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ARTICLE I.

REMARKS ON SAILING BY THE WIND.

29. When it is desired to gain to windward as much as possible, without absolutely wishing to sail with the greatest velocity, let the direction of the coast under the lee be supposed to make with the absolute direction of the wind (which must as near as possible be known) an angle of 90 degrees; or, in the sea phrase, blowing dead on shore: let the angle A C E (fig. 10.) formed by the sail and the keel, be known to be 30 degrees, let the lee-way be also known to be ten degrees, the angle E C I between the sail and the course will consequently be 40 degrees, which you must take from the total angle V C L 90 degrees; then there will remain 50 degrees, the half of which, 25 degrees, is to be taken for the absolute angle of incidence V C E, and for its equal I C L; so that the ship A B will go 55 degrees from the wind when she is close hauled, and will consequently recede as much as possible from the point D on the coast, the direction of which makes an angle of 90 degrees with the absolute direction of the wind V K.

But, if the situation C L, (fig. II.) of the point D, from which you wish to move, made an obtuse angle V C L, with the positive direction of the wind V M; then, the tangent of the apparent angle of incidence V C E must be made double the angle of obliquity E C I which the sail makes with the course, at the same time that the angle I C L, of the course and the coast shall be made equal to the angle V C E, formed by the real direction of the wind V K, and of the sail: so that two considerations must at once be attended to. For example: the angle A C E, formed by the sail and the keel, is 30 degrees; then, according to the first principle, it will be necessary that the apparent angle of incidence V C E, would be 49° 6'; and, if the difference between the apparent and real direction of the wind be 10 °, there will be 59° 6' for the angle which the sail E V, makes with the real direction of the wind V M: so that the angle L C I, of the course and the object stood from, must be found also to be 59° 6', and the total angle L C V, will then be 148° 12', adhering to the two principles of sailing with the greatest velocity, and of getting to windward of the point D, as much as possible, at the same time; while the angle L C V, formed by the apparent direction of the wind and that of the coast from which the ship moves, will be only 138° 12'. The yawing and the different velocities of the ship render the angle formed by the two directions of the wind, (the real and the apparent,) more or less open. If the ship has more velocity at the same time, or if the course approaches more to the direction of the wind, it will appear by the vanes that the wind draws forward, and the angle of the two directions of the wind will augment. If the ship falls off, and yet still preserves the same velocity, or if her velocity decreases without altering her course, the wind will seem by the vanes to draw aft, and the angle of the two directions will diminish; so that, whenever the ship shall have velocity or run obliquely to the wind, there will always be a difference between its real and its apparent direction. In short, if the ship run exactly before the wind, or have no motion at all, there will be then no other but the real direction of the wind shown by the vanes: but happen how it will, in oblique courses this is however certain, that the sails are always struck by the absolute direction of the wind; because, their position being once fixed by the braces and bowlines, it can no more change, but continues as steady as the real direction of the wind; for it is the vanes only which, being moveable, fix themselves in a middle direction between the absolute tendency of the wind and the course of the ship; whence we may easily conclude, as we did before, that the apparent direction of the wind shown by the vanes, is

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a medium between the respective velocities of the ship and of the wind; since that direction necessarily partakes more of the greater velocity than of the less: so that, if the ship runs East, with the wind at South, having the fourth part of the velocity of the wind, the vanes will show S. S. E. 4° 30' s for the apparent direction of the wind.

ARTICLE II.

THEOREM OF M. BOUGUER.

The velocity and real direction of the wind is C M(fig. II.) suppose the ship A B, of which E F is the sail situated at pleasure, to draw the course C I while the particles of air run in the direction C M: if, from the point I, be drawn I K, parallel to the sail E F, till it cuts the direction of the wind, V K, in the point K, there will be given the three points C, I, K, through which draw the circumference of a circle C I L K, and that circumference will show the extent of the forces acting on the ship, at the same time, in following the course C I, provided her sail be always trimmed in the same manner with respect to her keel.

DEMONSTRATION.

The apparent or relative velocity of the wind is represented by I M (fig. II.) in the course C I; and as I K is parallel to the sail E F, the angle M I K is equal to the apparent angle of incidence V C E. But to be more explicit: the wind strikes the sail with its apparent or relative velocity I M, (and not with its absolute velocity, because of the motion of the ship,) and with an angle of incidence M I K=V C E: so that, if the ship runs close hauled or perpendicular to the direct wind V C, I M will become in both cases stronger than the absolute velocity; because the ship will either approach to the source of the wind, or not recede from it. But the impulse on the sail is proportional to the square of the velocity I M, multiplied by the square of the sine of the angle of incidence M I K, equal to the angle V C E (§ 3. & 7.) and the proportion M K: sine K I M : : sine M K I which furnishes us the triangle K I M, shows us that M K x sine M K I=M I x sine K I M; squaring the two products, and substituting the sine of the angle V K I in the room of the sine of the angle M K I which is equal to it, since they are the supplement of each other, we shall then have this other equation: (sine V K I)2 x (M KO)2 = (sine K I M)2 x (M I)2; whence it follows, that instead of expressing the actual impulse of the wind upon the sail by the square of I M, multiplied by the square of the sine of the angle K I M, it may be expressed by the square of M K, multiplied by the square of the sine of the angle V K I, or of its equal V C E, formed by the absolute direction of the wind V M, and the sail E F.

We must not forget to be very attentive to this; viz. that the impulse of the wind upon the sails is in equilibrium with the effort of the water on the bows, or that they are exactly equal and contrary when the ship is come to an uniformity of motion (§ 9.) as here we suppose her to be. Besides, the impulse of the water on the bows is proportional or equal to the square of the velocity of sailing C I, (§ 3.) so that the square of the velocity of sailing C I, is equal to the actual impulsion of the wind upon the sail expressed by the square of K M, multiplied by the square of the sine of the angle V C E; and if s be supposed equal to the sine of V C R, or of V K I, we then always find (C I)2=(S)2 x (K M)2. The

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The first term in this equation represents the impulse of the water on the bows, and the second expresses the effort of the wind upon the sails; and, if the square roots of the one and of the other be taken, it will be found C I = S x K M; that is to say, that the very velocity of sailing C I, will be continually equal or proportional to the product of K M by the sine S of the angle V C E or C K I. The proportion between these quantities depends on the density of the two fluids, and on the magnitude of the surfaces struck: but it will be the same in all the different courses.

The different velocities of sailing C I, have a constant and given proportion with the products S x C K and C I x sine C I K; for the triangle C I K gives S : C I:: sine C I K : C K, which forms this equation, S x C K = sine C I K x C I; and all the angles C, I, K, are constant and known, since they are equal, being alternate to that which the sail makes with the course. But, as the velocity C I bears a continual and constant proportion with the product S x K M, and as it bears also a constant proportion with S x C K, it follows that S x K M : S x C K :: K M : K C; so that the point K always divides C M in the same proportion: the point K is then invariable when the sail, as well as the lee way, are both the same; (which never happens, however, as will be made appear (§ 47.) hereafter) but, in admitting those two hypotheses, which never can deviate from the truth but in respect to the lee way, which is always variable in the same ship, according to the different circumstances of wind, sea, velocity, sail, and course, it ought then to be concluded that all the points, 1, &c. will be situated on the circumference of a circle; for, without that, the angles C I K equal to those which are formed by the course and the sail, and which are supported on the same chord C K, would not be equal.

COROLLARY.

Admitting therefore, (fig. II.) that the velocities are continually proportional to the sines (whatever they be) of the angles V C E, which the sail makes with the absolute direction of the wind, provided the sail be always trimmed in the same manner with respect to the keel, and that, in the triangle C I K, the side C K and the angle C I K are constant, and the velocities of sailing C I are proportional to the sine of the angle C K I equal to the angle of incidence V C E; it follows, that, all the other conditions being the same, the more the sine of the angle V C E is augmented, the greater will the rate of sailing be; so that, if you want to carry it to the greatest rapidity, you have only to make a right angle of the angle V C E formed by the absolute or real direction of the wind with the sail; then the velocity C I will no longer be a simple chord in the circle C K I, but a diameter. This holds good for all the ships which have but one sail set; but, whenever they shall have several, the greatest velocity will be when the apparent angle of incidence of the wind upon the sail makes a right angle with the course; because then the sails will easily make with the apparent wind, an angle, of which the tangent will be double that of the angle they make with the course, without their becalming one another; while, at the same time, the ship will receive all the absolute impulse of the wind, because she does not recede from it, and it is the time when the greatest surface of sail is exposed to its impulse. The same advantage of the greatest velocity will still be had, when the apparent direction of the wind makes an angle of an hundred degrees with the course; and in this situation, the velocity will in some degree be increased. In a word, whenever the after sails do not becalm those forward, the ship's rapidity may always be increased, by trimming the sails as directed (n. 28.) but when the sails take the wind from one another, an increase of velocity can no longer be pretended to.

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We are now going to demonstrate the exactness of the rule given before. (§ 29.) When it is required to get off shore, or recede from a given right line with all possible expedition, or to keep absolutely as close to the wind as the ship will lie; C M (fig. II.) is the absolute direction of the wind; the circle C K L I marks all the points at which the ship can arrive with the same sail, the same disposition, without alteration of lee-way, and at the same time; and C L is the right line from which she is to move. Knowing the angle that line makes with the absolute direction of the wind V M, it is evident that the point I of the circumference, where the course ought to end, is in the middle of the arc C I L, of which C L is the chord: and all the points from one part to the other of C I, where the ship can come to at the same time, are less distant from C L, since D I, perpendicular to C L, divides it into two equal parts, and is the longest of all the perpendiculars which can be drawn from the circumference C I L; but the point I, cannot be taken without rendering the angle L C I equal to the angle C K I, which itself is equal to the angle V C E.

ARTICLE III.

A TABLE OF THE SITUATION OF THE SAILS TO RUN WITH THE GREATEST VELOCITY.

M. BOUGUER.

 Angles of theapparent directionof the wind andcourse Angles of the sailswith the keel. Angles of Apparentincidence of thewind on the sails D. M. D. M. D. M. 180, 00 90, 00 90, 00 176, 15 87, 30 88, 45 174, 37 86, 25 88, 12 172, 30 85, 00 87, 30 168, 44 8z, 30 86, 14 164, 58 80, 00 84, 58 161, 10 77, 30 83, 40 157, 22 75, 00 82, 20 153, 33 72, 30 81, 03 149, 41 70, 00 79, 41 145, 48 67, 30 78, 18 141, 53 65, 00 76, 53 137, 55 62, 30 75, 25 133, 54 60, 00 73, 54 129, 50 57, 30 72, 20 125, 42 55, 00 70, 42 121, 31 52, 30 66, 01 117, 14 50, 00 67, 14 112, 53 47, 30 65, 23 108, 26 45, 00 63, 26 a .. .. .. .. .. .. b 103, 53 42, 30 61, 23 99, 13 40, 00 59, 13 94, 25 37, 30 56, 55 89, 28 35, 00 54. 28 84, 23 32, 30 51, 53 79, 05 30, 00 49, 06 73, 39 27, 30 46, 09 68, 00 25, 00 43, 00

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N. B. The foregoing TABLE can be of no great service, except in the eight last circumstances under the line a, b; because, in all the cases mentioned above that line, the sails will cover one another too much.

OBSERVATIONS.

When a fast sailing ship (such as will, on a direct course, or right before it, take a third or a fourth part of the velocity of the wind) comes to run with the same quantity, or more sail, on a perpendicular to the apparent direction of the wind, then she acquires a greater rapidity of sailing with respect to the velocity of the wind; the angle made by the two directions, the apparent and the absolute, is at that time very considerable; it may be from 18° to 22° 30'; and if the ship hauls quite close by the wind, the angle will still be nearly the same; for, her velocity diminishes: but, as it is in sailing by the wind that it is most essential to know the greatness of the angle between the two directions of the wind, let the angle between the directions of the ship's head on the different tacks be observed, without paying any regard to the lee-way, but just to the exact point on which the ship stands, before and after going about, when strictly by the wind, neither too much to leeward nor to windward; and when you have determined that angle, from two or three observations, halve it, and then you will have the angle formed by the keel and the absolute direction of the wind; by which you will know the quantity she will come to upon the different tacks, and will never be deceived with respect to the lying on astern having gone about: a mistake pretty commonly made by those who pay attention only to the apparent direction of the wind, which always makes with the real one an angle more or less open in a compound ratio of the greatest velocity with the greatest obliquity of the course of the ship, with respect to the direction and the absolute velocity of the wind; things which vary in all ships, because they have not all the same advantage of sailing with the same rapidity in similar circumstances.

CHAPTER IV.

OF THE SAILS WHICH ARE BEFORE THE CENTER OF GRAVITY OF A SHIP.

30. THE sails which are before the center of gravity of a ship, are the sprit-sail, sprit-sail top-the jib, the fore-top-mast stay-sail, and the fore stay-sail.

Besides these sails there are, on the foremast, the foresail, fore-topsail, fore-topgallant-sail, and fore-topgallant-royal-sail, with their respective studdingsails. Now these four last sails may be regarded as only one large sail, wide at the foot and tapering towards the head, and which can be reduced, as occasion requires, either by taking in the royal, or by reefing the fore-top-sail, or even taking it quite in, if necessary, to have the fore-sail only set; or by hauling the fore-sail up, if nothing but the top-sail is wanted. It must notwithstanding be observed also, that the different parts of the whole sail may, in certain cases, be worked differently the one from the other; as, for example, in reeving the top-sail, or in taking in either the one or the other. But, when you want to set them to work all together, either for making a course, or performing some evolution, they must all be braced and trimmed in the same form, and with the greatest uniformity possible. Therefore, whatever we shall say concerning one of them in any operation, is to be understood to be the same with respect to all the rest.

The main-stay-sail, the main-top-mast stay-sail, the middle-stay-sail, and the main-top-gallant stay-sail, are likewise sails of the fore-part of the ship's center of gravity.

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ARTICLE I.

OF THE EFFECT OF THE FORE-AND-AFT-SAILS, WHICH ARE ON THE FORE PART OF THE CENTER OF GRAVITY OF A SHIP.

31. THE jib and stay-sails being of a triangular figure, their center of gravity is easily found; and that point is to be considered as the part, in all these sails, on which the whole effort of the wind is united, when they are exposed to its impulse, in whatever way it strikes them.

The particular effort of each fore-and-aft sail being on the fore part of the center of gravity of the ship, it follows that the total effort of all these sails must be there too; and that, if the ship was in a perfect equilibrium with respect to the wind, before her sails were set, she will lose it immediately after (§ II.) they will make the fore part of the ship obey the wind, whenever it strikes them perpendicularly or obliquely. For, it must be observed, that almost all these sails have their tacks; in the middle of the ship, and their sheets lead to the sides; so that they make with the keel a very acute angle: whence it is easy to conceive, that the perpendicular which would be raised on the exterior surface of these sails, in the direction of their effort to leeward, from their center of gravity, would differ but very little in the lateral direction from a perpendicular to the keel. From this we may therefore conclude, that these sails would have but very little effect to accelerate the rapidity of sailing with respect to their position, if it was not demonstrated that they are very advantageous in going by the wind. They make the ship steer well, and are particularly useful when a ship gripes much: and, when they do not take the wind out of any of the lower sails, they ought to be used, particularly when one is obliged to sail by the wind, or to run not very large. The jib and fore-top-mast staysail must be preferred, because they are at all times useful when they can receive the wind; for, by their position, they can take the wind out of any of the other sails, and their particular effect in veering is considerable, not only on account of their great surface, but because they act before the point, on which the ship turns, with a very long arm of a lever (§ 17.) On the other hand, all the sails draw the ship a-head in raising her: for the direction of their effort ascends obliquely towards the horizon; therefore, they do not make her plunge in the water, which is an advantage peculiar to them. Experience has confirmed their utility on all occasions when they can be employed without taking the wind out of the other sails.

ARTICLE II.

OF THE EFFECT OF THE FORE-SAIL, FORE TOPSAIL, FORE TOP-GALLANT-SAIL, AND SPRIT-SAIL, IN THEIR DIFFERENT SITUATIONS.

32. WHEN the sail A B (fig. 12.) is trimmed close to the wind which blows from the point V, it is impelled in the direction C D (§ 7.) with a force expressed by the square of the sign of incidence, and composed of the two effects C E and E D (§ 19.) But, as the center of effort of that sail A B is on the fore part of the center of gravity of the ship H, and as its power C D is always decomposed between those two effects C E and E D, it follows, that the effect of this sail is to cause the ship to bear away; while it keeps up at the same time, and even augments, the rapidity of sailing.

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33. If the fore-sail A B received the impulse of the wind perpendicularly, it would still produce the effects of bearing away, and augmenting the rate of sailing, for the reasons just given above, but more effectually would it do so (§29.) on account of the increase of the impulse of the wind upon the sail.

34. IT follows, from what has been said, that when the sails upon the fore-mast are full, on the same side they are tacked, being braced obliquely to the keel, there is always one part of their effort, in proportion to their obliquity, which acts to make the ship bear away; while the other part of their effort acts at the same time to accelerate or keep up the rate of her sailing.

35. When the sails A B of the fore-mast (fig. 13.) are situated obliquely with respect to the keel, and receive the wind in them, on the side of the sheet B, they act upon the ship in bringing her up to the wind, because their effort D G being discomposed, as customary, the lateral part D F carries the fore part of the ship towards the source of the wind V, in carrying her from D to F.

REMARKS.

36. IN general, when the yards are square or perpendicular to the keel, it is evident that they will act on the ship, only by impelling her right in the direction of the keel, from stern to head, or from head to stern, with more or less velocity, in proportion to the impulse of the fluid which strikes them.

37. When the sails A B on the fore-mast (fig. 14.) receive the impulse of the wind V, on their surfaces forward, they will make the ship both go a-stern and sail off; because the direction C E of their effort, being turned towards the after-part, serves as a diagonal to the parallelogram F D, which, by discomposing it, will show us those two effects C F and C D, the first of which takes its direction with that of the keel from forward aft; while the second takes it in a lateral direction in making the ship to turn.

38. When the wind blows between the keel and the yard, the ship comes to, until the point G (fig. 14.) is in the direction of the wind V. But, as soon as this is done, it is evident that she falls off; for the point G recedes farther and farther from the direction of the wind. Whence we may remark, that, as soon as the weather part of the sail catches a-back, on the tack side, the angle of incidence of the wind on it goes continually increasing, till the ship has fallen off so much, that her sail becomes perpendicular to its direction: and, if the vessel continues to fall off, then the angle of incidence diminishes more and more, till the sail is parallel to the course of the wind which comes from the tack B, or, as it is called, shivering.

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CHAPTER V.

OF THE SAILS WHICH ARE ABAFT THE CENTER OF GRAVITY OF A SHIP.

39. THE main-sail, main-top-sail, main-top-gallant-sail, and main-top-gallant-royal-sail, and their respective studding-sails; the mizen-stay-sail, mizen-top-mast-stay-sail the mizen-course, mizen-top-sail, mizen-top-gallant-sail, and mizen-top-gallant-royal-sail; are all sails, which are placed abaft the center of gravity of a ship, which is also abaft the point round which the total effort of the sails is placed.

ARTICLE. I.

OF THE EFFECT OF THE FORE-AND-AFT SAILS ABAFT THE CENTER OF GRAVITY OF A SHIP.

40. THE center of effort of these sails being abaft the center of gravity of the ship, it follows that they always force the after-part of the ship to leeward, and consequently contribute to bring her to the wind, as soon as they receive its impulse; for, that movement of the after-part of the ship cannot happen, without the head approaching to the direction of the wind.

The fore-and-aft sails being in general situated very obliquely, it follows, consistently with principles, that they are very advantageous for sailing by the wind. Therefore, we must not neglect augmenting them: observing, at the same time, that they do not take the wind out of one another, nor becalm the principal sails. They are only to fill up the space between the masts fore and aft, in sailing near the wind, in order that no wind may be lost.

ARTICLE II.

OF THE EFFECT OF THE SQUARE SAILS OF THE MAIN-MAST, AND OF THE MIZEN-TOP-SAIL, THEIR DIFFERENT OBLIQUITIES.

41. AS we have already demonstrated (§ 19.) that when the sail A B (fig. 15.) is trimmed obliquely to the keel, it produces evidently two effects on the ship; it must therefore follow, that, in dissolving its power C D, we shall find its compound effects, the one C F, in the direction of the keel which produces the velocity, and the other C E lateral, which (in forcing the after-part of the ship to leeward, by its action on the point C abaft the center of gravity G of the ship) occasions her to come to the wind; for that motion of the stern from C to E cannot take place, unless the fore-part acts contrarily in coming towards the point from which the wind blows, V.

42. If the sails A B were more or less oblique to the keel, they would still have the same effects of keeping up the ship's velocity, and bringing her to the wind. And, if they received its impulse perpendicularly, it would still be the same thing, producing those two effects, however, with greater efficacy than in any other situation with respect to the wind, because then they receive its greatest possible impulse for the time.

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43. When the sails A B (fig. 16.) of which the center of effort C is abaft the center of gravity of the ship, receive the impulse of the wind V on the sheet side, being placed obliquely to the keel, they will cause the ship to fall off, by forcing the after-part from C to I, towards V, the source of the wind, while they will, at the same time, keep up the velocity C I. For, this motion of the after-part E towards V, cannot be executed without the fore-part E going, as it moves off, in a contrary direction; and she will continue to fall off till the keel E H be right in the direction of the wind V C, or right aft; then the ship will come to the wind, as shown in the two preceding articles.

It may be remarked that, in this movement of the, ship the angle of incidence goes continually increasing till the wind is perpendicular to the sails.

44. When the sails A B (fig. 17.) of which the center of effort is abaft the center of gravity G, shall receive the impulse of the wind V on their forward surfaces, they will make the ship come to the wind, and go a a-stern at the same time. For the direction of their effort C D may be dissolved between the two efforts C F, in the direction of the keel, from forward to aft, and C E lateral and perpendicular to the keel; so that the after-part C H is forced to leeward from C to E, while the fore-part I approaches, by a contrary motion, the point of the wind V. In this case, therefore, the ship comes to, and goes a-stern.

45. When the ship is so far come to the wind, that the fore-part I (fig. 17.) has come into its direction, it is evident, that she will fall off more and more; for, that point I will constantly move from the point of the wind V; therefore, it is demonstrated that, in this case, the sine of incidence is continually decreasing more and more, till it is reduced to nothing. But, if the direction of the wind had made an obtuse angle V C B, the sine of incidence would have augmented until the direction of the wind had been perpendicular to the sails; and it is at that moment only it would have begun to diminish, as we have shewn before.

CHAPTER VI.

OF THE EQUILIBRIUM NECESSARY TO BE KEPT IN PRACTICE, BETWEEN THE SAILS BEFORE AND ABAFT THE CENTER OF GRAVITY OF A SHIP, IN ORDER THAT THE SAILING MAY BE THE MOST DIRECT AND THE MOST RAPID.

46. AFTER having demonstrated the different effects of the sails both before and abaft the center of gravity, it is clear that if either the head or after sails only were set, in sailing by the wind, the ship would not only steer badly, but consequently sail not so fast, as she could under the same quantity of surface, differently disposed. For, if the ship be supposed (fig. 18.) to be under her head sails, and one half be retrenched and set on the masts abaft, it will evidently appear that the velocity C T they produced is the same, since the direction and the velocity of the wind act always in the same manner on the same quantity of surfaces; the only difference which will be found, is, that the primitive effect is divided, and acts now on the points C, C, E before and abaft the center of gravity of the ship. It is not the same with respect to the effect C D, which acted only the head of the vessel in the first disposition of the sails, because that effect being now divided on the after-masts, it is diminished one half C E forward, by reason of that force being transporting aft, where, balancing the effect of falling-off produced by the head-sails, it keeps the ship to the wind; by equality in the movements (§ 34 & 42.) I say that it balances, because, when the weather permits, we may at any time either increase or

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diminish the sails, so as to preserve an equilibrium between their powers, and fix the ship on her course. When this point of equilibrium is obtained, we then possess the most advantageous disposition the sails can have for the vessel to run with the greatest celerity; provided that they have been trimmed in the most favourable manner to receive (§ 28.) the greatest impulse of the wind.

This equilibrium between the powers of the sails forward and aft, is likewise advantageous with respect to the rudder; because, as there is less occasion to use it to regulate the movements of the ship, its surface opposes itself but little, and less often, to the shock of the water, which glides along the ship's bottom. It is then of the greatest importance, in endeavouring to increase the ship's way, to combine, as much as possible, the reciprocal effect of the sails fore and aft; either in setting them to the wind, or in disposing more advantageously, forward or abaft, a greater or a less quantity of sails, according as the ship is more or less inclined to fall off, or come to; in order to make as little use of the helm as possible; the whole power of which, however, at the time of performing any evolution, must be put in action, as we shall make appear hereafter.

OBSERVATIONS.

47. When there is an equilibrium between the sails fore and aft, the resistance of the water from A to B (fig. 18.) on the bows is equal to the power of the sails, whether it passes through the center of gravity H of the ship, or through another point of the axis, more or less forward or aft; then a ship, thus situated, finds no more difficulty to veer than to come to the wind, with respect to the resistance of the water under her lee; since all things are equal, viz. the resistance of the water upon the bottom to leeward, and the impulse of the wind upon the sails. But it must be considered that the power composed of those of all the sails united, acts upon the ship according to the direction B A, perpendicular to their surfaces, the origin of which is the point H, a middle between all the effects C G of the sails fore and aft, which ought to correspond exactly with the resistance of the water from A to B: so that the ship is pushed to leeward of the course I K, which she holds into the direction B A of the effort of her sails; but the resistance which she finds from the water on the lee-side of her bottom, from A to B, sets her to rights again by its opposition, which is greater by reason of the greater facility she finds in dividing the fluid with her stem, than with her side; so that she runs on the true course N R, which approaches nearer to that on which she steers than B A. Therefore the angle K H R of the lee-way is proportional to the greater or less resistance the ship finds laterally from the fluid under her lee; which resistance depends intirely on the more or less facility she finds in dividing the water with her bows; so that the lee-way can never be considerable but when close hauled; for this reason, it is not much taken notice of when the course is less oblique than the wind on the beam. We might pursue this reasoning still further, from an experienced fact, which will prove that the lee-way depends, not only on the form of vessels, but still more on their greater or less velocity, and seldom, or never, on the intire disposition of their sails more or less oblique to the keel, as some authors have advanced. For, when a fine sailing vessel is trimmed sharp, with all her sails set, in a very light breeze, with which she scarcely obeys the rudder, the lee-way is considerable, though the sea be perfectly smooth. This great lee-way is made by the ship, because the vessel being only gently impelled, and with little force, the water, not being shocked with violence, offers little resistance, and she is then carried easily by her sails in the direction of their effort B A: and, if we consider the side of the ship, in the act of sailing, presenting a very great surface of sails above the water, it will visibly appear the lee-way will become still more perpendicular to the keel. But, let the

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wind begin to freshen, then the rapidity of sailing augments considerably; the ship shocks the water with a force expressed by the square of six or nine knots of velocity from B to A (fig. 18.) in the space of an hour, while the water repels her effort in a contrary direction: the water repels then in the ratio of this square to the square of her full velocity, and now no longer yields with facility (§ 4); the lee-way is suddenly diminished, and is reduced to five or six degrees, and sometimes less, if the rapidity of sailing continues to increase; if, at the very time when the ship has acquired already a very great velocity, she be kept away 12° or 15°, or even 22° 30', without altering the sails, their obliquity remaining the same, the ship should then fall off in the same proportion, according to the opinion of those who have written on the theory of the working of ships. This, notwithstanding never happens; the velocity augments, because the sails then receive the wind with a greatest sine of incidence, and thereby acquire more power, while the bows continue to be still shocked by the fluid in the same parts, and with the same sine of incidence; so that the lee-way diminishes again, because the water makes a greater resistance from the increase of velocity; and that resistance is greater on the ship's side than on her bows, which is less exposed to the shock. Whence it must be concluded that the lee-way, in the same ship, does not depend alone on the disposition of her sails, and that in different ships it is always dissimilar, from their not having the same form, or their sails not trimming equally in the same oblique courses; and because, in short, they have different velocities with the same weather, and under the same sails. Which proves, in a word, that the leeway is always in a proportion compounded of the velocity of the ship; of her form, which gives her more or less resistance on her side than on the bows; and of her sails, trimmed more or less obliquely.

To return to the consideration of the action of the water on the bottom from A to B (fig. 18. ), it must be remarked, that it acts forward, and that it must consequently contribute very much to the tendency which almost all ships have to come to the wind, whenever the after-sails are in the smallest degree more powerful than those forward: for, the shock of the water is then a power which is to be added to that of the impulse of the after-sails, since this action of the fluid is so much the stronger as it acts before the center of gravity of the ship at the point M (fig. 18.), in impelling the fore part towards the wind, which always makes ships difficult to wear, because all the effort A B of the water's resistance upon the bows is opposed to this movement, in forcing this part to windward with a very great effort. It is not therefore to be wondered at when ships veer with difficulty or slowly, especially such as have a large cut-water; because there are two forces acting one against the other, and the force which comes from the sail must surmount (§ 18.) that which comes from the water; which will always easily happen, whenever, in suppressing some of the after-sails, those forwards shall be disposed favourably enough to produce that effect; and when the rudder is used at the same time; the power of which is considerable, whether the ship goes a-head or a-stern rapidly. But if the ship, being abandoned to her own proper movements in an oblique course, had on a sudden all her sails suppressed, the vessel would come to the wind, should even the rudder never be used; because the water, acting on the fore part of her bottom more on one side than the other, impels the head to windward against the smaller resistance, until its power is entirely destroyed by the total cessation of the ship's velocity.

When the ship runs so large that the after-sails becalm part of those forward, this is again another reason for the ship having an inclination to come to the wind; for, the sails forward receive a much less impulse from the wind than in a course more oblique; because the sails abaft, by increasing in their power, prevent those forward from having as much wind as their surfaces would take, since all

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the lee parts of these sails become useless for the moment, being becalmed by the weather part of those on the main mast; so that the power of the sails forward diminishes, while that of the after-sails increases; for the sine of incidence is greater. The ship ought then, for these reasons, to have more inclination to come to the wind; but, regard must be paid to the direction of the power of the sails in general, which now approaches nearer the direction of the keel: so that the greatest part of their effort is in that direction, while their force in the lateral one diminishes.

It should be farther observed, that when the ship has as much sail as the weather will permit her to carry, that is the moment of the greatest velocity of sailing, providing that the sails having at the same time received the most favourable disposition, an exact equilibrium has also been placed between those afore and those aft, so that there should be little occasion for the use of the rudder.

APPLICATION.

48. One may readily discover, from what precedes, how to distinguish the degree of quickness with which different operations ought to be performed. For example, being obliged to run for a road-stead, the wind being large, and to let go an anchor as soon as come to it, it is evident this ought to be executed but under little sail, which should be all on the part before the center of gravity; because, in the first place, a ship has always velocity enough when she sails large; secondly, because she is to overcome the effort A B (fig. 18.) of the water which opposes her movement. If, on the contrary, being obliged to come to the wind in anchoring, as many sails as can conveniently be managed at that moment may be set, because that movement of the ship is always very quick, and as soon as the sails are taken a-back, the rapidity of the ship's way diminishes, and in a few moments entirely ceases, whereas it always augments when the ship falls off.

ARTICLE I.

REMARKS ON THE EFFECT OF THE MAIN-SAIL.

49. IN the use of the sails, attention should be paid to the effect of the main-sail, which perhaps may not be that of bringing the ship to the wind; for, if the ship be too much loaded a-stern, the center of gravity H (fig. 18.) of the ship might be abaft the main-mast, and then the direction of the effort of that sail, quitting the point C before the center of gravity, ought to make the ship fall off in lieu of keeping her up to the wind. But, for this to happen, the ship must be either very ill constructed, or very badly loaded; or, in short, there must be great error in the position of her masts. Notwithstanding the main-sail may always be made to assist the ship in veering, though the center of gravity H be (as it is almost always) before the effort C of the main-sail; yet, to do it, the effect of that sail need only be changed, by making it to pass before the center of gravity of the ship: which will suddenly happen, if, when close hauled, the main sheet be let go a-main, because the weather part of the sail being fixed forward by the tack, its effect is likewise before the center of gravity of the ship, though it has lost in that part much of its power, in becoming less exposed to the impulse of the wind; while the lee part, bellying out more, can receive a great impulsion of the wind, which will strike it more and more perpendicularly as the ship falls off with more and more rapidity. In this case, it may happen, that if the direction of the effort C G of the main-sail do

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not pass before the center of gravity H of the ship, it comes so near that point, that it may be said to have no longer the effect of an after-sail.

ARTICLE II.

OF THE RUDDER.

50. THE rudder is a machine known to all the marine world; it is supported by the sternpost, to which it is affixed by braces and pintles, which operate as hinges. It acts by means of a lever, called a tiller, which enters nearly horizontally into the ship, passing under the upper or middle deck transom; so that if, instead of leaving the rudder exactly in a right line with the keel, and as it were a prolongation of it, it be turned to one side or the other, as B D (fig. 19.), it receives an immediate shock from the water which glides along the ship's bottom, in running aft from A to B; and this fluid impels it towards the opposite side, while it continues in that situation, so that the stern, to which the rudder is confined, receives the same movement; and, the ship receiving an impulse sideways, her stern turns accordingly from B to b, on any point whatever C (§ 18.), while her head passes from A to a. It must be observed, that the water strikes the rudder obliquely, and only with that part of its motion which acts according to the sine of incidence, in impelling it in the direction N P with a force which depends not only on the rapidity of sailing, but also on the greatness of the sine of incidence: a force which is consequently in the compound ratio of the square of the greater or less velocity of the ship's motion, and of the square of the larger or smaller sine of incidence, which depends upon various circumstances. So that, if the vessel runs three or four times more swiftly, the absolute shock of the water upon the rudder will be nine or sixteen times stronger under the same angle of incidence, and will be augmented in a greater proportion, if the sine of incidence be increased. This impulsion, or, what is the same, the power of the helm, is always very feeble, when it is compared with the whole weight of the vessel; but it acts with a very long arm of a lever, which occasions it to work very advantageously in turning the ship; for the helm is fixed at a very great distance from the center of gravity G, as well as from the point C, upon which the ship is supposed to turn, with respect to the point of percussion B: and if the direction P N of the impression of the water upon the rudder be prolonged, it is evident that it will pass perpendicularly at the point R, widely distant from the center of gravity G; therefore the absolute effort of the water is very powerful. It is not therefore surprising, that this machine impresses the ship with a considerable circular movement, by forcing the stern from B to b, and the head from A to a, and even much farther, when the velocity of the ship is preserved; because the effect of the helm always keeps pace with the rapidity of the ship's way.

51. Amongst all the obliquities which may be given to the rudder, there is one situation which is more favourable than any of the others, to make it produce with more rapidity the effect of turning the ship, in order to change her course. To be convinced of this, we have only to consider that, if the obtuse angle A B D (fig. 19. ) were to be lessened, the impulse of the water on the rudder would augment, at the same time that it would more oppose the sailing of the ship, since the angle of incidence would be more open, and would present a greater surface (§ 7.) to the shock of the water, by opposing its passage more perpendicularly: but then the direction N P of the effort of the helm upon the ship would pass at a smaller distance from the center of

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gravity G towards R, and would less approach the perpendicular N L, according to which, it is absolutely necessary that the power should act with greater effect to turn the ship. Therefore, it is evident that, if the obtuse angle A B D were too much lessened, the greater shock of the water could not counterbalance the loss occasioned by the distance between the direction N P and N L, or by the great obliquity which would be given to the same direction N P of the absolute effort of the helm with the keel A B. If, on the other hand, the angle A B D were made more obtuse, the direction N P of the effort of the rudder would become more advantageous to turn the ship since it would approach more the perpendicular N L, and since the prolongation of N P would augment G R, by passing at a greater distance from the center of gravity G. But the rudder would then be struck too obliquely; for the angle of incidence would be more acute; so that it would only present a small part of its breadth to the shock of the water, and would of course receive but a faint impulsion. All this proves that the greatest distance G R from the center of gravity G will not counterbalance the too great obliquity of the shock of the water. Whence it must be concluded, that when the water strikes the rudder too obliquely or too perpendicularly, a great deal of the impulsion, or of the effect it should produce, is lost. Therefore, between these two extremes, there is a middle position, which must be the most favourable.

52. The diagonal N P of the rectangle I L (fig. 19.) represents the absolute direction of the effort of the water upon the rudder: N I expresses the portion of this effort which opposes the ship's head-way, or which forces her a-stern in the direction of the keel. It is easy to perceive that this portion N I of the whole power of the helm contributes little to turn the vessel; for, if I N were prolonged, it would be seen that its direction passes at a very small distance G V from the center of gravity G, and that the arm of the lever B N = G V, to which the force is as it were affixed, is at most equal only to one half of the breadth of the rudder. But, it is not so with respect to the relative force N L, which acts perpendicularly to the keel. If the first force N I is almost useless, and even hurtful, by retarding the velocity; the second N L is capable of a very great effect, since it is applied at a great distance from the center of gravity G of the ship, and acts on the arm of a lever G E, which is very long. Thus, it appears, that, between the two effects N L and N I which result from the absolute effort N P., there is one which is always opposing the ship's head-way, contributing but little therefore to the motion of her turning; whilst the other alone produces that movement of evolution, without retarding her velocity.

53. Geometricians have determined the most advantageous angle made by the helm with a line prolonged from the keel, and fixed it at 54° 44', on a presumption that the ship is not wider at her floating line than at her keel. But, as that supposition is absolutely false, since all vessels augment their breadth from the keel upwards to the extreme breadth where the floating line, or highest water-line, is terminated; it follows, that this angle is too large by a certain number of degrees. For the rudder is shocked by the water, at the height of the floating line, more perpendicularly than at the keel, since the fluid exactly follows the outlines of the bottom: so that one could almost say, that a particular position of the helm might be required for each different sine of incidence upwards from the keel. But, as a middle position may be taken between all those points, we need only consider the angle formed by the sides of the ship and her axis at the highest water-line, in order to determine afterwards the middle point, and the middle angle of incidence. It appears, from Mr. Bouguer's Traite de la Manoeuvre, Sect. I. Liv. II. that, in most ships, the angle of the rudder with the prolonged line of the keel should be made to be 46° 40'. Without following the

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calculations of that able geometrician, we shall perhaps be able to explain what: he has discussed in a more abstruse manner.

54. When it is required to turn the ship by means of the rudder, and, at the same time, keeping the head-way as much as possible, it is evident that the angle 54° 44', which some have determined to be the most favourable with the line of the keel prolonged, is in that case too open; because the water strikes the rudder with too great a line of incidence, and which is equal to that of the angle which it makes with the line prolonged from the keel below. Above, the shock of the water is almost perpendicular to the rudder, on account of the width of the ship's sides, as has been shewn before. But if the rudder opposes the fluid by making only with the line prolonged from the keel an angle of 45° 1', the impulse, by becoming weaker will be less opposed to the ship's head-way; and the direction N P (fig. 19.) of the absolute effort of the water on the rudder, approaching nearer to the lateral perpendicular N L, will be more advantageously placed; since the prolongation of the absolute effort passes at a greater distance G R from the center of gravity G. On the other hand, experience every day shows us that ships steer well, when they do not even make the angle D B E more than 35°. If this angle be made 45°, as we require it, and then we should discompose the absolute effort N P, we have the side N I equal to the other side N L of the same square; so that the part of the total power which opposes the head-way is only equal, in this case, to that which produces the movement of rotation: instead of which, if D B E. were 54° 44', N I would become much greater than N L, in proportion to the lines of the angles which are opposed to them in the triangles P I N or P L N, and the ship would consequently lose much more of her velocity than in the first situation of the rudder, to which we shall confine ourselves, as being that which is best adapted to the generality of vessels, but which nevertheless must be occasionally altered, according as they shall make an angle more or less open with their sides a-stern.*

The angle of the rudder with the keel may always be determined with sufficient precision, by observing the rule we have prescribed (§ 28.) for the determination of the angle of the sails.

55. As the water often strikes the rudder with a very great force, the tiller has a certain length, in order to lessen the labour of the helmsman.

But, to lighten his labour still more, there is in most ships, on the quarter-deck, directly over the extremity of the tiller a vertical wheel (fig. 19.) which has the effect of a capstern, and which is connected with the tiller by means of ropes and blocks (See Practice of Rigging). So that, if the wheel be turned either one way or the other, the extremity of the tiller approaches towards one of the sides of the ship, and exposes the rudder to the shock of the fluid.

56. The longer a lever is, the more effect it has when it acts with the same power: therefore, the longer the spokes of the wheel are, in proportion to the radius of the cylinder round which the tiller rope is wound, the more advantage the helmsman will have; for, if the spokes of the wheel be three or four times longer than the radius of the cylinder, the helmsman will act with three or four times more force, since he works on a lever which is three or four times longer than the radius of the cylinder, the extremity of which is supposed to be the fulcrum of the lever on which he works. So that, if he employs a force of 30 pounds weight, he will produce an effect of 90 or 120 pounds, by the disposition of the wheel alone. On the other hand, the impulse of the water is collected in the middle of the rudder's breadth, which is very narrow, compared with the length of the tiller; therefore, the effort of the water is very little distant from the point of support,; upon which it turns: whereas the tiller forms the arm of a lever 10 or 15 times longer, which still increases the power of

* It may be taken as a general position that the most advantageous angle will always be formed between 35° and 45°,

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the helmsman in a similar proportion to that which exists between the length of the tiller and that of the lever on which the impulse of the water acts. This force is therefore 10 or 15 times stronger; and the effort of 30 pounds, which before gave the helmsman a power of 90 or 120 pounds, will become one of 900 or 1800 pounds on the rudder. This advantage proceeds from the water's acting on a very short arm of a lever, while the helmsman works on one very powerful, in comparison; and, because this lever is moved by a wheel which multiplies its force. This demonstration ought to remove all surprise at the prodigious effect of the rudder, when its mechanism is not attended to; for we have only to consider the pressure of the water, which acts at a very great distance from the center of gravity G of the ship, as well as from the point C upon which she is supposed to turn (§ 15.) and there will easily be perceived the difference which exists between the effort of the water against the helmsman, and the effect of that same impulsion against the ship. With respect to the helmsman, the water acts with the arm of a lever N B very short, of which B is the fulcrum: on the contrary, with respect to the ship, the impulse of the water is exerted in a direction N P, which passes perpendicularly at a very great distance from the center of gravity G, in acting on a very long lever E G, which renders the action of the rudder very powerful in turning the ship: so that, if in a large ship, the rudder receives an impulse from the water 2700 or 2800 pounds (as very often happens provided that the ship sail at the rate of 9 or 12 knots, and that this power, applied at E, be 100 or 110 feet from the center of gravity G) it will act upon the vessel, to turn her, with a power equal to 270,000 or 308,000 pounds, while the helmsman need not act with a greater power than 30 pounds on the spokes of the wheel.

57. It is proper to remark, that the great length which is given to the tiller, in order to facilitate the work of the helmsman, is an obstacle to the play of the rudder; since that length hinders its presenting itself sufficiently to the shock of the water to produce all the effect which might attend it. For, this inconvenience does not, in most ships, allow the angle B D E (fig. 19) to be more open than 30°; whereas it should be 45° as we have before shewn. But, as this most favourable determination has not yet come into use, and the coarse dimensions commonly given the tiller have always been followed, we shall endeavour to propose something better for practice.

It must be considered, that if the tiller were shorter, the rudder would have more play, because its extremity, in describing the arc of a smaller circle, would occasion the rudder to make an angle more open, with the keel prolonged: and this new augmentation would be so much the more advantageous, as it would approach nearer to the angle of 45°. And as, in all ships, the length of the tiller might certainly be cut a fifth shorter, or perhaps more, it is evident that, thereby, the angle of the rudder and the keel prolonged might be rendered very near 45°, which would increase its force in a proportion of nearly 3 to 5, since the square of the sine of incidence of 45° is to the square of the sine of incidence of 30° :: 5 : 3, or thereabouts. This augmentation of the impulse is often of the greatest importance, especially when ships are of a large size, as their motions are but slow on account of their length.

If the tiller be shortened, the helmsman will be obliged to employ more force in proportion to the length taken from the lever on which he works: but this loss may be repaired by the facility with which the helm will be handled, if the diameter of the cylinder of the wheel be considerably lessened, augmenting at the same time the length of its axis, without diminishing that of its spokes, which ought on the contrary to be lengthened as much as possible, and two more turns of the tiller rope should be wound round the barrel.

These forces would be still multiplied, if two sheaves were fixed in the end of the tiller, in two mortises which might be made for that propose, and which might work on an iron pin passing

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through their centers, taking care to have the end of the tiller stoutly hooped with iron, in order to strengthen it; then the tiller rope might be reeved through the blocks which are for that purpose on each side the ship, thence through the two sheaves at the end of the tiller, and the standing part to be affixed close to the blocks on each side. By these means nothing would be lost with respect to the force; because if the lever be shorter, the forces which cause its action are likewise multiplied in proportion.

58. after what has been said respecting the helm, it is easy to conceive, that the greater the ship's velocity is, the more powerful is the action of the rudder, since it acts against the water with a force which increases as the square of the velocity of the fluid (§ 3.) whether the ship has head-way, or stern-way; observing always, that in these two circumstances the effects are contrary; for, if the ship goes a-stern, the rudder will be struck from I to N (fig. 19.); and, instead of being pushed from N to P, it will be so from N to R; so that the stern being moved in the same direction, the head will take a contrary one, and move towards the same side as the tiller B F.

59. It should be observed, in the use of the rudder, that there is one part of its effort which impedes the ship's sailing when it is struck by the water which runs rapidly along the ship's bottom. If it makes an angle of 45° with the keel prolonged, it receives only half the impulsion it would if acted upon perpendicularly; because the absolute impulse diminishes from two causes: (§ 7.) The surface which opposes the shock of the water is reduced to a less extent than it was at first, and the angle of incidence diminishes likewise: so that by this, the impulse has diminished one half. Considering next, that the impulsion N P, which remains (fig. 19.) it will appear that there is only one part N I which is opposed to the sailing (§ 54), and which is less than N P in the proportion as the sine total is to the sine of 45° the measure of the angle of incidence V N B equal to N P I; for the angle V N L is right, as well as the angle P N B; so that, if you take away the common angle L N B, the two angles P N L and V N B will remain equal between themselves; but, as the angle I P N is equal to its alternate angle P N L, it follows that I P N is always equal to V N B, whether the angle made by the rudder be more or less open with the keel prolonged. So that, if the surface of the rudder which receives the shock be 80 feet square superficies, it will first be reduced, by its being exposed to the course of the fluid, to an effort of 40 feet surface, then to 28 or 29, because, in the first place, there is only one part of the velocity of the water which contributes to the shock, and that is proportional to the relation of the square of the sine total to that of the sine of incidence; and, secondly, because out of the absolute impulse N P, which results from this last oblique shock, there is only a part N I which opposes the velocity of the ship proportional to the absolute N P, in the same relation as there is between the sine total and the sine of incidence; that is to say, that when the rudder makes, in the largest ships an angle of 45° it impedes the ship's rapidity of sailing, in the direction of the keel, with an effort N I equivalent to the impulsion which a surface of 28 or 29 feet square might receive if it were exposed perpendicularly to the shock of the water. So that, if the ship sails 12 knots an hour, or 19 feet a second, the effort of the rudder N I, which opposes the ship's way, will be 12,499 or 12,945 pounds; salt water weighing 1/35th more than fresh.

60. It follows, from all that has been said of the rudder, that it ought to be employed as little as possible; that is to say, the ship and her sails ought to be so disposed, that the smallest motion of this machine may bring her to her course, if she deviates from it, or make her perform any evolution which may be thought proper.

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ARTICLE III.

THE TIME EMPLOYED BY DIFFERENT VESSELS TO PERFORM THE SAME EVOLUTION, IS PROPORTIONED TO THEIR LENGTHS.

61. ALL that serves to produce motion in ships, has more force in large than in small ones; but the difficulty which large ships have to receive the motion, is greater, in a greater proportion, than that which opposes the motion of small ships. For, if the dimensions and machines which compose a large vessel, are twice as large as those which constitute a small one, (solidities being in ratio of their cubes,) the first will be eight times as great. Yet the obstacle which the large will oppose to its being put in motion will be two and thirty times as great as that of the small one. For, if both ships were considered as divided into an equal number of vertical sections, those of the large would appear to have four times as much surface as those of the small, besides that they would be twice as thick, since the dimensions are in general twice as large; consequently they will have eight times the solidity; which answers already to the relative effort of the rudder and sails.

Further, the parts of the large ship are twice as distant from the center of gravity as those of the small one, since those distances are proportional to the other simple dimensions of the two ships, So that if the evolution be supposed of the same number of degrees, the stern and head of the large ship will have to describe arcs twice as large as the small one; and this greater velocity being multiplied by the solidity of the parts of the large ship, which is eight times as great as that of the small one, the product will give 16 times more motion; the resistance will act consequently 16 times as much on the large as on the small; and as that resistance operates on the arm of a lever twice as long, the total resistance of the large ship will be 32 times as great. Thence it follows, that if the forces which act on the large ship be augmented only in proportion to her solidity, she will have still four times more difficulty than the small one to get into motion: and therefore the large ship, instead of making in the same time an angle of rotation as great as the small one, will only make an angle of one fourth, or three times less. Now, that the great ship should describe an angle of rotation equal to the other vessel, it will require only thrice as much time: but that angle, or the velocity with which the ship obeys the impulse of her rudder and sails, will follow the laws of acceleration, since the velocity acquired in the first instant is continually augmenting in arithmetical progression; so that the time which similar vessels of different sizes take in performing the same evolution, will be in proportion to their lengths. But the heavier body parts with its velocity not so readily as the lighter body, because the resistance of the mass is greater, being three times heavier than that of the small ship; which being moved with thrice the facility, is also brought to rest with the same degree of ease. So that, if a vessel 100 feet long takes four minutes to perform an evolution, a similar vessel of 150 feet will take six minutes or thereabouts to perform the same circular movement. For as 100:150 :: 4:6.

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CHAPTER VII.

OF THE HEIGHT OF MASTS.

THE correct height for the masts of ships is still a problem which remains to be solved for the builders. The most skilful of them have not paid attention enough to the solutions and determinations which are contained in the Works of the late Mr. BOUGUER on that subject. It seems, on the contrary, as if they had endeavoured to deviate, as much as possible, from the true principles in that respect, by raising the masts a great deal more than they were formerly, although they were already much too high, as the learned Author I have just mentioned has asserted. An experience, confirmed by repeated observations, has convinced me of this truth; viz. that "as soon as a ship inclines, her velocity diminishes in proportion as her inclination increases." This principle has been verified on different vessels, and at different times by several officers; and in various oblique courses. I had no share in those various experiments, and therefore cannot be suspected of partiality: but, as they have always proved, to those who have made them, that the present mode of masting is generally too high, I will not hesitate a moment longer to deliver here an epitome of my own experiments on that subject.*

Having all the sails out, and being hurried on by a strong gale, I have ordered all the top-gallant-sails, the studding and stay sails, to be taken in, without the ship losing the least perceptible degree of her velocity; nay, I have seen it sometimes to increase by a twentieth, and that at a time when the ship ran already at the rate of nine or twelve knots an hour.

These trials, which I have made with care, and which were performed so quickly, that the wind should not have time to increase or diminish in strength, are sufficient to prove the necessity of lowering the center of effort of the sails in general, and consequently all the masts. These experiments have been repeated in augmenting the number of sails, sometimes at the risque of fatiguing the masts; and it has always been found that the velocity did not increase, when the ship was more inclined; but that she laboured more and more in all her parts, as her movements became stronger and the concussions of her pitchings rougher, although the sea was not more swelled. At other times, when the ship inclined pretty much, though the wind was not quite strong enough to hurt the masts, I have lessened the number of sails; and it happened that the ship, after that suppression of the top-sails, was easier in her movements, steered better, and was, in short; more quiet, though the swells of the sea were still the same; an attention which must not be neglected in these kinds of observations, which should be often repeated before a positive decision. However, we do not recommend any diminution in the surface of the sails, in lessening their height: but, it will often happen that we shall rather recommend to increase it upon the whole. For that which is lost in height, may be regained by the width. There will even result, from that operation, another advantage: the top-sails, by this reform, being shorter and, thereby, proportionally wider than the lower sails, will be more easily cut to their shape; and their sides being formed with lines exactly strait, the sail will be the more tight, by which a much greater effect on the ship will be produced. The masts being shorter, and the sails

* We have thought it proper to give the reasoning of M. Bourde upon this subject although the practice of high-masting prevails in the British Navy.

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wider, with less fall, the surface will be the same: but the effort of that surface will, with the same wind, act on shorter levers, the fulcrum of which will not be altered; therefore, it will operate at a shorter distance from that fulcrum; and therefore much less will be the power which makes the ship incline: and the ship, being more upright, will sail with more velocity, because her water-lines will be then more advantageous than when she heels. On the other hand, the sails being less inclined, they present a wider surface to, and receive a stronger impulsion from, the wind; an advantage which must always produce an increase of swiftness and a decrease of drift. Add to this the real advantages of trimming the sails better, of working them with more ease, of rendering the masting in general more solid, and more capable of resistance in bad weather, as well as in battle.

But, how must we determine the height for the masts? or, in other words, how much they are to be shortened? The Treaties on the perfect masting and working of ships, by M. BOUGUER, teach us that method. It is from those Treatises I have imbibed the notions of my principles on that subject. But, in order to give a previous idea of that inquiry, and to engage the builders and seamen to bring to perfection this part both of the building and working of ships; upon which, almost as much as from their bottom, their steerage undoubtedly depends, I will subjoin here what M. BRUE, a learned and studious Officer, made me conceive on that subject.

"That masting," said he, "is absolutely perfect, when the center of effort of the sails is precisely opposite to, or at the same height as, or parallel with, the point velique. What is the point velique ? It is that point in a perpendicular, (raised from the center of gravity of the horizontal surface of the ship at the floating line,) which is intersected by the direction of the absolute impulse of the sea on the head of the vessel. This is the point-velique in direct courses."

It is clear; no great effort of imagination is necessary to conceive this principle, which appears so evident, that it may be surprising why it has not yet been made use of. For this point once known, the center of effort of the sails will be so too; and their right height, as well as that of the masts, will be determined. A little more calculation, and an attention to the plan of the ship will be necessary, in order to find out that absolute direction of the effort of the impulsion of the water on the bows. But that should not prevent the enquiry. On the contrary, it should be an additional inducement to those who, building such good vessels as we are now possessed of, and which might still be of a more advantageous form, will be desirous to make them more perfect, by masting them more advantageously. This would undoubtedly be the case; for several vessels have had their masts cut shorter, and the practice has been attended with decided success. These facts, which could be attested by many able seamen, will always speak highly in favour of this principle; although, when that shortening was made, the sails were not widened in proportion.

"But," continues M. BRUE, "in carrying this inquiry farther than it ever was, the intersection of the two above-mentioned lines, (viz. that of the absolute impulse of the water on the bows, and that of the perpendicular at the center of gravity of the surface of the floating line of the ship,) cannot take place unless in a direct course; and, as soon as the course becomes oblique, they no longer meet. The center of gravity of the floating line's surface of the ship passes to the leeward of its axis, on account of the inclination which always occurs in that sort of course; and the direction of the shock of the fluid, which then takes its origin a little to leeward also of the bow, passes, in its prolongation, to windward, without meeting the perpendicular at the center of gravity of the floating line's surface;" (which is easily conceived, if we represent to our imagination the horizontal edge of that floating line's surface but ever so little inclined;) " whence it results that

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"no point-velique will be found in any course but a direct one: which is true; unless we could fancy such a ship as would neither drive nor incline in an oblique course: but that is not possible; and hence no perfect mode of masting could be discovered in the last case of the oblique course."

This is true, strictly speaking: for, in each instant of a course a different point of the bow is struck by the water; which is owing to the pitching of the ship, the continual variations in the strength of the wind, and the greater or smaller inclination produced by the rolling motion of the ship.

"But," says again M. BRUE, " the point-velique, relative to these various circumstances, varies therefore in the proportion of the almost infinite variety of those circumstances, which accompany the course of a ship, that is to say, according to all the degrees of drift, all the degrees of inclination on either board, forward or abaft; as many times, in short, as there are new points of the bow either struck, or no longer struck, the point-velique ascends or descends.

"I pass over the minute examination I could make of each particular cause which contributes to lower that point from its utmost height, which is in the direct course, to its lowest, which takes place in the most oblique course, accompanied with the greatest lateral inclination of the ship: and there is no method to get out of that common road which is pursued in determining the dimensions of the masts, but that of attending to the following considerations; viz. Such a ship being intended for such a latitude, the wind she is most commonly to expect there, will be nearly of such a strength, and generally oblique to her course by so many degrees: so that her most common drift will be nearly so many degrees, and her lateral inclination so many, &c. To give her, therefore, the most suitable masting, making her relatively perfect, we must seek for her point-velique in what situation we shall think most convenient, and there place the center of effort of her sails."

All this reasoning tends evidently to the shortening of all the masts, and proves the necessity of doing it, at the same time as it determines their height. The most difficult point, in that operation, is to find out the direction of the absolute impulsion of the water on the bows, when the ship steers a course close hauled and one with the wind on the beam, with such an inclination as the ship could be supposed to have in either of these two courses; when the wind would allow to have four square-sails set, together with the mizen top-sail. Considering these two suppositions of the wind on the beam, and close hauled, it will be easy to determine the height of the masts proper for that double situation; because, if the gale blows harder, one may lessen the number of sails; if weaker, one may increase it by adding stay-sails, top-gallant sails, jib, &c: if the wind gets more aft, then the surface of the sails may be increased again by adding the studding and top-gallant royal sails: finally, it is very clear that top-gallant and top-gallant royal sails will always be of service when the center of effort of the sails should ascend.

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CHAPTER VIII.

OBSERVATIONS ON THE DIFFERENT INCLINATIONS GIVEN To THE MASTING OF SHIPS, WITH RESPECT TO THE WATER-LINE.

THE masts are hardly ever stepped in the same manner in all ships. This, too, is one of those things which are rather regulated by custom than reason. Some will have them perpendicular, while others chuse to have them rake forward, and others aft. Each party bring, to support their opinion, reasons drawn from some experiments which chance has sometimes rendered specious.

In this respect, we should rely on the judgement of the builder, who ought to know the qualities of his ship even before he puts her on the stocks. If one has not an opportunity of taking directly from him the necessary instructions, it is proper to observe that, if the masts are made to rake forward, the direction of effort of the sails will be inclined towards the bottom, obliquely with the horizon; which will consequently make the head of the ship plunge whenever they receive a strong impulse from the wind; and this may diminish the head-way of the ship, while it increases the celerity of the pitching: the sails also will be with more difficulty trimmed, especially when close hauled, since the bracing of the yards will be more confined. Therefore, the only advantage which can be drawn from this oblique masting of ships, is only to render the ships more ready to fall off.

If the masts are perpendicular the direction of effort of the sails will be horizontal, always supposing the ship to be in an upright position. Therefore, this effort not being discomposed, it will preserve a much greater action, and the ship will sail with the greatest velocity she is capable of.

If the masts rake aft, the ship will be more ready to come to the wind, because the sails will be a little more aft: these will also be more easy to trim sharp because the braces will not be so much confined. As this position of the sails will raise obliquely above the horizon the direction of the effort on the ship, it follows that, by their power, the ship will be eased away from the water: for, it is certain that she will not prolong her course, unless she heels too much; therefore, she will rise more lightly over the waves, pitch less, keep better the wind, and tack quicker. This is nearly all that can be said in respect to practise.

CHAPTER IX.

OF THE TENSION OF SAILS, AND THEIR TENDENCY TO FIX THEMSELVES PERPENDICULARLY TO THE DIRECTION OF THE WIND.

I. IT is clear that sails are never perfectly flat. But every one is not persuaded that the more extended is the sail, the greater impulsion it receives from the wind, which strikes it perpendicularly, and the more effectually, of course, the sail acts on the vessel. It is astonishing that any seamen should be of opinion that a bag must be left at the foot of the sail, to lodge the wind in. A hauled-

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down top-sail has as much cloth displayed in it as when hoisted up and well extended. It forms then, by its convexity, a considerable kind of bag, in which the wind may play at ease; and it is observed that the rapidity of the sailing decreases very much; whence we must necessarily conclude, that the impulse of the wind must have greatly diminished, since the sail produces no longer the same effect upon the vessel. To know demonstratively the cause of that diminution in the impulse of the wind, we have only to pay attention to the air which acts against the foot and the head of the sail; for that part of the wind, which strikes at the head, makes an effort to re-act towards the foot against that which, having struck at the same instant at the foot, endeavours to re-act towards the head. From this shock results (though the air escapes at each side) a compression in the sail. But, after having acted inwardly in the same manner as if it were shut up, it finds itself more and more compressed by that which succeeds to the first; and, though it escapes by the sides, it is evident that it tries to extend, and that it impels consequently with an equal power, all the parts of the sail perpendicularly; and this is the cause of the sail taking the form of a circle's arc. Therefore the sail will produce no greater effect than if it had no greater height than the space contained between the two yards: it may not even, strictly speaking, have that whole effect; for, that sort of whirlwind, which is made in the center, by the re-action of the wind which strikes the upper and lower parts, cannot fail to diminish the shock of those particles, which succeeding the former, would have struck the sail with all their primitive power; instead of which, this power is now almost intirely destroyed by this barrier which opposes for a while their passage. To which may be added, that the sail having the form of the arc of a circle, very little wind can strike it perpendicularly; and that it must, of course, have much less effect than another sail, of the same height and width, which should be very exactly stretched out.

The sails of a ship should be cut in such a form, as to present as flat a surface as possible.

II. The center of effort of the impulse of the wind upon the sails, exposed perpendicularly to the course of the wind, answers exactly to the center of gravity of the surface, struck in that direct situation. But, as soon as it is presented obliquely to the course of the fluid, and kept so, the center of effort of the total impulse will pass on the weather side of its center of gravity; because the particles of air which at first met the surface, have been re-acted, and by that re-action, they stop part of the passage to the succeeding ones, which diminishes of course both the strength of the shock and the impulse they would have communicated to the sail, if their movement had not been interrupted. But, this deviation of re-action in the first particles of air which have struck, is repeated afterwards. For, all those which succeed them while the surface is kept obliquely to the wind, continue to re-act to leeward: so that, from the first vertical line (taken from the windward side) out of all those which form together the surface, there is a continual series of obstacles which change the shock of immediate and succeeding particles, and which alters it so much the more as they ought to strike the parts of the sail most to leeward, and so much the less as they will strike those which are most to windward. Therefore, the leeward side of sails, obliquely exposed to the wind, is always less struck than that which stands to windward. Whence it results that the center of effort of the absolute impulse of the wind on the sail, is lodged in the weather side of the sail, (for it is supposed to be equally divided in two,) since that is the part which receives more impulsion. Therefore, the center of effort is also to windward of the center of gravity of the surface; and the removal of this center of effort towards the wind, is in proportion to the impulse received on the weather side of the sail, and that received on the lee side. The truth of this assertion is continually demonstrated by daily experience of ships at sea. The sails are carried by the yards and by the masts, which divide them perpendicularly into two equal parts, from top to bottom, through their center of gravity. When, being placed obliquely to the wind, they are left at liberty, without being confined by their braces or bowlines, they immediately range themselves

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perpendicularly to the course of the wind, because their weather side receives more impulse than the lee side; and there they remain constantly, unless their position be altered; because all their parts are struck equally, and an equilibrium is kept among them; for, the power of the wind, whether it increases or decreases, acts always the same on them all.

This proof, which shews the difference between the center of gravity and the center of effort in the sails, requires much attention in the use of that knowledge in practice. For example, in the middle of the yards on their aft side, there might be fixed a cleat, or bolster, which, in oblique courses, pushing them to leeward, would ease them off from the shrouds, and facilitate their bracing in carrying their center of gravity, as well as the center of the absolute effort, a little to leeward; which operation would of course draw that center of gravity nearer to the axis of the ship, from which it is so essential to remove it as little as possible.

CHAPTER X.

GENERAL OBSERVATIONS ON THE EFFECT OF MORE OR LESS SURFACE OF SAILS EXPOSED, IN VARIOUS WEATHERS, To THE WIND.

I. WHEN a ship, with a certain quantity of sail has acquired the utmost velocity with the power which then puts her in motion, it is certain that, if the surface of the sails is either increased or diminished, the rapidity of the head-way will likewise augment or lessen in a very complicated ratio. In order to find out the degree of impulsion of the wind on the sails, multiply their surface by the square of the excess of the velocity of the wind on that of the ship, or, which is the same thing, by the square of the apparent velocity of the wind. Then, a second multiplication of that product is to be made by the square of the sine of the angle of absolute incidence, or, in the second case, by the square of the sine of apparent incidence. And this second product will give the degree of the absolute impulse of the wind on the sails, in the actual state which we have supposed.

In order to find in what ratio the surface of the sails is to be augmented to make the ship acquire a certain degree of velocity above that which she possessed under a supposed particular quantity of sail, it must first be known by how much the velocity of the wind exceeds that of the ship: then, knowing how many degrees her head-way is wished to be accelerated, the sails must be increased in the ratio of the squares of the two velocities of the ship; viz. that which was known before the alteration of the sails, and that which she is afterwards to acquire. But, as the ship recedes so much the more from the action of the impulse of the wind as her velocity increases, it is evident the surface of the sails must be increased also in the ratio of the square of the two excesses, that is, the different excess of the wind over that of the ship both before and after the increase of the sails: then, the ship will acquire the wished-for velocity;

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provided no other cause happens to oppose it, as we have already hinted before, and as we shall have an opportunity to shew more particularly hereafter.

Suppose the wind has 12 degrees velocity, and the ship, under a certain set of sails, has 3; the velocity of the wind, in the direct course, will exceed that of the ship only by 9 degrees. If the velocity of the ship is intended to equal the third part of that of the wind, and to have therefore 4 degrees for head-way; then the sails are, to this effect, to be increased in the ratio of the squares of the squares of the two velocities 9 to 16, because the resistance of the water on the bows will increase in that very proportion. But, in the first case, the velocity of the wind exceeded that of the ship by 9 degrees, while in the second case it exceeds it no more than 8. Hence it results that the impulse of the wind on the sails has diminished in the ratio of the two squares 81 to 64: and, in order to repair that loss in the impulsion of the wind, the expansion of the sails is also to be increased in that last ratio of 64 to 81: then the ship will be able to run with the degree of velocity defined.

II. When the masting is perfect, that is to say, when the ship is masted according to the point-velique, she will rise from the water parallel to herself by a certain quantity relative to her velocity, and she will rise always more and more in proportion as she acquires new degrees of velocity in her head-way. Because she is moved by forces which stand exactly and continually in equilibrium with the action of the water on her bows, the inclination of which forward contributes so much the more to that rising out of the water as it is more remote from the perpendicular. For, then, the vertical impulsion will have more power, since it acts more directly on a very oblique bow than it would on a vertical one. This reasoning may be as exactly applied to the direct impulsion, the absolute effort of which may be decomposed, since it acts less against the velocity of the sailing on an oblique bow than on a vertical one, while the other part of its action joins with the vertical impulse to raise the head of the ship, which shocks the water with very great strength when she is arrived to a great velocity, and which water opposes her so much the more as it is shocked with violence. So it is easy to conclude that, in any ship whatever, the more rapid the head-way is the more parallel to herself she rises above the water, if the center of effort of her sails is at the same height as the point-velique: for the point of the bows, on which may be considered as united the action of the water which opposes its progress, may be taken also as the point of bearing. So that all the sails acting from abaft to forward on different points of the axis of the ship, (she being considered as a lever in the direction of her length), they raise the after part of that point, and place it on a level with the elevation of the bows; which never can happen, if the center of effort of the sails is above or below the point-velique. If it is placed above, the power of the sails, acting on too long levers, will raise the after part of the point of bearing of the bows above the level of the elevation of the ship's head. If it is placed below, the power of the sails, acting on too short levers, the after part of the ship will remain plunged in the water, without being able to rise on a level with the bows. Therefore, in either of the two cases, when the center of effort of the sails is either above or below the point-velique, the ship, however well-built, will lose some of the qualities of sailing, either in her readiness to obey the helm, or in her steadiness to carry sail, especially if she is over-masted: for, in this last case, she will gripe, incline easily, and lose much of her head-way, since her bows will plunge in the fluid, or, rather, her stern will rise too much out of it; which will diminish; the action of the water on the rudder and increase it on the bows. In the last case, an inconvenience of which ship builders seldom, if ever, have been guilty, the ship will be slow to obey, and her head-way will be slackened, because she will never present her most advantageous water-lines to the fluid, nor have a sufficient

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surface of sails, as, although their width is the same, their height is not so. The point of perfection then is this, viz. when the center of effort of the sails is placed at the height of the point-velique.

III. The next proposition will appear a paradox to many seamen. But, it is no less self-evident.