The Śulba-Sūtras and the rope

The constructions in this chapter are very old. Ancient mathematicians from different civilizations, including India, knew exact procedures for constructing perpendiculars and perpendicular bisectors.

In India, the earliest known texts containing these methods are the Śulba-sūtras — geometric texts of the Vedic period dealing with the construction of fire altars for rituals. They are part of one of the six Vedāṅgas, a term that literally means 'limbs of the Vedas'. Altars had to be built to exact shapes and areas, which is precisely why the geometry had to be exact.

The Śulbas contain the very methods we developed for the perpendicular and the perpendicular bisector. But they use a different kind of compass from yours: a ROPE.

A rope tied to a peg (a खूंटी) does both jobs a construction needs. Held at one end and swung, it draws circles or arcs, exactly as a compass does. Stretched between two points, it forms a straight line, exactly as a ruler does.

Here is a rope construction of the perpendicular bisector, from the Kātyāyana-Śulbasūtra 1.2. Let XY be the segment, drawn on the ground. Fix a small pole or peg vertically into the ground at X and at Y.

Take a sufficiently long rope and make a loop at each end. Ignoring the parts taken up by the loops, fold the rope in half to find and mark its MIDPOINT. Fasten the two loops to the pegs at X and Y.

Now pull the rope's midpoint out to one side of XY until both halves are fully stretched, and mark that position A. Do the same on the other side and mark B. AB is the required perpendicular bisector. Why does it work? Because the two halves of the rope are equal, A is the same distance from X as from Y — and so is B. Two points equidistant from X and Y give the perpendicular bisector, which is exactly the fact we proved.