120 Statics* 



q :/:?:: sin ABG : sin BAG : sin AGB ; 



that is, when two forces only act upon a body to retain it in 

 equilibrium upon a plane ; if we imagine two other planes to 

 which the forces are perpendicular, these two forces and the 

 pressure upon the given plane, are represented each by the 

 sine of the angle comprehended between the planes to which the 

 two other forces are perpendicular. 



204. Since the ratios which we have established, take place 

 Fig.l 12. whatever be the nature of the two forces^) and q, they will hold 



true when one of the forces p for example is gravity ; in this case 

 the plane BG is horizontal, and the intersection BG is called the 

 base, and AL, perpendicular to BG, the height of the plane. 



205. Since by article 202, 



q : p : p : : sin EFD : sin CFD : sin EFC, 



we have 



q : p : : sin EFD : sin CFD, 



: : sin HFp : sin HF q ; 



if, therefore, knowing the weight p, the power q, and the angle 

 HFp, which the direction of the weight p makes with the per- 

 pendicular to the plane, we would determine the angle which 

 the direction of the power q must make with the same perpen- 

 dicular, we shall obtain it by the above proportion, which 

 gives 



But, when an angle is determined by its sine, there is no reason 

 for taking as the value of this angle, the angle itself found in the 

 Trig. 13. tables, rather than its supplement. Accordingly, the same weight 

 may be supported upon the same plane, by the same power, di- 

 rected in two different ways. These two directions must there- 

 fore be such that the two angles HF q, HF q, which they form 

 with the perpendicular FH, may be supplements to each other. 

 Now if we produce the perpendicular HF, toward /, the greater 

 of these two angles HF q is the supplement of q FI ; therefore, 

 since it must also be the supplement of the smaller angle HF q^ 

 it follows that q FJ is equal to the smaller angle HF q. Hence: 



