Modern Mathematics in old Sanskrit books

Module 4

Pythagoras Theorem

Three Parts

• Part 1: Four Quotes.

• Part 2: Five Sanskrit passages.

• Part 3: Six remarks.

Pythagoras formula

In a triangle ABC, if the angle at B is a right angle, then AC2 = AB2 + BC2.

• If c is the largest side in a right angled triangle, and if a and b are the other two sides, then c2

= a2 + b2.

A

B C

Pythagoras

• Pythagoras of Samos was a Greek philosopher, mathematician and founder of a religious movement called Pythagoreanism.

Quote

• “I am convinced that everything has come down to us from the banks of Ganga(Ganges), -Astronomy, Astrology, metempsychosis,etc.”

William Thibaut

George Frederick William Thibaut(March 20, 1848–1914) was an Indologist notable for his contributions to the understanding of ancient Indian mathematics and astronomy.

Thibaut was born in Germany, worked briefly in England, and then in 1875 was appointed Professor at the Government Sanskrit College , Varanasi

Quote

The geometrical theorem I-47 of Sulba sutra which tradition ascribes to Pythagoras, was solved by Hindus atleast two centuries earlier.

- Dr. Thibaut,

Jour.Asiatic society of Bengal

(1875),p.227.

Quote

• “Many of the vedic contributions to mathematics have been neglected or worse. When it first became apparent that there was geometry contained within works that were not of Greek origin, historians and mathematical commentators went to great length to try and claim that this geometry was Greek-influenced.”

--Ian G.Pearce.

Jim Loy

• One of the most visited sites on the internet.

• “The proof by Legendre was probably originally devised by an ancient Hindu mathematician”.

Voltaire France 1694-1774 Writer, philosopher

Thibaut Germany 1848-1914 Indologist

Ian G.Pearce U.K. Now Historian of Mathematics

Jim Loy U.S.A. Now Multifaceted,Computer expert.

Part 1 ends.

Part 2 starts.

Names of this theorem

• Bhuja koti karna nyaya

• Hypotenuse theorem

• Diagonal square theorem

• Sulva theorem

• Bodhayana theorem.

Five Sanskrit books

• Sulvasutra (of Bodhaayana and of Apastamba)

• Aryabhateeyam (of Aryabhata)

• Brahmasphutasiddhanta (of Brahmagupta)

• Trishatika (of Sridhara)

• Lilavati (of Bhaskara)

This is a statement in Chapter I of Bodhaayana’s Sulvasutra.

Deergha chaturasrasya akshnayaa rajjuh paarshvamaanee tiryangmaanee cha

yat pruthagbhuute kurutah tat ubhayam karoti

Source Book

Translation: In a rectangle, the square of the diagonal is equal to the sum of the squares of the adjacent sides.

Technical terms:

Dirgha caturasra = rectangle.

Akshnayaa = along the diagonal.

Parshvam &tiryak = two adjacent sides.

Karoti = yields a square.

Usages of the word akshnaya(this word seems to be available only

in vedic literature.)word meaning book

akshnayavan One who goes transversely Rigveda

akshnaya 1.Across 2.in a crooked way Satapata Brahmanam

Akshnayadesam Interim region --do--

Akshnayaapaccedanam Cutting across Sulvasutra

akshnayaastomeeya Name of something Taittiriya samhita

Quote

• “This (passage from Sulbasutra) appears to be referring to a rectangle, although some interpretations consider this to refer to a square. … The text seems to be quite open to unequal sides. If this refers to a rectangle, it is the earliest recorded statement of the Pythagorean theorem”. – Wikipedia.

Quote

• “Sulba sutras include unarguable evidence of the use of Pythagoras theorem and Pythagorean triples, predating Pythagoras (c.572-497B.C.) and evidence of a number of geometrical proofs”. – Ian G.Pearce., May 2002.

Aryabhateeyam

•

Whatever is the square of the base and the square of the perpendicular side, that (together) is the square of the hypotenuse.

From Brahma gupta’s work

•

• Hypotenuse is the square root of the sum of the squares of the base and the perpendicular side.

• C = √(a^2+b^2)

-- Brahma sphuta siddhanta.

Sridhara

•

Take the two quantities namely base and the perpendicular side. Square them and add. Take the square root. It becomes the length of the hypotenuse.

-- Trisatika.

Bhaskara

•

•

Given a side, the other side perpendicular to it is called koti. This terminology applies to both the triangles and quadrilaterals. The square root of the sum of the squares of these two is the hypotenuse.

-Lilavati, kshetra vyavahara.

A page from Lilavati describing Pythagoras theorem.

Time

• Bodhayana

• Aryabhata

• Brahmagupta

• Sridhara

• Bhaskara

• 800B.C.

• 476A.D.

• 628A.D.

• 8th century A.D.

• 1114A.D.

Six remarks

• Pythagoras theorem is very important. It is the first among “the seventeen equations that changed the world”.

• It is probably the only theorem having more than 200 proofs.

• Some proofs are available in Sanskrit books. One of the oldest proofs is due to Bhaskara.

• Visit of Pythagoras to India is a controversial topic.

• Dating of Sulvasutras is a rich topic of investigation.

• Pythagorean triples form an interesting related topic.