68 LECTURE VII. 



machine is, however, much stronger than such a cyhnder would be, and does 

 not require so great a curvature in the ropes employed. (Plate IV". Fig. 51.) 



The laws of the efjuilibrium of puUies have been referred, by some writers 

 on mechanics, to those of the lever ; but the comparison is both unnecessary 

 and imperfect; in the simple case of two equal weights attached to a thread 

 passing over a single pulley, which is the only one that allows us to recur to 

 the properties of the lever, the conditions of equilibrium are axioraatically 

 evident, without any further reasoning; and in more complicated cases, the cal- 

 culations proceed on perfectly different grounds. We are, therefore, to con- 

 sider a pulley as a cylinder, moving on an axis, merely in order to change 

 the direction of a thread, without friction ; for whatever is demonstrable of 

 pullies or their combinations, would be equally true of as many perfectly smooth 

 grooves, which do not bear the most distant analogy to the lever. 



Now when the direction of a thread is altered, by passing over any perfectly 

 smooth surface, it communicates the whole force acting on it; for the resist- 

 ance of a surface, without friction, can only be in a direction perpendicular 

 to itself and to the thread, and the operation of any force remains undisturbed 

 by a resistance which is always in a direction perpendicular to it, 



A fixed pulley, therefore, has no effect in gaining a mechanical advantage ; 

 but by means of a moveable pulley, it is obvious that a weight may be sup- 

 ported by two forces, each equivalent to half the weight, applied in a vertical 

 direction to the extremities of the thread; and these forces may be derived 

 from two weights, if the thread be made to pass over two fixed pullies in a 

 proper position ; and if one of the ends be attached to a fixed point, and the 

 other remain connected to its weight, the equilibrium will continue unimpair- 

 ed, each portion of the thread still supporting one half of tlie original weight; ' 

 so that, by means of a single moveable pulley, one body may retain in equili- 

 brium another of double its weight. (Plate IV. Fig 52, 53.) 



The modes of arranging pullies are very various, but the advantage which 

 they procure may always be estimated, from the consideration that every part 

 of the same thread must be equally stretched; and where there is only one 

 thread, the weight will be divided equally among all the portions which help 



