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

The following applet demonstrates the method for constructing a square found in Section 1.8 of the Śulba-sūtra of Baudhāyana (BSS 1.8). Slide \(E\) to change the side length of the desired square. The translated text is in black, with corresponding interpretation below it in blue. Click on “Go” to advance through the construction, and “Reset” when the construction is completed.

Baudhāyana provided a method for constructing a square in BSS 1.8 [Sen and Bag, p. 78]:

Now another (method). Ties are made at both ends of a cord of length equal to the measure increased by its half (so that the whole length of the cord is divided into three parts of half the measure each). In the third [extended] part on the western side a mark is given at a point shorter by one-sixth (of the third part); this is the nyañcana. Another mark is made at the desired point for fixing the corners. With the two ties fastened to the two ends of the east-west line (prsthyā), the cord is stretched towards the south by the nyañcana, and the western and eastern corners [of the square] are fixed by the desired mark.

Baudhāyana’s construction of a square in BSS 1.8 using only ropes and stakes in the ground makes implicit use of the converse of the Pythagorean Theorem (pre-Pythagoras) to ensure that all angles are indeed right. To begin the construction, a rope of length equal to the desired side length of the square is laid along the east-west line. The desired length can be adjusted by moving \(E\) in the applet. Thus, our goal is to construct a square with side length \(EW.\)

In step 1 of the applet, the rope is extended to three-halves of the desired length. That is,

\[WZ={\frac{3}{2}}EW,\]

while

\[EZ={\frac{1}{2}}EW.\]

Next a mark is made on the rope “at a point shorter by one-sixth (of the third part); this is the nyañcana...” So in step 2 of the applet we have defined point \(N\) on the rope so that \(EN\) is one-twelfth of the original side length. Notice that the “third” part is segment \(EZ\) which is half of the desired side length, and one-sixth of one-half is indeed one-twelfth: \[EN={\frac{1}{6}}EZ={\frac{1}{6}}\cdot{\frac{1}{2}}EW={\frac{1}{12}}EW.\]

Moreover,

\[NZ=EZ-EN={\frac{1}{2}}EW-{\frac{1}{12}}EW={\frac{5}{12}}EW.\]

In step 3, the midpoint of segment \(EW\) is marked. This will be important in the construction later.

In step 4, we see use of the converse of the Pythagorean Theorem. The applet simulates fixing stakes in the ground at point \(W\) and \(E.\) Imagine taking the rope of length \(WZ\) and attaching its ends to \(W\) and \(E,\) respectively. Pull point \(N\) to the south until the rope is taut. It should be noted that when point \(N\) is pulled directly south, there will be a unique point at which the rope is taut. We denote this point \(Q_1\) in the applet. So, in the applet, when \(N\) is pulled south until it is taut, point \(Z\) coincides with \(E,\) and \(N\) coincides with \(Q_1.\) Thus, \(EQ_1=ZN\) and \(Q_1W=NW.\) Now we claim that triangle \(EQ_1W\) is indeed a right triangle. To see this, note that \[EQ_1=ZN=\frac{5}{12}EW,\] \[Q_1W=NW=EW+EN=EW+\frac{1}{12}EW=\frac{13}{12}EW,\] and therefore:

\[EW^2+EQ_1^2=EW^2+\frac{25}{144}EW^2=\frac{169}{144}EW^2=\left(\frac{13}{12}EW\right)^2=Q_1W^2,\]

or \[EW^2+EQ_1^2 =Q_1W^2.\]

Applying the converse of the Pythagorean Theorem, we infer that angle \(WEQ_1\) is indeed right.

The remaining points \(P_1, Q_2,\) and \(P_2\) are constructed in analogous fashion as shown in the applet, with angles \(WEP_1, EWQ_2,\) and \(EWP_2\) all being right. However, it is important to note that the four points \(Q_1, P_1, Q_2,\) and \(P_2\) are not the four corners of the square, as \(EQ_1, EP_1, WQ_2,\) and \(WP_2\) are \(5/12\)ths of the desired square side length (just a little less than half the desired side length).

In step 5 of the applet, we finally fix the four corners of the square. This is where point \(M\) is used. Take a rope of length \(EW\) (the desired square side length). Recall \(M\) was the midpoint of this rope. Attach both ends of the rope to point \(E\) and pull \(M\) south so that the rope passes through point \(Q_1.\) Now the point where \(M\) ends up determines corner \(A\) of the square. Indeed, this is a simple step, as we are just extending segment \(EQ_1\) to a segment with half the desired side length. The remaining corners \(B,\) \(C,\) and \(D\) are fixed in analogous fashion to obtain square \(ABDC,\) as desired. The fact that angles \(WEQ_1, WEP_1, EWQ_2,\) and \(EWP_2\) are right allow us to infer the angles at \(A,\) \(B,\) \(C,\) and \(D\) are right because sides \(BD\) and \(AC\) are parallel to the east-west line by construction.