Archimedes ran out into the street naked because he was excited and completely forgot himself - He wanted to tell everyone his new discovery. It may be untrue that he ran out onto the street with no clothes on - But who knows?.
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According to popular legend, Archimedes ran out onto the street naked because he had discovered a solution to a problem he had been working on for some time - how to determine whether a crown was made of pure gold or if some other metal had been added. He was so excited by this realization that he forgot his clothes and ran to share his discovery with others. However, it is important to note that this story may not be historically accurate.