More Classroom Activities Based on Ancient Indian Rope Geometry - Constructing a rectangle

The following applet demonstrates the method for constructing a rectangle of given dimensions found in Section 1.6 of the Śulba-sūtra of Baudhāyana (BSS 1.6). In the applet, slide \(E\) to change the side length of the desired rectangle and \(A\) to change the desired width (in step 3). The translated text is in black, with corresponding interpretation in blue below it. Click on “Go” to advance through the construction, and “Reset” when the construction is completed.

In BSS 1.6, Baudhāyana presented a method to construct a rectangle using only ropes and stakes in the ground [Sen and Bag, p. 78]:

When (the construction of) a rectangle is desired, two poles are fixed on the ground at a distance equal to the desired length. (This makes the east-west line.) Two poles one on each side of the (two above mentioned) poles are fixed at equal distances (along the east-west line). A cord equal in length to the breadth (of the rectangle) is taken, its two ends are tied and a mark is given at the middle. With the two ties fastened to the end poles (on either side of the pole) in the east, the cord is stretched to the south by the mark; at the mark (where it touches the ground) a sign is given. Both the ties are now fastened to the middle (pole at the east end of the prāci), the cord is stretched towards the south by the mark over the sign (previously obtained) and a pole is fixed at the mark. This is the south-east corner. In this way are explained the north-east and the two western corners (of the rectangle).

Here is an explanation of the method quoted above. First, we begin with two stakes in the ground at points \(E\) and \(W.\) The length of \(EW\) shall be the length of the desired rectangle and can be adjusted in the applet by sliding point \(E.\)

In step 2 of the applet, two points \(Q_1\) and \(P_1\) are fixed at equal distance from \(E\) to the left and right, respectively, and collinear to the east-west line. The distance is arbitrary and can be adjusted by sliding point \(P_1\) in the applet. This is easy to do with ropes in the ground since it just involves extending segment \(EW\) and measuring appropriate distances.

In step 3, a rope of length equal to the desired width is cut, and its midpoint is marked. This is represented by segment \(AB\) with midpoint \(N.\) To adjust the desired width, slide point \(A\) in the applet. In step 4, we attach the ends of the rope \(A\) and \(B\) to points \(Q_1\) and \(P_1,\) respectively, and pull \(N\) south until the rope is taut. We mark point \(S_1\) where \(N\) ends up. It is important to note that \(S_1\) is not a corner of the rectangle. It is merely an auxiliary point that ensures angle \(WES_1\) is a right angle. To see that \(WES_1\) is a right angle, note that triangles \(EQ_1S_1\) and \(EP_1S_1\) are congruent by construction, so that angles \(Q_1ES_1\) and \(P_1ES_1\) are equal. They must sum to a straight angle of 180 degrees and hence they are both 90 degrees.

Now corner \(G\) is marked by attaching both ends of rope \(AB\) to point \(E\) and pulling on \(N\) until the rope is taut. Where point \(N\) ends up is the location of \(G.\) The remaining three corners are found in a similar fashion, resulting in rectangle \(CDGF.\) By construction, \(EG\) and \(DE\) are half of the desired side width, and hence \(DG\) has length equal to the desired width. Similarly, \(CF\) also has the desired width as length. Because angles \(WED,\) \(WEG,\) \(EWC,\) and \(EWF\) are all 90 degrees, so are all the required angles at \(C,\) \(D,\) \(F,\) and \(G.\) This can be seen by observing \(CD\) and \(FG\) are parallel to \(EW\) by construction.

It is interesting to note that this method does not rely on the Pythagorean Theorem, and can be used to construct a square when \(AB\) has the same length as \(EW.\)