OnceakingaskedArchmedestodeterminewhetherornothiscrownwasreallymadeofgoldorofsomeothermaterial.Archimedesusedreasoningbetweenthevolumeandweightofgoldagainstthevolumeandweightofsilver,whichwasthesuspectedsubstitutematerialinthecrown.Theonlyproblemhehadwasthathedidnotknowhowtofindthevolumeofanirregularobject.Onenightwhilehewastakingabathherealizedthatifhewouldfillthebathtubcompletelywithwaterandthencatchallofthewaterthatspilledovertheedgeoncehegotinthatthatwouldbethevolumeofhisbody.Hewassoexcitedbythisdiscoverythatheranoutintothestreetsnakedyelling,"Eureka!(I'vefoundit!)"Whenheputthecrownintothewateritdisplacedmorewaterthanthevolumeofgold,meaningthatthegoldsmithmixedsilverwiththegoldwhenmakingthecrown.
有一次国王让阿基米德判断他的王冠是否是用纯金做的。阿基米德用金和银的体积与重量之间的关系理论来推理，（因为国王怀疑有人用银代替黄金做假了）但是问题是阿基米德不知道该怎么来测量不规则的王冠的体积。一天晚上，阿基米德洗澡时发现水往外溢出，他突然想到他的自身的体积正好是溢出的水的体积。他非常兴奋得衣服都没有穿就从水盆里跳出来走到街上大喊到：我找到办法了！当他把王冠放到水里，他发现这溢出的水比相同重量的黄金放到水里溢出的水多，他知道了这顶王冠是用黄金和银子混合做的。
