56 Statics. 



98. It would be easy to deduce from what precedes an easy 

 method of finding the centre of gravity of the surface and of the 

 solidity of any cylinder or prism. Indeed it is evident that this 

 centre must be the middle of the line that passes through the 

 centres of gravity of the two opposite bases, since bodies of this 

 form, being composed of laminae or material planes, perfectly 

 equal and similar to the base, may be considered as so many 

 equal weights uniformly distributed upon this line. 



99. To find the centre of gravity G of a triangular pyramid 

 Fig. 39. SABC, we draw from the vertex to the centre of gravity F of the 



base, the straight line SF, and take in this line reckoning from F, iht 

 part FG = J FS. 



To show that G is the centre of gravity required, from the 

 middle D of the side AB, we draw DC, DS, to the opposite ver- 

 tices C, S, of the pyramid, and having taken DF = DC, and 

 DE = | DS, the points F, E, are respectively the centres of 

 93 gravity of the two triangles ABC, ASB. 



This being supposed, if we consider the pyramid as compos- 

 ed of material planes, parallel to ABC, the line SF, which 

 passes through the point F, of the base, will pass through a 

 similarly point placed in each of the parallel planes or strata. 



408. ' Thus the particular centres of gravity of the several parallel 

 planes will all be in the line SF. For the same reason the par- 

 ticular centres of gravity of the several planes parallel to ABS, 

 of which we may suppose the pyramid in like manner com- 

 posed, are all in the line EC. Accordingly, the centre of gravi- 

 ty of the pyramid is the point G, where the two lines SF, EC, 

 situated in the plane SDC, intersect each other. Now if we 

 draw FE, it will be parallel to CS, since DF being a third of DC, 

 and DE a third of DS, these two lines are cut proportionally. 



199. ' The two triangles FEG, GCS, are therefore similar ; and the two 

 triangles DFE, DCS, are also similar ; whence 



FG : GS : : FE : CS : : DF : DC : : 1 : 3 ; 

 that is, FG is a third of GS, and consequently a fourth of FS. 



100. As any solid may be decomposed into triangular pyr- 

 amids, knowing the centre of gravity of a triangular pyramid, it 



