Vedic Science:Veda Ganita Query: Find Lotus length & water depth

[Hemanth Kumar G]

Hemanth Kumar G has sent you a link to a blog:

How to interpret this Formula? I request interested ones to reply to this post with their solutions.

Blog: Vedic Science
Post: Veda Ganita Query: Find Lotus length & water depth
Link: http://veda-vijnana.blogspot.com/2012/09/veda-ganita-query-find-lotus-length.html

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[Nityanand Misra]

Dear Sh. Hemanth Kumar

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.

Anyway कोयादूर्ध्वम् is incorrect, it is तोयादूर्ध्वम् and here is the complete correct verse (a beautiful मन्दाक्रान्ता on top of it!!)

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

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).

I leave it to the poets and scholars to give the answer in मन्दाक्रान्ता.

Thanks, Nityanand

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Nityānanda Miśra
Member, Advisory Council, Jagadguru Rambhadracharya Handicapped University
Chitrakoot, Uttar Pradesh, India
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|| आत्मा तत्त्वमसि श्वेतकेतो ||
(Thou art from/for/of/in That Ātman, O Śvetaketu)
     - Ṛṣi Uddālaka to his son, Chāndogyopaniṣad 6.8.7, The Sāma Veda

[narayanan er]

Dear Sh. Nityanand,

I am neither a poet nor a mathematician. Thank you for putting verse 112 of Bhaskara's Bijaganita. You are well a mathematician and provided the progression and answer too. I take this challenge to make a Mandakranta with the answer to the question you have already provided. Being too week in mathematics, I can hardly calculate anything like this. It is not easy to bring that environment of birds, lotus, breeze and waves like Bhaskara brought for us. Still, here is my answer:

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

Puzzles like this are always interesting. But mathematics is utterly beyond my reach. You can teach me by putting verse 113 of Bhaskara's Bijaganita answered. Then I may try for a Sardulavikridita answer. Thanks, Sh. Nityanand.

Regards,

Narayanan

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From: Nityanand Misra <nmi...@gmail.com>
To: heman...@gmail.com
Cc: bvpar...@googlegroups.com
Sent: Saturday, 1 September 2012 6:47 AM
Subject: Re: {भारतीयविद्वत्परिषत्} Vedic Science:Veda Ganita Query: Find Lotus length & water depth

[Nityanand Misra]

Dear Sh. Narayanan

That's a beautiful verse you composed, and going by it, nobody would agree you are not a poet! I am humbled. Thanks also for correcting me - I thought the verse was from Lilavati but you are right - it is from BIjaganitam.

As you requested, please see the solution to the Bijaganitam's 113th verse. The second monkey jumps to a height of 50 hand-lengths.

Same concepts are used - Pythagoras Theorem, Binomial Expansion and Linear Equation in One Variable (एकवर्णसमीकरणम्). Note that भास्कराचार्य uses the technical term श्रुतिपथ which means the "path of the hypotenuse" of a right-angled triangle. The Hindi medium mathematics books in Uttar Pradesh use the term कर्ण for the hypotenuse.

Now I will eagerly wait for your शार्दूलविक्रीडितम्. As a request, please use the अङ्कानां वामतो गतिः principle and Samskrita names of numbers (e.g. ब्रह्म=1, अक्षि=2, अग्नि=3, श्रुति=4, बाण = 5, रस=6, द्वीप=7, वसु=8, ग्रह=9).

Finally, thanks a lot to Sh. Hemanth Kumar and you - you made me remember my high-school Mathematics teacher from Kendriya Vidyalaya (Junagadh), Sh. Pawan Kumar Gupta of Mainpuri, Uttar Pradesh, who gave us many problems of Bhaskara from Lilavati and Bijaganitam to solve in classes 9th and 10th. May he have a long life.

Thanks, Nityanand

2012/9/1 narayanan er <drerna...@yahoo.com>

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निराशीर्निर्ममो भूत्वा युध्यस्व विगतज्वरः।। (भ.गी.)
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[Nityanand Misra]

Typo, it is verse 113 and not 115.

2012/9/1 Nityanand Misra <nmi...@gmail.com>

For everybody's benefit, here is the verse 115 of Bijaganitam:

वृक्षाद्धस्तशतोच्छ्रयाच्छतयुगे वापीं कपिः कोऽप्यगादुत्तीर्याथ परो द्रुतं श्रुतिपथात्प्रोड्डीय किञ्चिद्द्रुमात्
जातैवं समता तयोर्यदि गतावुड्डीयमानं कियद्विद्वन् चेत्सुपरिश्रमोऽस्ति गणिते क्षिप्रं तदाचक्ष्व मे

Thanks, Nityanand

[Nityanand Misra]

For everybody's benefit, here is the verse 115 of Bijaganitam:

वृक्षाद्धस्तशतोच्छ्रयाच्छतयुगे वापीं कपिः कोऽप्यगादुत्तीर्याथ परो द्रुतं श्रुतिपथात्प्रोड्डीय किञ्चिद्द्रुमात्
जातैवं समता तयोर्यदि गतावुड्डीयमानं कियद्विद्वन् चेत्सुपरिश्रमोऽस्ति गणिते क्षिप्रं तदाचक्ष्व मे

Thanks, Nityanand

On Sat, Sep 1, 2012 at 6:52 PM, Nityanand Misra <nmi...@gmail.com> wrote:

[narayanan er]

Dear Sh. Nityanand,
I understand that what Pythagoras says: in a right angled triangle, the square on the hypotenuse is equals to the sum of the squares on other two sides. Bhaskara says:
वृक्षाद्धस्तशतोच्छ्रयाच्छतयुगे वापीं कपिः कोऽप्यगा-

दुत्तीर्याथ परो द्रुतं श्रुतिपथात् प्रोड्डीय किञ्चिद् द्रुमात्।
जातैवं समता तयोर्यदि गतावुड्डीयमानं कियद्

विद्वन् चेत् सुपरिश्रमोऽस्ति गणिते क्षिप्रं तदाचक्ष्व मे।।११३।। (बीजगणितम्)

So, this my Sardulavikridita answer (actual answer is yours) to it:

आद्यो हस्तशतद्वयं पथिगमो वृक्षेण चोर्ध्वं शतं
वाप्या वृक्षशिखान्तगस्त्रिशतं हस्तैश्च शाखामृगः।
कर्णेनोच्चखबाणहस्तसुमितः प्रोड्डीयमानः कपि-
र्विद्वान्नाप्यपरिश्रमोऽस्मि गणिते तत्त्वं ब्रवीमि श्रुतम्।।

Here ख is zero and बाण is five to make खबाण is fifty. Bhaskara used श्रुति (karna) for hypotenuse, not for four Vedas.

Regards,
Narayanan
 

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From: Nityanand Misra <nmi...@gmail.com>

To: drerna...@yahoo.com
Cc: "heman...@gmail.com" <heman...@gmail.com>; "bvpar...@googlegroups.com" <bvpar...@googlegroups.com>
Sent: Saturday, 1 September 2012 4:31 PM

[Nityanand Misra]

Dear Sh. Narayanan

Very beautiful and nice. Two doubts:

1. The second foot is one syllable too short - did you miss a syllable while typing? By changing  त्रिशतं to त्रिशतकं or to खखतपं (ख=0, ख=0, तप=अग्नि=3) or शतत्रयं with the त as लघु would make it perfect.
   
2. In the last foot, is  विद्वान्नाप्यपरिश्रमोऽस्मि to be split as विद्वान् न अपि अपरिश्रमः अस्मि? 

Regards, Nityanand

2012/9/2 narayanan er <drerna...@yahoo.com>

[narayanan er]

Dear Sh. Nityanand,

आद्यो हस्तशतद्वयं पथिगमो वृक्षेण चोर्ध्वं शतं

वाप्या वृक्षशिखान्तगस्त्रिशततो हस्तैश्च शाखामृगः।

कर्णेनोच्चखबाणहस्तसुमितः प्रोड्डीयमानः कपि-
र्विद्वान्नाप्यपरिश्रमोऽस्मि गणिते तत्त्वं ब्रवीमि श्रुतम्।।

I corrected the second foot, as a tasilanta was intended. Thanks for pointing out the typo error, which was not intended. I did not check it after typing. There had current while preparing the verse but, had no current at home this morning while I was sending, but there had a fifteen minutes UPS back-up. So, I finished the sending with the help of monitor light.
Regarding the final foot, the prose order is: (अहं) विद्वान् न, गणिते अपरिश्रमः अपि अस्मि। श्रुतं तत्त्वं ब्रवीमि।
Regards,
Narayanan

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From: Nityanand Misra <nmi...@gmail.com>
To: drerna...@yahoo.com
Cc: "heman...@gmail.com" <heman...@gmail.com>; "bvpar...@googlegroups.com" <bvpar...@googlegroups.com>

Sent: Sunday, 2 September 2012 9:50 AM