52 EQUILIBRIUM OF AN IMMERSED SOLID. 



He concluded at once that he could completely solve the king's prob- 

 lem, by weighing the crown in water. Overjoyed with his discovery, 

 he ran directly into the street, crying out, " Eureka ! Eureka ! " 



The two books of Archimedes which have come down to us were first 

 found in old Latin MS. by Nicholas Tartaglia, and edited by him in 

 1537. These books contain the solutions of a number of problems on 

 the equilibrium of paraboloids, and various problems relating to the 

 equilibrium of portions of spherical bodies. 



The authenticity of these books is confirmed by the fact that they 

 are referred to by Strabo, who not only mentions their title, but also 

 quotes from the first book. 



25. Conditions of Equilibrium of an Immersed 

 Solid. Let v denote the volume and p the density of the 

 solid; v' the volume and p' the density of the displaced fluid: 

 the weights of the solid and of the displaced fluid will be 

 respectively gpv and gp'v' ; then, if the solid rest in equilib- 

 rium in the fluid, we shall have 



gpv = gpv 



(1) 



If we suppose the solid to be entirely immersed, the vol- 

 umes v and v' will be equal, and the densities p and p' must 

 also be equal if the solid remains in equilibrium, having no 

 tendency either to ascend or descend. 



But if the weight of the immersed solid be greater than 

 that of the fluid displaced, we shall have 



gpv > gp'v, 



and the solid will be urged downwards by a force equal to 

 gpv gp'v. 



If, on the contrary, the weight of the solid be less than 

 that of the fluid, we shall have 



gpv < gp'v ; 



and the solid will be urged upwards by a force equal to 

 gp'v gpv. 



