12 / ANTIQUITY TO THE RENAISSANCE

Pappus of Alexandria

active c. 320 CE·Alexandria · Egypt

Put three pegs on each of two straight rails. Six strings, joined in a crossing pattern, hide a third straight line.

01 / PAPPUS’S THEOREM

Find the hidden straight line

Join the pegs in pairs, A to b and B to a, and mark where each pair of strings crosses.

Slide any peg or tilt a rail. The crossings move, yet they never leave one straight line.

Three pegs on each rail. String A to b and B to a, and mark where they cross.

Every crossing is computed from the strings · drag any peg along its rail, or tilt a rail

THE IDEAThree points on one line and three on another produce three aligned crossings when connected in this pattern.

Worked in Alexandria in the fourth century CE.
His Mathematical Collection preserved and extended earlier geometry.
A theorem associated with his work later became important in projective geometry.

Pappus’s writings preserve material from works that would otherwise be lost.

This theorem concerns points meeting and lining up, rather than a particular length or angle.