284 



THE INCLINED PLANE, WEDGE, AND SCREW. 



inclined plane, and the other expressed by W E upon the resisting plane. 

 These pressures respectively have the same proportion to the entire weight as 

 WFandWE have to W D, or as D E and W E have to W D, because D E 

 is equal to W F. Now the triangle WEDis in all respects similar to the 

 triangle ABC, the one differing from the other only in the scale on which it 

 is constructed. Therefore the three lines A C, C B, and B A, are in the same 

 proportion to each other as the lines W E, E D, and W D. Hence A B has 

 to A C the same proportion as the whole weight has to the pressure directed 

 toward B, and A B has to B C the same proportion as the whole weight has 

 to the pressure on the inclined plane. 



We have here supposed the weight to be sustained upon the inclined plane, 

 by a hard plane fixed at right angles to it. But the power necessary to sus- 

 tain the weight will be the same, in whatever way it is applied, provided it act 

 in the direction of the plane. Thus a cord may be attached to the weight, and 

 stretched toward A, or the hands of men may be applied to the weight below 

 it, so as to resist its descent toward B. But in whatever way it be applied, 

 the amount of the power will be determined in the same manner. Suppose the 

 weight to consist of as many pounds as there are inches in A B, then the power 

 requisite to sustain it upon the plane will consist of as many pounds as there 

 are inches in A C, and the pressure on the plane will amount to as many pounds 

 as there are inches in B C. 



From what has been stated, it may easily be inferred that the less the ele- 

 vation of the plane is, the less will be the power requisite to sustain a given 

 weight upon it, and the greater will be the pressure upon it. Suppose the in- 

 clined plane A B to turn upon a hinge at B, and to be depressed so that its 

 angle of elevation shall be diminished, it is evident that as this angle decreases, 

 the height of the plane decreases, and its base increases. Thus, when it takes 

 the position B A', the height A' C is less than the former height A C, while the 

 base B C is greater than the former base B C. The power requisite to support 

 the weight upon the plane in the position B A' is represented by A' C, and 

 is as much less than the power requisite to sustain it upon the plane A B, as 

 the height A 7 C is less than the height AC. On the other hand, the pressure 

 upon the plane in the position B A' is as much greater than the pressure upon 

 the plane B A, as the base B C is greater than the base B C. 



The power of an inclined plane, considered as a machine, is therefore esti- 

 mated by the proportion which the length bears to the height. This power is 

 always increased by diminishing the elevation of the plane. 



Roads which are not level may be regarded as inclined planes, and loads 

 drawn upon them in carriages, considered in reference to the powers which 

 impel them, are subject to all the conditions which have been established for 

 inclined planes. The inclination of the road is estimated by the height cor- 

 responding to some proposed length. Thus it is said to rise one foot in fifteen, 

 one foot in twenty, &c, meaning that if fifteen or twenty feet of the road be 

 taken as the length of an inclined plane, such as A B, the corresponding 

 height will be one foot. Or the same may be expressed thus : that if fifteen 

 or twenty feet be measured upon the road, the difference of the levels of the 

 two extremities of the distance measured is one foot. According to this method 

 of estimating the inclination of roads, the power requisite to sustain a load upon 

 them (setting aside the effect of friction) is always proportional to that eleva- 

 tion. Thus, if a road rise one foot in twenty, a power of one ton will be suffi- 

 cient to sustain twenty tons, and so on. 



On a horizontal plane, the only resistance which the power has to overcome, 

 is the friction of the load with the plane, and the consideration of this being 

 for the present omitted, a weight once put in motion would continue moving 



