242 OF SPECIFIC GRAVITY AND THE WEIGHING OF SOLED BODIES. 



bodies are the same, it follows also, that the weight of the cylinder is 

 equal to three times the weight of the cone ; consequently, by the 

 principles of the lever, the length of the arm AF, is three times the 

 length of the arm BF; for it is a well known property in the doctrine 

 of mechanics, that when two bodies of different weights are in equi- 

 librio on the opposite arms of a straight lever : 



The lengths of the arms are to each other, reciprocally as 

 the weights of the suspended bodies. 



Now, suppose the cone LKI, which is obviously equal in magnitude 

 to CDE, to be abstracted from the cylinder, and to have its place sup- 

 plied by another cone of half the specific gravity as the former ; then 

 it is evident, that if the cone CDE is suffered to retain its magnitude, 

 it will preponderate and cause the cylinder to ascend ; it is therefore 

 necessary, in order that the equilibrium shall'not be disturbed, to 

 diminish the magnitude, and consequently the weight of the equili- 

 brating cone; and for the purpose of assigning the quantity of 

 diminution, 



Put m m the magnitude of the conical body CDE, 



m'~ the magnitude of the cylindrical body GHIK, 



m"-=. the magnitude of the remaining portion cab, 



w 1=1 the weight which the cone loses in the fluid, 



w' =. the weight lost by the cylinder, 



w" the weight lost by the remaining cone cab, 



s =n the specific gravity of the fluid, and 



s' =i the specific gravity of the cone and cylinder. 



Then, since the weight which a body loses by being immersed in a 

 fluid, is to its whole weight, as the specific gravity of the fluid is to 

 the specific gravity of the body, we have 



s' : s : : m s' : w ; 



therefore, by equating the products of the extremes and means, it is 

 wmmsi=. the weight lost by the cone ; but according to the 

 principles of mensuration, the magnitude of a cylinder is equal to 

 three times the magnitude of a cone of the same base and altitude ; 

 consequently, we have 



m' nr 3m, 



and for the weight lost by the cylinder, we get 



s' : s : : 3ms 1 : w' ; 



Horn which, by equating the product of the extremes and means, we 

 obtain 



w/zr 3ms~m's, 



