Himie Koshevoy has a version of this story in his “Treasure Jest of Best Puns.”

When Alexander The Great was waging war on the entire known world of his time, it chanced that he received a slight spear-wound on his wrist. Wrapping an old cloth around it, he continued the battle. After victory was his, one of his aides noticed that the dried blood on the rag around Alexander The Great’s wrist was lining up on it in such a way that if one looked at it in the light of the sun it resembled a sundial and you could tell the correct time!

So, they called it Alexander’s Rag-Time-Band!