अस्मद्गुरुभ्यो नमः -> Salute to our gurus!
Today, I'll take up रेखागणितम्
रेखागणितम् or Science of Geometry in संस्कृतम् was written by जगन्नाथ साम्राट् (1652–1744) under the order of राजा जयसिंह of Jaipur
In रेखागणितम् are similarities with Arabic-work by Nasir Eddin, raising the question whether it was an original work or just a translation..
Author says in opening verses saying of another work of his ‘सिद्धान्त साम्राट्’ that it is a translation of Arabic-language-book-मिजास्ती
Although Pandit जगन्नाथ साम्राट् defends saying it was orally transferred through generations in his family…
...evident from the primitive language in the grantha But, it is generally agreed that both these works are translations from Arabic.
However, शिल्पशास्त्रम् - the science of Geometry, misunderstood as for only sculpting, was first cultivated in India, ..
.. imported to Greece and other countries, and probably was lost in time.
Which raises the next question whether geometry itself originated in India or Greece?
The point of contact of classical-geometry is Alexandrian-Geometry and Sulba-Sutras.
Alexandrian-mathematicians Hero (215 BC) the earliest from Greece, and Pythogoras are known to have drawn parallels to Sulba-sutras but,..
.. the Sulba-Sutras are part of Srauta-sutras of Bodhayana and Apastamba (date debated 800-200BC).
However, Yajurveda+Taittiriya-BrahmaNa lay strict rules for construction of altars, bricks, arrangement, etc.Sulba-sutras are prior to them!
Anyway, mere translation/original work comprised from a collection of oral-renditions through generations, the merit needs to be appreciated
So let's star the journey!
In the author’s words, any book dealing with geometry or solid-geometry cannot be clear unless "theory of numbers" and terms used.
Here are some basic definitions:
Number अङ्क
Unit रूप
Even सम
Odd विषम
Prime-number प्रथमाङ्क
Multiplier गुणक
Multiplicand गुण्य
Point बिन्दु
Line रेखा
Plane क्षेत्र
Arm/Side भुज, बाहु
Square-number वर्गाङ्क or just वर्ग
Cube-number घनाङ्क or just घन
Proportional सजातीय
Perfect-number पूर्णाङ्क
.. goes on .. lets take more of them as they appear
Some Geometry-specific definitions before we take a couple of astonishing Theorems ..
Ready?
Definition #1: बिन्दु : point
यः पदार्थो दर्शनयोग्यो विभागानर्ह: स बिन्दुशब्दवाच्यः|
#SanskritAppreciationHour
Lets do the पदच्छेद and W2W
यः Which पदार्थो object दर्शन-योग्यो capable of being seen विभागानर्ह: not eligible for division (into parts) ..
स that वाच्यः is to be called as/by बिन्दुशब्द the word-बिन्दु nka 'point'
kna - now-known-as :)
Interesting definition isn't it? Apt too, I suppose! A point is one, that is capable of being seen (minutest) & cannot be divided further!
As usual, after the first definition, here is Quiz1: Can you split विभागानर्ह:, what sandhi?
OK, next definition:
चतुर्भुजम् : Quad-Arm kna Rectangle :)
Definition#2
चतुर्भुजम् : यस्य बाहुचतुष्टयम् समानं कोणचतुष्टयमपि समानं तञ्चतुरश्रं समकोणं समचतुर्भुजम् ज्ञेयम् |
#SanskritAppreciationHour
पदच्छेद and W2W now: यस्य Whose बाहु arms चतुष्टयम् all-four समानं are equal कोण angle चतुष्टयमपि all-four also are समानं are equal
.. तञ्चतुरश्रं that quad-cornered figure (from अश्रि) is ज्ञेयम् to be known as समकोणं equi-angular समचतुर्भुजम् equi-armed (figure)
समचतुर्भुजम् equi-quad-armed (figure):)
Definition clear? Question?
Quiz2: What sandhi is तञ्चतुरश्रं ?
Now to real stuff, प्रथमोsध्यायः एकानविंशतितम क्षेत्रम् Pg88 of रेखागणितम् vol1
Note क्षेत्रम् here means property/theorem
Greatest/Shortest sides of त्रिभुजम्–Tri-Arm
तत्र त्रिभुजे योsधिककोणस्तत्सम्मुखभुजोsपि महान् भवति योsल्पकोणस्तत्सम्मुखभुजोsपि लघुर्भवति |
त्रिभुजम्–Tri-Arm NKA => Triangle :)
Don't bother about the long compounds, lets split them!
{ Now to real stuff, प्रथमोsध्यायः एकोनविंशतितम क्षेत्रम् Pg88 of रेखागणितम् vol1
Note क्षेत्रम् here means property/theorem
पदच्छेद and W2W:
तत्र There त्रिभुजे in Tri-Arm
Let’s break the compound: योsधिककोणस्तत्सम्मुखभुजोsपि => य:+अधिक+कोण:+ तत्+सम्मुख+भुज:+अपि
य:which अल्प meagre/smaller कोण: angle तत् that(it’s) सम्मुख opposite भुज: arm/side अपि also भवति will be लघु: small!
Easy isn't it?
#SanskritAppreciationHour
Meaning .. य:which अधिक greater कोण: angle तत् that(it’s) सम्मुख opposite भुज: arm/side अपि also भवति will be महान् great!
Similarly- योsल्पकोणस्तत्सम्मुखभुजोsपि => य:+ अल्प+कोण:+ तत्+सम्मुख+भुज:+अपि
य:which अल्प meagre/smaller कोण: angle तत् that(it’s) सम्मुख opposite भुज: arm/side अपि also भवति will be लघु: small!
Complete meaning: Arm opposite to the largest angle is the largest side, and arm-opposite to the smallest-angle is the shortest!
Orally, easy to pass it on to generations!
तत्र त्रिभुजे योsधिककोणस्तत्सम्मुखभुजोsपि महान् भवति योsल्पकोणस्तत्सम्मुखभुजोsपि लघुर्भवति |
Of course, need to decipher the meaning!
Quick theorem:
तृतीयोsध्यायः एकविंशतितम क्षेत्रम्
वृत्तद्वयस्य संस्पर्शः एकस्मिन्नेव चिन्हे भवति नान्यत्र |
#SanskritAppreciationHour
BTW, these are non-intersecting-planes. Lets do पदच्छेद and W2W:
द्वयस्य two वृत्त circles’ संस्पर्शः touch भवति happens एकस्मिन्नेव only-at-one चिन्हे point नान्यत्र not-anywhere-else!
That was easy to understand, wasn't it?
BTW, Quiz3: What sandhi is: एकस्मिन्नेव
Shall we take another interesting theorem now?
द्वितीयोsध्यायः द्वादशक्षेत्रम् from Pg156
यत्त्रिभुजमधिककोणरूपमस्ति तत्कोणसम्मुखभुजस्य वर्गोsवशिष्टभुजद्वयवर्गयोगादाधिको भवति|
Again, quite easy when we break it, lets do the पदच्छेद and W2W ..
यत्त्रिभुजमधिककोणरूपमस्ति => यत्+त्रिभुजम्+अधिक+कोण+रूपम्+अस्ति
यत्+त्रिभुजम्+अधिक+कोण+रूपम्+अस्ति यत् In which त्रिभुजम् Tri-Arm अस्ति there is अधिक+कोण greater-angle => Obtuse-Angle रूपम् appearance
तत्+कोण+सम्मुख+भुजस्य तत् that कोण angle’s सम्मुख opposite भुजस्य Arm’s वर्गः+अवशिष्ट+भुज+द्वय+वर्गात्+अधिकः वर्ग: square भवति will be …
अवशिष्ट remaining भुज+द्वय two-arm’s वर्ग squares अधिकः more than योगात् sum
As you can see, just breaking it up makes it a cake-walk!
Attaching a pic showing the proof .. pic.twitter.com/OdpfhY3OXp
(https://twitter.com/haritirumalai/status/4318039727901450250)
Of course, the proof draws root from रज्जु-सूत्र of बोधायन-सुलभ-सूत्राणि rather than plagiarizing from आचार्य-Pythogoras :)
That’s I have for today .. time's up as well!
Thanks for those who attended, hope you’ve enjoyed and are encouraged to look into रेखागणितम्/शिल्पशास्त्रम् in #Sanskrit!
I often ask myself, what did #Sanskrit not have? I’ll hang-around for a bit if any interactions, over-to रोहिणी महोपाध्याया @rohinibakshi
BTW, Quiz-Answers:
Quiz1: विभागानर्ह: =>विभाग अनर्ह: => सवर्णदीर्घ-सन्धिः @PnNamboo
Quiz2: तञ्चतुरश्रं => तम् चतुरश्रम् => अनुनासिक-सन्धिः
Quiz3: एकस्मिन्नेव => एकस्मिन् एव => ङमुडागम-सन्धिः
@RohiniBakshi Thanks for the opportunity!
Have a great weekend everyone! And see you next week for more #Sanskrit with #SanskritAppreciationHour
