After this battle, the only battle in which a Chimaera was defeated, the heroic wizard was so badly injured and exhausted that he fell from his winged horse to his death (FB).
Although the date is not given in the book Fantastic Beasts and Where to Find Them, the incident seems to be a reference to the legendary defeat of a Chimaera in Lycia, Asia Minor, by the hero Bellerophon riding Pegasus, the winged horse. In the legend, Bellerophon won the battle only by flying directly toward the Chimaera with a large lump of lead on the end of his spear. He forced the lead into the mouth of the monster, whose own firey breath melted the metal and choked it.
Given this similarity to the incident described, we are placing this battle in ancient Greece.