Yuktibhāṣā (Malayalam: യുക്തിഭാഷ; "Rationale in the Malayalam/Sanskrit language") also known as Gaṇitanyāyasaṅgraha ("Compendium of astronomical rationale"), is a major treatise on mathematics and astronomy, written by Indian astronomer Jyesthadeva of the Kerala school of mathematics in about AD 1530. The treatise is a consolidation of the discoveries by Madhava of Sangamagrama, Nilakantha Somayaji, Parameswara, Jyeshtadeva, Achyuta Pisharati and other astronomer-mathematicians of the Kerala school. Yuktibhasa is mainly based on Nilakantha's Tantra Samgraha. It is considered an early text on some of the foundations of calculus and predates those of European mathematicians such as James Gregory by over a century. However, the treatise was largely unnoticed beyond Kerala, as the book was written in the local language of Malayalam. However, some have argued that mathematics from Kerala were transmitted to Europe (see Possible transmission of Keralese mathematics to Europe).
The work was unique for its time, since it contained proofs and derivations of the theorems that it presented; something that was not usually done by any Indian mathematicians of that era. Some of its important developments in analysis include: the infinite series expansion of a function, the power series, the Taylor series, the trigonometric series for sine, cosine, tangent and arctangent, the second and third order Taylor series approximations of sine and cosine, the power series of π, π/4, θ, the radius, diameter and circumference, and tests of convergence.
legions of shadows
surround me now
welcome
they say
it’s ok
you’re here to stay
they’re sent for us
they’re sent for us
and somehow we came
years of a lifetime
they’re all blurring into one
they’re sent for us
they’re sent for us
and somehow we came
there’s no more sent
no more sent
they’re sent for us
trouble and pain
they were streamed all away
set us free from the chains
tah-tah yeah tah-tah
trouble and pain
I wanna see you wash all away
set us free from the chains
they’re sent for us
they’re sent for us
and somehow we came
there’s no more sent
I saw you closing up inside,
I must’ve tried a thousand times
To light the darkness in your eyes
While you were falling.
Now we’re heading out of town
‘Cause there’s nothing left for you now.
And you swept them through my hands
As you were falling.
I never get it right,
I never see you fight.
Though you [...]
You’re breaking in two,
Can’t clue you together.
So what can I do?
I’ve got a thousand things to say,
I saw you drifting [...] from the page.
You’re the only one that keeps me safe.
I’m calling, I’m calling.
Gotta keep, gotta keep holding on.
Gotta keep, gotta keep holding on.
Holding on.
I never get it right,
I never see you fight.
Though you [...]
You’re breaking in two,
Can’t clue you together.
So what can I do?
I never can’t get through.
I never can’t get through.
Why do we build these walls so tall?
Just to climb back over them,
Climb back over them.
I never get it right,
I never see you fight.
Though you [...]
You’re breaking in two,
Can’t clue you together.
So what can we do?
I never get it right,
I never see you fight.
Though you [...]
You’re breaking in two,
Can’t clue you together.
So what can we do?
I never can’t get through.