A stretched rope as compass and straightedge: the Śulba-Sūtra constructions

0:00You have just built two things with a compass and a straight edge. A line that cuts a segment in half and meets it square, and a right angle at a point you were handed. Both of those constructions are far older than the compass. They were done outdoors, at the size of a courtyard, by people laying out altars. And the tool they used was a rope. Not as a picturesque substitute for the real thing. The rope came first.

0:30A compass is the tabletop version of a rope, and not the other way round. The instructions survive in a set of texts called the Shulba Sutras. They are not geometry books, and that matters. They are building manuals. They tell you how to lay out an altar to a specification: this shape, this area, these proportions, corners that are actually square. The geometry is in them because you cannot build to a specification without it.

1:04Which is worth saying plainly, because it is the usual reason mathematics gets written down anywhere. Somebody needed something to come out right. The rope construction we are about to follow is credited to one of those texts, the Katyayana Shulbasutra. So what can a rope actually do? Tie one end to a peg. Pull the rest tight, and walk. The far end traces a circle. Every point of that circle is the same distance from the peg, because the rope did not change length while you were walking.

1:41And that is the whole of what a compass gives you. One distance, held, while the thing turns. The rope enforces exactly the same condition. It just does it with tension instead of with a hinge. A rope also does a second job, which a compass cannot do at all. Pull it tight between two points and it lies along the straight line between them. Let it go slack and it does not. It sags into a curve, and it rules nothing.

2:13So a rope is a compass and a straight edge in one object, and which of the two it is depends only on whether it is tight. Slack is the only way this construction can go wrong. And slack is visible from across a field, which is more than you can say for a wobbling hand. Here is the problem, on the ground. Two pegs, driven in. Call them X and Y, and stretch a line between them. Twenty-four units of it, say.

2:46You want the line that cuts that in half and crosses it square. You have no ruler, no protractor, and no compass. You have rope, and you have pegs. That turns out to be enough. And watch for the moment where the rope does something your hand would otherwise have had to be trusted for. Take a length of rope and make a loop at each end, so it can be dropped over a peg.

3:12Now find its middle. Fold it in half, and mark the fold. But fold what, exactly? Not the whole rope. The rope that has gone into the two loops does not count. It is tied up. It is not part of what stretches between the pegs. So you fold the working part, the part that will actually come under tension, and you mark the middle of that. This is the step every reader skips as fussiness.

3:41It is not fussiness. It is the entire guarantee. Here is what happens if you skip it. Say the working rope is twenty-six units, and the two loops are not the same size. Two units at one end, six at the other. Fold the whole thing end to end, and your mark lands in the middle of thirty-four, not in the middle of twenty-six. Measured from the pegs, the two halves come out fifteen and eleven.

4:11They still add to twenty-six. The fold has simply split it unevenly. And when you pull that mark away from the line, the point you reach is not on the centre line at all. It misses by thirteen sixths. More than two units, on pegs only twenty-four apart. So the loops are not a detail. Excluding them is what makes the two halves equal, and equal halves are the only thing this construction has.

4:42Done properly, then. Drop one loop over X. Drop the other over Y. Now take the middle mark and pull it away from the line until both halves are fully tight. Both of them. Not one tight and the other nearly tight. Where the mark ends up, drive a peg, and call it A. And notice what you did not do. You did not choose where A goes. The rope chose. You only pulled until it stopped you.

5:13Now the same thing on the other side. Same rope, same pegs, same mark. Pull it the other way until both halves are tight again, and mark that point B. Two points. And you are holding a thing that pulls straight. Stretch it from A to B, and there is your line. That is the whole construction. Two pegs, one rope, one fold, two pulls. Nothing was measured, and nothing was read.

5:42Now why it has to work, which is a different question from whether it just did. How far is A from X? One half of the rope. How far is A from Y? The other half of the rope. And those two are equal, because you folded it. How far is B from X, and from Y? The same two halves over again. So there it is. One length, four times.

6:07Which is exactly the situation you have already argued your way through. Four equal lengths force the two big triangles to match on all three sides, and that forces the crossing to cut the segment in half and to sit square on it. The corner up at A is not the right angle, incidentally. It is nowhere near one. The right angle is down where the two lines cross. Two things this does not depend on.

6:38First, the length of the rope. Take a longer one, working length thirty, or forty, or seventy-four, and the point climbs to nine units up, or sixteen, or thirty-five. Different points every time, and every one of them on the same line. All the rope has to do is beat the gap. At exactly twenty-four it lies flat along the ground and gives you nothing. Second, the size of the whole thing. Double it, or multiply it by a hundred, and every equality in that argument holds unchanged.

7:16Which is the point. This was being done at the size of a courtyard, not the size of a page. An argument that never mentioned a size cannot break when you change one. Two more rope problems, and the argument you already have answers both. First, a right angle at a marked point of a line. Peg the point. Use one rope length to step the same distance out each way, which leaves your point exactly in the middle of the two marks you just made.

7:50Then run the construction you have just watched, on those two marks. Step nine, take a rope whose half is fifteen, and it reaches twelve above the ground. Second, cutting an angle in half. From the corner, step the same rope length out along each arm. Two marks, both the same distance from the corner. Now every point that is equally far from those two marks lies on the line that halves the angle, and the corner is one of them, which is why that line runs through it.

8:25The same idea a third time. Equal distances, and a line through two of them. So what was the compass ever for? Convenience. A hinge holds a distance more conveniently than tension does, indoors, on paper. But the argument never mentions hinges. It does not mention rope either. It mentions one thing. That two lengths are equal. So anything that holds a distance fixed while it turns will serve. A compass. A rope and a peg. A string and a pin. A stick pivoted on a nail.

9:04A construction is not defined by what you happen to be holding. It is defined by the condition the thing you are holding will not let you break.