Anon he finds him
Striking too short at Greeks; his antique sword,
Rebellious to his arm, lies where it falls,
Repugnant to command: unequal match'd,
Pyrrhus at Priam drives; in rage strikes wide;
But with the whiff and wind of his fell sword
The unnerved father falls. Then senseless Ilium,
Seeming to feel this blow, with flaming top
Stoops to his base, and with a hideous crash
Takes prisoner Pyrrhus' ear: for, lo! his sword,
Which was declining on the milky head
Of reverend Priam, seem'd i' the air to stick:
So, as a painted tyrant, Pyrrhus stood,
And like a neutral to his will and matter,
But, as we often see, against some storm,
A silence in the heavens, the rack stand still,
The bold winds speechless and the orb below
As hush as death, anon the dreadful thunder
Doth rend the region, so, after Pyrrhus' pause,
Aroused vengeance sets him new a-work;
And never did the Cyclops' hammers fall
On Mars's armour forged for proof eterne
With less remorse than Pyrrhus' bleeding sword
Now falls on Priam.
Publius Vergilius Maro, known to us as Virgil, was among the greatest Roman poets, known most particularly for his epic The Aeneid. In the vein of Homer’s equally celebrated Iliad and Odyssey, it tells the story of Aeneas, detailing his escape from Troy, his love affair with Dido and his eventual founding of what would become Rome. Among Virgil’s other works were The Eclogues and The Georgics.
Pyrrhus, more often known as Neoptolemus, was the son of Achilles and Deidamia. Achilles’ mother, the sea goddess Thetis, wanted to protect him from having to fight in the Trojan War, so she disguised him as a girl and hid him in the kingdom of Skyros. There, he fell for the princess Deidamia and she eventually gave birth to a son. Since Achilles’ name was Pyrrha while he was in his female disguise, the child was called Pyrrhus. This son goes on to be instrumental in the Trojan War - no more than his father before him - and had a reputation for being vicious. Shakespeare is drawing on that reputation in the passages he quotes in Hamlet.
Priam was the mythical king of Troy. His son Paris was caught up in the spat between competing goddesses that led to his affair with Helen, and this led to the Trojan War. Eventually nearly all of Priam’s children - among them Paris, Hector, and many, many others - are killed, as is Priam himself. The sack of Troy was massively important in the classical imagination, and figures heavily in much of the literature that has survived from classical Greece and Rome.
Ilium is another name for Troy - it is from this name that we get the title Iliad. (It literally means ‘the poem about Ilium’…) Its location has been variously disputed throughout history.
Dido was the mythical founder and queen of Carthage. She is most famous thanks to her depiction in Virgil’s Aeneid, wherein she is sometimes also called Elissa. She is also the subject of an early English opera by Henry Purcell - his Dido and Aeneas remains one of the most popular operas of the English baroque.
Carthage was a major city-state in North Africa, an almost-permanent enemy of Rome. Virgil’s depiction of its fiery foundress Dido contains many elements designed to reflect Roman history and politics, as a kind of origin-story for the conflict between the two powers. As Dido dies, she enjoins the spirit of the place to rise up in permanent vengeance and enmity between her city and that of Aeneas - and indeed this certainly came to pass. The most famous episodes in the conflict were the Punic Wars, and perhaps the most famous character of all is Hannibal, the Carthaginian warrior who marched an army - including elephants - through Spain and across the Alps into Italy. He never managed to reach Rome, but is remembered as one of the greatest military strategists in history.
The Cyclops (whose name means ‘round-eyed’) were a race of giants. They forged Zeus’ thunderbolt, and Poseidon’s trident, among other things. In Homer, one of their number became the most famous Cyclops of all when he was tricked by Odysseus. Although he is now synonymous with the word Cyclops, his name was actually Polyphemus.
Shakespeare seldom uses the Greek names for any of the gods - particularly when he’s writing something that is based on a poem by Virgil or Ovid. Mars is the Roman version of the classical god of War.