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The worship of Apollo Smintheus (or Apollo Sminthius) extended only to Asia Minor and not the Greek mainland. Alexandria in Troas was the center of this cult. This is one of the strongest arguments for the thesis that the origin of the Apollo cult was Asia minor.
After the fall of Troy the Greeks start to spread to the East. They settled on the Aegean islands and the western coast of Asia Minor. The worship of Apollo in this region had a curious origin. When the old Teukri under their king Teucer came from Crete to the coast of Asia Minor, the oracle told them to stay there where they could see their enemies creeping out of the ground. When they came to Hamaxitos, a city in this region, the mice creeping out of the ground gnawed on their shields during the night. So they saw the oracle of the god fulfilled, settled down and erected a statue of Apollo with a mouse laying at his feet, which in the Aeolian dialect was called Smintha. (Ovid Met. II, 5685)
There are known two different versions of Apollo Smintheus depictions:
1. A cult statue where he stands facing front holding a mouse in his hand. This version is characteristic of Alexandria in Troas. The fact that the statue is held by the Genius of the city may be an allusion that the temple of Apollo got governmental benefits. (Pat Lawrence)
2. A cult statue where Apollo is standing left and has a mouse under his foot. In Chryse there was a statue made by Scopas, showing exactly this position. This statue is depicted on coins.
The meaning of the epithet 'Smintheus' is interpreted different ways:
1. The origin of the name is the city of Sminthe in Troas, where Apollo was worshippedin pre-Hellenic times. So Apollo Smintheus = Apollo from Sminthe.
2. In the Aeolian dialect 'smintha' means 'mouse'. So Apollo Smintheus = the mice-god. The mouse in ancient times was a symbol of prophetic power because it was thought mice were inspired by the exhaling coming out of the ground.
3. Apollo the mice-killer. The Greek already had recognized the mice as vermin and worshipped Apollo as protector against mice.
The second explanation seems the most probable. It is unlikely the Greeks might have identified mice as carriers of plagues because they would have been incorrect. It is the rat flea that carries plague. The mouse is innocent.
The first mention of Apollo Smintheusis found in Homer's Ilias I, 39. The beginning of the Ilias describes how Apollo strikes the Greeks with a plague because Agamemnon has raped Chrysis, the daughter of Apollo's priest Chryses, and so has humiliated his priest.
The old man, afraid, obeyed his words, walked off in silence,
along the shore by the tumbling, crashing surf.
Some distance off, he prayed to Lord Apollo,
Leto's fair-haired child:
"God with the silver bow,
protector of Chryse, sacred Cilla, 40
mighty lord of Tenedos, Sminthean Apollo,
hear my prayer: If I've ever pleased you
with a holy shrine, or burned bones for you— 
bulls and goats well wrapped in fat—
grant me my prayer. Force the Danaans
to pay full price for my tears with your arrows."
So Chryses prayed. Phoebus Apollo heard him.
He came down from Olympus top enraged,
carrying on his shoulders bow and covered quiver,
his arrows rattling in anger against his arm. 50
So the god swooped down, descending like the night.
He sat some distance from the ships, shot off an arrow—
the silver bow reverberating ominously.
First, the god massacred mules and swift dogs, 
then loosed sharp arrows in among the troops themselves.
Thick fires burned the corpses ceaselessly.
(Translation by Ian Johnston, http://www.mala.bc.ca/~johnstoi/homer/iliad1.htm)
Some more information under https://www.forumancientcoins.com/board/index.php?topic=22524.msg149938#msg149938
Der kleine Pauly
Please add updates or make corrections to the NumisWiki text version as appropriate.
Apollo Sminthius. -- Amongst his various surnames, and distinctive appellations, Sminthius was assigned to Apollo. According to some writers, this was because he destroyed or drove away mice which had over-run the town of Sminthe or Smithium, on the coast of the Troad. Out of gratitude, a temple was built there for his worship.
The following types of Apollo occur on colonial imperial coins, with Latin legends:
Besides thos of Alexandria Troas and Apamea, above noticed, Apollo appears on coins struck in the colony of Caesarea Palestinae, under Hadrian, Antonius Pius, and Aurelius; of Corinth under Comodus; of Patrae under Nero, Domitian, Marcus Aurelius and Commodus; of Deltum under Maximus Caesar and Gordian III; and of Tyre under Trebonianus Gallus and Gallienus.
The attributes most often included with images of Apollo are the laurel and lyre. On a third Brass of Maximus (son of Maximiinus) struck at Deultum, Apollo stands holding a laurel branch in the right hand and placing with his left a lyre on a tripod. Before his feet is a lit altar. Apollo, says Vaillant, ii. 145, bears the laurel as consecrated to him on account of his reputed gift of foretelling events. The laurel tree, according to the Greeks, confered divine inspiration. The tripod is also symbolic of his oracular power.
"Whilst thus I sang, inflam'd with nobler fire,
On a second brass of Commodus, struck in the colony of Patrae, Apollo stands in a female dress, with his bow in the right hand, opposite him stands Venus, holding up a shield with both hands. Apollo and Venus were half brother and sister, the offspring of Jupiter, the former by Latona, the latter by the nymph Dione.
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