of the wind on the different sails balance the energies of the water on the different parts of the hull.

The seaman has two principal tasks to perform. The first is to keep the ship steadily in that course which will bring her farther on in the line of her intended voyage. This is frequently very different from that line, and the choice of the best course is sometimes a matter of considerable difficulty. It is sometimes possible to shape the course precisely along the line of the intended voyage; and yet the intelligent seaman knows that he played in will arrive sooner, or with greater safety, at his port, by taking a different course; because he will gain more by increasing his speed than he loses by increasing the distance. Some principle must direct him in the selection of this course. This we must attempt to lay before the reader.

Having chosen such a course as he thinks most advantageous, tageous, he must set such a quantity of sail as the strength of the wind will allow him to carry with safety and effect, and must trim the sails properly, or to adjust their positions to the direction of the wind, that they may have the greatest possible tendency to impel the ship in the line of her course, and to keep her steadily in that direction.

His other task is to produce any deviations which he sees proper from the present course of the ship; and to produce these in the most certain, the safest, and the most expeditious manner. It is chiefly in this movement that the mechanical nature of a ship comes into view, and it is here that the superior address and resource of an expert seaman is to be perceived.

Under the article Sailing some notice has been taken of the first task of the seaman, and it was there shown how a ship, after having taken up her anchor and fitted her sails, accelerates her motion, by degrees which continually diminish, till the increasing resistance of the water becomes precisely equal to the diminished impulse of the wind, and then the motion continues uniformly the same so long as the wind continues to blow with the same force and in the same direction.

It is perfectly consonant to experience that the impulse of fluids is in the duplicate ratio of the relative velocity. Let it be supposed that when water moves one foot per second, its perpendicular pressure or impulse on a square foot is $m$ pounds. Then, if it be moving with the velocity $V$ estimated in feet per second, its perpendicular impulse on a surface $S$, containing any number of square feet, must be $m SV^2$.

In like manner, the impulse of air on the same surface may be represented by $n SV^2$; and the proportion of the impulse of these two fluids will be that of $m$ to $n$. We may express this by the ratio of $q$ to $1$, making

$$\frac{m}{n} = q.$$

M. Bouguer's computations and tables are on the supposition that the impulse of sea-water moving one foot per second is 23 ounces on a square foot, and that the impulse of the wind is the same when it blows at the rate of 24 feet per second. These measures are all French. They by no means agree with the experiments of others; and what we have already said, when treating of the Resistance of Fluids, is enough to show us that nothing like precise measures can be expected. It was shown as the result of a rational investigation, and confirmed by the experiments of Buat and others, that the impulsions and resistances at the same surface, with the same obliquity of incidence and the same velocity of motion, are different according to the form and situation of the adjoining parts. Thus the total resistance of a thin board is greater than that of a long prism, having this board for its front or bow, &c.

We are greatly at a loss what to give as absolute measures of these impulsions.

1. With respect to water. The experiments of the French academy on a prism two feet broad and deep and four feet long, indicate a resistance of 0.973 pounds avoirdupois to a square foot, moving with the velocity of one foot per second at the surface of still water.

Mr Boat's experiments on a square foot wholly immersed in a stream were as follows:

- A square foot as a thin plate: 1,81 pounds. - Ditto as the front of a box one foot long: 1,42 pounds. - Ditto as the front of a box three feet long: 1,29 pounds.

The resistance of sea-water is about $\frac{1}{3}$ greater.

2. With respect to air, the varieties are as great. The resistance of a square foot to air moving with the velocity of one foot per second appears from Mr Robin's experiments on 16 square inches to be on a square foot:

- Chevalier Borda's on 16 inches: 0.001596 pounds, - on 81 inches: 0.002042 pounds, - Mr Roufe's on large surfaces: 0.002291 pounds.

Precise measures are not to be expected, nor are they necessary in this inquiry. Here we are chiefly interested in their proportions, as they may be varied by their mode of action in the different circumstances of obliquity and velocity.

We begin by recurring to the fundamental proposition concerning the impulse of fluids, viz., that the absolute pressure is always in a direction perpendicular to the impelled surface, whatever may be the direction of the stream of fluid. We must therefore illustrate the doctrine, by always supposing a flat surface of sail stretched on a yard, which can be braced about in any direction, and giving this sail such a position and such perpendicular extent of surface that the impulse on it may be the same both as to direction and intensity with that on the real sails. Thus the consideration is greatly simplified. The direction of the impulse is therefore perpendicular to the yard. Its intensity depends on the velocity with which the wind meets the sail, and the obliquity of its stroke. We shall adopt the constructions founded on the common doctrine, that the impulse is, as the square of the sine of the inclination, because they are simple; whereas, if we were to introduce the values of the oblique impulses, such as they have been observed in the excellent experiments of the Academy of Paris, the constructions would be complicated in the extreme, and we could hardly draw any consequences which would be intelligible to any but expert mathematicians. The conclusions will be erroneous, not in kind but in quantity only; and we shall point out the necessary corrections, so that the final results will be found not very different from real observation.

If a ship were a round cylindrical body like a flat tub, floating on its bottom, and fitted with a mast compared fail in the centre, she would always fail in a direction perpendicular to the yard. This is evident. But she is an oblong body, and may be compared to a chaff, whose length greatly exceeds its breadth. She is so shaped, that a moderate force will push her through the water with the head or stern foremost; but it requires a very great force to push her sidewise with the same velocity. A fine sailing ship of war will require about 12 times as much force to push her sidewise as to push her head foremost. In this respect therefore she will very much resemble a chaff whose length is 12 times its breadth; and whatever be the proportion of these resistances in different ships, we may always substitute a box which shall have the same resistances headwise and sidewise.

Let $EFGH$ (fig. 1.) be the horizontal section of such such a box, and AB its middle line, and C its centre. In whatever direction this box may chance to move, the direction of the whole resistance on its two sides will pass through C. For as the whole stream has one inclination to the side EF, the equivalent of the equal impulses on every part will be in a line perpendicular to the middle of EF. For the same reason, it will be in a line perpendicular to the middle of FG. These perpendiculars must cross in C. Suppose a mast erected at C, and YC y to be a yard hoisted on it carrying a sail. Let the yard be first conceived as braced right athwart at right angles to the keel, as represented by Y' y'. Then, whatever be the direction of the wind abaft this sail, it will impel the vessel in the direction CB. But if the sail has the oblique position Y y', the impulse will be in the direction CD perpendicular to CY, and will both push the vessel ahead and sidewise: For the impulse CD is equivalent to the two impulses CK and CI (the sides of a rectangle of which CD is the diagonal). The force CI pushes the vessel ahead, and CK pushes her sidewise. She must therefore take some intermediate direction a b, such that the resistance of the water to the plane FG is to its resistance to the plane EF as CI to CK.

The angle b CB between the real course and the direction of the head is called the Leeway; and in the course of this dissertation we shall express it by the symbol x. It evidently depends on the shape of the vessel and on the position of the yard. An accurate knowledge of the quantity of leeway, corresponding to different circumstances of obliquity of impulse, extent of surface, &c., is of the utmost importance in the practice of navigation; and even an approximation is valuable. The subject is so very difficult that this must content us for the present.

Let V be the velocity of the ship in the direction C b, and let the surfaces FG and F'E be called A' and B'. Then the resistance to the lateral motion is $mV^2 \times B' \times \text{fine}^2 bCB$, and that to the direct motion is $mV^2 \times A' \times \text{fine}^2 bCK$, or $mV^2 \times A' \times \text{col}^2 bCB$. Therefore these resistances are in the proportion of $B' \times \text{fine}^2 x$ to $A' \times \text{col}^2 x$ (representing the angle of leeway b CB by the symbol x).

Therefore we have CI : CK, or CI : ID = A': col. $x$: B': fine $x$, = A': B': tan. $x$ = A : B : tan. $x$.

Let the angle YCB, to which the yard is braced up, be called the Trim of the sails, and expressed by the symbol b. This is the complement of the angle DCI. Now CI : ID = rad. : tan. DCI, = 1 : tan. DCI, = 1 : cotan. b. Therefore we have finally 1 : cotan. b = A': B': tan. $x$, and A': cotan. b = B': tan. $x$, and tan. $x$ = $\frac{A}{B}$ cot. b. This equation evidently affords the mutual relation between the trim of the sails and the leeway in every case where we can tell the proportion between the resistances to the direct and broadside motions of the ship, and where this proportion does not change by the obliquity of the course. Thus, suppose the yard braced up at an angle of 35° with the keel. Then cotan. 35° = 1.732 very nearly. Suppose also that the resistance sidewise is 12 times greater than the resistance headwise. This gives

\[ A' = 1 \quad \text{and} \quad B' = 12. \]

Therefore $1.732 = 12 \times \text{tangent}^2 x$, and $\text{tangent}^2 x = \frac{1.732}{12} = 0.14434$, and tan. $x = 0.3799$, and $x = 20^\circ 48'$, very nearly two points of leeway.

This computation, or rather the equation which gives room for it, supposes the resistances proportional to the squares of the fines of incidence. The experiments of the Academy of Paris, of which an abstract is given in the article Resistance of Fluids, show that this supposition is not far from the truth when the angle of incidence is great. In the present case the angle of incidence on the front FG is about 70°, and the experiments just now mentioned show that the real resistances exceed the theoretical ones only $\frac{1}{5}$. But the angle of incidence on EF is only 25° 48'. Experiment shows that in this inclination the resistance is almost quadruple of the theoretical resistances. Therefore the lateral resistance is assumed much too small in the present instance. Therefore a much smaller leeway will suffice for producing a lateral resistance which will balance the lateral impulse CK, arising from the obliquity of the sail, viz. 30°. The matter of fact is, that a pretty good sailing ship, with her sails braced to this angle at a medium, will not make above five or six degrees leeway in smooth water and easy weather; and yet in this situation the hull and rigging present a very great surface to the wind, in the most improper positions, so as to have a very great effect in increasing her leeway. And if we compute the resistances for this leeway of five degrees by the actual experiments of the French Academy on the angle, we shall find the result not far from the truth; that is, the direct and lateral resistances will be nearly in the proportion of CI to ID.

It results from this view of the matter, that the leeway is in general much smaller than what the usual theory affirms.

We also see, that according to whatever law the resistances change by a change of inclination, the leeway depends on the direction of the impulse of the wind; and this depends solely on the position of the sails with respect to the keel, whatever may be the direction of the wind. This is a very important observation, and will be frequently referred to in the progress of the present investigation. Note, however, that we are here considering only the action on the sails, and on the same sails. We are not considering the action of the wind on the hull and rigging. This may be very considerable; and it is always in a lee direction, and augments the leeway; and its influence must be to much the more sensible as it bears a greater proportion to the impulse on the sails. A ship under courses, or close-reefed topsails and courses, must make more leeway than when under all her canvas trimmed, to the same angle. But to introduce this additional cause of deviation here would render the investigation too complicated to be of any use.

This doctrine will be considerably illustrated by attending to the manner in which a lighter is tracked along a canal, or swung to its anchor in a stream. The time by track rope is made fast to some staple or bolt E on the deck (fig. 2.), and is passed between two of the timberheads of the bow D, and laid hold of at F on shore. The men or cattle walk along the path FG, the rope keeps... keeps extended in the directions DF, and the lighter arranges itself in an oblique position AB, and is thus dragged along in the direction ab, parallel to the side of the canal. Or, if the canal has a current in the opposite direction ba, the lighter may be kept steady in its place by the rope DF made fast to a post at F. In this case, it is always observed, that the lighter swings in a position AB, which is oblique to the stream ab. Now the force which retains it in this position, and which precisely balances the action of the stream, is certainly exerted in the direction DF; and the lighter would be held in the same manner if the rope were made fast at C amidship, without any dependence on the timberheads at D; and it would be held in the same position, if instead of the single rope CF, it were riding by two ropes CG and CH, of which CH is in a direction right ahead, but oblique to the stream, and the other CG is perpendicular to CH or AB. And, drawing DI and DK perpendicular to AB and CG, the strain on the rope CII is to that on the rope CG as CI to CK. The action of the rope in these cases is precisely analogous to that of the sail y Y; and the obliquity of the keel to the direction of the motion, or to the direction of the stream, is analogous to the leeway. All this must be evident to anyone accustomed to mechanical dispositions.

A most important use may be made of this illustration. If an accurate model be made of a ship, and if it be placed in a stream of water, and ridden in this manner by a rope made fast at any point D of the bow, it will arrange itself in some determined position AB. There will be a certain obliquity to the stream, measured by the angle Bo b; and there will be a corresponding obliquity of the rope, measured by the angle FCB. Let y CY be perpendicular to CF. Then CY will be the position of the yard, or trim of the sails corresponding to the leeway b CB. Then, if we shift the rope to a point of the bow distant from D by a small quantity, we shall obtain a new position of the ship, both with respect to the stream and rope; and in this way may be obtained the relation between the position of the sails and the leeway, independent of all theory, and susceptible of great accuracy; and this may be done with a variety of models suited to the most usual forms of ships.

In farther thinking on this subject, we are persuaded that these experiments, instead of being made on models, may with equal ease be made on a ship of any size. Let the ship ride in a stream at a mooring D (fig. 3.) by means of a float hawser BCD from her bow, having a spring AC on it carried out from her quarter. She will swing to her moorings, till she ranges herself in a certain position AB with respect to the direction ab of the stream; and the direction of the hawser DC will point to some point E of the line of the keel. Now, it is plain to anyone acquainted with mechanical dispositions, that the deviation BE b is precisely the leeway that the ship will make when the average position of the sails is that of the line GEH perpendicular to ED; at least this will give the leeway which is produced by the sails alone. By heaving on the spring, the knot C may be brought into any other position we please; and for every new position of the knot the ship will take a new position with respect to the stream and to the hawser. And we persist in saying, that more information will be got by this train of experiments than from any mathematical theory: for all the theories of the impulses of fluids must proceed on physical postulates with respect to the motions of the filaments, which are exceedingly conjectural.

And it must now be farther observed, that the subdivision which we have made of an oblong parallelopiped for a ship, although well fitted to give us clear notions of the subject, is of small use in practice: for it is body next to impossible (even granting the theory of oblique only useful) to make this subdivision. A ship is of a form which is not reducible to equations; and therefore on the action of the water on her bow or broadside can only be had by a most laborious and intricate calculation for almost every square foot of its surface. (See Beszout's Cours de Mathemat. vol. v. p. 72, &c.) And this must be different for every ship. But, which is more unlucky, when we have got a parallelopiped which will have the same proportion of direct and lateral resistance for a particular angle of leeway, it will not answer for another leeway of the same ship; for when the leeway changes, the figure actually exposed to the action of the water changes also. When the leeway is increased, more of the lee-quarter is acted on by the water, and a part of the weather-bow is now removed from its action. Another parallelopiped must therefore be discovered, whose resistances shall suit this new position of the keel with respect to the real course of the ship.

We therefore beg leave to recommend this train of experiments to the notice of the Association for the Improvement of Naval Architecture as a very promising method for ascertaining this important point. And we proceed, in the next place, to ascertain the relation between the velocity of the ship and that of the wind, modified as they may be by the trim of the sails and the obliquity of the impulse.

Let AB (figs. 4, 5, and 6.) represent the horizontal position of a ship. In place of all the drawing falls, that is, the falls which are really filled, we can always substitute one fall of equal extent, trimmed to the same angle the ship with the keel. This being supposed attached to the wind yard DCD, let this yard be first of all at right angles astern to the keel, as represented in fig. 4. Let the wind blow in the direction WC, and let CE (in the direction WC continued) represent the velocity V of the wind. Let CF be the velocity v of the ship. It must also be in the direction of the ship's motion, because when the sail is at right angles to the keel, the absolute impulse on the sail is in the direction of the keel, and there is no lateral impulse, and consequently no leeway. Draw EF, and complete the parallelogram CFEe, producing eC through the centre of the yard to w. Then w C will be the relative or apparent direction of the wind, and Ce or FE will be its apparent or relative velocity: For if the line Ce be carried along CF, keeping always parallel to its first position, and if a particle of air move uniformly along CE (a fixed line in absolute space) in the same time, this particle will always be found in that point of CE where is is intersected at that instant by the moving line C; so that if Ce were a tube, the particle of air, which really moves in the line CE, would always be found in the tube Ce. While Ce is the real direction of the wind, Ce will be the position of the vane at the mast head, which will therefore mark the apparent direction of the wind, or its motion relative to the moving ship.

We may conceive this in another way. Suppose a cannon-shot fired in the direction CE at the passing ship, and that it passes through the mast at C with the velocity of the wind. It will not pass through the off-side of the ship at P, in the line CE; for while the shot moves from C to P, the point P has gone forward, and the point P is now in the place where P was when the shot passed through the mast. The shot will therefore pass through the ship's side in the point P, and a person on board seeing it pass through C and P will say that its motion was in the line CP.

Thus it happens, that when a ship is in motion the apparent direction of the wind is always ahead of its real direction. The line WC is always found within the angle WCB. It is easy to see from the construction, that the difference between the real and apparent directions of the wind is much the more remarkable as the velocity of the ship is greater: For the angle WCE depends on the magnitude of E or CF, in proportion to CE. Persons not much accustomed to attend to these matters are apt to think all attention to this difference to be nothing but affectation of nicety. They have no notion that the velocity of a ship can have any sensible proportion to that of the wind. "Swift as the wind" is a proverbial expression; yet the velocity of a ship always bears a very sensible proportion to that of the wind, and even very frequently exceeds it. We may form a pretty exact notion of the velocity of the wind by observing the shadows of the summer clouds flying along the face of a country, and it may be very well measured by this method. The motion of such clouds cannot be very different from that of the air below; and when the pressure of the wind on a flat surface, while blowing with a velocity measured in this way, is compared with its pressure when its velocity is measured by more unexceptionable methods, they are found to agree with all desirable accuracy. Now observations of this kind frequently repeated, show that what we call a pleasant brisk gale blows at the rate of about 10 miles an hour, or about 15 feet in a second, and exerts a pressure of half a pound on a square foot. Mr Smeaton has frequently observed the sails of a windmill, driven by such a wind, moving faster, nay much faster, towards their extremities, so that the sail, instead of being pressed to the frames on the arms, was taken aback, and fluttering on them. Nay, we know that a good ship, with all her sails set and the wind on the beam, will in such a situation sail above ten knots an hour in smooth water. There is an observation made by every experienced seaman, which shows this difference between the real and apparent directions of the wind very distinctly. When a ship that is sailing briskly with the wind on the beam tacks about, and then sails equally well on the other tack, the wind always appears to have shifted and come more ahead. This is familiar to all seamen. The seaman judges of the direction of the wind by the position of the ship's vanes. Suppose the ship sailing due west on the starboard tack, with the wind apparently N. N. W., the vane pointing S. S. E. If the ship put about, and stands the reef on the larboard tack, the vane will be found no longer to point S. S. E. but perhaps S. S. W., the wind appearing N. N. E., and the ship must be nearly close-hauled in order to make an easterly course. The wind appears to have shifted four points. If the ship tacks again, the wind returns to its old quarter. We have often observed a greater difference than this. The celebrated astronomer Dr Bradley, taking the amusement of sailing in a pinnace on the river Thames, observed Bradley on this, and was surprised at it, imagining that the change this subject of wind was owing to the approaching to or retiring from the shore. The boatmen told him that it always happened at sea, and explained it to him in the best manner they were able. The explanation struck him, and set him a musing on an astronomical phenomenon which he had been puzzled by for some years, and which he called the aberration of the fixed stars. Every star changes its place a small matter for half a year, and returns to it at the completion of the year. He compared the streams of light from the star to the wind, and the telescope of the astronomer to the ship's vane, while the earth was like the ship, moving in opposite directions when in the opposite points of its orbit. The telescope must always be pointed ahead of the real direction of the star, in the same manner as the vane is always in a direction ahead of the wind; and thus he ascertained the progressive motion of light, and discovered the proportion of its velocity to the velocity of the earth in its orbit, by observing the deviation which was necessarily given to the telescope. Observing that the light shifted its direction about 40", he concluded its velocity to be about 11,000 times greater than that of the earth; just as the intelligent seaman would conclude from this apparent shifting of the wind, that the velocity of the wind is about triple that of the ship. This is indeed the best method for discovering the velocity of the wind. Let the direction of the vane at the mast-head be very accurately noticed on both tacks, and let the velocity of the ship be also accurately measured. The angle between the directions of the ship's head on these different tacks being halved, will give the real direction of the wind, which must be compared with the position of the vane in order to determine the angle contained between the real and apparent directions of the wind or the angle ECE; or half of the observed shifting of the wind will show the inclination of its true and apparent directions. This being found, the proportion of EC to FC (fig. 6.) is easily measured.

We have been very particular on this point, because since the mutual actions of bodies depend on their relative motions only, we should make prodigious mistakes if we estimated the action of the wind by its real direction and velocity, when they differ so much from the relative or apparent.

We now resume the investigation of the velocity of the ship (fig. 4.), having its sails at right angles to the keel, and the wind blowing in the direction and with the velocity CE, while the ship proceeds in the direction of the keel with the velocity CF. Produce EE, angle to which is parallel to BC, till it meet the yard in g, and draw FG perpendicular to EG. Let a represent the angle WCD, contained between the sail and the real direction of the wind, and let b be the angle of trim DCB. CE the velocity of the wind was expressed by V, and CF the velocity of the ship by v.

The absolute impulse on the sail is (by the usual theory theory) proportional to the square of the relative velocity, and to the square of the fine of the angle of incidence; that is, to $FE^2 \times \text{fin.}^2$ w CD. Now the angle GFE = w CD, and EG is equal to FE $\times$ fin. GFE; and EG is equal to $EG - gG$. But $EG = EC \times \text{fin.}$. ECg, $= V \times \text{fin.}$. a; and $gG = CF$, $= v$. Therefore $EG = V \times \text{fin.}$. a$-v$, and the impulse is proportional to $V \times \text{fin.}$. a$-v^2$. If S represent the surface of the sail, the impulse, in pounds, will be $nS(V \times \text{fin.}$. a$-v^2)$.

Let A be the surface which, when it meets the water perpendicularly with the velocity $v$, will sustain the same pressure or resistance which the bows of the ship actually meets with. This impulse, in pounds, will be $mAv^2$. Therefore, because we are considering the ship's motion as in a state of uniformity, the two pressures balance each other; and therefore $mAv^2 = nS(V \times \text{fin.}$. a$-v^2)$, and $\frac{m}{n}A = S(V \times \text{fin.}$. a$-v^2)$;

therefore $\sqrt{\frac{m}{n}}A \times v = \sqrt{S} \times V \times \text{fin.}$. a$-v\sqrt{S}$, and

$v = \frac{\sqrt{S} \times \text{fin.}$. a$-V \times \text{fin.}$. a}{\sqrt{\frac{m}{n}}A + \sqrt{S}} = \frac{V \times \text{fin.}$. a}{\sqrt{\frac{m}{n}}A + \sqrt{S}} = \frac{V \times \text{fin.}$. a}{\sqrt{\frac{m}{n}}A + \sqrt{S}} + 1.$

We see, in the first place, that the velocity of the ship is (ceteris paribus) proportional to the velocity of the wind, and to the fine of its incidence on the sail jointly; for while the surface of the sail S and the equivalent surface for the bow remains the same, $v$ increases or diminishes at the rate same with $V \times \text{fin.}$. a$-v$. When the wind is right aft, the fine of $a$ is unity,

and then the ship's velocity is $\frac{V}{\sqrt{\frac{m}{n}}A + \sqrt{S}} + 1.$

Note, that the denominator of this fraction is a common number; for $m$ and $n$ are numbers, and $A$ and $S$ being quantities of one kind, $\frac{A}{S}$ is also a number.

It must also be carefully attended to, that $S$ expresses a quantity of sail actually receiving wind with the inclination $a$. It will not always be true, therefore, that the velocity will increase as the wind is more abaft, because some sails will then be less effective. This observation is not, however, of great importance; for it is very unusual to put a ship in the situation considered hitherto; that is, with the yards square, unless she be right before the wind.

If we would discover the relation between the velocity and the quantity of sail in this simple case of the wind right aft, observe that the equation $v = \frac{V}{\sqrt{\frac{m}{n}}A + \sqrt{S}} + 1.$ gives us $\sqrt{\frac{m}{n}}A + v = V$, and $\sqrt{\frac{m}{n}}A = V - v$, and $\frac{m}{n}A = V - v^2$, and $\frac{n}{m}A = \frac{v^2}{(V-v)^2}$; and because $n$ and $m$ and $A$ are constant quantities, $S$ is proportional to $\frac{v^2}{(V-v)^2}$, or the surface of sail is proportional to the square of the ship's velocity directly, and to the square of the relative velocity inversely. Thus, if a ship be sailing with one-eighth of the velocity of the wind, and we would have her sail with one-fourth of it, we must quadruple the sail. This is more easily seen in another way. The velocity of the ship is proportional to the velocity of the wind; and therefore the relative velocity is also proportional to that of the wind, and the impulse of the wind is as the square of the relative velocity. Therefore, in order to increase the relative velocity by an increase of sail only, we must make this increase of sail in the duplicate proportion of the increase of velocity.

Let us, in the next place, consider the motion of a ship whose sails stand oblique to the keel.

The construction for this purpose differs a little from the former, because, when the sails are trimmed to any other position DCB (fig. 5, and 6.), there must be a deviation from the direction of the keel, or a leeway to the keel BC b. Call this $x$. Let CF be the velocity of the ship, fig. 5, and FG perpendicular to EG; also draw FH perpendicular to the yard; then, as before, EG, which is in the subductive ratio of the impulse on the sail, is equal to $EG - GG$. Now $EG$ is, as before, $= V \times \text{fin.}$. a, and $GG$ is equal to FH, which is $= CF \times \text{fin.}$. FCH, or $= v \times \text{fin.}$. ($b+x$). Therefore we have the impulse $= nS(V \times \text{fin.}$. a$-v \times \text{fin.}$. ($b+x$)$^2$.

This expression of the impulse is perfectly similar to that in the former case, its only difference consisting in the subductive part, which is here $v \times \text{fin.}$. ($b+x$) instead of $v$. But it expresses the same thing as before, viz. the diminution of the impulse. The impulse being reckoned solely in the direction perpendicular to the sail, it is diminished solely by the sail withdrawing itself in that direction from the wind; and as $GE$ may be considered as the real impulsive motion of the wind, $GE$ must be considered as the relative and effective impulsive motion. The impulse would have been the same had the ship been at rest, and had the wind met it perpendicularly with the velocity $GE$.

We must now show the connection between this impulse and the motion of the ship. The sail, and consequently the ship, is pressed by the wind in the direction CI perpendicular to the sail or yard with the force impulse which we have just now determined. This (in the state of the ship of uniform motion) must be equal and opposite to the action of the water. Draw IL at right angles to the keel. The impulse in the direction CI (which we may measure by CI) is equivalent to the impulses CL and LI. By the first the ship is impelled forward, and by the second she is driven sidewise. Therefore we must have a leeway, and a lateral as well as a direct resistance. We suppose the form of the ship to be known, and therefore the proportion is known, or discoverable, between the direct and lateral resistances corresponding to every angle $x$ of leeway. Let A be the surface whose perpendicular resistance is equal to the direct resistance of the ship corresponding to the leeway $x$, that is, whose resistance is equal to the resistance really felt by the ship's bows in the direction of the keel when she is sailing with this leeway; and let B in like manner be the surface whose perpendicular resistance is equal to the actual resistance to the ship's motion in the direction LI, perpendicular to the keel. (N.B. This is not equivalent to A and B' adapted to the rectangular box, but to A' col. $x$ and B' fin. $x$.) We have therefore therefore \( A : B = CL : LI \), and \( LI = \frac{CL \cdot B}{A} \). Also, because \( CI = \sqrt{CL^2 + IP^2} \), we have \( A : \sqrt{A^2 + B^2} = CL : CI \), and \( CI = \frac{CL \cdot \sqrt{A^2 + B^2}}{A} \). The resistance in the direction LC is properly measured by \( m \cdot A \cdot v^2 \), as has been already observed. Therefore the resistance in the direction IC must be expressed by \( m \cdot \sqrt{A^2 + B^2} \cdot v^2 \); or (making C the surface which is equal to \( \sqrt{A^2 + B^2} \)), and which will therefore have the same perpendicular resistance to the water having the velocity \( v \) it may be expressed by \( m \cdot C \cdot v^2 \).

Therefore, because there is an equilibrium between the impulse and resistance, we have \( m \cdot A \cdot v^2 = n \cdot S \cdot (\text{V} \cdot \sin a - v \cdot \sin b + x)^2 \) and \( \frac{m}{n} \cdot C \cdot v^2 = q \cdot C \cdot v^2 = S \cdot (\text{V} \cdot \sin a - v \cdot \sin b + x)^2 \), and \( \sqrt{q} \cdot \sqrt{C} \cdot v = \sqrt{S} \cdot (\text{V} \cdot \sin a - v \cdot \sin b + x) \).

Therefore \( v = \frac{\sqrt{S} \cdot \text{V} \cdot \sin a}{\sqrt{q} \cdot \sqrt{C} + \sqrt{S} \cdot \sin b + x} = \frac{\text{V} \cdot \sin a}{\sqrt{q} \cdot \sqrt{C} + \sin b + x} = \frac{\sqrt{S} \cdot \text{V} \cdot \sin a}{\sqrt{q} \cdot \sqrt{C} + \sin b + x} \).

Observe that the quantity which is the coefficient of V in this equation is a common number; for \( \sin a \) is a number, being a decimal fraction of the radius 1, \( \sin b + x \) is also a number, for the same reason. And since \( m \) and \( n \) were numbers of pounds, \( \frac{m}{n} \) or \( q \) is a common number. And because C and S are surfaces, or quantities of one kind, \( \frac{C}{S} \) is also a common number.

This is the simplest expression that we can think of for the velocity acquired by the ship, though it must be acknowledged to be too complex to be of very prompt use. Its complication arises from the necessity of introducing the leeway \( x \). This affects the whole of the denominator; for the surface C depends on it, because \( C = \sqrt{A^2 + B^2} \), and A and B are analogous to \( A' \cdot \cotan b \cdot x \) and \( B' \cdot \sin b \cdot x \).

But we can deduce some important consequences from this theorem.

While the surface S of the sail actually filled by the wind remains the same, and the angle DCB, which in future we shall call the trim of the sails, also remains the same, both the leeway \( x \) and the substituted surface C remains the same. The denominator is therefore constant; and the velocity of the ship is proportional to \( \sqrt{S} \cdot \text{V} \cdot \sin a \); that is, directly as the velocity of the wind, directly as the absolute inclination of the wind to the yard, and directly as the square root of the surface of the sails.

We also learn from the construction of the figure that FG parallel to the yard cuts CE in a given ratio. For CF is in a constant ratio to EG, as has been just now demonstrated. And the angle DCF is constant. Therefore \( CF \cdot \sin b \), or FH or \( G \cdot \sin b \), is proportional to EG, and OC to EC, or EC is cut in one proportion, whatever may be the angle ECD, so long as the angle DCF is constant.

We also see that it is very possible for the velocity of the ship on an oblique course to exceed that of the wind. This will be the case when the number

\[ \frac{\sin a}{\sqrt{q} \cdot \sqrt{C} + \sin b + x} \]

exceeds unity, or when \( \sin a \) is greater than \( \sqrt{q} \cdot \sqrt{C} + \sin b + x \). Now this may easily be by sufficiently enlarging \( S \) and diminishing \( b + x \). It is indeed frequently seen in fine sailors with all their sails set and not hauled too near the wind.

We remarked above that the angle of leeway \( x \) affects the whole denominator of the fraction which expresses the velocity. Let it be observed that the angle JCL is the complement of LCD, or of \( b \). Therefore, \( CL : LI \), or \( A : B = 1 : \tan ICL \), \( ICL = 1 : \cotan b \), \( B = A' \cdot \cotan b \). Now A is equivalent to \( A' \cdot \cotan b \cdot x \), and thus \( b \) becomes a function of \( x \). C is evidently fo, being \( \sqrt{A^2 + B^2} \). Therefore before the value of this fraction can be obtained, we must be able to compute, by our knowledge of the form of the ship, the value of A for every angle \( x \) of leeway. This can be done only by resolving her bows into a great number of elementary planes, and computing the impulses on each and adding them into one sum. The computation is of immense labour, as may be seen by one example given by Bouguer. When the leeway is but small, not exceeding ten degrees, the substitution of the rectangular prism of one determined form is abundantly exact for all leeways contained within this limit; and we shall soon feel reason for being contented with this approximation. We may now make use of the formula expressing the velocity for solving the chief problems in this part of the seaman's task.

And first let it be required to determine the best position of the sail for standing on a given course \( ab \), when to determine the direction and velocity of the wind, and its angle with the course WCF, are given. This problem has been often exercised the talents of the mathematicians ever since the days of Newton. In the article Pneumatics we found the solution of one very nearly related to it, namely, on a given course, to determine the position of the sail which would produce the greatest impulse in the direction of the course. The solution was to place the yard CD in such a position that the tangent of the angle FCD may be half of the one half of the tangent of the angle DCW. This will indeed be the best position of the sail for beginning the motion; but as soon as the ship begins to move in the course are direction CF, the effective impulse of the wind is diminished, and also its inclination to the sail. The angle DCW diminishes continually as the ship accelerates; for CF is now accompanied by its equal \( E \), and by an angle EC \( e \) or WC \( w \). CF increases, and the impulse on the sail diminishes, till an equilibrium obtains between the resistance of the water and the impulse of the wind. The impulse is now measured by \( CE \times \sin a \cdot e \cdot CD \) instead of \( CE \times \sin a \cdot ECD \), that is, by \( EG \times \sin a \cdot E \).

This introduction of the relative motion of the wind renders the actual solution of the problem extremely difficult. difficult. It is very easily expressed geometrically: Divide the angle \( \omega CF \) in such a manner that the tangent of DCF may be half of the tangent of DC \( \omega \), and the problem may be constructed geometrically as follows.

Let WCF (fig. 7.) be the angle between the sail and course. Round the centre C describe the circle WDFY; produce WC to Q, so that CO = \( \frac{1}{2} \) WC, and draw QY parallel to CF cutting the circle in Y; bisect the arch WY in D, and draw DC. DC is the proper position of the yard.

Draw the chord WY, cutting CD in V and CF in T; draw the tangent PD cutting CF in S and CY in R.

It is evident that WY, PR, are both perpendicular to CD, and are bisected in V and D; therefore (by reason of the parallels QY, CF) \( 4 : 3 = QW : CW \), \( = YW : TW, = RP : SP \). Therefore PD : PS = 2 : 3, and PD : DS = 2 : 1. Q.E.D.

But this division cannot be made to the best advantage till the ship has attained its greatest velocity, and the angle \( \omega CF \) has been produced.

We must consider all the three angles, \( a, b, \) and \( x \), as variable in the equation which expresses the value of \( v \), and we must make the fluxion of this equation \( = 0 \); then, by means of the equation \( B = A \cdot \cotan b \), we must obtain the value of \( b \) and of \( \dot{b} \) in terms of \( x \) and \( \dot{x} \).

With respect to \( a \), observe, that if we make the angle WCF = \( \rho \), we have \( \rho = a + b + x \); and \( \rho \) being a constant quantity, we have \( a + b + x = 0 \). Substituting for \( a, b, \dot{a}, \dot{b} \), their values in terms of \( x \) and \( \dot{x} \), in the trigonometrical equation \( = 0 \), we readily obtain \( x \), and then \( a \) and \( b \), which solves the problem.

Let it be required, in the next place, to determine the course and trim of the sails most proper for plying to windward.

In fig. 6. draw FP perpendicular to WC. CF is the motion of the ship; but it is only by the motion PC that she gains to windward. Now CP is \( = CF \times \text{cofin}.(a + b + x) \). This must be rendered a maximum, as follows.

By means of the equation which expresses the value of \( v \) and the equation \( B = A \cdot \cotan b \), we exterminate the quantities \( v \) and \( b \); we then take the fluxion of the quantity into which the expression \( v \cdot \text{cofin}.(a + b + x) \) is changed by this operation. Making this fluxion \( = 0 \), we get the equation which must solve the problem. This equation will contain the two variable quantities \( a \) and \( x \) with their fluxions; then make the coefficient of \( \dot{x} \) equal to \( v \), also the coefficient of \( \dot{a} \) equal to \( v \). This will give two equations which will determine \( a \) and \( x \), and from this we get \( b = \rho - a - x \).

Should it be required, in the third place, to find the best course and trim of the sails for getting away from a given line of coast CM (fig. 6.), the process perfectly resembles this last, which is in fact getting away from the sails for a line of coast which makes a right angle with the wind. Therefore, in place of the angle WCF, we must substitute the angle WCM \( = \pm WCF \). Call this angle \( e \). We must make \( v \cdot \text{cofin}.(e - a - b - x) \) a maximum. The analytical process is the same as the former, only \( e \) is here a constant quantity.

These are the three principal problems which can be solved by means of the knowledge that we have obtained of the motion of the ship when impelled by an oblique sail, and therefore making leeway; and they may be considered as an abstract of this part of M. Bouguer's work. We have only pointed out the process for this solution, and have even omitted some things taken notice of by M. Bezout in his very elegant compendium. Our reasons will appear as we go on. The learned reader will readily see the extreme difficulty of the subject, and the immense calculations which are necessary even in the simplest cases, and will grant that it is out of the power of any but an expert analyst to derive any use from them; but the mathematician can calculate tables for the use of the practical seaman. Thus he can calculate the best position of the sails for advancing in a course \( 90^\circ \) from the wind, and the velocity in that course; then for \( 85^\circ, 80^\circ, 75^\circ \), &c. M. Bouguer has given a table of this kind; but to avoid the immense difficulty of the process, he has adapted it to the apparent direction of the wind. We have inferred a few of his numbers, suited to such cases as can be of service, namely, the sails for when all the sails draw, or none stand in the way of advancing others. Column 1st is the apparent angle of the wind in any course; column 2nd is the corresponding angle of the sails and keel; and column 3rd is the apparent angle of the sails and wind.

| \( \omega CF \) | DCB | \( \omega CD \) | |-------------|-----|-------------| | 103° 53' | 42° 30' | 61° 23' | | 99° 13' | 40° | 59° 13' | | 94° 25' | 37° 30' | 56° 55' | | 89° 28' | 35° | 54° 28' | | 84° 23' | 32° 30' | 51° 53' | | 79° 66' | 30° | 49° 06' | | 73° 39' | 27° 30' | 46° 09' | | 68° | 25° | 43° |

In all these numbers we have the tangent of \( \omega CD \) double of the tangent of DCF.

But this is really doing but little for the seaman. Intuitively the apparent direction of the wind is unknown to him till the ship is sailing with uniform velocity; and he is still uninformed as to the leeway. It is, however, of service to him to know, for instance, that when the angle of the vanes and yards is 56 degrees, the yard should be braced up to 37° 30', &c.

But here occurs a new difficulty. By the construction of a square-rigged ship it is impossible to give the yards that inclination to the keel which the calculation requires. Few ships can have their yards braced up to 37° 30'; and yet this is required in order to have an incidence of 56°, and to hold a course 94° 25' from the apparent direction of the wind, that is, with the wind apparently 4° 25' abaft the beam. A good sailing ship in this position may acquire a velocity even exceeding that of the wind. Let us suppose it only one half of this velocity. We shall find that the angle WC \( \omega \) is in this case about 29°, and the ship is nearly going 123° from the wind, with the wind almost perpendicular to the sail; therefore this utmost bracing up of the sails is only giving them the position suited to a wind broad on the quarter. It is impossible therefore to comply with the demand of the mathematician, and the seaman must be contented to employ a less favourable disposition of his sails in all cases where his course does not lie at least eleven points from the wind. Let us see whether this restriction, arising from necessity, leaves anything in our choice, and makes one course preferable to another. We see that there are a prodigious number of courses, and these the most usual and the most important, which we must hold with one trim of the sails; in particular, failing with the wind on the beam, and all cases of plying to windward, must be performed with this unfavourable trim of the sails. We are certain that the smaller we make the angle of incidence, real or apparent, the smaller will be the velocity of the ship; but it may happen that we shall gain more to windward, or get sooner away from a lee-coast, or any object of danger, by failing slowly on one course than by failing quickly on another.

We have seen that while the trim of the sails remains the same, the leeway and the angle of the yard and course remains the same, and that the velocity of the ship is as the sine of the angle of real incidence, that is, as the sine of the angle of the sail and the real direction of the wind.

Let the ship AB (fig. 8.) hold the course CF, with the wind blowing in the direction WC, and having her yards DCD braced up to the smallest angle BCD which the rigging can admit. Let CF be to CE as the velocity of the ship to the velocity of the wind; join FE and draw Cw parallel to EF; it is evident that FE is the relative motion of the wind, and w CD is the relative incidence on the sail. Draw FO parallel to the yard DC, and describe a circle through the points COF; then we say that if the ship, with the same wind and the same trim of the same drawing sails, be made to fail on any other course Cf, her velocity along CF is to the velocity along Cf as CF is to Cf; or, in other words, the ship will employ the same time in going from C to any point of the circumference CFO.

Join fO. Then, because the angles CFO, cFO are on the same chord CO, they are equal, and fO is parallel to dC, the new position of the yard corresponding to the new position of the keel ab, making the angle dCB=DCB. Also, by the nature of the circle, the line CF is to Cf as the sine of the angle CFO to the sine of the angle COf, that is (on account of the parallels CD, OF and Cd, Of), as the sine of WCD to the sine of WCD. But when the trim of the sails remains the same, the velocity of the ship is as the sine of the angle of the sail with the direction of the wind; therefore CF is to Cf as the velocity on CF to that on Cf, and the proposition is demonstrated.

Let it now be required to determine the best course for avoiding a rock R lying in the direction CR, or for withdrawing as fast as possible from a line of coast PQ. Draw CM through R, or parallel to PQ, and let m be the middle of the arch CmM. It is plain that m is the most remote from CM of any point of the arch CmM, and therefore the ship will recede farther from the coast PQ in any given time by holding the course Cm than by any other course.

This course is easily determined; for the arch CmM =360° - (arch CO + arch OM), and the arch CO is the measure of twice the angle CFO, or twice the angle DCB, or twice b+x, and the arch OM measures twice the angle ECM.

Thus suppose the sharpest possible trim of the sails to be 35°, and the observed angle ECM to be 70°; then CO+OM is 70°+140° or 210°. This being taken from 360°, leaves 150°, of which the half Mm is 75°, and the angle MCm is 37° 30'. This added to ECM makes ECM 107° 30', leaving WCm=72° 30', and the ship must hold a course making an angle of 72° 30' with the real direction of the wind, and WCD will be 37° 30'.

This supposes no leeway. But if we know that under all the fail which the ship can carry with safety and advantage she makes 5 degrees of leeway, the angle DCm of the fail and course, or b+x, is 40°. Then CO+OM=220°, which being taken from 360° leaves 140°, of which the half is 70°, =Mm, and the angle MCm =35°, and ECM=105°, and WCm=75°, and the ship must lie with her head 70° from the wind, making 5 degrees of leeway, and the angle WCD is 35°.

The general rule for the position of the ship is, that the line on shipboard which bisects the angle b+x may also bisect the angle WCM, or make the angle between the course and the line from which we wish to withdraw equal to the angle between the fail and the real direction of the wind.

It is plain that this problem includes that of plying to windward. We have only to suppose ECM to be 90°; then, taking our example in the same ship, with the same trim and the same leeway, we have b+x=40°. This taken from 90° leaves 50° and WCm=90°-25=65°, and the ship's head must lie 65° from the wind, and the yard must be 25° from it.

It must be observed here, that it is not always eligible to select the course which will remove the ship fastest from the given line CM; it may be more prudent to remove from it more securely though more slowly. In such cases the procedure is very simple, viz. to shape the course as near the wind as is possible.

The reader will also easily see that the propriety of these practices is confined to those courses only where the practicable trim of the sails is not sufficiently sharp. Whenever the course lies so far from the wind that it is impossible to make the tangent of the apparent angle of the wind and fail double the tangent of the fail and course, it should be done.

These are the chief practical consequences which can be deduced from the theory. But we should consider how far this adjustment of the sails and course can be performed. And here occur difficulties so great as to make it almost impracticable. We have always supposed the position of the surface of the fail to be distinctly observable and measurable; but this can hardly be affirmed even with respect to a fail stretched on a yard. Here we supposed the surface of the fail to have the same inclination to the keel that the yard has. This is by no means the case; the fail assumes a concave form, of which it is almost impossible to affirm the direction of the mean impulse. We believe that this is always considerably to leeward of a perpendicular to the yard, lying between CI and CE (fig. 6.). This is of some advantage, being equivalent to a sharper trim. We cannot affirm this, however, with any confidence, because it renders the impulse on the weather-leech of the fail so exceedingly feeble as hardly to have any effect. In failing close to the wind the ship is kept to near that the weather-leech of the fail is almost ready to receive the wind edgewife, and to flutter or shiver. The most effective or drawing fails with a side-wind, especially when when plying to windward, are the stayfails. We believe that it is impossible to say, with any thing approaching to precision, what is the position of the general surface of a stayfail, or to calculate the intensity and direction of the general impulse; and we affirm with confidence that no man can pronounce on these points with any exactness. If we can guess within a third or a fourth part of the truth, it is all we can pretend to; and after all, it is but a guess. Add to this, the fails coming in the way of each other, and either becalming them or sending the wind upon them in a direction widely different from that of its free motion. All these points we think beyond our power of calculation, and therefore that it is in vain to give the seaman mathematical rules, or even tables of adjustment ready calculated; since he can neither produce that medium position of his fails that is required, nor tell what is the position which he employs.

This is one of the principal reasons why so little advantage has been derived from the very ingenious and promising disquisitions of Bouguer and other mathematicians, and has made us omit the actual solution of the chief problems, contenting ourselves with pointing out the process to such readers as have a relish for these analytical operations.

But there is another principal reason for the small progress which has been made in the theory of seamanship: This is the error of the theory itself, which supposes the impulsions of a fluid to be in the duplicate ratio of the sine of incidence. The most careful comparison which has been made between the results of this theory and matter of fact is to be seen in the experiments made by the members of the Royal Academy of Sciences at Paris, mentioned in the article RESISTANCE OF FLUIDS. We subjoin another abstract of them in the following table; where col. 1st gives the angle of incidence; col. 2nd gives the impulsions really observed; col. 3rd the impulsions had they followed the duplicate ratio of the sines; and col. 4th the impulsions, if they were in the simple ratio of the sines.

| Angle | Impulsion Observed | Impulsion as Sine² | Impulsion as Sine | |-------|-------------------|--------------------|------------------| | 90 | 1000 | 1000 | 1000 | | 84 | 980 | 980 | 995 | | 78 | 958 | 957 | 978 | | 72 | 928 | 925 | 951 | | 66 | 845 | 835 | 914 | | 60 | 771 | 750 | 866 | | 54 | 693 | 655 | 809 | | 48 | 615 | 552 | 743 | | 42 | 543 | 448 | 660 | | 36 | 480 | 346 | 587 | | 30 | 440 | 250 | 500 | | 24 | 424 | 165 | 407 | | 18 | 414 | 96 | 309 | | 12 | 406 | 43 | 208 | | 6 | 400 | 11 | 105 |

Here we see an enormous difference in the great obliquities. When the angle of incidence is only six degrees, the observed impulse is forty times greater than the theoretical impulse; at 12° it is ten times greater; at 18° it is more than four times greater; and at 24° it is almost three times greater.

No wonder then that the deductions from this theory and the deductions from it are so unlike what we familiarly observe. We took notice of this when we were considering the leeway of a rectangular box, and thus saw a reason for admitting an incomparably smaller leeway than what would result from the laborious computations necessary by the theory. This error in theory has as great an influence on the impulsions of air when acting obliquely on a sail; and the experiments of Mr Robins and of the Chevalier Borda on the oblique impulsions of air are perfectly conformable (as far as they go) to those of the academicians on water. The oblique impulsions of the wind are therefore much more efficacious for preluding the ship in the direction of her course than the theory allows us to suppose; and the progress of a ship plying to windward is much greater, both because the oblique impulsions of the wind are more effective, and because the leeway is much smaller, than we suppose. Were not this the case, it would be impossible for a square-rigged ship to get to windward. The impulse on her sails when close hauled would be so trifling that she would not have a third part of the velocity which we see her acquire; and this trifling velocity would be wasted in leeway; for we have seen that the diminution of the oblique impulsions of the water is accompanied by an increase of leeway. But we see that in the great obliquities the impulsions continue to be very considerable, and that even an incidence of six degrees gives an impulse as great as the theory allows to an incidence of 40°. We may therefore, on all occasions, keep the yards more square; and the loss which we sustain by the diminution of the very oblique impulse will be more than compensated by its more favourable direction with respect to the ship's keel. Let us take an example of this. Suppose the wind about two points before the beam, making an angle of 68° with the keel. The theory affirms 43° for the inclination of the wind to the sail, and 15° for the trim of the sail. The perpendicular impulse being supposed 1000, the theoretical impulse for 45° is 465. This reduced in the proportion of radius to the fine of 25°, gives the impulse in the direction of the course only 197.

But if we ease off the lee-braces till the yard makes an angle of 50° with the keel, and allows the wind an incidence of no more than 18°, we have the experimented impulse 414, which, when reduced in the proportion of radius to the fine of 50°, gives an effective impulse 317°. In like manner, the trim 56°, with the incidence 12°, gives an effective impulse 337°; and the trim 62°, with the incidence only 6°, gives 353°.

Hence it would at first sight appear that the angle DCB of 62° and WCD of 6° would be better for holding a course within five points of the wind than any more oblique position of the sails; but it will only give a greater initial impulse. As the ship accelerates, the wind apparently comes ahead, and we must continue to brace up as the ship freethens her way. It is not unusual for her to acquire half or two thirds of the velocity of the wind; in which case the wind comes apparently ahead more than two points, when the yards must be braced up to 35°, and this allows an impulse no greater than about 7°. Now this is very frequently observed. observed in good ships, which in a brisk gale and smooth water will go five or six knots close hauled, the ship's head six points from the wind, and the sails no more than just full, but ready to shiver by the smallest luff. All this would be impossible by the usual theory; and in this respect these experiments of the French academy gave a fine illustration of the seaman's practice. They account for what we should otherwise be much puzzled to explain; and the great progress which is made by a ship close hauled being perfectly agreeable to what we should expect from the law of oblique impulsion deducible from these so often mentioned experiments, while it is totally incompatible with the common theory, should make us abandon the theory without hesitation, and strenuously set about the establishment of another, founded entirely on experiments. For this purpose the experiments should be made on the oblique impulsions of air on as great a scale as possible, and in as great a variety of circumstances, so as to furnish a series of impulsions for all angles of obliquity. We have had four or five experiments on this subject, viz. two by Mr Robins and two or three by the Chevalier Borda. Having thus got a series of impulsions, it is very practicable to raise on this foundation a practical institute, and to give a table of the velocities of a ship suited to every angle of inclination and trim; for nothing is more certain than the resolution of the impulse perpendicular to the sail into a force in the direction of the keel, and a lateral force.

We are also disposed to think that experiments might be made on a model very nicely rigged with sails, and trimmed in every different degree, which would point out the mean direction of the impulse on the sails, and the comparative force of these impulses in different directions of the wind. The method would be very similar to that for examining the impulse of the water on the hull. If this can also be ascertained experimentally, the intelligent reader will easily see that the whole motion of a ship under sail may be determined for every case. Tables may then be constructed by calculation, or by graphical operations, which will give the velocities of a ship in every different course, and corresponding to every trim of sail. And let it be here observed, that the trim of the sail is not to be estimated in degrees of inclination of the yards; because, as we have already remarked, we cannot observe nor adjust the latter sails in this way. But, in making the experiments for ascertaining the impulse, the exact position of the tacks and sheets of the sails are to be noted; and this combination of adjustments is to pass by the name of a certain trim. Thus that trim of all the sails may be called 45°, whose direction is experimentally found equivalent to a flat surface trimmed to the obliquity 45°.

Having done this, we may construct a figure for each trim similar to fig. 8, where, instead of a circle, we shall have a curve COM'F', whose chords CF', CF', &c. are proportional to the velocities in these courses; and by means of this curve we can find the point m', which is most remote from any line CM from which we wish to withdraw; and thus we may solve all the principal problems of the art.

We hope that it will not be accounted presumption in us to expect more improvement from a theory founded on judicious experiments only, than from a theory of the impulse of fluids, which is found to incongruous with observation, and of whose fallacy all its authors, from Newton to D'Alembert, entertained strong suspicions. Again, we beg leave to recommend this view of the subject to the attention of the Society recommended for the Improvement of Naval Architecture, and to the Society Should these patriotic gentlemen entertain a favourable opinion of the plan, and honour us with their correspondence, we will cheerfully impart to them our naval notions of the way in which both these trains of experiments may be prosecuted with success, and results obtained in which we may confide; and we content ourselves at present with offering to the public these hints, which are not the speculations of a man of more science, but of one who, with a competent knowledge of the laws of mechanical nature, has the experience of several years service in the royal navy, where the art of working of ships was a favourite object of his scientific attention.

With these observations we conclude our discussion of Means employed to consider the means that are employed to prevent or produce any deviations from the uniform rectilinear course, which has been selected.

Here the ship is to be considered as a body in free course, convertible round her centre of inertia. For whatever may be the point round which she turns, this motion may always be considered as compounded of a rotation round an axis passing through her centre of gravity or inertia. She is impelled by the wind and by the water acting on many surfaces differently inclined to each other, and the impulse on each is perpendicular to the surface. In order therefore that she may continue steadily in one course, it is not only necessary that the impelling forces, estimated in their mean direction, be equal and opposite to the resisting forces estimated in their mean direction; but also that these two directions may pass through one point, otherwise she will be affected as a log of wood is when pushed in opposite directions by two forces, which are equal indeed, but are applied to different parts of the log. A ship must be considered as a lever, acted on in different parts by forces in different directions, and the whole balancing each other round that point or axis where the equivalent of all the resisting forces passes. This may be considered as a point supported by this resisting force and as a sort of fulcrum; therefore, in order that the ship may maintain her position, the energies or momenta of all the impelling forces round this point must balance each other.

When a ship sails right abreast the wind, with her yards square, it is evident that the impulses on each side of the ship's keel are equal, as also their mechanical momenta round sailing right any axis passing perpendicularly through the keel. So before the actions of the water on her bows. But when she enters from sails on an oblique course, with her yards braced up on those on either side, she sustains a prelude in the direction CM' when (fig. 5.) perpendicular to the sail. This, by giving her a lateral prelude LI, as well as a prelude CL ahead, causes her to make leeway, and to move in a line CB inclined to CB. By this means the balance of action on the two bows is destroyed; the general impulse on the lee-bow is increased; and that on the weather-bow is diminished. minished. The combined impulse is therefore no longer in the direction BC, but (in the state of uniform motion) in the direction IC.

Suppose that in an instant the whole sails are annihilated and the impelling pressure CI, which precisely balanced the resisting pressure on the bows, removed. The ship tends, by her inertia, to proceed in the direction Cb. This tendency produces a continuation of the resistance in the opposite direction IC, which is not directly opposed to the tendency of the ship in the direction Cb; therefore the ship's head would immediately come up to the wind. The experienced seamen will recollect something like this when the sails are suddenly lowered when coming to anchor. It does not happen solely from the obliquity of the action on the bows; it would happen to the parallelepiped of fig. 2, which was sustaining a lateral impulse B·fin. x, and a direct impulse A·col. x. These are continued for a moment after the annihilation of the tail: but being no longer opposed by a force in the direction CD, but by a force in the direction Cb, the force B·fin. x must prevail, and the body is not only retarded in its motion, but its head turns towards the wind. But this effect of the leeway is greatly increased by the curved form of the ship's bows. This occasions the centre of effort of all the impulsions of the water on the lee side of the ship to be very far forward, and this so much the more remarkably as she is sharper afore. It is in general not much abaft the foremast. Now the centre of the ship's tendency to continue her motion is the same with her centre of gravity, and this is generally but a little before the mainmast. She is therefore in the same condition nearly as if she were pushed at the mainmast in a direction parallel to Cb, and at the foremast by a force parallel to IC. The evident consequence of this is a tendency to come up to the wind. This is independent of all situation of the sails, provided only that they have been trimmed obliquely.

This tendency of the ship's head to windward is called gripping in the seaman's language, and is greatest in ships which are sharp forward, as we have said already. This circumstance is easily understood. Whatever is the direction of the ship's motion, the absolute impulse on that part of the bow immediately contiguous to B is perpendicular to that very part of the surface. The more acute, therefore, that the angle of the bow is, the more will the impulse on that part be perpendicular to the keel, and the greater will be its energy to turn the head to windward.

Thus we are enabled to understand or to see the propriety of the disposition of the sails of a ship. We see her crowded with sails forward, and even many sails extended far before her bow, such as the spritail, the bowsprit-topmast, the fore-topmast stay-sail, the jib, and flying jib. The sails abaft are comparatively smaller. The sails on the mizzenmast are much smaller than those on the foremast. All the staysails hoisted on the mainmast may be considered as head sails, because their centres of effort are considerably before the centre of gravity of the ship; and notwithstanding this disposition, it generally requires a small action of the rudder to counteract the windward tendency of the lee-bow. This is considered as a good quality when moderate; because it enables the seaman to throw the sails aback, and stop the ship's way in a moment, if she be in danger from any thing ahead; and the ship which does not carry a little of a weather helm, is always a dull sailer.

In order to judge somewhat more accurately of the action of the water and sails, suppose the ship A.B the water (fig. 9.) to have its sails on the mizzenmast D, the mainmast E, and foremast F, braced up or trimmed alike, and that the three lines Di, Ee, Ff, perpendicular to the sails, are in the proportion of the impulses on the sails. The ship is driven ahead and to leeward, and moves in the path a.Cb. This path is so inclined to the line of the keel that the medium direction of the resistance of the water is parallel to the direction of the impulse. A line CI may be drawn parallel to the lines Di, Ee, Ff, and equal to their sum; and it may be drawn from such a point C, that the actions on all the parts of the hull between C and B may balance the momenta of all the actions on the hull between C and A. This point may justly be called the centre of effort, or centre of resistance. We cannot determine this point effort for want of a proper theory of the resistance of fluids. Nay, although experiments like those of the Parisian academy should give us the most perfect knowledge of the intensity of the oblique impulsions on a square foot, we should hardly be benefited by them: for the action of the water on a square foot of the hull at p, for instance, is so modified by the intervention of the stream of water which has struck the hull about B, and glided along the bow Bo p, that the prelure on p is totally different from what it would have been were it a square foot or surface detached from the rest, and presented in the same position to the water moving in the direction bC. For it is found, that the resistances given to planes joined so as to form a wedge, or to curved surfaces, are widely different from the accumulated resistances, calculated for their separate parts, agreeably to the experiments of the academy on single surfaces. We therefore do not attempt to ascertain the point C by theory; but it may be accurately determined by the experiments which we have so strongly recommended; and we offer this as an additional inducement for prosecuting them.

Draw through C a line perpendicular to CI, that is, to be parallel to the sails; and let the lines of impulse of the trimmed three sails cut it in the points i, k, and m. This line by experiments im may be considered as a lever, moveable round C, and acted on at the points i, k, and m, by three forces. The rotary momentum of the sails on the mizzenmast is Di × iC; that of the sails on the mainmast is Ee × kC; and the momentum of the sails on the foremast is Ff × mC. The two first tend to press forward the arm Ci, and then to turn the ship's head towards the wind. The action of the sails on the foremast tends to pull the arm Cm forward, and produce a contrary pre-rotation. If the ship under these three sails keeps steadily in her course, without the aid of the rudder, we of the sails must have Di × iC + Ee × kC = Ff × mC. This is very possible, and is often seen in a ship under her mizen-topmast, main-topmast, and fore-topmast, all parallel to one another, and their surfaces duly proportioned by reefing. If more sails are set, we must always have a similar equilibrium. A certain number of them will have their efforts directed from the larboard arm of the lever im lying to leeward of CI, and a certain number will have their efforts directed from the starboard arm lying to windward of CI. The sum of the products of each of the first set, by their distances from C, must be equal. equal to the sum of the similar products of the other feet. As this equilibrium is all that is necessary for preserving the ship's position, and the cessation of it is immediately followed by a conversion; and as these states of the ship may be had by means of the three square sails only, when their surfaces are properly proportioned—it is plain that every movement may be executed and explained by their means. This will greatly simplify our future discussions. We shall therefore suppose in future that there are only the three topsails set, and that their surfaces are so adjusted by reefing, that their actions exactly balance each other round that point C of the middle line AB, where the actions of the water on the different parts of her bottom in like manner balance each other. This point C may be differently situated in the ship according to the leeway she makes, depending on the trim of the sails; and therefore although a certain proportion of the three surfaces may balance each other in one state of leeway, they may happen not to do so in another state. But the equilibrium is evidently attainable in every case, and we therefore shall always suppose it.

It must now be observed, that when this equilibrium is destroyed, as, for example, by turning the edge of the mizen-topmast to the wind, which the seamen call "diverting the mizen-topmast," and which may be considered as equivalent to the removing the mizen-topmast entirely, it does not follow that the ship will round the point C, this point remaining fixed. The ship must be considered as a free body, still acted on by a number of forces, which no longer balance each other; and she must therefore begin to turn round a spontaneous axis of conversion, which must be determined in the way set forth in the article ROTATION. It is of importance to point out in general where this axis is situated. Therefore let G (fig. 10.) be the centre of gravity of the ship. Draw the line qGv parallel to the yards, cutting Dd in q, Ee in r, CI in t, and Ff in v. While the three sails are set, the line qv may be considered as a lever acted on by four forces, viz. Dd, impelling the lever forward perpendicularly in the point q; Ee, impelling it forward in the point r; Ff, impelling it forward in the point t; and CI, impelling it backward in the point u. These forces balance each other both in respect of progressive motion and of rotatory energy: for CI was taken equal to the sum of Dd, Ee, and Ff; so that no acceleration or retardation of the ship's progress in her course is supposed.

But by taking away the mizen-topmast, both the equilibriums are destroyed. A part Dd of the accelerating force is taken away; and yet the ship, by her inertia or inherent force, tends, for a moment, to proceed in the direction Cp with her former velocity; and by this tendency exerts for a moment the same pressure CI on the water, and sustains the same resistance IC. She must therefore be retarded in her motion by the excess of the resistance IC over the remaining impelling forces Ee and Ff, that is, by a force equal and opposite to Dd. She will therefore be retarded in the same manner as if the mizen-topmast were still set, and a force equal and opposite to its action were applied to G the centre of gravity, and she would soon acquire a smaller velocity, which would again bring all things into equilibrium; and she would stand on in the same course, without changing either her leeway or the position of her head.

But the equilibrium of the lever is also destroyed. It is now acted on by three forces only, viz. Ee and Ff, impelling it forward in the points r and v, and IC impelling it backward in the point t. Make rv : rv = Ee + Ff : Ff, and make op parallel to CI and equal to Ee - Ff. Then we know, from the common principles of mechanics, that the force op acting at o will have the same momentum or energy to turn the lever round any point whatever as the two forces Ee and Ff applied at r and v; and now the lever is acted on by two forces, viz. IC, urging it backwards in the point t, and op urging it forwards in the point o. It must therefore turn round like a floating log, which gets two blows in opposite directions. If we now make IC - op : op = t : x, or IC - op : IC = t : op, and apply to the point x a force equal to IC - op in the direction IC; we know by the common principles of mechanics, that this force IC - op will produce the same rotation round any point as the two forces IC and op applied in their proper directions at t and o. Let us examine the situation of the point x.

The force IC - op is evidently = Dd, and op = Ee + Ff. Therefore ot : tx = Dd : op. But because, when all the sails were filled, there was an equilibrium round C, and therefore round t, and because the force op acting at o is equivalent to Ee and Ff acting at r and v, we must still have the equilibrium; and therefore we have the momentum Dd × qt = op × ot. Therefore ot : tx = Dd : op, and tx = ot. Therefore the point x is the same with the point q.

Therefore, when we shiver the mizen-topmast, the rotation of the ship is the same as if the ship were at rest, and a force equal and opposite to the action of the mizen-topmast were applied at q or at D, or at any point in the line Dq.

This might have been shown in another and shorter way. Suppose all sails filled, the ship is in equilibrium. This will be disturbed by applying to D a force opposite to Dd; and if the force be also equal to Dd, it is evident that these two forces destroy each other, and that this application of the force dD is equivalent to the taking away of the mizen-topmast. But we choose to give the whole mechanical investigation; because it gave us an opportunity of pointing out to the reader, in a case of very easy comprehension, the precise manner in which the ship is acted on by the different sails and by the water, and what share each of them has in the motion ultimately produced. We shall not repeat this manner of procedure in other cases, because a little reflection on the part of the reader will now enable him to trace the modus operandi through all its steps.

We now see that, in respect both of progressive motion and of conversion, the ship is affected by shivering the sail D, in the same manner as if a force equal and opposite to Dd were applied at D, or at any point in the line Dd. We must now have recourse to the principles established under the article ROTATION.

Let p represent a particle of matter, r its radius vector, or its distance pG from an axis passing through the centre of gravity G, and let M represent the whole quantity of matter of the ship. Then its momentum of inertia is \( \int p \cdot r \) (see ROTATION, No. 18.) The ship, impelled in the point D by a force in the direction Dd, will begin to turn round a spontaneous vertical axis, passing through a point S of the line qG, which which is drawn through the centre of gravity G, perpendicular to the direction dD of the external force, and the distance GS of this axis from the centre of gravity is

\[ \frac{p}{M \cdot G} q^2 \] (see Rotation, No 96.), and it is taken on the opposite side of G from q, that is, S and q are on opposite sides of G.

Let us express the external force by the symbol F. It is equivalent to a certain number of pounds, being the pressure of the wind moving with the velocity V and inclination a on the surface of the sail; and may therefore be computed either by the theoretical or experimental law of oblique impulses. Having obtained this, we can ascertain the angular velocity of the rotation and the absolute velocity of any given point of the ship by means of the theorems established in the article Rotation.

But before we proceed to this investigation, we shall consider the action of the rudder, which operates precisely in the same manner. Let the ship AB (fig. 11.) have her rudder in the position AD, the helm being hard a-starboard, while the ship falling on the starboard tack, and making leeway, keeps on the course ab. The lee surface of the rudder meets the water obliquely. The very foot of the rudder meets it in the direction DE parallel to ab. The parts farther up meet it with various obliquities, and with various velocities, as it glides round the bottom of the ship and falls into the wake. It is absolutely impossible to calculate the accumulated impulse. We shall not be far mistaken in the deflection of each contiguous filament, as it quits the bottom and glides along the rudder; but we neither know the velocity of these filaments, nor the deflection and velocity of the filaments gliding without them. We therefore imagine that all computations on this subject are in vain. But it is enough for our purpose that we know the direction of the absolute pressure which they exert on its surface. It is in the direction Dd, perpendicular to that surface. We also may be confident that this pressure is very considerable, in proportion to the action of the water on the ship's bows, or of the wind on the sails; and we may suppose it to be nearly in the proportion of the square of the velocity of the ship in its course; but we cannot affirm it to be accurately in that proportion, for reasons that will readily occur to one who considers the way in which the water falls in behind the ship.

It is observed, however, that a fine faller always flows well, and that all movements by means of the rudder are performed with great rapidity when the velocity of the ship is great. We shall see by and by, that the speed with which the ship performs the angular movements is in the proportion of her progressive velocity: For we shall see that the squares of the times of performing the evolution are as the impulses inversely, which are as the squares of the velocities. There is perhaps no force which acts on a ship that can be more accurately determined by experiment than this. Let the ship ride in a stream or tide-way whose velocity is accurately measured; and let her ride from two moorings, so that her bow may be a fixed point. Let a small tow-line be laid out from her stern or quarter at right angles to the keel, and connected with some apparatus fitted up on shore or on board another ship, by which the strain on it may be accurately measured; a person conversant with mechanics will see many ways in which this can be done. Perhaps the following may be as good as any: Let the end of the tow-line be fixed to some point as high out of the water as the point of the ship from which it is given out, and let this be very high. Let a block with a hook be on the rope, and a considerable weight hung on this hook. Things being thus prepared, put down the helm to a certain angle, so as to cause the ship to sheer off from the point to which the far end of the tow-line is attached. This will stretch the rope, and raise the weight out of the water. Now heave upon the rope, to bring the ship back again to her former position, with her keel in the direction of the stream. When this position is attained, note carefully the form of the rope, that is, the angle which its two parts make with the horizon. Call this angle a. Every person acquainted with these subjects knows that the horizontal strain is equal to half the weight multiplied by the cotangent of a, or that 2 is to the cotangent of a as the weight to the horizontal strain. Now it is this strain which balances and therefore measures the action of the rudder, or De in fig. 11. Therefore, to have the absolute impulse Dd, we must increase De in the proportion of radius to the secant of the angle b which the rudder makes with the keel. In a great ship falling six miles in an hour, the impulse on the rudder inclined 35° to the keel is not less than 3000 pounds. The surface of the rudder of such a ship contains near 80 square feet. It is not, however, very necessary to know this absolute impulse Dd, because it is its part De alone which measures the energy of the rudder in producing a conversion. Such experiments, made with various positions of the rudder, will give its energies corresponding to these positions, and will settle that long disputed point, which is the best position for turning a ship. On the hypothesis that the impulsions of fluids are in the duplicate ratio of the fines of incidence, there can be no doubt that it should make an angle of 54° 44' with the keel. But the form of a large ship will not admit of this, because a tiller of a length sufficient for managing the rudder in falling with great velocity has not room to deviate above 35° from the direction of the keel; and in this position of the rudder the mean obliquity of the filaments of water to its surface cannot exceed 40° or 45°. A greater angle would not be of much service, for it is never for want of a proper obliquity that the rudder fails of producing a conversion.

A ship will stay in rough weather for want of a sufficient progressive velocity, and because her bows are miffes stays, beat off by the waves: and there is seldom any difficulty in wearing the ship, if she has any progressive motion. It is, however, always desirable to give the rudder as much influence as possible. Its surface should be enlarged (especially below) as much as can be done confidently with its strength and with the power of the steermen to manage it; and it should be put in the most favourable situation for the water to get at it with great velocity; and it should be placed as far from the axis of the ship's motion as possible. These points are obtained by making the stern-post very upright, as has always been done in the French dockyards. The British ships have a much greater rake; but our builders are gradually adopting the French forms, experience having

The action of the rudder similar to that of the sails, and very great.

Thus the action and effects of the sails and of the rudder are perfectly similar, and are to be considered in the same manner. We see that the action of the rudder, though of a small surface in comparison of the sails, must be very great: For the impulse of water is many hundred times greater than that of the wind; and the arm \( qG \) of the lever, by which it acts, is incomparably greater than that by which any of the impulsions on the sails produces its effect; accordingly the ship yields much more rapidly to its action than she does to the lateral impulse of a sail.

Observe here, that if \( G \) were a fixed or supported axis, it would be the same thing whether the absolute force \( Dd \) of the rudder acts in the direction \( De \), or its transverse part \( De \) acts in the direction \( De \), both would produce the same rotation; but it is not so in a free body. The force \( Dd \) both tends to retard the ship's motion and to produce a rotation: It retards it as much as if the same force \( Dd \) had been immediately applied to the centre. And thus the real motion of the ship is compounded of a motion of the centre in a direction parallel to \( Dd \), and of a motion round the centre. These two constitute the motion round \( S \).

As the effects of the action of the rudder are both more remarkable and somewhat more simple than those of the sails, we shall employ them as an example of the mechanism of the motions of conversion in general; and as we must content ourselves in a work like this with what is very general, we shall simplify the investigation by attending only to the motion of conversion. We can get an accurate notion of the whole motion, if wanted for any purpose, by combining the progressive or retrograde motion parallel to \( Dd \) with the motion of rotation which we are about to determine.

In this case, then, we observe, in the first place, that the angular velocity (see Rotation, No. 22.) is

\[ \frac{Dh \cdot qG}{\int \rho r^2} \]

and, as was shown in that article, this velocity of rotation increases in the proportion of the time of the forces uniform action, and the rotation would be uniformly accelerated if the forces did really act uniformly. This, however, cannot be the case, because, by the ship's change of position and change of progressive velocity, the direction and intensity of the impelling force is continually changing. But if two ships are performing similar evolutions, it is obvious that the changes of force are similar in similar parts of the evolution. Therefore the consideration of the momentary evolution is sufficient for enabling us to compare the motions of ships actuated by similar forces, which is all we have in view at present.

The velocity \( v \), generated in any time \( t \) by the continuance of an invariable momentary acceleration (which is all that we mean by saying that it is produced by the action of a constant accelerating force), is as the acceleration and the time jointly. Now what we call the angular velocity is nothing but this momentary acceleration. Therefore the velocity \( v \) generated in the time \( t \) is

\[ v = \frac{F \cdot qG}{\int \rho r^2} \cdot t \]

The expression of the angular velocity is also the expression of the velocity \( v \) of a point situated at the distance \( r \) from the axis \( G \).

Let \( x \) be the space or arch of revolution described in the time \( t \) by this point, whose distance from \( G \) is \( r \). Then \( x = v \cdot t = \frac{F \cdot qG}{\int \rho r^2} \cdot t^2 \), and taking the fluent \( x = \frac{F \cdot qG}{\int \rho r^2} \cdot t^3 \). This arch measures the whole angle of rotation accomplished in the time \( t \). These are therefore as the squares of the times from the beginning of the rotation.

Those evolutions are equal which are measured by equal arches. Thus two motions of 45 degrees each are equal. Therefore because \( x \) is the same in both, the quantity \( \frac{F \cdot qG}{\int \rho r^2} \cdot t^2 \) is a constant quantity, and \( t^2 \) is reciprocally proportional to \( \frac{F \cdot qG}{\int \rho r^2} \), or is proportional to \( \frac{\sqrt{\int \rho r^2}}{\sqrt{F \cdot qG}} \). That is to say, the times of the similar evolutions of two ships are as the square root of the momentum of inertia directly, and as the square root of the momentum of the rudder or sail inversely. This will enable us to make the comparison easily. Let us suppose the ships perfectly similar in form and rigging, and to differ only in length \( L \) and \( l \); \( P \cdot R^2 \) is to \( \int \rho r^2 \) as \( L^3 \) to \( l^3 \).

For the similar particles \( P \) and \( p \) contain quantities of matter which are as the cubes of their linear dimensions, that is, as \( L^3 \) to \( l^3 \). And because the particles are similarly situated, \( R^2 \) is to \( r^2 \) as \( L^2 \) to \( l^2 \). Therefore \( P \cdot R^2 : p \cdot r^2 = L^5 : l^5 \). Now \( F \) is to \( f \) as \( L^2 \) to \( l^2 \).

For the surfaces of the similar rudders or sails are as the squares of their linear dimensions, that is, as \( L^2 \) to \( l^2 \). And, lastly, \( G \cdot q \) is to \( g \cdot q \) as \( L \) to \( l \), and therefore \( F \cdot G \cdot q : f \cdot g \cdot q = L^3 : l^3 \). Therefore we have \( T^2 : \)

\[ T^2 = \frac{\int P \cdot R^2}{F \cdot G \cdot q} : \frac{\int p \cdot r^2}{f \cdot g \cdot q} = \frac{L^5}{l^5} : \frac{L^3}{l^3} = L^2 : l^2, \quad \text{and } T : t = L : l. \]

Therefore the times of performing similar evolutions with similar ships are proportional to the lengths of the similar ships when both are falling equally fast; and since the similar evolutions are similar, and the forces vary similarly in their their different parts, what is here demonstrated of the smallest incipient evolutions is true of the whole. They therefore not only describe equal angles of revolution, but also similar curves.

A small ship, therefore, works in less time and in less room than a great ship, and this in the proportion of its length. This is a great advantage in all cases, particularly in wearing, in order to fall on the other tack close-hauled. In this case she will always be to windward and ahead of the large ship, when both are got on the other tack. It would appear at first sight that the large ship will have the advantage in tacking. Indeed the large ship is farther to windward when again trimmed on the other tack than the small ship when she is just trimmed on the other tack. But this happened before the large ship had completed her evolution, and the small ship, in the mean time, has been going forward on the other tack, and going to windward. She will therefore be before the large ship's beam, and perhaps as far to windward.

We have seen that the velocity of rotation is proportional, ceteris paribus, to $F \times G q$. F means the absolute impulse on the rudder or sail, and is always perpendicular to its surface. This absolute impulse on a sail depends on the obliquity of the wind to its surface. The usual theory says, that it is as the square of the sine of incidence; but we find this not true. We must content ourselves with expressing it by some yet unknown function $\phi$ of the angle of incidence $\alpha$, and call it $\phi \alpha$; and if $S$ be the surface of the sail, and $V$ the velocity of the wind, the absolute impulse is $n V^2 S \times \phi \alpha$. This acts (in the case of the mizen-topmast, fig. 10.) by the lever $q G$, which is equal to $DG \times \text{cof.} DG q$, and $DG q$ is equal to the angle of the yard and keel; which angle we formerly called $b$. Therefore its energy in producing a rotation is $n V^2 S \times \phi \alpha \times DG \times \text{cof.} b$. Leaving out the constant quantities $n$, $V^2$, $S$, and $DG$, its energy is proportional to $\phi \alpha \times \text{cof.} b$. In order, therefore, that any sail may have the greatest power to produce a rotation round $G$, it must be so trimmed that $\phi \alpha \times \text{cof.} b$ may be a maximum. Thus, if we would trim the sails on the foremast, so as to pay the ship off from the wind right ahead with the greatest effect, and if we take the experiments of the French academicians as proper measures of the oblique impulses of the wind on the sail, we will brace up the yard to an angle of 48 degrees with the keel. The impulse corresponding to 48° is 615, and the cosine of 48° is 669. These give a product of 411435. If we brace the sail to 54.44°, the angle assigned by the theory, the effective impulse is 403274. If we make the angle 45°, the impulse is 408774. It appears then that 48° is preferable to either of the others. But the difference is inconsiderable, as in all cases of maximum a small deviation from the best position is not very detrimental. But the difference between the theory and this experimental measure will be very great when the impulses of the wind are of necessity very oblique. Thus, in tacking ship, as soon as the headails are taken aback, they serve to aid the evolution, as is evident: But if we were now to adopt the maxims inculcated by the theory, we should immediately round in the weather-braces, so as to increase the impulse on the sail, because it is then very small; and although we by this means make yard more square, and therefore diminish the rotatory momentum of this impulse, yet the impulse is more increased (by the theory) than its vertical lever is diminished.

Let us examine this a little more particularly, because it is reckoned one of the nicest points of seamanship to aid the ship's coming round by means of the headails; and experienced seamen differ in their practice in this manoeuvre. Suppose the yard braced up to 48°, which is as much as can be usually done, and that the sail flutters (the bowlines are usually let go when the helm is put down), the sail immediately takes aback, and in a moment we may suppose an incidence of 6 degrees. The impulse corresponding to this is 400, (by experiment), and the cosine of 48° is 766. This gives 306400 for the effective impulse. To proceed according to the theory, we should brace the yard to 70°, which would give the wind (now 34° on the weather-bow) an incidence of nearly 36°; and the sail an inclination of 26° to the intended motion, which is perpendicular to the keel. For the tangent of 26° is about $\frac{1}{4}$ of the tangent of 36°. Let us now see what effective impulse the experimental law of oblique impulsions will give for this adjustment of the sails. The experimental impulse for 36° is 480; the cosine of 70° is 342; the product is 164160, not much exceeding the half of the former. Nay, the impulse for 36°, calculated by the theory, would have been only 346, and the effective impulse only 118332. And it must be further observed, that this theoretical adjustment would tend greatly to check the evolution, and in most cases would entirely mar it, by checking the ship's motion ahead, and consequently the action of the rudder, which is the most powerful agent in the evolution; for here would be a great impulse directed almost aft.

We were justified, therefore, in saying, in the beginning of this article, that a seaman would frequently find himself baffled if he were to work a ship according to the rules deduced from M. Bouguer's work; and we see by this instance of what importance it is to have the oblique impulsions of fluids ascertained experimentally. The practice of the most experienced seaman is directly the opposite to this theoretical maxim, and its success greatly confirms the usefulness of these experiments of the academicians so often praised by us.

We return again to the general consideration of the rotatory motion. We found the velocity $v = \frac{F \cdot q \cdot G}{\int p \cdot r^2}$. It is therefore proportional, ceteris paribus, to $q \cdot G$. We have seen in what manner $q \cdot G$ depends on the position and situation of the sail or rudder when the point $G$ is fixed. But it also depends on the position of $G$. With respect to the action of the rudder, it is evident that it is so much the more powerful as it is more remote from $G$. The distance from $G$ may be increased either by moving the rudder farther aft or farther forward. And as it is of the utmost importance that a ship answer her helm with the greatest promptitude, those circumstances have been attended to which distinguished fine steering ships from such as had not this quality; and it is in a great measure to be ascribed to this, that, in the gradual improvement of naval architecture, the centre of gravity has been placed far forward. Perhaps the notion of a centre of gravity did not come into the thoughts of the rude builders in early times; but they observed that those boats and ships steered best which had... had their extreme breadth before the middle point, and consequently the bows not so acute as the stern. This is so contrary to what one would expect, that it attracted attention more forcibly; and, being somewhat mysterious, it might prompt to attempts of improvement, by exceeding in this singular maxim. We believe that it has been carried as far as is compatible with other essential requisites in a ship.

We believe that this is the chief circumstance in what is called the trim of a ship; and it were greatly to be wished that the best place for the centre of gravity could be accurately ascertained. A practice prevails, which is the opposite of what we are now advancing. It is usual to load a ship so that her keel is not horizontal, but lower abaft. This is found to improve her steerage. The reason of this is obvious. It increases the acting surface of the rudder, and allows the water to come at it with much greater freedom and regularity; and it generally diminishes the gripping of the ship forward, by removing a part of the bows out of the water. It has not always this effect; for the form of the harping aloft is frequently such, that the tendency to grip is diminished by immersing more of the bow in the water.

But waving these circumstances, and attending only to the rotatory energy of the rudder, we see that it is of advantage to carry the centre of gravity forward. The same advantage is gained to the action of the after sails. But, on the other hand, the action of the head sails is diminished by it; and we may call every sail a headail whose centre of gravity is before the centre of gravity of the ship; that is, all the sails hoisted on the bowsprit and foremast, and the staysails hoisted on the mainmast; for the centre of gravity is seldom far before the mainmast.

Suppose that when the rudder is put into the position AD (fig. 11.), the centre of gravity could be shifted to g, so as to increase qG, and that this is done without increasing the sum of the products pr². It is obvious that the velocity of conversion will be increased in the proportion of qG to gg. This is very possible, by bringing to that side of the ship parts of her loading which were situated at a distance from G on the other side. Nay, we can make this change in such a manner that ∫pr² shall even be less than it was before, by taking care that everything which we shift shall be nearer to g than it was formerly to G. Suppose it all placed in one spot m, and that m is the quantity of matter to shifted, while M is the quantity of matter in the whole ship. It is only necessary that m · G² shall be less than the sum of the products pr² corresponding to the matter which has been shifted. Now, although the matter which is easily moveable is generally very small in comparison to the whole matter of the ship, and therefore can make but a small change in the place of the centre of gravity, it may frequently be brought from places so remote that it may occasion a very sensible diminution of the quantity ∫pr², which expresses the whole momentum of inertia.

This explains a practice of the seamen in small wherries or skiffs, who in putting about are accustomed to place themselves to leeward of the mast. They even find that they can aid the quick motions of these light boats by the way in which they rest on their two feet, sometimes leaning all on one foot, and sometimes on the other. And we have often seen this evolution very sensibly accelerated in a ship of war, by the crew running suddenly, as the helm is put down, to the lee-bow. And we have heard it asserted by very expert seamen, that after all attempts to wear ship (after lying-to in a storm) have failed, they have succeeded by the crew collecting themselves near the weather fore-throats the moment the helm was put down. It must be agreeable to the reflecting seaman to see this practice supported by undoubted mechanical principles.

It will appear paradoxical to say that the evolution may be accelerated even by an addition of matter to the ship; and though it is only a piece of curiosity, our readers may wish to be made sensible of it. Let m be the additional matter, placed in some point m lying beyond G from g. Let S be the spontaneous centre of conversion before the addition. Let v be the velocity of rotation round g, that is, the velocity of a point whose distance from g is r, and let ρ be the radius vector, or distance of a particle from g. We have (Rotation, No. 22.) \( v = \frac{F \cdot qg}{\int p \cdot r^2 + m \cdot mg} \). But we know (Rotation, No. 23.) that \( \int p \cdot r^2 = \int p \cdot r^2 + M \cdot G \cdot s^2 \). Therefore \( v = \frac{F \cdot qg}{\int p \cdot r^2 + M \cdot G \cdot s^2 + m \cdot mg} \). Let us determine G, g and mg and qg.

Let mG be called z. Then, by the nature of the centre of gravity, \( M + m : M = G : g \), \( m = \frac{M}{M + m} \), and \( g = \frac{m}{M + m} \). In like manner, \( M \cdot G \cdot s^2 = \frac{M \cdot m}{M + m} \cdot s^2 \). Now \( M \cdot m + M \cdot m = \frac{M \cdot m}{M + m} \cdot s^2 \). Therefore \( M \cdot G \cdot s^2 + m \cdot g \cdot s^2 = \frac{M \cdot m}{M + m} \cdot s^2 \). Let n be \( \frac{m}{M + m} \), then \( M \cdot G \cdot s^2 + m \cdot g \cdot s^2 = M \cdot n \cdot s^2 \). Also \( G \cdot g = n \cdot s^2 \), being \( \frac{m}{M + m} \cdot s^2 \). Let qG be called c: then \( q \cdot g = c + n \cdot s^2 \). Also let Sg be called e.

We have now for the expression of the velocity \( v = \frac{F \cdot (c + n \cdot s^2)}{\int p \cdot r^2 + M \cdot n \cdot s^2} \), or \( v = \frac{F}{M} \times \frac{c + n \cdot s^2}{\int p \cdot r^2 + M \cdot n \cdot s^2} \). But \( \frac{\int p \cdot r^2}{M} = c \cdot e \). Therefore, finally, \( v = \frac{F}{M} \times \frac{c + n \cdot s^2}{c \cdot e + n \cdot s^2} \). Had there been no addition of matter made, we should have had \( v = \frac{F}{M} \times \frac{c}{c \cdot e} \). It remains to show, that \( s^2 \) may be so taken that \( \frac{c}{c \cdot e} \) may be less than \( \frac{c + n \cdot s^2}{c \cdot e + n \cdot s^2} \). New, if c be to s as c to s², that is, if \( s^2 \) be... be taken equal to \( e \), the two fractions will be equal. But if \( x \) be less than \( e \), that is, if the additional matter is placed anywhere between \( S \) and \( G \), the complex fraction will be greater than the fraction \( \frac{c}{e} \), and the velocity of rotation will be increased. There is a particular distance which will make it the greatest possible, namely, when \( x \) is made \( = \frac{1}{n} (\sqrt{c^2 + nc} - c) \), as will easily be found by treating the fraction \( \frac{c + n}{c + n x} \), with \( x \), considered as the variable quantity, for a maximum.

In what we have been saying on this subject, we have considered the rotation only in as much as it is performed round the centre of gravity, although in every moment it is really performed round a spontaneous axis lying beyond that centre. This was done because it afforded an easy investigation, and any angular motion round the centre of gravity is equal to the angular motion round any other point. Therefore the extent and the time of the evolution are accurately defined.

From observing that the energy of the force \( F \) is proportional to \( qG \), an inattentive reader will be apt to conceive the centre of gravity as the centre of motion, and the rotation as taking place, because the momenta of the sails and rudder, on the opposite sides of the centre of gravity, do not balance each other. But we must always keep in mind that this is not the cause of the rotation. The cause is the want of equilibrium round the point \( C \) (fig. 10.), where the actions of the water balance each other. During the evolution, which consists of a rotation combined with a progressive motion, this point \( C \) is continually shifting, and the unbalanced momenta which continue the rotation always respect the momentary situation of the point \( C \). It is nevertheless always true that the energy of a force \( F \) is proportional (ceteris paribus) to \( qG \), and the rotation is always made in the same direction as if the point \( G \) were really the centre of conversion. Therefore the mainail acts always (when oblique) by pushing the stern away from the wind, although it should sometimes act on a point of the vertical lever through \( C \), which is ahead of \( C \).

These observations on the effects of the sails and rudder in producing a conversion, are sufficient for enabling us to explain any case of their action which may occur. We have not considered the effects which they tend to produce by inclining the ship round a horizontal axis, viz., the motions of rolling and pitching. See Rolling and Pitching. To treat this subject properly would lead us into the whole doctrine of the equilibrium of floating bodies, and it would rather lead to maxims of construction than to maxims of manoeuvre.

M. Bouguer's Traité du Navire and Euler's Scientia Nautica are excellent performances on this subject, and we are not here obliged to have recourse to any erroneous theory.

It is easy to see that the lateral pressure both of the wind on the sails and of the water on the rudder tends to incline the ship to one side. The sails also tend to press the ship's bows into the water, and, if they were kept from advancing, would press them down considerably. But by the ship's motion, and the prominent form of her bows, the resistance of the water to the fore part of the ship produces a force which is directed upwards. The sails also have a small tendency to raise the ship, for they constitute a surface which in general separates from the plumb-line below. This is remarkably the case in the staysails, particularly the jib and fore-topmast staysail. And this helps greatly to soften the plunges of the ship's bows into the head seas. The upward pressure also of the water on her bows, which we just now mentioned, has a great effect in opposing the immersion of the bows which the sails produce by acting on the long levers furnished by the masts.

M. Bouguer gives the name of point velique to the point \( V \) (fig. 12.) of the mast, where it is cut by the line \( CV \), which marks the mean place and direction of the whole impulse of the water on the bows. And he observes, that if the mean direction of all the actions of the wind on the sails be made to pass also through this point, there will be a perfect equilibrium, and the ship will have no tendency to plunge into the water or to rise out of it; for the whole action of the water on the bows, in the direction \( CV \), is equivalent to, and may be resolved into the action \( CE \), by which the progressive motion is resisted, and the vertical action \( CD \), by which the ship is raised above the water. The force \( CE \) must be opposed by an equal force \( VD \), exerted by the wind on the sails, and the force \( CD \) is opposed by the weight of the ship. If the mean effort of the sails presses above the point \( V \), the ship's bow will be pressed into the water; and if it passes below \( V \), her stern will be pressed down. But, by the union of these forces, the will rise and fall with the sea, keeping always in a parallel position. We apprehend that it is of very little moment to attend to the situation of this point. Except when the ship is right abreast the wind, it is a thousand chances to one that the line \( CV \) of mean resistance does not pass through any mast; and the fact is, that the ship cannot be in a state of uniform motion on any other condition but the perfect union of the line of mean action of the sails, and the line of mean action of the resistance. But its place shifts by every change of leeway or trim; and it is impossible to keep these lines in one constant point of intersection for a moment, on account of the incessant changes of the surface of the water on which the floats. M. Bouguer's observations on this point are, however, very ingenious and original.

We conclude this dissertation, by describing some of Chief evolutions described hitherto is intended for the instruction of the artificer, by making him sensible of the mechanical procedure. The description is rather meant for the amusement of the landsman, enabling him to understand operations that are familiar to the seaman. The latter will perhaps smile at the awkward account given of his business by one who cannot hand, reef, or steer.

To tack Ship.

The ship must first be kept full, that is, with a very sensible angle of incidence on the sails, and by no means hugging the wind. For as this evolution is chiefly performed by the rudder, it is necessary to give the ship a good velocity. When the ship is observed to luff up of herself, that moment is to be watched for beginning the evolution, because she will by her inherent force continue this motion. The helm is then put down. When the officer calls out Helm's alee, the fore-sheet, fore-top bowline, jib, and flag sail sheets forward. ward are let go. The jib is frequently hauled down. Thus the obstacles to the ship's head coming up to the wind by the action of the rudder are removed. If the mainsail is set, it is not unusual to clue up the weather side, which may be considered as a headail, because it is before the centre of gravity. The mizen must be hauled out, and even the sail braced to windward. Its power in paying off the stern from the wind confines with the action of the rudder. It is really an aerial rudder. The sails are immediately taken aback. In this state the effect of the mizen-topail would be to obstruct the movement, by prelling the stern contrary way to what it did before. It is therefore either immediately braced about sharp on the other tack, or lowered. Bracing it about evidently tends to pay round the stern from the wind, and thus afflit in bringing the head up to the wind. But in this position it checks the progressive motion of the ship, on which the evolution chiefly depends. For a rapid evolution, therefore, it is as well to lower the mizen-topail. Meantime, the headails are all aback, and the action of the wind on them tends greatly to pay the ship round. To increase this effect, it is not unusual to haul the fore-top bowline again. The sails on the mainmast are now almost becalmed; and therefore when the wind is right ahead, or a little before, the mainail is hauled round and braced up sharp on the other tack with all expedition. The stayail sheets are now shifted over to their places for the other tack. The ship is now entirely under the power of the headails and of the rudder, and their actions confine to promote the conversion. The ship has acquired an angular motion, and will preserve it, so that now the evolution is secured, and the sails off space from the wind on the other tack. The farther action of the rudder is therefore unnecessary, and would even be prejudicial, by causing the ship to fall off too much from the wind before the sails can be shifted and trimmed for falling on the other tack. It is therefore proper to right the helm when the wind is right ahead, that is, to bring the rudder into the direction of the keel. The ship continues her conversion by her inherent force and the action of the headails.

When the ship has fallen off about four points from the wind, the headails are hauled round, and trimmed sharp on the other tack with all expedition; and although this operation was begun with the wind four points on the bow, it will be fix before the sails are braced up, and therefore the headails will immediately fill. The after-sails have filled already, while the headails were inactive, and therefore immediately check the farther falling off from the wind. All sails now draw, for the stayail sheets have been shifted over while they were becalmed or shaking in the wind. The ship now gathers way, and will obey the smallest motion of the helm to bring her close to the wind.

We have here supposed, that during all this operation the ship preserves her progressive motion. She must therefore have described a curve line, advancing all the while to windward. Fig. 13. is a representation of this evolution when it is performed in the complete manner. The ship standing on the course E, with the wind blowing in the direction WF, has her helm put hard alee when she is in the position A. She immediately deviates from her course, and describing a curve, comes to the position B, with the wind blowing in the direction WF of the yards, and the square-sails now shiver. The mizen topsail is here represented braced sharp on the other tack, by which its tendency to aid the angular motion (while it checks the progressive motion) is distinctly seen. The main and foresails are now shivering, and immediately after are taken aback. The effect of this on the headails is distinctly seen to be favourable to the conversion, by pushing the point F in the direction Fi; but for the same reason it continues to retard the progressive motion. When the ship has attained to the position C, the mainail is hauled round and trimmed for the other tack. The impulse in the direction Fi still aids the conversion and retards the progressive motion. When the ship has attained a position between C and D, such that the main and mizen topsail yards are in the direction of the wind, there is nothing to counteract the force of the headails to pay the ship's head off from the wind. Nay, during the progress of the ship to this intermediate position, if any wind gets at the main or mizen topsails, it acts on their anterior surfaces, and impels the after parts of the ship away from the curve a b c d, and thus aids the revolution. We have therefore said, that when once the sails are taken fully aback, and particularly when the wind is brought right ahead, it is scarce possible for the evolution to fail; as soon therefore as the main topsail (trimmed for the other tack) shivers, we are certain that the headails will be filled by the time they are hauled round and trimmed. The stayails are filled before this, because their sheets have been shifted, and they stand much sharper than the square-sails; and thus everything tends to check the falling off from the wind on the other tack, and this no sooner than it should be done. The ship immediately gathers way, and holds on in her new course d G.

But it frequently happens, that in this conversion the ship loses her whole progressive motion. This sometimes happens while the sails are shivering before they are taken fully aback. It is evident, that in this case there is little hopes of success, for the ship now lies like a log, and neither sails nor rudder have any action. The ship drives to leeward like a log, and the water acting on the lee side of the rudder checks a little the driving of the stern. The head therefore falls off again, and by and by the sails fill, and the ship continues on her former tack. This is called missing stays, and it is generally owing to the ship's having too little velocity at the beginning of the evolution. Hence the propriety of keeping the sails well filled for some little time before. Rough weather, too, by raising a wave which beats violently on the weather-bow, frequently checks the first luffing of the ship, and beats her off again.

If the ship lose all her motion after the headails have been fully taken aback, and before we have brought the wind right ahead, the evolution becomes uncertain, but by no means desperate; for the action of the wind on the headails will presently give her sternway. Suppose this to happen when the ship is in the position C. Bring the helm over hard to windward, so that the rudder shall have the position represented by the small dotted line ef. It is evident, that the resistance of the water to the stern-way of the rudder acts in a favourable direction, pushing the stern outwards. In the mean time, the action of the wind on the headails pushes the head in the opposite direction. These actions To wear Ship.

When the seaman feels that his ship will not go about head to wind, but will miss stays, he must change his tack the other way; that is, by turning her head away from the wind, going a little way before the wind, and then hauling the wind on the other tack. This is called wearing or veering ship. It is most necessary in stormy weather with little sail, or in very faint breezes, or in a disabled ship.

The process is exceedingly simple; and the mere narration of the procedure is sufficient for showing the propriety of every part of it.

Watch for the moment of the ship's falling off, and then haul up the mainail and mizen, and shiver the mizen topail, and put the helm a-weather. When the ship falls off sensibly (and not before), let go the bowlines. Ease away the fore-sheet, raise the fore-tack, and gather aft the weather fore-sheet, as the lee-sheet is eased away. Round in the weather-braces of the fore and main-masts, and keep the yards nearly bisecting the angle of the wind and keel, so that when the ship is before the wind the yards may be square. It may even be of advantage to round in the weather-braces of the main-topail more than those of the head-falls; for the mainail is abaft the centre of gravity. All this while the mizen-topail must be kept shivering, by rounding in the weather-braces as the ship pays off from the wind. Then the main topail will be braced up for the other tack by the time that we have brought the wind on the weather-quarter. After this it will be full, and will aid the evolution. When the wind is right aft, shift the jib and stay-fall sheets. The evolution now goes on with great rapidity; therefore briskly haul on board the fore and main tacks, and haul out the mizen, and let the mizen-stayfall as soon as they will take the wind the right way. We must now check the great rapidity with which the ship comes to the wind on the other tack, by righting the helm before we bring the wind on the beam; and all must be trimmed sharp fore and aft by this time, that the headfalls may take and check the coming-to. All being trimmed, stand on close by the wind.

We cannot help losing much ground in this movement. Therefore, though it be very simple, it requires much attention and rapid execution to do it with as little loss of ground as possible. One is apt to imagine at first that it would be better to keep the headfalls braced up on the former tack, or at least not to round in the weather-braces so much as is here directed. When the ship is right afore the wind, we should expect assistance from the obliquity of the head-falls; but the rudder being the principal agent in the evolution, it is found that more is gained by increasing the ship's velocity, than by a smaller impulse in the headfalls more favourably directed. Experienced seamen differ, however, in their practice in respect of this particular.

To box-haul a Ship.

This is a process performed only in critical situations, as when a rock, a cliff, or some danger, is suddenly seen right ahead, or when a ship misses stays. It requires the most rapid execution.

The ship being close-hauled on a wind, haul up the mainail mainail and mizen, and shiver the top-fails, and put the helm hard a-lee altogether. Raise the fore-tack, let go the head bowlines, and brace about the headails sharp on the other tack. The ship will quickly lose her way, get stern-way, and then fall off, by the joint action of the headails and of the inverted rudder. When she has fallen off eight points, brace the after-fails square, which have hitherto been kept shivering. This will at first increase the power of the rudder, by increasing the stern-way, and at the same time it makes no opposition to the conversion which is going on. The continuation of her circular motion will presently cause them to take the wind on their after surfaces. This will check the stern-way, stop it, and give the ship a little head-way. Now shift the helm, so that the rudder may again act in conjunction with the headails in paying her off from the wind. This is the critical part of the evolution, because the ship has little or no way through the water, and will frequently remain long in this position. But as there are no counteracting forces, the ship continues to fall off. Then the weather-braces of the after-fails may be gently rounded in, so that the wind acting on their hinder surfaces may both push the ship a little ahead and her stern laterally in conjunction with the rudder. Thus the wind is brought upon the quarter, and the headails shiver. By this time the ship has acquired some headway. A continuation of the rotation would now fill the headails, and their action would be contrary to the intended evolution. They are therefore immediately braced the other way, nearly square, and the evolution is now completed in the same manner with wearing ship.

Some seamen brace all the sails aback the moment that the helm is put hard a-lee, but the after-fails no more aback than just to square the yards. This quickly gives the ship stern-way, and brings the rudder into action in its inverted direction; and they think that the evolution is accelerated by this method.

There is another problem of seamanship deserving of our attention, which cannot properly be called an evolution. This is lying-to. This is done in general by laying some sails aback, so as to stop the head-way produced by others. But there is a considerable address necessary for doing this in such a way that the ship shall lie easily, and under command, ready to proceed in her course, and easily brought under weigh.

To bring-to with the fore or main topsail to the mast, brace that fail sharp aback, haul out the mizen, and clap the helm hard a-lee.

Suppose the fore topsail to be aback; the other sails shoot the ship ahead, and the lee-helm makes the ship come up to the wind, which makes it come more perpendicularly on the sail which is aback. Then its impulse soon exceeds those on the other sails, which are now shivering, or almost shivering. The ship stands still awhile, and then falls off, so as to fill the after-fails, which again shoot her ahead, and the process is thus repeated. A ship lying-to in this way goes a good deal ahead and also to leeward. If the main topsail be aback, the ship shoots ahead, and comes up till the diminished impulse of the drawing sails in the direction of the keel is balanced by the increased impulse on the main-topsail. She lies a long while in this position, driving slowly to leeward; and she at last falls off by the beating of the water on her weather-bow. She falls off but little, and soon comes up again.

Thus a ship lying-to is not like a mere log, but has a certain motion which keeps her under command. To get under weigh again, we must watch the time of falling off; and when this is just about to finish, brace about briskly, and fill the sail which was aback. To aid this operation, the jib and fore-topmast stay-sail may be hoisted, and the mizen brailed up: or, when the intended course is before the wind or large, back the foretopsail sharp, shiver the main and mizen topsail, brail up the mizen, and hoist the jib and fore-topmast stay-sails altogether.

In a storm with a contrary wind, or on a lee shore, a ship is obliged to lie-to under a very low sail. Some sail is absolutely necessary, in order to keep the ship steadily down, otherwise she would kick about like a cork, and roll too deep as to strain and work herself to pieces. Different ships behave best under different sails. In a very violent gale, the three lower stay-sails are in general well adapted for keeping her steady, and distributing the strain. This mode seems also well adapted for wearing, which may be done by hauling down the mizen-stay-sail. Under whatever sail the ship is brought to in a storm, it is always with a fitted sail, and never with one laid aback. The helm is lashed down hard a-lee; therefore the ship shoots ahead, and comes up till the sea on her weather-bow beats her off again. Getting under weigh is generally difficult; because the ship and rigging are lofty abaft, and hinder her from falling off readily when the helm is put hard a-weather. We must watch the falling off, and assist the ship by some small headail. Sometimes the crew get up on the weather fore-throds in a crowd, and thus present a surface to the wind.

These examples of the three chief evolutions will enable those who are not seamen to understand the propriety of the different steps, and also to understand the other evolutions as they are described by practical authors. We are not acquainted with any performance in our language where the whole are considered in a connected and systematic manner. There is a book on this subject in French, called Le Manoeuvrier, by M. Burde de Ville-Huet, which is in great reputation in France. A translation into English was published some years ago, said to be the performance of the Chevalier de Saufeuil a French officer. But this appears to be a bookeller's puff; for it is undoubtedly the work of some person who did not understand either the French language, or the subject, or the mathematical principles which are employed in the scientific part. The blunders are not such as could possibly be made by a Frenchman not versant in the English language, but natural for an Englishman ignorant of French. No French gentleman or officer would have translated a work of this kind (which he professes to think so highly of) to serve the rivals and foes of his country. But indeed it can do no great harm in this way; for the scientific part of it is absolutely unintelligible for want of science in the translator; and the practical part is full of blunders for want of knowledge of the French language.

We offer this account of the subject with all proper respect and diffidence. We do not profess to teach; but by pointing out the defects of the celebrated works of M. Bouguer, and the course which may be taken to remove them, while we prefer much valuable knowledge which they contain, we may perhaps excite some persons to apply to this subject, who, by a combination of what is just in M. Bouguer's theory, with an experimental doctrine of the impulses of fluids, may produce a treatise of seamanship which will not be confined to the libraries of mathematicians, but become a manual for seamen by profession.