Congratulations, Pat Costello!

Pat’s doctoral training was in quadratic forms.

Brahmagupta

• Brahmashutasiddhanta (628 CE)
• Like most other text, contains substantial material on astronomy/astrology.
• First surviving text to distinguish between:
  • “manifest” computation (arithmetic)
  • “unmanifest” computation (algebra, more or less)
• Impressive contributions in several areas, including
  • geometry
  • number theory (“square-nature” problems, AKA quadratic forms!)

So He Was Good …

… and he was only too happy to say so:

• “If [Aryabhata’s] constants give the correct results for conjunctions and eclipses, they must be considered as accidental as the letters cut into wood by weevils.” (BrShSi I.62)
• “Men who have learned from the works of Srisena, Visnucandra and Aryabhata cannot face me in any debate, like the deer who sees the lion.” (BrShSi I.63)

(Translation modified from Satya Prakash, see [2].)

Square-Nature Problem

BrShSi 18.64-65:

Put down twice the square-root of a given square multiplied by a multiplier and increased or diminished by an arbitrary number. The product of the first pair multiplied by the multiplier, with the product of the last pair, is the last computed.

The sum of the thunderbolts is the first. The additive is equal to the product of the additives. The two square roots, divided by the additive or the subtractive, are the additive rupas.

In Modern Parlance

Suppose that

• \((x,y) = (\alpha_1, \beta_1)\) satisfies \(Nx^2 + k_1 = y^2\)
• \((x,y) = (\alpha_2, \beta_2)\) satisfies \(Nx^2 + k_2 = y^2\)

Then two solutions of \(Nx^2 + k_1k_2 = y^2\) are:

\[(x,y) = (\alpha_1\beta_2 \pm \alpha_2\beta_1,\beta_1\beta_2 \pm N\alpha_1\alpha_2).\]

A Modern Proof

Assume that:

• \(N\alpha_1^2 + k_1 = \beta_1^2\)
• \(N\alpha_2^2 + k_2 = \beta_2^2\)

Consider “Brahmagupta’s Identity”:

\[(a^2+nb^2)(c^2+nd^2)=(ac+nbd)^2+n(ad-bc)^2.\]

Substitute: \(n = -N, a = \beta_1, b = \alpha_1, c = \beta_2, d = \alpha_2\).

\[(\beta_1-n\alpha_1^2)(\beta_2-N\alpha_2^2)=(\beta_1\beta_2-N\alpha_1\alpha_2)^2-N(\beta_1\alpha_2-\alpha_1\beta_2)^2.\]

\[ \begin{aligned} (\beta_1-n\alpha_1^2)(\beta_2-N\alpha_2^2)=(\beta_1\beta_2-N\alpha_1\alpha_2)^2-N(\beta_1\alpha_2-\alpha_1\beta_2)^2,\\ k_1k_2=(\beta_1\beta_2-N\alpha_1\alpha_2)^2-N(\beta_1\alpha_2-\alpha_1\beta_2)^2,\\ k_1k_2+N(\beta_1\alpha_2-\alpha_1\beta_2)^2=(\beta_1\beta_2-N\alpha_1\alpha_2)^2,\\ N(\alpha_1\beta_2- \beta_1\alpha_2)^2 + k_1k_2=(\beta_1\beta_2-N\alpha_1\alpha_2)^2. \end{aligned} \]

For \(+\) versions, note that \((\alpha_1, -\beta_1)\) also solves \(Nx^2+k_1=y^2\).

“Brahmagupta Identity”

\[(a^2+nb^2)(c^2+nd^2)=(ac+nbd)^2+n(ad-bc)^2.\]

Not mentioned by Brahmagupta. Also not mentioned in any extant pre-modern Indian proofs (that I know of).

But it inspired Manjul Bhargava as a child.

The Kerala School

• Flourished circa 1380-1600 CE.
• Founder and most important member: Madhava of Samgamagrama
  • Infinite series formulas for trig functions
  • correction terms for partial-sum approximations
• Madhava and succesors develop concise and accurate trig tables.
• Successors develop astronomy, write illuminating commentaries.
• Nilakantha Somayagin (circa 1450-1550) is in this line.

Aryabhatiyabhasya

• Commentary on the Aryabhatiya of Aryabhata (400’s CE)
• One of the earliest commentaries to offer detailed proofs of most propositions.

Aryabhata 22

\[1^2+2^2+\ldots+n^2=\frac{n(n+1)(2n+1)}{6}.\] \[1^3+2^3+\ldots+n^3=\left(\frac{n(n+1)}{2}\right)^2.\]

Nilakantha’s Proof (Cubes)

Part of it has been translated by Kim Plofker (in [1]).

My aim is to fill in the omitted passage.

References/Resources

1. The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, ed. V Katz. Princeton University Press, 2007.
2. Brahmagupta and His Works, Satya Prakash. Indian Inst. Astronomical & Sanskrit Research, New Delhi, 1968.
3. Mathematics in India, Kim Plofker. Princeton University Press, 2009.
4. The Arybhatiya of Aryabhatacarya, with the Bhasya of Nilakanthasomasutvan, ed. K. Sambasiva Sastri, Trivandrum, 1930.