VOL. XXXVI.] IHILOSOPHICAL TRANSACTIONS. 34Q 



by pushing upwards against b, or any where between b and d, will make him- 

 self lighter, or be overpoised by the weight w, which before only counterpoised 

 the weight of his body and the scale. 



If the common centre of gravity of the scale e, and the man supposed to 

 stand in it be at k, and the man by thrusting against any part of the beam, 

 cause the scale to move outwards so as to carry the said common centre of 

 gravity to x; then instead of be, l1 will become the line of direction of the 

 compound weight, whose action will be increased in the ratio of lc to bc. 

 This is what has been explained by several writers of mechanics; but no one 

 has considered the case when the scale is kept from flying out, as here by the 

 post GG, which keeps it in its place, as if the strings of the scale were become 

 inflexible. Now to explain this case, let us suppose the length bd of half the 

 brachium bc to be equal to 3 feet, the line be to 4 feet, the line ed of 5 feet 

 to be the direction in which the man pushes, df and pe to be respectively 

 equal and parallel to be and bd, and the whole or absolute force with which 

 the man pushes, equal to 10 stone. Let the oblique force ed, = 10 stone, 

 be resolved into the two ef and eb, or its equal pd, whose directions are at 

 right angles to each other, and whose respective quantities, or intensities, are 

 as 6 and 8, because ef and be are in that proportion to each other, and to 

 ED. Now since ep is parallel to bdca, the beam, it no ways affects the beam 

 to move it upwards; and therefore there is only the force represented by fd, 

 or 8 stone, to push the beam upwards at d. For the same reason, and because 

 action and re- action are equal, the scale will be pushed down at e with the 

 force of 8 stone also. Now since the force at e pulls the beam perpendicu- 

 larly downwards from the point b, distant from c the whole length of the 

 brachium bd, its action downwards will not be diminished, but may be ex- 

 pressed by 8 X Bc: whereas the action upwards against d will be half lost, by 

 reason of the diminished distance from the centre, and is only to be expressed 

 by 8 X 4-bc ; and when the action upwards to raise the beam is subtracted 

 from the action downwards to depress it, there will still remain 4 stone to push 

 down the scale; because 8 X bc — 8 X -^-bc = 4bc. Consequently a weight 

 of 4 stone must be added at the end a to restore the equilibrium. Therefore 

 a man, &c. pushing upwards under the beam between b and d, becomes 

 heavier, q. e. d. 



On the contrary, if the scale should hang at f from the point d, only 3 feet 

 from the centre of motion c, and a post gg hinders the scale from being pushed 

 inwards towards c ; then if a man in this scale f pushes obliquely against b 

 with the oblique force abovementioned; the whole force, for the reasons 

 before given, will be reduced to 8 stone, which pushes the beam directly up- 



