Why it must be so: reasoning about lines

So far we have described lines, drawn them, counted them and measured them. This layer asks a harder question: how do we know? How can we be certain that two lines never meet twice, or that the three perpendicular bisectors of every triangle pass through one point, when nobody can check every triangle that could ever be drawn?

The answer, invented in ancient Greece and India and polished over two thousand years, is proof: start from a few statements everyone accepts, and reason step by step to new ones. Along the way you will meet the most famous controversy in the history of geometry, a rule about parallel lines that took more than 2,000 years to understand.

Around 300 BCE, in the city of Alexandria in Egypt, a teacher named Euclid wrote the Elements, thirteen books that organised all the geometry known at the time. Its big idea was not any single fact but its structure: a short list of definitions and starting rules, followed by hundreds of results, each proved only from the rules and the results before it.

The Elements opens with definitions that should feel familiar from the Understand layer. In a well-known English translation: "A point is that which has no part." "A line is breadthless length." "The ends of a line are points." Notice that Euclid's "line" is what we now call a segment or a curve, and his "straight line" is closer to our segment that can be extended. Words change over 2,300 years; ideas survive.

A postulate (or axiom) is a starting statement accepted without proof. Euclid chose five about geometry, plus some "common notions" about quantities in general, such as things equal to the same thing are equal to each other and the whole is greater than the part.

A proof is a chain of statements, each justified by a postulate, a definition or something already proved. One powerful style is proof by contradiction: assume the opposite of what you want, and show that it leads to something impossible.

In Investigate you found that n points on a line make n × (n − 1) ÷ 2 segments. Here is why it must be so, for every n.

The handshake argument. Each segment is decided by a pair of points. Each of the n points can pair with the (n − 1) others, giving n × (n − 1) ordered pairs. But the pair (A, B) and the pair (B, A) are the same segment, so every segment has been counted exactly twice. Divide by 2. The same argument counts the handshakes when n people all shake hands once, which is where the name comes from.

In Understand we measured the gap between parallel lines along a perpendicular. Why a perpendicular? Because the perpendicular segment from a point to a line is the shortest of all segments from that point to the line. Here is a proof using a fold, with no formulas at all.

The first four postulates are short and obviously true. The fifth is long, complicated and talks about lines meeting "eventually", far away where nobody can check. For centuries mathematicians felt it did not belong among the starting rules and tried to prove it from the other four.

Every attempt failed. Most accidentally assumed something equivalent to the fifth postulate along the way, such as "the angles of a triangle add to 180°", "rectangles exist" or "parallel lines stay the same distance apart". Each of those turns out to be just the fifth postulate in disguise.

Here are some statements that turned out to be the parallel postulate in disguise. Each one can be proved from the other four postulates plus the parallel postulate, and each one, if assumed, is enough to prove the parallel postulate back.

On the surface of a ball, what is the "straightest" path between two points? Stretch a rubber band tightly between two points on a football: it settles along a great circle, a circle whose centre is the centre of the ball. The equator is a great circle; so is every line of longitude together with its partner on the far side.

Great circles behave like lines in some ways: they are the shortest paths, and they never bend left or right as you walk along them. But any two great circles meet, and in two points, exactly opposite each other. Lines of longitude are all perpendicular to the equator, yet they all meet at the North and South Poles. So on a sphere there are no parallel "lines" at all.

Long before Euclid, priests in India needed exact geometry for building fire altars (vedi) of precise shapes and sizes. Their rules survive in the Sulba Sutras, texts of the first millennium BCE attached to the names of Baudhayana, Apastamba, Katyayana and Manava. Baudhayana's is the oldest, usually dated to roughly 800–500 BCE; Katyayana's is the latest. Sulba (or shulba) means cord or rope.

The tools were a stretched cord (rajju) and pegs (sanku). A cord pulled tight between two pegs gives a straight segment, exactly the idea of "two points fix a line". Marks on the cord allowed equal lengths to be copied, like a divider.

Only the latest of these texts, the Katyayana Sulvasutra, actually writes down a rule for finding the directions; the earlier ones start from an east–west line without saying how to get it.

Draw a line on squared graph paper and walk along it from one crossing point of the grid to the next. You might go 2 squares across and 1 square up, again and again. That repeated step describes the line's steepness, often called its slope or gradient: here 1 up for every 2 across, a slope of 1 ÷ 2.

This gives quick tests that work without a protractor. Parallel lines have the same step (the same slope). Perpendicular lines have steps that are the first one turned through a quarter turn: (2 across, 1 up) turned becomes (1 back, 2 up). And three points are collinear exactly when the steps between them are in the same ratio.

Careful thinkers ask what happens at the edges of a definition. The answers are sometimes a matter of convention: an agreement about how to use a word, rather than a fact to be discovered.

In Investigate the segment count appeared as 1 + 2 + 3 + … + (n − 1). In this chapter's first proof it appeared as n × (n − 1) ÷ 2. Why are these equal? The answer uses a trick often told about the young Carl Friedrich Gauss.