Can a shape be written as numbers?

That time, that place

Until then mathematics ran in two channels: geometry for figures, algebra for numbers. Geometric problems were solved by drawing with straightedge and compass, numerical ones by manipulating expressions. The two were strangers.

Why this question

Descartes's idea is simple: lay two lines at right angles so that any point on the plane can be written as a pair of numbers.

Then a point becomes a pair, and a line or curve becomes a single equation those pairs satisfy.

What was found

A circle is the set of points at equal distance from a center. In coordinates that becomes one equation: x squared plus y squared equals the radius squared.

Now a problem about figures can be solved by calculation, and an equation can be seen as a shape. What Oresme had sketched became a system.

And something larger follows. A drawing runs to three directions; a list of numbers can run to any length. What cannot be drawn can still be handled. Mathematics begins to come loose from the visible.

Terms on this card

Function