360 PHILOSOPHICAL TRANSACTIONS. [anNO 175Q. 



The result was according to N^ 5, being nearly the same as the plane sails, in 

 their best position : but being turned round in their sockets, so that every part of 

 each sail stood at an angle of 3°, and afterwards of 6°, greater than before, that 

 is, their extremities being moved from g^ to 12° and \5°, the products were ad- 

 vanced to 518 and 527 respectively. Now from the small difference between 

 those two products, we may conclude, that they were nearly in their best po- 

 sition, according to N° 7, or some angle between that and N° 6: but from these, 

 as well as the plane sails and others, we may also conclude, that a variation in 

 the angle of a degree or two makes very little difference in the effect, when the 

 angle is near upon the best. 



It is to be observed, that a sail inclined by the preceding rule will expose a 

 convex surface to the wind : whereas the Dutch, and all our modern mill- 

 builders, though they make the angle to diminish, in receding from the centre 

 towards the extremity, yet constantly do it in such manner, as that the surface 

 of the sail may be concave towards the wind. In this manner the sails made use 

 of in N° 8, 9, 10, 11, 12, and 13, were constructed ; the middle of the sail 

 making an angle with the extreme bar of 12° ; and the greatest angle (which 

 was about -^ of the radius from the centre) of 15° with it. Those sails being 

 tried in various positions, the best appears to be that of N° 11, where the ex- 

 tremities stood at an angle of 7-^° with the plane of motion, the product being 

 639 t greater than that of those made by the theorem in the ratio of 9 : 11, 

 and double to that of N° 1 ; and this was the greatest product that could be pro- 

 cured without an augmentation of surface. Hence it appears, that " when the 

 wind falls on a concave surface, it is an advantage to the power of the whole, 

 though every part, taken separately, should not be disposed to the best 

 advantage." 



Having thus obtained the best position of the sails, or manner of weathering, 

 as it is called by workmen, the next point was to try what advantage could be 

 made by an addition of surface on the same radius. For this purpose, the sails 

 made use of had the same weather as those N° 8 to 1 3, with an addition to the 

 leading side of each of a triangular cloth, whose height was equal to the height 

 of the sail, and whose base was equal to half the breadth : of consequence the 

 increase of surface on the whole was a 4th part, or as 4:5. Those sails, by 

 being turned round in their sockets, were tried in 4 different positions, specified 

 in N° 14, 15, 16, and 17 ; whence it appears that the best was when every part 

 of the sail made a greater angle by 2-f°, with the plane of the motion, than 

 those without the addition, as appears by N° 15, the product being 820 : this 

 exceeds 639 more than in the ratio of 4 : 5, or that of the increase of cloth. 

 Hence it appears, that " a broader sail requires a greater angle ; and that when 



