A point, a line, a circle

Before the Elements can prove anything, it has to hand you the tools. The whole of plane geometry — every triangle, every square, the Pythagorean theorem itself — is built from three objects and five permitted moves. Pick them up.

Three objects, defined before anything is claimed

Euclid opens with definitions: words fixed in meaning so that later claims have something to be made of. Three of them carry the whole book — a point (Def. 1, “that which has no part”), a straight line (Def. 4), and a circle (Def. 15, every point equidistant from a centre).

In Byrne’s language — which this whole teardown borrows — we don’t label points with letters. Colour is the label. The given figure is red, construction lines are dashed purple, filled areas are amber, and the constant ink carries outlines and right-angle marks. You can flip on classical A, B, C letters any time with the toggle.

Place, draw, sweep

Here are the five moves, live. Place a point anywhere; draw a segment, ray, or full line through two points; and sweep a circle from a centre out to a point it must pass through. Then switch to Move and drag any point — everything you built on top of it follows, recomputed in dependency order.

The collapsing compass

Notice you never typed a radius. Euclid’s compass is a collapsing one: a circle is defined by a centre and a point it runs through (Postulate 3). Lift it off the paper and it forgets its span. That sounds like a crippling limitation — you cannot “carry” a length across the page — yet Proposition I.2 shows the collapsing compass can copy any length anyway. The restriction costs nothing, and it keeps the foundation honest: a radius is always some distance you already constructed, never a number from nowhere.

The rules of the game: postulates & common notions

Two kinds of givens sit beneath every proof. Postulates are the geometry-specific powers you’re granted without proof — mostly permissions to draw. Common notions are self-evident truths about magnitude in general, true of numbers and weights as much as lengths. The difference matters: a postulate is something you accept because you’re doing geometry; a common notion you’d accept in any quantitative reasoning at all.

Click through them. Watch the fifth postulate — the parallel postulate — which Euclid distrusted enough to avoid until Proposition I.29. Everything before it is “neutral geometry,” true even on a hyperbolic plane. We’ll return to that watershed in Chapter 6.

Two answers, one, or none

One subtlety powers the rest of the book. When two circles meet, they meet in two points, or — if they just kiss — one, or, if they’re too far apart, none. A line meets a circle the same way. Drag the circles below and watch the count flip 2 → 1 → 0. That innocent “two points” is the deepest engineering problem in dynamic geometry — which of the two is the one you meant? — and it is exactly what the next chapter, Euclid’s very first proof, forces us to answer.

You have the alphabet. Next: the first word.