246 



THE LEVER AND WHEELWORK. 



elements is perpendicular to the surface, and supported by its resistance ; the 

 other is parallel to the surface, and supported by the power. The proportion, 

 therefore, of the power to the weight will always depend on the obliquity of the 

 surface to the direction of the weight. 



Under this class of machines come the inclined plane, commonly so called, 

 the wedge, the screw, and various others. 



In order to simplify the development of the elementary theory of machines, 

 it is expedient to omit the consideration of many circumstances, of which, 

 however, a strict account must be taken before any practically useful applica- 

 tion of that theory can be attempted. A machine, as we must for the present 

 contemplate it, is a thing which can have no real or practical existence. Its 

 various parts are considered to be free from friction : all surfaces which move 

 in contact, are supposed to be infinitely smooth and polished. The solid parts 

 are conceived to be absolutely inflexible. The weight and inertia of the ma- 

 chine itself are wholly neglected, and we reason upon it as if it were divested 

 of these qualities. Cords and ropes are supposed to have no stiffness, to be 

 infinitely flexible. The machine, when it moves, is supposed to suffer no re- 

 sistance from the atmosphere, and to be in all respects circumstanced as if it 

 were in vacua. 



It is scarcely necessary to state that, all these suppositions being false, none 

 of the consequences deduced from them can be true. Nevertheless, as it is the 

 business of art to bring machines as near to this state of ideal perfection as 

 possible, the conclusions which are thus obtained, though false in a strict sense, 

 yet deviate from the truth in but a small degree. Like the first outline of a 

 picture, they resemble, in their general features, that truth to which, after many 

 subsequent corrections, they must finally approximate. 



After a first approximation has been made on the several false suppositions 

 which have been mentioned, various effects, which have been previously neg- 

 lected, are successively taken into account. Roughness, rigidity, imperfect 

 flexibility, the resistance of air and other fluids, the effects of the weight and 

 inertia of the machine, are severally examined, and their laws and properties 

 detected. The modifications and corrections, thus suggested as necessary to 

 be introduced into our former conclusions, are applied, and a second approxi- 

 mation, but still only an approximation, to truth is made. For, in investigating 

 the laws which regulate the several effects just mentioned, we are compelled 

 to proceed upon a new group of false suppositions. To determine the laws 

 which regulate the friction of surfaces, it is necessary to assume that every 

 part of the surfaces of contact is uniformly rough ; that the solid parts which 

 are imperfectly rigid, and the cords which are imperfectly flexible, are consti- 

 tuted throughout their entire dimensions of a uniform material ; so that the im- 

 perfection does not prevail more in one part than another. Thus all irregular- 

 ity is left out of account, and a general average of the effects taken. It is 

 obvious, therefore, that by these means we have still failed in obtaining a re- 

 sult exactly conformable to the real state of things ; but it is equally obvious 

 that we have obtained one much more comformable to that state than had 

 been previously accomplished, and sufficiently near it for most practical pur- 

 poses. 



This apparent imperfection in our instruments and powers of investigation is 

 not peculiar to mechanics ; it pervades all departments of natural science. In 

 astronomy, the motions of the celestial bodies, and their various changes and 

 appearances, as developed by theory, assisted by observation and experience, 

 are only approximations to the real motions and appearances which take place 

 in nature. It is true that these approximations are susceptible of almost unlim- 

 \ ited accuracy ; but still they are, and ever will continue to be, only approxima- 



