Concept

Method of exhaustion

Pinning a quantity down by bounding it above and below with pieces that can be made as small as needed. It is what Archimedes used for the circle and it is the ancestor of the limit, arriving two thousand years before one was defined.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A disc unrolled into a triangle. A disc cut into 12 concentric rings, and the same rings straightened and stacked. The longest is the outer circumference; the shortest is nearly a point; the stack is a triangle.

A circle unrolled into a triangle

Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.

geometry · Circle area $A square cut into 7 pieces and a remainder. A square divided by cutting off a fixed fraction of what is left, over and over, so that the pieces are the terms of a geometric series and the uncut corner is the tail.$

The sum that fits in one square

Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.

analysis · Geometric series

Named alongside it

The objects these essays reach for when they reach for this one.

Dissection Limit Archimedes Area Circle area Convergence Convergence rate Geometric series Harmonic series Pi Power series Real numbers

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