In 'The Republic', Plato defines justice as a state of harmony where each individual performs the specific function for which they are best suited by nature. This functional specialization ensures that the soul and the state operate in balance, with the ruling class, warriors, and producers each fulfilling their designated roles without interference, thereby achieving a virtuous and orderly society.

Eudoxus of Cnidus, a student of Plato, was a brilliant mathematician and astronomer. He developed the theory of concentric spheres to explain planetary motion. Furthermore, he is credited with refining the calendar by estimating the solar year to be 365 days and six hours, a calculation that demonstrated the advanced observational capabilities of the Platonic school during that era.

Historical records and scholarly consensus generally place Plato's death at approximately 80 years of age. Born around 428/427 BCE and dying around 348/347 BCE, his long life allowed him to establish the Academy and produce a vast body of influential philosophical work. This age is widely cited in biographical accounts of his life and career in ancient Athens.

Menaechmus was a prominent Greek mathematician and geometer who studied under Eudoxus of Cnidus. Eudoxus was a student of Archytas and Plato, and he significantly influenced the development of Greek mathematics and astronomy. Menaechmus is best known for his discovery of conic sections, which he used to solve the problem of doubling the cube.

Plato, in his seminal work 'The Republic', argued that justice in the state is achieved when each class performs its specific function according to its nature—rulers (wisdom), guardians (courage), and producers (temperance). This functional specialization ensures harmony and order within the ideal state, reflecting his belief that individuals are best suited for roles that align with their inherent capacities and education.

Plato's concept of justice in 'The Republic' is not limited to the inward aspect of the soul. He views justice as both an internal harmony of the soul's parts and an external social arrangement where each class performs its specific function, thereby maintaining the stability and order of the ideal state.

In Plato's 'The Symposium', the comic playwright Aristophanes provides a famous speech describing the origin of love. He recounts a myth where humans were once spherical beings with two faces and four limbs, who were cut in half by Zeus, resulting in the eternal human search for one's missing soulmate.

Anytus represents the traditional Athenian view that virtue is not a specialized skill requiring professional teachers like the Sophists. Instead, he argues that any respectable Athenian gentleman, through the influence of the city's laws and customs, is capable of imparting virtue to the youth, thereby rejecting the intellectual elitism of the Sophists.

In his seminal work 'The Republic', Plato draws an analogy between the human soul and the state. He argues that the state is essentially the individual writ large, meaning that the virtues of justice, wisdom, and temperance must exist in both the individual and the political community for either to be considered harmonious and well-ordered. This macro-micro correspondence is central to his theory of the ideal state.

The 'Statesman' (also known as Politicus) is a Socratic dialogue written by Plato. It follows the 'Sophist' and precedes the 'Laws'. In this work, Plato explores the qualities required for a true statesman, the nature of political expertise, and the classification of different forms of government. It is a critical text for understanding Plato's evolving views on governance, law, and the role of the expert ruler in a well-ordered society.