In the 'Meno', Plato explores the nature of virtue and the possibility of teaching it. During the discussion, Socrates introduces a method of hypothesis, which involves working backward from a known truth to verify the premises of an argument. This method is an early precursor to modern logical and mathematical proof techniques, demonstrating Plato's interest in structured reasoning and the attainment of certain knowledge.

The Greeks placed a strong emphasis on the concept of justice from a moral standpoint. They believed that individuals should act in accordance with their duty (arete) and follow the principles of the Golden Mean, which promotes virtues like courage, moderation, and wisdom. While the Greeks acknowledged the importance of social, legal, and political aspects, moral justice was considered the foundation for these other forms of justice.

Socrates was the primary mentor of Plato. Much of what we know about Socratic philosophy comes from Plato's dialogues, where Socrates is the central character. This intellectual lineage is essential to understanding the development of ethics and political theory in ancient Greece, as Plato expanded upon Socratic methods to develop his own comprehensive philosophical system.

In 'The Republic', Plato explores the 'Idea of the Good' as the highest form of reality. He posits that this concept serves as the ultimate source of truth, beauty, and justice. For Plato, the philosopher-king must grasp this intellectual form to govern the ideal state effectively, as it provides the necessary moral and rational foundation for all human knowledge and societal organization.

Menaechmus was a Greek mathematician and a pupil of Plato. He is historically recognized for his discovery of the conic sections, which he used to solve the problem of doubling the cube. His work laid the mathematical foundations that would later be expanded upon by figures like Apollonius of Perga.

The Timaeus is widely recognized for its profound incorporation of Pythagorean ideas regarding cosmology, mathematics, and the nature of physical reality. This dialogue explores the creation of the universe through geometric and mathematical principles, reflecting key Pythagorean themes that were highly influential in Plato's later metaphysical thought.

Plato's 'Republic' is the primary text where he defines justice as the harmony of the soul and the state. He argues that justice exists when each class—the rulers, the guardians, and the producers—performs its specific function without interference. While his later works like 'Statesman' and 'Laws' discuss governance and legal structures, the foundational theory of justice as a tripartite virtue is most explicitly developed in the 'Republic'.

Paul Shorey, a distinguished American classical scholar, authored 'What Plato Said' in 1933. The book provides a comprehensive and influential analysis of Plato's dialogues, defending the unity of Plato's philosophical thought against those who argued for significant contradictions between his early and late works. It remains a highly regarded text for students and scholars of ancient Greek political philosophy and Platonic studies.

Plato's 'Republic' is a comprehensive dialogue that examines the nature of justice in both the individual soul and the ideal city-state. By exploring these concepts, Plato integrates ethics, moral philosophy, and political structure, making all the provided options central to the work's overarching argument.

Eudoxus of Cnidus was a student of Plato at the Academy. He is famous for his mathematical contributions and his development of a geocentric model using concentric spheres to explain the motion of celestial bodies. This work laid the foundational geometric framework that later astronomers, including Ptolemy, adapted into the system of epicycles to account for planetary observations.