80 Review of 0. Gregory^ Treatise on Mechanics. 



the parallel forces, and the other parallel to them. The 

 proposition taken in this sense, is simply a corollory from 

 that of the composition and resolution of forces, and as it 

 is that, which is fundamental in the demonstration of the 

 principle of the lever, and other mechanical powers, some 

 illustration of it may not be unacceptable to the reader. 



Let AB be a right ^ C b 



line, perpendicular to ..„— z.^i-. '• 



which two weights or ' -"■r----T" -•-"-""'•- ' 



parallel forces, W, w, A i r,---'-'^' — -^ — ""~~'~^ B 



are supposed to act,W^ ^^ #a; 



and to be kept in equilibrio by the reaction of the prop P, 

 the pressure on the prop would then be equal to the re- 

 sultant of the two forces W, w. This supposed pressure 

 on the prop P, when AB has no thickness, or vertical di- 

 mension, or becomes a mere mathematical line, would ev- 

 idently be nothing; for the forces W, w, communicate no 

 force, either longitudinally, in the direction AB, or late- 

 rally in that of CP. This therefore is an impossible con- 

 dition. Now if we suppose AB to be a bar, or physical 

 substance of a given or definite breadth aA, or feB, then 

 the weights W, w, may be sustained by the strength of co- 

 hesion of the particles of the bar, on the line, or surface 

 CP, but the action of these is not in the direction AB, but 

 oblique to the parallel forces, and the equivalent or joint 

 action of all these cohesive forces will be represented in 

 direction by some oblique line, drawn to some point c, 

 from AB. Now in these directions are the weights W, iv, 

 sustained by the cohesion of the bar at the line CP, and 

 each is an oblique action resolvible into two others, viz. 

 W X AP-|-Pc, and w X BP+Pc, whereof W X P c-fa; X Pc, 

 are annihilated by the reaction of the prop, and there re- 

 main only W X AP and w X BP : but when AB is in equili- 

 brio about P, these forces are equal, and WxAP=zoXBP, 

 or W : w: :PB : AP, or the length of the arms of the lever 

 are inversely, as the weights or forces. 



If the resolved parts of the forces, viz. AP and BP, 

 which are in one continu(^d line, be what was intended by 

 their action on a straight line, even in this sense the propo- 

 sition is incorrect ; for the forces in the direction of AP, 

 and BP do not act in the line AB, at P, but at c, and pre- 

 vent the bar from turning about the point P. 



