PI day - What Aryabhatta said about Pi

by indophile Fri Mar 14, 2014 9:27 am

Let us remember on Pi day Aryabhatta, the mathematician who mentioned what Pi is in a simple verse, First the verse in original Sanskrit followed by English transliteration.

चतुरधिकम् शतम्, अष्टगुणम् द्वाषष्टि
तथा सहस्राणाम् अयुत द्वय विष्कम्भस्य
आसन्न: वृत्त

चतुरधिकम् शतम् = 100+4 = 104; अष्टगुणम् = 104 * 8 = 832; द्वाषष्टि = 62; तथा = thus; सहस्राणाम् = 62 * 1000 = 62,000; अयुत = ten thousand fold;द्वय = by two; विष्कम्भस्य = of the diameter of a circle; आसन्न: = approximately; वृत्त = circle; परिणाह: = circumference

(((100+4) * 8 + (62 * 1000)) / 20000) * Diameter = Circumfrence of a circle, approximately.

In English:
Chaturadhikam shatam, ashTaguNam davaashashTi
Tathaa sahasraguNam ayuta davaya vishkambhasya
Aasannah vRtta

Chaturadhikam shatam = 100+4 = 104; ashTaguNam = 104 * 8 = 832; davashashTi = 62; tatha = thus / that is; sahasraguNam = 62 * 1000 = 62,000; ayuta = ten thousand fold; davaya = by two; vishkambhasya = of the diameter of a circle; aasannah = approximately; vRtta = circle; pariNaahah = circumference

(62,832 / 20,000) * Diameter = Circumfrence of a circle, approximately
or
3.1416 * D = Circumfrence of a circle, approximately

Aryabhatta didn't give a name for the approximate 3.1416 quantity however.

by MaxEntropy_Man Fri Mar 14, 2014 9:56 am

the real towering intellectual of ancient indian mathematics is madhava of sangamagrama (present day irinjilakudi in kerala).

read about the madhava-leibniz series formula for pi:

http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

that cirumference formulation useful as it is, is not sexy mathematics like the series solutions. this one converges, but converges slowly. ramanujan came up with much more astounding ones later.

by MaxEntropy_Man Fri Mar 14, 2014 10:01 am

here is ramanujan's:

$PI day - What Aryabhatta said about Pi 3cca05380343cb6f78d776d5e40b3a9d$

computes eight additional digits for each extra term.

by rawemotions Sat Mar 15, 2014 8:32 pm

MaxEntropy_Man wrote:here is ramanujan's:

$PI day - What Aryabhatta said about Pi 3cca05380343cb6f78d776d5e40b3a9d$

computes eight additional digits for each extra term.

Didn't know this one. The series goes to Infinity. To how many digits of Pi, has this Sum of the series been verified ?

by rawemotions Sat Mar 15, 2014 8:35 pm

MaxEntropy_Man wrote:the real towering intellectual of ancient indian mathematics is madhava of sangamagrama (present day irinjilakudi in kerala).

read about the madhava-leibniz series formula for pi:

http://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80

that cirumference formulation useful as it is, is not sexy mathematics like the series solutions. this one converges, but converges slowly. ramanujan came up with much more astounding ones later.

Totally agree. He predates newton by about 200 years and his work came pretty close to calculus. Anyway he is one of the few Indian mathematicians, that the West has grudgingly recognized. The newton and Leibnitz series is now called Madhava-Newton and Madhava-Leibnitz series. The sad thing is that, most kids studying today, still study older textbooks and do not know about this.

by Idéfix Sat Mar 15, 2014 11:24 pm

indophile wrote:Let us remember on Pi day Aryabhatta, the mathematician who mentioned what Pi is in a simple verse, First the verse in original Sanskrit followed by English transliteration.

चतुरधिकम् शतम्, अष्टगुणम् द्वाषष्टि
तथा सहस्राणाम् अयुत द्वय विष्कम्भस्य
आसन्न: वृत्त

चतुरधिकम् शतम् = 100+4 = 104; अष्टगुणम् = 104 * 8 = 832; द्वाषष्टि = 62; तथा = thus; सहस्राणाम् = 62 * 1000 = 62,000; अयुत = ten thousand fold;द्वय = by two; विष्कम्भस्य = of the diameter of a circle; आसन्न: = approximately; वृत्त = circle; परिणाह: = circumference

(((100+4) * 8 + (62 * 1000)) / 20000) * Diameter = Circumfrence of a circle, approximately.

In English:
Chaturadhikam shatam, ashTaguNam davaashashTi
Tathaa sahasraguNam ayuta davaya vishkambhasya
Aasannah vRtta

Chaturadhikam shatam = 100+4 = 104; ashTaguNam = 104 * 8 = 832; davashashTi = 62; tatha = thus / that is; sahasraguNam = 62 * 1000 = 62,000; ayuta = ten thousand fold; davaya = by two; vishkambhasya = of the diameter of a circle; aasannah = approximately; vRtta = circle; pariNaahah = circumference

(62,832 / 20,000) * Diameter = Circumfrence of a circle, approximately
or
3.1416 * D = Circumfrence of a circle, approximately

Aryabhatta didn't give a name for the approximate 3.1416 quantity however.

Thanks for posting this!

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