RUDDER. 



this angle, they have prefumed that the ihip is as narrow at 

 her floating line, or at the line defcribed by the furface of 

 the water round her bottom, as at the keel ; whereas all 

 veflels increafe in breadth from the keel upward to the 

 extreme breadth, where the floating line, or the higheit 

 water-line, is terminated ; and, therefore, the angle above 

 dated is too large. For the rudder is impreiled by the wa- 

 ter, at the height of the floating line, more directly than at 

 the keel, becaul'e the fluid exactly follows the horizontal 

 outlines of the bottom ; fo that a particular pofition of the 

 helm might be fuppofed necelfary for each different inci- 

 dence which it encounters from the keel upwards. But as 

 a middle pofition may be taken between all thefe points, it 

 will be fufficient to confider the angle formed by the fides 

 of the fhip, and her axis, or the middle line of her length, 

 at the furface of the water, in order to determine afterwards 

 the mean point, and the mean angle of incidence. The 

 angle 54 44', it is faid, is too open, and very unfavourable 

 to the (hip's head-way, becaufe the water acts upon the 

 rudder there with too great a fine of incidence, as being 

 equal to that of the angle which it makes with the line pro- 

 longed from the keel below ; but above, the fhock of the 

 water is almoit perpendicular to the rudder, becaufe of the 

 breadth of the bottom, or that of the (hip's fides. If then 

 the rudder is only oppofed to the fluid, by making an angle 

 of 45 J , or 45 1', with the line prolonged from the keel, the 

 imprellion, by becoming weaker, will be lefs oppofed to the 

 (hip's head-way, and the direction N P {Jig. 4.) of the ab- 

 lolute effort of the water upon the rudder, approaching 

 nearer to the lateral perpendicular N L, will be placed 

 more advantageoufly, fince the prolongation of the abiolute 

 effort paffes at a greater dillance G R from the centre of 

 gravity of the fliip. To which it is added, that experience 

 daily teitifies, that a (hip fleers well, when the rudder makes 

 the angle D B E no more than 35 . If this angle be made 

 45 , and the abfolute effort N P be difcompofed, we (hall 

 have N I equal to the other iide N L of the fame fquare ; fo 

 that the part of the whole power which oppofes the h. ad- 

 way of the (hip would be only equal to that which pro- 

 duces the movement of rotation ; inftead of which, if D B E 

 were 54 44', N I would become much greater than N L, 

 in proportion to the fines of the angles which are oppofed 

 to them in the triangles PIN or PLN, and the (hip 

 would lofe much more of her velocity than in the firft iitua- 

 tion of the rudder, which is thought to be belt adapted to 

 veflels in general, fubject, however, to occafional altera- 

 tion, as they fhall make an angle more or lefs open with 

 their fide' a-ftern. Hence it has been concluded, as a ge- 

 neral pofition, thit the moll advantageous angle will always 

 be found between 35 and 45 . From Bouguer's treatife 

 (ubi infra), it appears, that, in moft fhips, the angle of the 

 rudder with the prolonged line of the keel (hould be 46 40'. 

 L.. Euler recommends an obliquity fomewhat lefs than 54 

 44', for the greatelt action of the rudder; and eltablifhes 

 this rule, that an obliquity of about 48 will, m general, 

 produce the bell effort. 



Tliofe who duly confider the action of the helm and tiller 

 (for an account of which, fee thofe articles) will eafily 

 conceive, that the greater the fhip's velocity is, the more 

 powerful is the action of the rudder, fince it acts againll the 

 water with a force which increafes as the fquare of the ve- 

 locity of the fluid, whether the (hip has head-way, or flern- 

 way ; observing always, that in thefe two circumftances the 

 effects are contrary ; for, if the (hip goes a-ftern, the rudder 

 will be ftruck from I to N [jig. 4.) ; and, inftead of being 

 pufhed from N to P, it will be fo from N to R ; fo that the 



9 



flern being moved in the fame direction, the head will take 

 a contrary one, and move towards the fame fide as the 

 tiller B F. 



It fhould be obferved, in the ufe of the rudder, that there 

 is one part of its effort which impedes the fhip's failing, 

 when it is flruck by the water which runs rapidly along the 

 fhip's bottom. If it makes an angle of 45 with the keel 

 prolonged, it receives only half the impullion it would, if 

 acted upon perpendicularly ; becaufe the abfolute impulfe 

 diminifhes from two caufes. The furface which oppofes the 

 fhock of the water is reduced to a lefs extent than it was at 

 firft, and the angle of incidence diminifhes likewife ; fo that 

 by this, the impulfe has diminilhed one half. Confidering 

 next the impullion N P, which remains (Jig. 4), it will 

 appear that there is only one part N I which is oppofed 

 to the failing, and which is lefs than N P, in the proportion 

 as the fine total is to the fine of 45°, the meafure 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 ; fo that, if 

 you take away the common angle LNB, the two anpdes 

 P N L and V N B will remain equal between themfelves ; 

 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 lefs open 

 with the keel prolonged. So that, if the furface of the 

 rudder which receives the fhock be 80 feet fquare fuper- 

 ficies, it will firll be reduced, by its being expofed to the 

 courfe of the fluid, to an effort of 40 feet furface, then to 

 28 or 29 ; becaufe, in the firft place, there is only one part 

 of the velocity of the water which contributes to the fhock, 

 and that is proportional to the relation of the fquare of the 

 fine total to that of the fine of incidence ; and, fecondly, be- 

 caufe out of the abfolute impulfe N P, which refults from 

 this lad oblique fhock, there is only a part N I which op- 

 pofes the velocity of the (hip proportional to the abfolute force 

 N P, in the fame relation as there is between the fine total 

 and the line of incidence ; that is to fay, that when the 

 rudder makes, in the largeft fhips, an angle of 45 , it im- 

 pedes the fliip's rapidity of failing, in the direction of the 

 keel, with an effort N I, equivalent to the impulfion which 

 a furface of 28 or 29 feet fquare might receive, if it were 

 expofed perpendicularly to the fhock of the water. So 

 that, if the fhip fails 12 knots an hour, or 19 feet a fecond, 

 the effort of the rudder N 1, which oppofes the fhip'6 way, 

 will be 12,499 or I2 '945 pounds ; fait water weighing /^th 

 more than frefh. 



It follows, from all that has been faid of the rudder, that 

 it ought to be employed as little as poffible ; that is to fay, 

 the fhip and her lails ought to be fo difpofed, that the 

 fmalleil motion of this machine may bring her to her courfe, 

 if fhe deviates from it, or make her perform any evolution 

 which may be thought proper. 



The ingenious writer of the article Seamanjh'ip in the 

 Enc. Brit, iuggeits, that the theory of M. Bouguer and 

 fome other French mathematicians is founded on principles 

 that are erroneous. They aflume, as we have above fup- 

 pofed, that the impulfe of a fluid is in the proportion of the 

 fquare of the fine of the angle of incidence ; and alfo that 

 its action on any fmall portion, fuch as a fquare foot of the 

 fails or hull, is the fame as if that portion were detached 

 from the reft, and were expofed, tingle and alone, to the 

 wind or water in the fame angle. Both thefe principles, it 

 is affirmed, are erroneous ; and the error is very confiderable 

 in cafes which molt frequently occur in practice, that is, in 

 the fmall angles of inclination. The error of this theory, 

 efpecially in cafes of great obliquity, may be feen in the 



following 



