MECHANICS. 



nature to be introduced here and its 

 most striking results are spread over 

 departments of physical science, far 

 beyond the necessary limits of this trea- 

 tise. Nevertheless", even within our 

 narrow limits, the student will have nu- 

 merous instances of the truth and power 

 of this principle. 



(G.) Every machine, however com- 

 plex it be, must consist of some combi- 

 nation of the following simple machine?, 

 which are commonly called the mecha- 

 nic powers: 



1. THE LEVER, 



2. THE WHEEL AND AXLE, 



3. THE PULLEY, 



4. THE INCLINED-PLANE, 



5. THE WEDGE, 



6. THE SCREW. 



This classification of the ELEMENTS 

 OF MACHINERY, although veiy simple 

 when considered with respect to the 

 extent and power of the results which 

 spring from it, may be still further sim- 

 plified ; not because any of the six ma- 

 chines which we have just enumerated 

 admits of being resolved into more sim- 

 ple parts, but because some of them are 

 identical in principle, and different only 

 in appearance. We shall show here- 

 after that the wheel and axle is in fact 

 a lever, and that the wedge and screw 

 are only modifications of the inclined 

 plane; so that it follows, that all the 

 varieties of simple machines may be 

 reduced to three : 



1. THE LEVER, 



2. THE PULLEY, 



3. THE INCLINED-PLANE. 



CHAPTER II. Of the Lever. 



(".) A LEVER is sometimes defined 

 " an inflexible right line, void of gra- 

 vity, and turning on a certain point 

 as a centre." It is also defined " an 

 inflexible bar or rod resting upon a 

 fulcrum or prop, on which itls capable 

 of turning as on a centre." We shall, 

 however, take a more general view of 

 this machine, and consider it as any 

 solid body having a fixed axle on which 

 it is capable of turning, and round 

 which all its parts describe circles. In 

 considering such a machine as applica- 

 ble to mechanical purposes, we usually 

 conceive its axis to be placed at right 

 angles to the plane in which the power 

 and weight or resistance act. In order, 

 also, to simplify the investigation, we 

 shall, in the first instance, omit the 

 weight of the machine itself, or, what 



will amount to the same effect, we will 

 consider the fixed axle as passing 

 through the centre of gravity of the 

 machine, which will therefore rest indif- 

 ferently in any position. (Treatise I. 

 42.) 



r Let us suppose that AB D E {fig. i.J is 

 the section of a solid body, moveable on 

 a fixed axis, and taken in a plane perpen- 

 dicular to that fixed axis ; and suppose 

 the axis passes through the plane of 

 the section at C. The axis being sup- 

 posed horizontal, the section A B D E 

 will be vertical. Through C suppose 

 the horizontal line H C if to be drawn, 

 and let the weight W, to be sustained, 

 be applied at F, and the power P which 

 supports it be applied at G. Let us 

 consider, then, under what conditions P 

 can support W, conformably to the 

 principle of virtual velocities. If the 

 machine be put in motion round the 

 centre C, so that P shall descend and 

 W ascend ; the points G and F, to which 

 the power and weight are applied, will 

 commence to move through similar cir- 

 cular arcs, having C as their common 

 centre, and C G and C F as their radii. 

 These arcs, if taken of small magni- 

 tudes, will then be the spaces through 

 which the power and weight will move 

 in the vertical direction ; and whatever 

 be their magnitudes, they will be pro- 

 portional to the vertical motions of these 

 weights. But these arcs being similar, 

 are proportional to their radii; and 

 hence follows, what indeed is otherwise 

 abundantly evident, that the perpendi- 

 cular descent of P is to the correspond- 

 ing ascent of W, as the distances C G 

 and C F of the points, at which these 

 forces are applied, from the centre C. 

 These distances CG and CF may be 

 taken to represent the vertical velocities 

 of the power and weight j and if C G 



