The School of Athens shows the great thinkers of ancient Greece in one vast hall, talking, writing and arguing, though none of them ever stood in the same room. Raphael painted it for Pope Julius II between 1509 and 1511, and it stands for philosophy, one of four kinds of knowledge celebrated in the room. Plato and Aristotle stand at its center.
Plato, the older man, points a finger at the sky and carries his Timaeus. Aristotle, beside him, holds his Ethics and spreads a flat hand over the ground. The two of them walk toward the viewer, and the whole hall is arranged around that one disagreement.
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The thinkers in the crowd are a matter of record. The faces are another matter, and the most famous story about them is a lot younger than the fresco.
The story goes like this. In the foreground, a gloomy man rests his elbow on a block of marble and writes alone, away from the crowd. He is supposed to be Michelangelo, painted in at the last minute by Raphael as a tribute to the rival who was working on the Sistine Chapel ceiling nearby.
Art historians who went looking for the origin of that idea traced it to 1941, more than four centuries after the fresco, and to one Italian scholar, Deoclecio Redig de Campos. The man is real, and he was added late. But I would not repeat the Michelangelo story as fact, and the paint itself explains why.
What are Plato and Aristotle pointing at?
Plato points at the sky, and Aristotle points at the ground. The Vatican Museums name the books they carry: Plato holds the Timaeus, Aristotle the Ethics.
Plato taught that the things we touch are only copies of perfect forms that exist somewhere beyond them, so his finger goes up. Aristotle started from what can be seen and studied, so his palm faces down. The Timaeus is Plato's account of how the universe was made. The Ethics is Aristotle's book about how to live well.
Plato's face is often said to be Leonardo's. That is a belief, and the sources I checked give no document behind it. Behind the two men, the hall they walk out of is stranger than any of the philosophers.

What You'll Discover
Whose church the philosophers walk in.
Who sits alone with a pen.
Who is staring back at us.
What this room was built for.

