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On Floating Bodies

The story

This book's Greek text was lost for six hundred years. It survived only in a Latin translation made in the thirteenth century, and the world read Archimedes on water through that borrowed tongue until 1906 — when the Constantinople palimpsest (the same prayer book that kept the Method) gave back the Greek. So even this reading is a palimpsest: the translation the world used, with the original waiting beneath the prayers.

The book opens with one sentence about the character of a fluid: its parts, "lying evenly and being continuous," are such that "that part which is thrust the less is driven along by that which is thrust the more." Pressure, stated as a postulate. And then, the second proposition, quietly enormous: the surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the earth.

He proves the Earth is round from a cup of water. The argument is pure counterfactual: suppose the fluid's surface were not spherical. Then some parts of the fluid would be pressed more than others; and by the postulate, the less-pressed parts would be driven along by the more-pressed; and the water would move. "Therefore there will not be rest; which is contrary to the hypothesis." The water is at rest; therefore the Earth is a sphere. Twenty-two centuries before anyone photographed the planet, the shape of the world was derived from stillness.

Then the famous laws. A solid lighter than the fluid floats immersed exactly as deep as makes the weight of displaced fluid equal to its own weight. Pushed under, it is driven up by the difference. A heavier solid, weighed in the fluid, is lighter than its true weight by exactly the weight of the fluid it displaces. That last law is the honest Eureka: the legend says Archimedes leapt from the bath shouting, having measured the king's crown by displaced water — but the overflow method is clumsy, and the story comes from Vitruvius, not from Archimedes. The weighing-in-water method is what the book actually proves, and it is better: weigh the crown in air, weigh it in water, and the difference is the receipt.

And Book II goes where no one went again for eighteen centuries: the stability of a paraboloid floating in water, tilted at angles, proving which positions it returns from and which it capsizes in — the mathematics of a ship's hull, done for the love of the shape.

The lesson

Rest is the signature of a balanced account. The floating body sits exactly as deep as makes the ledger true — displaced weight equals solid weight — and the water's stillness is the proof that the books balance. Physics keeps double-entry records; equilibrium is a receipt.

And the largest facts can be proven from the smallest evidence, by asking what would happen if it were otherwise. A cup of water at rest, asked the right counterfactual, tells you the shape of the world.

The receipt