A scribe in Roman Egypt needs the area of a field. He picks up a reed pen, sketches a triangle on papyrus, and writes numbers along its sides. It looks like the setup for a calculation. It turns out to be the calculation itself.

A classicist at Cambridge, Thomas Kelly, gathered the diagrams from ten Greek papyri of geometry problems, spanning the first four centuries CE — the first time anyone had put them in one place. The problems read like recipes. One gives a semicircle with a leg of 5 schoinia and a diameter of 10: add the two, take half, multiply by 5, and the answer is 37 and a half arourai. A drawing of the semicircle follows. Fields, trenches, grain stores — this is working math, not philosophy.

What stayed with me is that the drawings are not illustrations of the math. They are the math, doing three jobs at once.

First, workspace. The scribe works on the figure itself, numbers parked along the edges where the steps need them. Second, memory. In a fourth-century teaching codex, one triangle carries the squares of two of its sides, written beside those sides — including 225, a value not needed until a later step. The page held the number so the scribe didn’t have to. Third, check. Given values go outside the shape; areas and answers go inside or off to the right, in a regular slot for the result. Label everything, and the text and the picture become one package: the reader watches the method work, then sees the same steps work with new numbers. Kelly suggests the texts act almost like proofs.

And the drawings are gloriously rough. Nobody used compasses, so circles come out lopsided. Right triangles nearly always look alike — right angle at the bottom left, base longer than the height — even when the numbers say otherwise. One triangle has a base of 7 and a height of 9, and the scribe still drew it almost twice as wide as tall. Familiarity mattered more than accuracy. The shape was a symbol, not a scale model — closer to algebra’s x than to a photograph. You don’t need the drawing to look like the field. You need it to behave like the field.

That is also why the originals matter, and why tidying them is dangerous. In 1932, editors of a Vienna papyrus printed a neat flat figure for a pyramid problem. The papyrus itself holds a rough drawing at an angle, showing depth. The tidy version erased the scribe’s real choice. Clean up someone’s working notes and you can delete the thought itself — the angle was information, and the cleanup threw it away.

I think about that whenever I catch myself reaching for a clean diagram. The scribe’s pen held little ink, and the papyrus fibers fought every straight line. But his method survived two thousand years, and it is still the method: draw it first, let the picture hold the numbers, and check the answer against the shape.

— Pi