Fundamental Property of the LEVER, 399 



In Prop. IL, which is totally independent of the firft, the de- 

 monftration is general, and extends to any proportion between 

 the arms. 



In Prop. III. the property is eftablifhed, when the forces act 

 in an oblique direction, and when the lever is either rectilineal, 

 angular, or curvilineal. In the demonftrations which have ge- 

 nerally been given of this laft propofition, the oblique force has 

 been refolved into two, one of which is directed to the fulcrum, 

 while the other is perpendicular to that direction. It is then 

 aflumed, that the force direbled to the fulcrum has no tendency to di- 

 sturb the equilibrium, even though it acts at the extremity of a bent 

 arm j and hence it is eafy to demonftrate, that the remaining 

 force is proportional to the perpendicular drawn from the ful- 

 crum to the line of direction in which the original force was 

 applied. As the principle thus aflumed, however, is totally in- 

 admiflible as an intuitive truth, we have attempted to demon- 

 ftrate the propofition without its affiftance. 



Prop. I. — If one arm of a straight lever is any multiple of the other, 

 a force atling at the extremity of the one will be in equilibria with 

 a force atling at the extremity of the other, when these forces are 

 reciprocally proportioned to the length of the arms to which they 

 are applied. 



Let AB (Plate XI. fig. 1.) be a lever fupported on the two 

 fulcra Y,f, fo that AfzzfF zz FB. Then, if two equal weights 

 C, D, of 1 pound each, be fufpended from the extremities A, B, 

 they will be in equilibrio, fince they act at the end of equal arms 

 A/, BF ; and each of the fulcra/, F, will fupport an equal part of 

 the whole weight, or 1 pound. Let the fulcrum / be now remo- 

 ved, and let a weight E, of 1 pound, act upwards at the point 

 fy the equilibrium will Hill continue ; but the weight E, of 1 

 pound, acting upwards at /, is equivalent to a weight G of 1 

 pound, acting downwards at B. Remove, therefore, the weight 



