When the Enterprise diverts to answer a medical emergency on an unknown planet, its crew discovers a group of natives of the star Sandara who claim to have traveled to Earth during the time of Plato and Socrates after their star supernovaed. When Greek civilization died, they left and settled on their current planet, which they call Platonius. The Platonians have very long life spans but are easily injured. They are also sadistic psychokinetics who amuse themselves by manipulating Alexander the Dwarf.
McCoy attempts to treat a minor cut their Philosopher-King Parman has sustained and which has become massively infected. However, Parman has a high fever and begins to become delirious. Unfortunately, his delirium translates into moving, shaking, and breaking of objects, including the Enterprise. Fortunately, when McCoy hypos Parman, the psychokinetic fits end. When Parman comes to, Parman, Queen Shirana (who is 2300 years old), Dianide, Eristethis, and the other Platonians begin to make life miserable for Kirk, McCoy, and Spock in order to convince McCoy to stay behind and act as their personal physician.
Parman first makes Kirk slap himself in the face repeatedly, then makes Kirk and Spock sing and perform silly songs for them. He also makes Spock laugh. When McCoy begs Parman not to continue forcing Spock to laugh, Parman makes him cry hysterically. In a final insult, Kirk acts as a whinnying horse while Alexander rides on his back.
Spock and Kirk question Alexander and find that the Platonians' powers developed 6 months 14 days after their arrival. Spock surmises that a substance found in the planet's food, is responsible for their mental powers, since the Platonians began consuming native foods after consuming their 2-3 months of supply. The chemical kironide is found to work in conjunction with pituitary growth hormone, explaining why the dwarf Alexander is unaffected.
The Platonians force Uhura and Nurse Chapel to beam down for their further amusement. They force Spock to serenade Chapel, and Kirk to kiss Uhura. Kirk and Spock also are forced to take a hot poker and a whip and pretend to use them on the women. Alexander tries to stab Parman, but is discovered and forced to turn the knife on himself. However, Kirk gives the Platonians a taste of their own medicine when he stops Alexander using the psychokinetic power he has acquired using a kironide injection given him by McCoy. Mind games between Parman and Kirk reveal Kirk to have the stronger power, and Parman backs down and promises to behave in the future. Kirk takes Alexander with him, leaving the Platonians to their own devices and warning that he can recreate the psychokinetic power if it should be needed.