The five postulates
Fix a string to a peg and walk around it, pulled tight. You drew a circle. Now try the moves.
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Step 1 of 5
- Postulate 1 — any two points join with a line.
- Postulate 2 — a line extends further, end to end.
- Postulate 3 — any centre, any radius, draws a circle.
- Postulate 4 — all right angles equal one another.
- Postulate 5 — angles under two right angles, so the lines meet.
Three that draw, two that claim
A right angle is 90°. Two right angles make 180°.
| # | The agreement | It gives |
|---|---|---|
| 1 | A line can be drawn from any point to any other point. | A joining move. |
| 2 | A line segment can be extended without end. | A lengthening move. |
| 3 | A circle can be drawn with any centre and any radius. | A round move. |
| 4 | All right angles are equal to one another. | A claim of equality. |
| 5 | If a line cuts two lines and the angles inside, on one side, add to less than two right angles, the two lines meet on that side. | A promise of meeting. |
The first three put something on the paper. Notice the word "any". Every point, every line, every size.
The last two draw nothing. Later reasoning uses them.
The long one
The first four fit in one short sentence each. The fifth needs a condition and a promise. The condition is the angle sum. The promise is that the lines meet. So the fifth is longer.