Blocks, &'c. ; and on the Polygon of Forces. 325 



iional to the portion AB of the vertical by construction. 

 -4/<o, the block BC, fi^. 3, is sustained in its position, by 

 a force acting aQ;ainst B, (being the reaction of the end B 

 of the block AB) proporional to the line SB, fig. 2, and ia 

 a parallel direction to SB; by a force acting against the 

 end C proportional to the lineSC, tig. 2, and in a direction 

 parallel to CS ; and by the weight of the block itself taken 

 in the construction proportional to the portion BC of the 

 vertical. And so for the other blocks. The short lines in 

 fig, 3, drawn across the ends of the blocks at A, B, C, D, 

 &c. show the section of the planes of abutment, and are 

 therefore at right angles to the directions of the forces 

 dieting there, or to SA, SB, SC, &c. fig. 2. 



Demonstration. 



Similar to the triangle AaB, fig. 1, make S^a, fig. 2, 

 which is therefore similar to though reverse of BaA, and 

 will represent the block AB in its position of equilibrium 

 in fig. 3. Through g, the representative point of the centre 

 of gravity of the block dra^v a vertical gov, which will be 

 parallel to ABG, and from o, where SA is intersected by 

 It, draw oa. By mechanics it is known that if a body Sa 

 be kept in equilibrio by two forces acting at its ends, the 

 directions of these forces must intersect each other in the 

 vertical line that passes through the centre of gravity of the 

 body: hence, if one force act on the body at S in the di- 

 rection SA crossing the vertical line at o, the other fcirce at 

 a must act. in the direction ao ; and these forces and the 

 weight of the block will respectively be proportional to the 

 lines A, a and An. The triangles A« a and Bfla, will 

 respectively be similar to, but reverse of those avg and 

 Svg: for the aufflc at A of the first of these trian(>;les, and 

 at B of the second, are, by construction, equal to that at a 

 of the third, and to t'iat at S of the fourth triangle; and 

 the angles at a of the two fanner triangles arc, because of 

 the parallelism of the verticals gov a.nd AG, respectively 

 equal to those at v of the two latter : whence Ac : Ai>:: 

 av : aS : : Ao : AS, and therefore SB is parallel to oa. 

 Wtieref'ire AB, BS and SA are respectively proportional 

 to Aa, ao, and o A, and conse(jueiitlv to the forces which 

 keep the block AB, fig. 3, in equilibrio, 



, The same reason- T BC "^ by subsijlnling T B, C, b, b 

 I iug will apply J CD 1 for the Ittters [^ C, D, c, c 

 ' to the block | 1^1^ ( A, B, a, a, the TD, E, d, c/ 

 LScc.J letters J he. 



X 3 and 



