all things floating on the surface of the sea or any water; a word much used in the commissions of water bailiffs.

FLOATING BODIES are those which swim on the surface of a fluid, the most interesting of which are ships and vessels employed in war and commerce. It is known to every seaman, of what vast moment it is to ascertain the stability of such vessels, and the positions they assume when they float freely on the surface of the water. To be able to accomplish this, it is necessary to understand the principles on which that stability and these positions depend. This has been done with great ingenuity by Mr Atwood, of whose reasoning the following is a summary account, taken from the Philosophical Transactions for 1796.

A floating body is pressed downwards by its own weight in a vertical line passing through its centre of gravity; and it is supported by the upward pressure of a fluid, which acts in a vertical line that passes through the centre of gravity of the part which is under the water; water; and without a coincidence between these two lines, in such a manner as that both centres of gravity may be in the same vertical line, the solid will turn on an axis, till it gains a position in which the equilibrium of floating will be permanent. From this it is obviously necessary to find what proportion the part immersed bears to the whole, to do which the specific gravity of the floating body must be known, after which it must be found by geometrical methods, in which positions the solid can be placed on the surface of the fluid, so that both centres of gravity may be in the same vertical line, when any given part of the solid is immersed under the surface. These things being determined, something is still wanting, for positions may be assumed in which the circumstances now mentioned concur, and yet the solid will assume some other position wherein it will permanently float. If the specific gravity of a cylinder be to that of the fluid on which it floats as 3 to 4, and its axis to the diameter of the base as 2 to 1: if it be placed on the fluid with its axis vertical, it will sink to a depth equal to a diameter and a half of the base; and while its axis is preserved in a vertical position by outward force, the centres of gravity of the whole solid and immersed part will remain in the same vertical line; but when the external force is removed, it will deviate from its upright position, and will permanently float with its axis horizontal. If we suppose the axis to be half the diameter of the base, and placed vertically, the solid will sink to the depth of three-eighths of its diameter, and in that position it will float permanently.

If the axis be made to incline to the vertical line, the solid will change its position till it permanently settles with its axis perpendicular to the horizon.

Whether a solid floats permanently, or oversets when placed on the surface of a fluid, provided the centre of gravity of the solid and that of the immersed part be in the same vertical line, it is said to be in a position of equilibrium, of which there are three kinds; the equilibrium of stability, in which the solid permanently floats in a given position; the equilibrium of instability, in which the solid spontaneously oversets, if not supported by external force; and the equilibrium of indifference, or the insensible equilibrium, in which the solid rests on the fluid indifferent to motion, without tendency to right itself when inclined, or to incline farther.

If a solid body floats permanently on the surface of a fluid, and external force be applied to turn it from its position, the resistance opposed to this inclination is termed the stability of floating. Some ships at sea yield to a given impulse of the wind, and suffer a greater inclination from the perpendicular than others. As this resistance to heeling, duly regulated, has been considered of importance in the construction of vessels, many eminent mathematicians have laid down rules for ascertaining the stability of ships from their known dimensions and weight, without recurring to actual experiment. Bouguer, Euler, Chapman, and others, have laid down theorems for this purpose, founded on the supposition that the inclinations of ships from their quiescent positions are evanescent, or very small in a practical point of view. But ships at sea have been found to heel 15°, 20°, or 30°, and therefore it may be doubted how far such rules are applicable in practice. If statics can be applied to naval architecture, it seems necessary that the rules should be extended to those cases in which the angles of inclination are of any magnitude, likely to occur in the practice of navigation. A solid body placed on the surface of a lighter fluid, at such a depth as corresponds to the relative gravities, cannot alter its position by the joint action of its own weight and the pressure of the fluid, except by turning on some horizontal axis passing through the centre of gravity; but, as many axes may be drawn through this point of the floating body, in a direction parallel to the horizon, and the motion of the solid regards only one axis, this must be determined by the figure of the body and the particular nature of the case. When this axis of motion is ascertained, and the specific gravity of the solid found, the positions of permanent floating will be determined, by finding the several positions of equilibrium through which the solid may be conceived to pass, while it turns round the axis of motion; and by determining in which of these positions the equilibrium is permanent, and in which of them it is momentary.

The whole of Mr Atwood's valuable paper relates to the theory of naval architecture, in so far as it is dependent on the laws of pure mechanics. If the proportions and dimensions adopted in the construction of individual vessels are obtained by exact geometrical measurement, and observations are made on the performance of these vessels at sea; a sufficient number of experiments of this nature, judiciously varied, are the proper grounds on which theory may be effectually applied, in reducing to system those hitherto unperceived causes, which contribute to give the greatest degree of excellence to vessels of every description. Naval architecture being reckoned among the practical branches of science, every voyage may be viewed in the light of an experiment, from which useful truths are to be deduced. But inferences of this nature cannot well be obtained, except by acquiring a thorough knowledge of all the proportions and dimensions of each part of the ship, and by making a sufficient number of observations on the qualities of the vessel, in all the varieties of situation to which a ship is commonly subject in the practice of navigation.

The following is an ingenious investigation of the same subject by Mr English, which we give in his own words.

"However operose and difficult (says he) the calculations necessary to determine the stability of nautical vessels may, in some cases, be, yet they all depend, says this author, upon the four following simple and obvious theorems, accompanied with other well known stereometrical and statistical principles.

"Theorem 1. Every floating body displaces a quantity of the fluid in which it floats, equal to its own weight; and consequently the specific gravity of the fluid will be to that of the floating body, as the magnitude of the whole is to that of the part immersed.

"Theorem 2. Every floating body is impelled downward by its own essential power, acting in the direction of a vertical line passing through the centre of gravity of the whole; and is impelled upward by the reaction of the fluid which supports it, acting in the direction of a vertical line passing through the centre of gravity of the part immersed; therefore, unless these two lines are coincident, the floating body thus impelled must revolve round an axis, either in motion or at rest, until the equilibrium is restored.

"Theorem 3. If by any power whatever a vessel be deflected..." Floating bodies deflected from an upright position, the perpendicular distance between two vertical lines passing through the centres of gravity of the whole, and of the part immersed respectively, will be as the stability of the vessel, and which will be positive, nothing, or negative, according as the metacentre is above, coincident with, or below, the centre of gravity of the vessel.

"Theorem 4. The common centre of gravity of any system of bodies being given in position, if any one of these bodies be moved from one part of the system to another, the corresponding motion of the common centre of gravity, estimated in any given direction, will be to that of the aforesaid body, estimated in the same direction, as the weight of the body moved is to that of the whole system.

"From whence it is evident, that in order to ascertain the stability of any vessel, the position of the centres of gravity of the whole, and of that part immersed, must be determined; with which, and the dimensions of the vessel, the line of floatation, and angle of deflection, the stability or power either to right itself or overturn, may be found.

"In ships of war and merchandise, the calculations necessary for the purpose become unavoidably very operose and troublesome; but they may be much facilitated by the experimental method pointed out in the New Transactions of the Swedish Academy of Sciences, first quarter of the year 1787, page 48.

"In river and canal boats, the regularity and simplicity of the form of the vessel itself, together with the compact disposition and homogenous quality of the burden, render that method for them unnecessary, and make the requisite calculations become very easy. Vessels of this kind are generally of the same transverse section throughout their whole length, except a small part in prow and stern, formed by segments of circles or other simple curves; therefore a length may easily be assigned such, that any of the transverse sections being multiplied thereby, the product will be equal to the whole solidity of the vessel. The form of the section ABCD is for the most part either rectangular, as in fig. 1. Plate CCXVIII. trapezoidal as in fig. 2. or mixtilinear as in fig. 3. in all which MM represents the line of floatation when upright, and EF that when inclined at any angle MXE; also G represents the centre of gravity of the whole vessel, and R that of the part immersed.

"If the vessel be loaded quite up to the line AB, and the specific gravity of the boat and burden be the same, then the point G is simply the centre of gravity of the section ABCD; but if not, the centres of gravity of the boat and burden must be found separately, and reduced to one by the common method, namely, by dividing the sum of the momenta by the sum of weights, or areas, which in this case are as the weights. The point R is always the centre of gravity of the section MMCD, which, consisting of different figures, must also be found by dividing the sum of the momenta by the sum of the weights as common. These two points being found, the next thing necessary is to determine the area of the two equal triangles MXE, MXF, their centres of gravity o, o, and the perpendicular projected distance n n of these points on the water line EF. This being done, through R and parallel to EF draw RT = a fourth proportional to the whole area MMCD, either triangle MXE or MXF, and the distance n n; through T, and at right angles to RT or EF, draw TS meeting the vertical axis of the vessel in S the metacentre; also through the points G, B, and parallel to ST, draw NGW and BV; moreover through S, and parallel to EF, draw WSV, meeting the two former in V and W; then SW is as the stability of the vessel, which will be positive, nothing, or negative, according as the point S is above, coincident with, or below, the point G. If now we suppose W to represent the weight of the whole vessel and burden (which will be equal to the section MMCD multiplied by the length of the vessel), and P to represent the required weight applied at the gunwale B to sustain the vessel at the given angle of inclination; we shall always have this proportion: as VS : SW :: W : P; which proportion is general, whether SW be positive or negative; it must only in the latter case be supposed to act upward to prevent an overturn.

"In the rectangular vessel, of given weight and dimensions, the whole process is so evident, that any farther explanation would be unnecessary. In the trapezoidal vessel, after having found the points G and R, let AD, BC be produced until they meet in K. Then, since the two sections MMCD, EFDC are equal; the two triangles MMK, EFK are also equal; and therefore the rectangle EK X KF = KM X KM = KM²; and since the angle of inclination is supposed to be known, the angles at E and F are given. Consequently, if a mean proportional be found between the sines of the angle at E and F, we shall have the following proportions:

"As the mean proportional thus found: sine ∠ F :: KM : KF, and in the said mean proportional: sine ∠ F :: KM : KE; therefore ME, MF become known; from whence the area of either triangle MXE or MXF, the distance n n, and all the other requisites, may be found.

"In the mixtilinear section, let AB = 9 feet = 108 inches, the whole depth = 6 feet = 72 inches, and the altitude of MM the line of floatation 4 feet or 48 inches; also let the two curvilinear parts be circular quadrants of two feet, or 24 inches radius each. Then the area of the two quadrants = 904.7808 square inches, and the distance of their centres of gravity from the bottom = 13.8177 inches very nearly, also the area of the included rectangle abce = 1440 square inches, and the altitude of its centre of gravity 12 inches; in like manner, the area of the rectangle AB cd will be found = 518.4 square inches, and the altitude of its centre of gravity 48 inches: therefore we shall have

\[ \begin{align*} \text{Momentum of the two quad.} & = 904.7808 \times 13.8177 = 12501.9866016 \\ \text{Moment of the rectan. abce} & = 1440 \times 12 = 17280 \\ \text{Moment of the rectan. AB cd} & = 518.4 \times 48 = 24832 \\ \end{align*} \]

Now the sum of the momenta, divided by the sum of the areas, will give \( \frac{27861.39866016}{75287808} = 37.006 \) inches, the altitude of G, the centre of gravity of the section. section ABCD above the bottom. In like manner, the altitude of R, the centre of gravity of the section MMCD, will be found to be equal

\[ \frac{1239379866616}{49367858} = 24'934 \text{ inches}; \text{and consequently their difference, or the value of GR} = 12'572 \text{ inches}, \text{will be found.} \]

Suppose the vessel to heel \(15^\circ\), and we shall have the following proportion; namely, As radius : tangent of \(15^\circ\) :: MX = 54 inches = ME or MF; and consequently the area of either triangle MXE or MXF = 390'663 square inches. Therefore, by theorem 4th, as 4936'7858 : 390'663 :: 72 = n = \( \frac{1}{2} \) AB : 5'6975 inches = RT; and again, as radius : sine of \(15^\circ\) :: 12'572 = GR : 3'1245 inches = RN; consequently RT - RN = 5'6975 - 3'1245 = 2'573 inches = SW, the stability required.

Moreover, as the sine of \(15^\circ\) : radius :: 5'6975 = RT : 22'013 = RS, to which if we add 24'934, the altitude of the point R, we shall have 46'947 for the height of the metacentre, which taken from 72, the whole altitude, there remains 25'033; from which, and the half width = 54 inches, the distance BS is found = 59'529 inches very nearly, and the angle SBV = 80° - 06' - 42''; from whence SV = 38'645 inches.

Again: Let us suppose the mean length of the vessel to be 40 feet, or 480 inches, and we shall have the weight of the whole vessel equal to the area of the section MMCD = 4936'7858 multiplied by 480 = 236995'4784 cubic inches of water, which weighs exactly 8'758 pounds avoirdupoise, allowing the cubic foot to weigh 62'5 pounds.

And finally, as SV : SW (i.e.) as 38'645 : 2'573 :: 85708 : 3760 + , the weight on the gunwale which will sustain the vessel at the given inclination. Therefore a vessel of the above dimensions, and weighing 38 tons 5 cwt. 28lbs. will require a weight of one ton 13 cwt. 64lbs. to make her incline \(15^\circ\).

In this example the deflecting power has been supposed to act perpendicularly on the gunwale at B; but if the vessel is navigated by sails, the centre velique must be found; with which and the angle of deflection, the projected distance thereof on the line SV may be obtained; and then the power calculated as above, necessary to be applied at the projected point, will be that part of the wind's force which causes the vessel to heel. And conversely, if the weight and dimensions of the vessel, the area and altitude of the sails, the direction and velocity of the wind be given, the angle of deflection may be found.

Floating Bridge. See Bridge. Flock Paper. See Paper.