Saturday, 1 September 2012

Veda Ganita Query: Find Lotus length & water depth

While going through manuscript by Sharnghyadhara, there was a shloka to demonstrate that one doesn't have to go near to an object for its measurement. By its gOchaara lakshana itself, whole knowledge can be attained. Based on these mathematical arguments he proves that its not required to go to a planet to know its Kirana, GOchaara, ShaaKha, Chalanaa rules. The mathematical formulas which calculate these kinds of parameters accurately are grouped under KhagOLa-Ganita. So the query is as follows:
There was a pond, it had blossomed a Lotus which stood by its stalk, bit high on top of water. This was the picture got when a person stands in the banks of pond. A mathematician if wishes can calculate the total length of Lotus-Stalk, depth of water from tip of stalk floating up, and other detail can be calculated as he wishes. It is the power of Beejaganita & Rekhaganita.

Veda Sutra - 
                                           अनारम्भणे तदवीरयेथा रणिस्थाने नारी देशे समुद्रे | 
                                           यदश्विना ऊहथुर्जं अस्तु शतपत्रं ऊनत्रयं अस्तुमेकं ||
Kavya Sutra - 

               चक्रक्रौञ्चाकुलितसलिले क्वापि दृष्टं तडागे तोयादूर्ध्वं कमलकलिकाग्रं वितस्तिप्रमाणम् ।
                मन्दं मन्दं चलितमनिलेनाहतं हस्तयुग्मे तस्मिन्मग्नं गणक कथय क्षिप्रमम्बुप्रमाणम् ||

We know that Trigonometry can be utilized. But this is bit higher than basic ones. So, how to interpret this?

Solution of Sri Nityanad Misraji - 

This problem from Lilavati can be solved without going into trigonometry or other higher concepts. All you need is the concepts of Pythagoras Theorem, Binomial Expansion and Linear Equation in One Variable. All three concepts were known at the time of Bhaskara I|. The Pythagoras Theorem is known from शुल्बसूत्र and Binomial Expansion coefficients can be known from what हलायुध refers as मेरुप्रस्तार, an old concept attributed to पिङ्गल.
वितस्तिप्रमाणम् = One hand-span
हस्तयुग्मे = lotus submerges in the pond at a distance of 2 hand-spans from its original location.

Refer the graphic for the solution, the depth of the water is 1.5 hand-spans. The Pythagorean triplet is (1.5,2,2.5) which corresponds to the simplest Pythagorean triplet of (3,4,5).

Solution in Mandakranta Vrutta by Narayanan E R :-

पद्मस्याग्रं स्पृशति सलिलं मानतश्चेद्वितस्तं
नित्यानन्दो गणनकुशलः संगणय्येति तत्त्वम्।
अर्धादर्धं प्रगतिरुदिता तावदूर्ध्वं तडागे
सार्धैका सा जलपरिमितिर्ह्युच्चलद्धस्तयुग्मा।।  

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