Book I
Foundations
Euclid does not begin with a theorem. He begins with a vocabulary, five permissions to draw, and five truths about equality. Everything that follows is licensed by these pages.
1
The straightedge
To draw a straight line from any point to any point.
Given two points, we may join them with a unique straight line.
This is the license to use a straightedge. Every diagram in the Elements begins here.
2
Produce the line
To produce a finite straight line continuously in a straight line.
A segment may be extended as far as we need, in either direction.
Constructions often require a line to continue past a point. Euclid never assumes a line is long enough — he produces it.
3
The compass
To describe a circle with any centre and distance.
Given a center and a radius, we may draw the circle they determine.
Together with Postulate 1 this is the entire toolkit: compass and straightedge. Euclid’s compass collapses when lifted — you cannot copy a length by moving the compass, only by construction.
4
Right angles
That all right angles are equal to one another.
Every right angle is the same size. There is one rightness.
This makes perpendicularity a universal standard, not a local accident of a particular figure.
5
The parallel postulate
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
If a transversal cuts two lines so that the interior angles on one side add up to less than 180°, those two lines will meet on that side.
The most famous sentence in mathematics. Euclid delayed using it until Proposition I.29. Deny it and you get hyperbolic or spherical geometry — entire other worlds.