How do we know that Harry was born on July 31? This has been established in many places in the books, perhaps most importantly in Dumbledore's discussion of the prophecy in OP37. However, we can first determine this from PS8.
In the Daily Prophet, Harry reads:
"GRINGOTTS BREAK-IN LATEST -- Investigations continue into the break-in at Gringotts on 31 July..."
He reacts in surprise:
"Hagrid!" said Harry, "that Gringotts break-in happened on my birthday! It might've been happening while we were there!"