MECHANICS. 



From all that we have stated, it fol- 

 lows, that the effort of any force to 

 turn a body round an axis, is to be 

 measured by multiplying the force by 

 the perpendicular from the axis on its 

 direction. The product thus obtained 

 is called the moment of the force round 

 that axis. This is a principle of such 

 extreme importance, that we shall deve- 

 lop it somewhat further. 



(!).) To establish satisfactorily the pro- 

 position, that the efficacy of a force to 

 turn a machine round an axis is mea- 

 sured by its moment, we ought to prove, 

 that if the moment be doubled or halved, 

 or increased, or decreased, in any pro- 

 portion, the efficacy of the force in 

 turning; the machine round the axle 

 is doubled or halved, or increased or 

 decreased in exactly the same propor- 

 tion : this may be very easily proved by 

 experiment. 



Let the weight W act perpendicularly 

 to the line C F. We shall assume as a 

 self-evident truth, that if the weight W 

 be doubled or halved, or increased or 

 decreased in any proportion, its effort 

 to turn the machine round C will be 

 doubled or halved, or increased or de- 

 creased in the same proportion. 



Let the power P at the distance C G 

 balance the weight TV at the distance 

 C F. Hence the product P x C G must 

 be equal to W x C F. Now, suppose 

 that the power P, or its distance C G 

 from the centre, or both, be so increased, 

 that the moment P x C G shall be 

 doubled, it is evident, that, in order to 

 preserve equilibrium, it will be necessary 

 that the moment W x C F shall be also 

 doubled ; and if the distance C F be 

 preserved, this can only be done by 

 doubling W. Hence the double mo- 

 ment P x C G will balance a double 

 weight acting at the same distance C F, 

 and therefore must have a double effect 

 in turning the machine round its centre. 

 In the same manner exactly it may be 

 proved, that in whatever manner the 

 moment P x C G may be varied by the 

 change of the power P, or the distance 

 C G, or both, the weight W must suffer 

 a proportional change, the distance C F 

 remaining unaltered : but the effort to 

 turn the machine round the centre is 

 in this case proportional to the weight 



(10.) We therefore conclude that the 

 effort of any force to turn a machine 

 round its axis, is rightly measured by 

 the moment of that force round that 

 axis. 



Hence, if several forces tend to turn 

 any body round its axis, they will sus- 

 tain it in equilibrium if the sum of the 

 moments of those forces which tend to 

 turn it round in one direction, be equal 

 to the sum of the moments of the forces 

 ichich tend to turn it round in the other 

 direction. For then, according to what 

 we have just proved, the sum of the 

 efforts which tend to turn the body 

 round in one direction, will be exactly 

 equal to the sum of the efforts which 

 tend to turn it round in the other di- 

 rection. 



This, which is the most general view 

 which can be taken of the lever, may be 

 illustrated by experiment as follows : 



Let a circular board be placed with its 

 plane vertical, and turning upon an 

 horizontal axle C (Jig. 3.) and let strings 



be attached to pins A, B, D, E, F ;* and 

 let these strings be drawn by weights in 

 different directions, passing over fixed 

 wheels as represented in the figure. Let 

 the board drawn by these strings settle 

 itself until it come to a state of equili- 

 brium. Then draw from the centre C 

 perpendiculars C o, C b, C d, C e, &c. 

 on the directions of the strings, and 

 measure the lengths of these lines. 

 Multiply the weights A', B', D' by the 

 perpendiculars C a, C b, C d, and the 

 products A' x C a, B' x C b, D' x Cd, 

 will express the effort of each weight to 

 turn the board round in one direction. 



Multiply the weights E', F', G', by the 

 perpendiculars C e, C/, Cg, and the 

 products E' x C e, F x C/, G' x Cg, 



will express the effort of each of these 

 weights to turn the board round in the 

 other direction. Now, it will be found 

 that the sum of the former products is 

 equal to the sum of the latter ; that is,* 



* We have endeavoured, as far as possible, in 

 these treatises on MECHANICS, to give the various 

 conditions iu a popular form, and divested of geome- 



