30

To deter

difficult. It is very easily expressed geometrically: Divide the angle w CF in such a manner that the tangent of DCF may be half of the tangent of DCw, 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 CQ=+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 w 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 = 9; then, by means of the equation B A cotan. b, we must obtain the value of b and of ¿ in terms of r and x. With respect to a, observe, that if we make the angle WCF=p, we have p=a+b+; and p being a constant quantity, we have a+b+x=o. Substituting for a, b, a and b, their values in terms of r and x, in the fluxionary equationo, 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 the trim of the sails most proper for plying to windward.

In fig. 6. draw FP perpendicular to WC. CF is the Problem I motion of the ship; but it is only by the motion PC mine the that she gains to windward. Now CP is = CF x Course and cosin. WCF, or v cosin. (a+b+x). This must be trim of the rendered a maximum, as follows.

sails most

By means of the equation which expresses the value proper for plying to of and the equation BA cotan. b, we exterminate adward the quantities v and b; we then take the fluxion of the quantity into which the expression v cos. (a+b+x) is changed by this operation. Making this fluxion =o, we get the equation which must solve the problem. This equation will contain the two variable quantities o and x with their fluxions; then make the coefficient of x equal to o, also the coefficient of a equal to o. This will give two equations which will determine a and x, and from this we get b-p-a-x.

31

Problem

termine the best course

Should it be required, in the third place, to find the To de best course and trim of the sails for getting away from a given line of coast CM (fig. 6.), the process perfectly and trim of resembles this last, which is in fact getting away from the satis for a line of coast which makes a right angle with the wind. Eetting a Therefore, in place of the angle WCF, we must substi

Bay from

a givel Line of

Podst 32 ObservaCoas on the preceding

problems.

tute the angle WCM+WCF. Call this angle e. We must make v cos. (e±u±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 obtain

ed 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° from the wind, and the velocity in that course; then for 850°, 800, 75°, &c. M. Bouguer has given a table of this kind; but to avoid the immense difficulty for finding of the process, he has adapted it to the apparent direc- the best tion of the wind. We have inserted a few of his num- position of the sails for bers, suited to such cases as can be of service, namely, when all the sails draw, or none stand in the way of in any advancing others. Column 1st is the apparent angle of the wind course.. and course; column 2d is the corresponding angle of the sails and keel; and column 3d is the apparent angle of the sails and wind.

In all these numbers we have the tangent of w CD double of the tangent of DCF.

33 M. Bou

guer's table

34

But this is really doing but little for the seaman. Inutility of The apparent direction of the wind is unknown to him these calcutill the ship is sailing with uniform velocity; and he is lations. 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 w 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 ou
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.
L 2
Let

Fig. 8,

35 To determine the

for avoid

Let us see whether this restriction, arising from necessity, leaves any thing 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, sailing 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 sailing slowly on one course than by sailing 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 ber 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 sail 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 f O. Then, because the angles CFO, cƒ O are on the same chord CO, they are equal, and ƒ O is parallel to dC d, the new position of the yard corresponding to the new position of the keel a b, making the angle d Cb=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 CO f, that is (on account of the parallels CD, OF and C d, Of), as the sine of WCD to the sine of WC d. 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 best course withdrawing as fast as possible from a line of coast PQ. ing a rock. Draw CM through R, or parallel to PQ, and let m be the middle of the arch Cm M. It is plain that m is the most remote from CM of any point of the arch Cm M, 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 Cm M 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 ta

ken from 360°, leaves 150°, of which the half M m is 75°, and the angle MC m is 37° 30'. This added to ECM makes ECm 107° 30', leaving WC m=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 sail which the ship can carry with safety and advantage she makes 5 degrees of leeway, the angle DC m of the sail and course, or b+x, is 40°. Then CO+OM=220°, which being taken from 360° leaves 140°, of which the half is 70°, M m, and the angle MC m = 35°, and EC m = 105°, and WC m = 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 sail and the real direction of the wind.

36

It is plain that this problem includes that of plying Corollaries, 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 WC n=90-25= 65, and the ship's head must lie 60° 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 cour-es only where the practicable trim of the sails is not sufficiently sharp. Whenever the course lies so far from the wind that it is possible to make the tangent of the apparent angle of the wind and sail double the tangent of the sail and course, it should be done.

37

These are the chief practical consequences which can The adjustbe deduced from the theory. But we should consider ment of the how far this adjustment of the sails and course can be sails suppoperformed. And here occur difficulties so great as to sed in the make it almost impracticable. We have always suppo-practicable. theory imsed the position of the surface of the sail to be distinctly observable and measurable; but this can hardly be affirmed even with respect to a sail stretched on a yard. Here we supposed the surface of the sail to have the same inclination to the keel that the yard has. This is by no means the case; the sail assumes a concave form, of which it is almost impossible to assign 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 sail so exceedingly feeble as hardly to have any effect. In sailing close to the wind the ship is kept so near, that the weather-leech of the sail is almost ready to receive the wind edgewise, and to flutter or shiver. The most effective or drawing sails with a side-wind, especially

when

when plying to windward, are the staysails. We believe that it is impossible to say, with any thing approaching to precision, what is the position of the general surface of a staysail, 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 sails 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 sails 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.

38 The theory But there is another principal reason for the small self eiro 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. Ist gives the angle of incidence; col. 2d gives the impulsions really observed; col. 3d the impulses, had they followed the duplicate ratio of the sines; and col. 4th the impalses, if they were in the simple ratio of the sines.

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.

icss.

39

No wonder then that the deductions from this theory and the deare so useless and so unlike what we familiarly observe, ductions from it useWe 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 pressing 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 impulses 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 impulses 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 assigns 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 sine 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 180, we have the experimented impulse 414, which, when reduced in the proportion of radius to the sine of 50°, gives an effective impulse317. In like manner, the trim 56°, with the incidence 12°, gives an effective impulse 337; and the trim 62°, with the incidence only 6o, gives 353.

Hence it would at first sight appear that the angleDCB of 62° and WCD of 6° would be better for holding a course within 6 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 freshens 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

4r

founded on judicious experiments only, than frem a theory of the impulse of fluids, which is found so inconsistent with observation, and of whose fallacy all its au hors, 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 recomFOR THE IMPROVEMENT OF NAVAL ARCHITECTURE mended to the Society Should these patriotic gentlemen entertain a favourable for the Imopinion of the plan, and honour us with their corre-provement spondence, we will cheerfully impart to them our no- of Naval tions of the way in which both these trains of experi- Architecments may be prosecuted with success, and results ob tained 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 mere science, but of one who, with a competent knowledge of the laws of mechanical nature, bas the experience of several years service in the royal navy, where the art of working of ships was a favourite object of his scientific at

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, Experi- founded entirely on experiments. For this purpose the ments pro- experiments should be made on the oblique impulsions per for esta- of air on as great a scale as possible, and in as great a Llishing anvariety of circumstances, so as to furnish a series of imother pulsions for all angles of obliquity. We have but four or five experiments on this subject, viz. two by Mr Robins, and two or three by the Chevalier Borda. Having thus gotten 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 of 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.

4C

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 lateen 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 40, whose direction is experimentally found equivalent to a flat surface trimmed to the obliquity 40°.

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', &e. 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

tention.

ture,

42

With these observations we conclude our discussion Means emof the first part of the seaman's task, and now proceed ployed to to consider the means that are employed to prevent or prevent or to produce any deviations from the uniform rectilineal viations produce decourse which has been selected.

from a

Here the ship is to be considered as a body in free course. space, 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.

43

When a ship sails right afore the wind, with her yards Impulses square, it is evident that the impulses on each side of the of a ship keel are equal, as also their mechanical momenta round sailing right before the any axis passing perpendicular through the keel. So wind differare the actions of the water on her bows. But when she ent from sails on an oblique course, with her yards braced up on those on either side, she sustains a pressure in the direction CI her when sailing ob (fig. 5.) perpendicular to the sail. This, by giving her liquely. a lateral pressure LI, as well as a pressure CL ahead, causes her to make leeway, and to move in a line C binclined 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.