Line

Euclid's second definition is as spare as his first: a line is breadthless length. The third says that the ends of a line are points. The fourth defines a straight line as one that lies evenly with the points on itself, a phrase that has occupied commentators for two thousand years and that nobody has improved on without quietly changing the subject.

Two of the five postulates are about lines, and they are permissions rather than descriptions. The first grants that a straight line may be drawn from any point to any point. The second grants that a finite straight line may be produced continuously in a straight line. Together they say that the straightedge exists and that nothing stops it. Note what they do not say: they do not say a line is infinite. Euclid works with finite segments that can always be extended, which is a weaker and more careful claim than infinity, and the difference matters for how his geometry behaves.

The fifth postulate, the parallel postulate, is also about lines, and it is the one that would not sit still. Attempts to derive it from the other four ran for two millennia and all failed. In the nineteenth century Lobachevsky, Bolyai and Gauss independently found that consistent geometries exist in which it is false. The line is therefore the element of Euclid's system through which the system was eventually shown not to be the only one available.

Dimensionally the line is the first extension. The point has no dimension; the line has one, so it admits direction, order and distance along itself, but nothing across. Everything in plane geometry that is not a point or a curve is built from segments of it.

Euclid's line, incidentally, need not be straight. His second definition covers curves as well, and straightness is specified separately.

The classical toolkit is two instruments, and the constraints placed on them matter as much as the instruments themselves. The compass draws a circle given a centre and a point on it. The straightedge draws the line through two given points. That is all. In particular the straightedge is unmarked: it carries no scale, so it cannot be used to measure, to transfer a distance, or to slide until two marks land on two given curves.

That last exclusion is the significant one. A construction that slides a marked ruler until two points on it fall on two given curves is called a neusis, and it is genuinely more powerful than compass and straightedge. Archimedes used neusis; so did Nicomedes and Pappus. With a marked ruler an arbitrary angle can be trisected, a cube can be doubled, and the regular heptagon and nonagon can be constructed exactly, all of which the unmarked tools provably cannot do. The Greeks knew these methods and used them, but treated them as a separate class of solution.

So the famous impossibilities of classical geometry are impossibilities relative to a chosen restriction, not limits on geometry itself. Doubling the cube and trisecting the angle become straightforward the moment a mark is allowed on the ruler. Squaring the circle does not, because that one fails for a deeper reason. Paper folding is more powerful still: origami constructions solve cubic equations, and will produce both the heptagon and the trisected angle.

Why the restriction was honoured is a question about Greek mathematical culture rather than about lines. Whatever the reason, it turned out to be the more interesting constraint, because it is the one sharp enough to be settled by proof. The theory that eventually answered these questions is algebraic. Each new point in a compass-and-straightedge construction lies in a field extension of degree one or two over the previous one, so every constructible length has degree a power of 2 over the rationals, and anything requiring an irreducible cubic is out of reach.