To illustrate this, he remarks that if the breadth of a vessel be regarded as variable, the burthen of the ship will vary as the cube of that dimension, and the velocity of sailing as the cube root of the same. The number of the crew also, which is proportional to the area of the 2 sails II' will vary as B'. Hence, by supposing two ships, whose capacities for burthen are in the ratio of eight to one, and their relative velocities as ten to eight ; if the larger vessel sail with a crew of twenty-four men, the smaller will require four. According to the capaci ties of the two ships, the latter ought to be navigable by three men. hence also we perceive, that in making small ships similar to large ones, the former will pos sess the worst sailing qualities, and will require a more numerous crew in proportion to their capacities than large ones.
Mr. Chapman also remarks it is possible to render a small ship navigable by a crew proportionate to its capa city; but that it cannot be done without diminishing the quantity of canvass, and then the sailing qualities of the vessel will be impaired. This fault may be remedied to a certain degree, by giving it less breadth ; but this would be attended with inconvenience. Hence we pre fer in small ships the property of sailing well, to having it in our power to diminish the crew.
The velocity also being in proportion to the quantity as the depth decreases, supposing at DI the same time the length and breadth to increase. This object may be attained more easily by adding to the length, but for the greater safety of the navigation it is more convenient to increase the breadth. This will ele vate the metaccntre. The sails also may, in such a ease, have an increased surface; hut the ship would require a more numerous crew.
Great and small ships moreover cannot, with the same form, sail with the same security; nor can we avoid the inconvenience of being obliged to have a more numerous crew in small ships than in large. Small ships, there fore, cannot have the same advantages as large ones, wben they are employed in the same trade.
As small ships lose in the quality of sailing, by being assimilated in form to large ones, so large vessels will improve this valuable property by being moulded simi lar to small ones. Mr. Chapman hence concludes, that it is proper to give to large ships forms similar to small ones, because they would thereby gain in the quality of sailing. But for merchant ships, where it is so much the more necessary to give great capacities in the water, as they are larger, they seldom want a superior quality of sailing, provided they are sufficiently stiff upon a wind not to be embayed on a lee shore. Added to this, in such a case, these ships would lose the advantage of sailing with a small crew, and as they cost more in con struction in proportion than small ones, it is necessary in their formation to endeavour to combine qualities most advantageous to the interests of their owners.
In the following table is recorded the values of cer tain essential elements in shipbuilding, derived from one primitive clement, the burthen, and which, owing its origin to Chapman, and deduced by him from a long course of experience, will be considered as very valu able.
The foregoing tables suggest some important re flections. It is evident that Chapman entertained the idea of connecting together all the essential elements of naval architecture, by means of empirical formu he, and of deriving all from some one primitive root or element. The bare conception of such an idea, marks the character of his mind in strong and origi nal colours; and the stepS he made towards its prac tical execution, stamp his name with double honour and renown.
The point to which naval architecture should con tinually approximate as a limit, ought to be the per fecting of the elements alluded to: obtaining for them more correct or appropriate coefficients; establish ing more completely their necessary relations, and throwing over the whole investigation a more accu rate and philosophical character. No method of pro cedure, we would remark, can be more consistent with the legitimate and proper objects of philosophi cal inquiry. In the language of the modern analysis, it would be regarding every element of a ship, as a function of some one primitive element, and, by means of properly prepared coefficients, deducing each one from it. There must, for example, be in every ship some relation between its length and displacement; so that by adopting one of them as its primitive term, the other by means of some multiplier or co-effi cient, ought to be deduced from it. The breadth, too, must be a function of the length, and therefore some function of the displacement, if that element (and perhaps it is the best) be adopted as the primitive one. The area of the main section may also be so connected with the breadth, as to be resolvable at first into terms of the length, and ultimately into that of the displacement itself. In like manner, may the area of the plane of flotation, the moment of stability, the place of the metacentre, the position of the cen tre of gravity, and indeed the value of every other el ement of a ship, be ultimately traced to the displace ment. 'Nile whole length on the water line be de noted by / the entire length of the same line between the rabbets, must be some multiple of the same di mension; and as the former may be shown to be a function of the displacement, so may the latter. So that the displacement, or some other appropri ate quantity, being assumed as a primitive element, every other element becomes connected with it; and no sooner is the relation of one part to another shown, than that relation becomes immediately connected with the element assumed. Hence adopting the lan guage of functions we may say with Lagrange, that if the displacement be denoted by the primitive func tionf D, in the series, f D, f" D,f" D, &c., the derived functions of the same series, f' D,f" D,f"' D, Stc., may represent, successively, the other elements of a ship.