.jpg)
Ancient Indian Mathematical texts are a treasure trove of interesting problems
Date : 18/05/2024
यदि समभुवि वेणुर्द्वित्रिपाणिप्रमाणो
गणक पवनवेगादेकदेशे स भग्न: ।
भुवि नृपमितहस्तेष्वंग लग्नं तदग्रम्
कथय कतिषु मूलादेष भग्न: करेषु ॥१५६॥
A 32 C (cubits, हस्त or forearm, 1 hasta = 20 inches approx) high bamboo pole standing on a level ground was broken by strong winds. The tip of the bamboo touched the ground 16 C from the foot of the pole. Then tell me the height of the point where the bamboo broke.
This interesting problem is from Bhaskarcharya's book Lilavati. For more information on Lilavati refer my earlier article "Completing the Square". https://www.indiachapter.in/user/article/2/36/81
To solve the above kind of problems, Bhaskaracharya had provided a method described in the following verses:
तत्कृत्योर्योगपदं कर्णो दो:कर्णवर्गयोर्विवरात् ।
मूलं कोटि: कोटिश्रुतिकृत्योरंतरात् पदं बाहु: ॥१४२॥
In a right triangle, the square root of the sum of squares of the base and the altitude is the hypotenuse, and the square root of the difference between the squares of the hypotenuse and the base (respectively altitude) is the altitude (respectively base).
This is popularly known as the Pythagoras theorem (560 BC). But this was stated earlier in Baudhayana's Sulbasutra (earlier than 800 BC).
To solve the current problem, Bhaskaracharya further provides us with the following method:
वंशाग्रमूलान्तरभूमिवर्गो वंशोध्दृतस्तेन पृथग्युतोनौ ।
वंशो तदर्धे भवत: क्रमेण वंशस्य खंडे श्रुतिकोटिरुपे ॥१५५॥
Figure 1
Suppose a bamboo BAC is broken at A and the part AC touches the ground at C. Refer Figure 1. Given the length X of the bamboo and the distance BC = Y, find X1 and X2 by:
Derivation:
Let's use this wisdom to solve our problem.
We have, X = 32, Y = 16
Hence, we have
Thus, the height of the point where the bamboo broke is 12 hastas.
Let us look at some more interesting problems from Bhaskaracharya's Lilavati which use Baudhayana's Sulbasutra for their solution:
अस्ति स्तंभतले बिलं तदुपरि क्रीडाशिखंडी स्थित:
स्तंभे हस्तनवोच्छ्रिते त्रिगुणितस्तंभप्रमाणान्तरे ।
दृष्ट्वाहिं बिलमाव्रजन्तमपतत् तिर्यक् स तस्योपरि ।
क्षिप्रं ब्रूहि तयोर्बिलात्कतिमितै: साम्येन गत्योर्युति: ॥१५८॥
A peacock was seated on a 9 C (cubits, हस्त or forearm, 1 hasta = 20 inches approx) high pole. At the foot of this pole was a snake burrow. The peacock saw a snake crawl towards the pole at a distance of 27 C from the pole. The peacock pounced on the snake at the same speed as the snake's crawl and caught it at a certain distance from the pole. Find the distance?
To solve this problem Bhaskaracharya had given the following sutra:
स्तंभस्य वर्गोऽहिबिलान्तरेण भक्त: फलं व्यालबिलान्तरालात् ।
शोध्यं तदर्धप्रमित: करै: स्यात् बिलाग्रतो व्यालकलापियोग: ॥१५७॥
Refer Figure 2. The peacock is perched at point B and it sees the snake at C. The peacock pounces and catches the snake at D. The peacock and the snake are moving at the same speed. Hence, they will move equal distances in equal time. That is, BD = BC = d.
Here, x and y are given. We have to find r.
Figure 2
Hence, the peacock caught the snake at a distance of 12 hastas from the pole.
Let us now solve one more interesting problem from Lilavati:
वृक्षाध्दस्तशतोच्छ्र्याच्छतयुगे वापीं कपि: कोऽप्यगात्
उत्तीर्याथ परो द्रुतं श्रुतिपथत्प्रोड्डीय किंचित द्रुमात् ।
जातैवं समता तयोर्यदि गतावुड्डीयमानं कियत्
विद्वंश्चेत्सुपरिश्रमोऽस्ति गणिते क्षिप्रं तदाचक्ष्व मे ॥१६३॥
There was a well at a distance of 200 cubits from a palm tree. Two monkeys were perched on this palm tree at a height of 100 C. One of the monkeys came down the tree and walked to the well. The other jumped up to a certain height on the tree and then pounced on the well along the hypotenuse. If both covered equal distances, find the height of the second monkey's jump.
To solve this problem Bhaskaracharya had given the following sutra:
द्विनिघ्नतालोच्छ्रितिसंयुतं यत्सरोऽन्तरं तेन विभाजिताया: ।
तालोच्छ्रितेस्तालसरोन्तरघ्न्या उड्डीयमानं खलु लभ्यते तत् ॥१६२॥
Figure 3
One of the monkey jumps from a point D on the palm tree to a higher point C on the tree and then onto the well B. Refer Figure 3. We are required to find x.
The other monkey climbs down from point D on the tree to point A and then from there he walks to the well.
The monkey had jumped up to a height of 50 hastas.
One might take an objection that the path of the peacock and jumping to the well by the monkey should be parabolic under the effect of the gravity. But for the sake of simplicity we can take them to be straight.
It is not that Bhaskaracharya was ignorant of gravity. Rather, he has given a sutra in Goladhyaya :
आकृष्टशक्तिश्च मही तया यत् खस्थं गुरुं स्वाभिमुखं स्वशक्त्तया ।
आकृष्यते तत्पततीव भाति समे समन्तात् क्व पतत्वियं खे ॥
Earth attracts inert bodies (lacking the ability to move) in space towards itself. The bodies appear to fall down on the earth. Since the space is homogeneous (alike in all directions), where will the earth fall?
Ancient Indian Mathematical texts are a treasure trove of such interesting and illustrative problems.
Image credits: Pixabay
Tags :
Note: Your email address will not be displayed with the comment.
.png)