56 LECTURE VII. 



chine is analogous to a very thin wedge, of which the thickness is only 

 equal to the difference of the distances of the threads, and which of course 

 acts with a great mechanical advantage. It might in some cases be more 

 convenient to make two cylindrical screws, of different kinds, at, different 

 parts of the same axis, rather than to perforate it. The friction of such 

 machines is, however, a great impediment to their operation. (Plate V. 

 Fig. 71.) 



In all the kinds of equilibrium that we have considered, and in all other 

 cases that can be imagined, it will be found that the forces, or rather 

 weights, opposed to each other, are so arranged that if they were put in 

 motion, their momenta in the direction of gravity would, in the first 

 instance, be equal and contrary, the velocity being as much greater as the 

 magnitude of the weight is smaller.* Thus, if an ounce weight, placed on 

 a lever, at the distance of four feet from the fulcrum, counterpoise a weight 

 of four ounces at the distance of one foot, the velocity with which the 

 ounce would descend, if the lever were moved, would be four times as 

 great as that with which the weight of four ounces would descend. A 

 single moveable pulley ascends with half the velocity of the end of the rope 

 which is drawn upwards, and acts with a force twice as great ; a block of 

 three shieves enables a weight to sustain another six times as great ; but 

 the velocity with which this weight ascends, is only one sixth of that with 

 which the smaller weight must descend. When a weight rests on an in- 

 clined plane, of w r hich the height is one half of the length, it may be 

 retained by the action of a weight of half its magnitude, drawing it up 

 the plane by means of a thread passing over a pulley. Here if the weight 

 ascended or descended along the oblique surface, its velocity, reduced to a 

 vertical direction, would be half as great as that of the smaller weight 

 which balances it. 



Some authors have considered this law as affording a fundamental de- 

 monstration of the conditions of equilibrium in all possible cases.t For 

 since, wherever two weights are in equilibrium, if one of them descended, 

 the other must ascend with an equal quantity of motion, it appears absurd 

 to suppose that the force of gravitation could produce these two equal and 

 contrary effects at the same time. But it is more satisfactory to trace, in 

 every case, the steps by which the immediate actions of the different 

 weights are enabled to oppose each other ; and the general law may then 

 be inferred, by induction, from the agreement of the particular results, in 

 confirmation of the general reasoning which tends to establish its truth. 



LECT. VII. ADDITIONAL AUTHORITIES. 



Mechanical Powers. Roberval's Paradox, Leupold, Theatrum St. 4 t. 17. Lud- 

 lam's Essays, 1770. 



Equilibrium. Varignon on Composition of Forces, Hist. etMem. de Paris, 1714, 

 280, H. 87. Riccati, Comm. Bon. ii. II. 305 ; III. 215 ; v. II. 186. Foncenex, 

 Miscel. Taurin. ii. II. 299. Euler, Hist, et Mem. de 1'Acad. de Berlin, 1762, p. 265. 



* Varro de Motu, Geneva, 1584, Th. 1. 



f Lagrange, Mecanique Analytique, 4to, 1788, and 2 vols. 4to, 1811. 



