This is a part of the series of lectures by Prof V Kannan from University of Hyderabad

1. Modern Mathematics in old
Sanskrit books
Module 5
Area formula for cyclic quadrilaterals

2. Quadrilateral
• A geometrical shape having four vertices and
four straight edges, without crossing.
• Squares, rectangles, rhombuses,
parallelograms, trapeziums, and others.

3. Area
• If a,b,c,d are the sides of a quadrilateral, can
the area be expressed in terms of a,b,c,d?
• No. Because different quadrilaterals with
same side-lengths can have different areas.
• For a quadrilateral with sides 1,1,2,2, if there
are right angles between unequal sides, the
area is 2; if there is a right angle between the
two sides of length 1, area is (1/2)+sq.rt.of 7.

4. Two quadrilaterals with sides 1,1,2,2
Area = 2 Area = ½ + √7
1
1
1
1
2
2
2
2

5. Cyclic quadrilaterals
• If the four vertices of a quadrilateral lie on a
circle, then that quadrilateral is said to be
cyclic. That circle is called the circum-circle of
that quadrilateral.
• In a cyclic quadrilateral, area is determined by
the lengths of the four sides. (In the two
examples of the previous slide, the former is
cyclic, the latter is not).

6. First version of the Formula
• Area = sq.rt. of (s-a)(s-b)(s-c)(s-d)
• Where s = (a+b+c+d)/2.
• Example: In a rectangle with length a and
breadth b, we have s = (a+b+a+b)/2 = a+b.
s-a = b, s-b = a, s-c = b, s-d = a.
Area = sq.rt.of b.a.b.a = a.b

7. Second version of the formula
• Area = one-fourth of the sq.rt.of
(a+b+c-d)(b+c+d-a)(c+d+a-b)(d+a+b-c)

8. Three parts of the lecture
• Part 1 : Four appreciations from foreigners.
• Part 2 : Two Samskrita passages.
• Part 3 : Remarks.

9. Four appreciaters
• Canner and Robertson
• Prof.Sachau
• Prof.Wallace
• A reviewer in Edinburgh Reviews

10. Quote 1
• “One can only marvel at this brilliant work of
Brahmagupta in 628.A.D.”
– Canner and Robertson.

11. Quote 2
• “There is one remarkable proposition namely
that which discovers the area of a triangle
when its three sides are known. This does not
seem to have been known to the ancient
Greek geometers”.
- Wallace, in Edin. Encyclopedia.

12. Quote 3
• “Brahmagupta occupies an important place in the
history of oriental culture. Brahmagupta taught
astronomy to the Arabs before they came to
know of Ptolemy’s works, since, reference to the
works Sindhind and Al-Arkand frequently occur in
Arabic literature.; these are the translations of
Brahmagupta’s works, Brahma(sphuta)siddhanta
and Khandakhayaka”.
- Prof.Sachau in the translation of Al-Biruni’s India.

13. About Sachau
• Carl Eduard Sachau (1845 - 1930) was a
German orientalist.
• He was an expert on Al-Biruni and wrote a
translation.
• he was appointed director of the new Seminar
of Oriental languages in 1887.

14. Quote 4
• “We confess that we did not expect to find it
(formula for area of a triangle) in the
Geometry of Hindusthan. We believe that it
was unknown to the Greek geometers, and
was, if we mistake not, first published by
Clavius in the sixteenth century”.
- A writer in Edinburgh review.

15. Corollary: Area of a triangle
• If a,b,c are the sides of a triangle, then its area
is the sq.rt.of s(s-a)(s-b)(s-c) where s =
a+b+c/2.
• Every triangle is a degenerate cyclic
quadrilateral, whose fourth side is 0.

16. Verification
• In a rectangle with sides a,b,a,b, diagonal
length is sq.rt.of) (a^2+b^2) (by Pythagoras
theorem).
• The area of the triangle (which is half of the
rectangle) is one-fourth of
((a+b+ √(a^2+b^2))((a+b-√(a^2+b^2))
((a-b+√ (a^2+b^2))((b-a+√(a^2+b^2))
On simplification this becomes ab/2.

17. About Brahmagupta
• Title: Ganita cakra chudaamani, meaning a
crown-jewel in the world of mathematicians.
• Born in 598 A.D.
• Father’s name: Jishnu.
• Place: Bhillamala in Gujarat.
• Author of two books: Brahma sphuta siddhanta
and Khanda khadyaka.
• The former book consists of 24 chapters and
1008 slokas.
• Bold in criticizing earlier experts.

18. Al Biruni praises Brahmagupta
• Brahmagupta was the “most distinguished
Indian astronomer”
• Note in the next five slides that Al Biruni does
not always praise Indians.

19. Al Biruni

20. First of five negative remarks on India
by Al Biruni
• “The uneducated among Hindus are much
more numerous than the educated”.

21. Second of five negative remarks on
India by Al Biruni
• “India itself is in comparison to the whole
inhabitable world, only a small matter”.

22. Third of five negative remarks on India
by Al Biruni
• “The number of those who differ from Hindus,
both in religion and law, is larger than the
number of those who agree with them”.

23. Fourth of five negative remarks on
India by Al Biruni
• They mix trivial results with profound results,
like pebbles with gems.

24. Fifth of five negative remarks on India
by Al Biruni
• For greeting elders, Indians have a peculiar
habit of falling down and kissing the earth in
front of them.

25. Part 1 (Four quotes of
appreciation) ends
Part 2 (Two Sanskrit passages) starts.

26. The first passage
-- Brahmasphuta siddhanta
= sides
= sum
= half
= quadruple
= decreased by
= product
= square root
= exactly.
• It is exactly the square
root of the product of
four quantities, each of
which is obtained by
subtracting one side from
the half-perimeter.

27. Brevity
Two drawbacks in the absence
of symbols
• Beauty gets lost.
• Brevity gets lost.
• Space taken while writing
down. (16 for Brahmagupta,
32 in one version with
mathematical symbols, 35
in another version with
mathematical symbols).
• Time taken while reading
out. (23 maatras in Sanskrit,
34 in mathematical
expression)
• Number of syllables. (16 in
Sanskrit, more than 35 in
the expressions with
symbols).

28. Three reasons for brevity in Sanskrit
• Prepositions (like of, with, from, to) are
merged with nouns.
• Compound words have hidden prepositions.
• Sometimes the number of letters decreases
when words are joined.

29. Second Sanskrit passage
- Mahaviracharya’s Ganitasarasangraha.

30. PART 3 STARTS : REMARKS
Part 2 ends: Two Sanskrit passages

31. Four remarks
• In Europe, who discovered this formula first?
• Omission of the word ‘cyclic’ by Brahmagupta.
• The corollary for triangles (area formula) is
attributed to Heron.
• Maximal area is for the cyclic ones.

32. First Remark
• Brahmagupta formula
was rediscovered in
Europe nearly a
thousand years later by
W.Snell in 1619 A.D. He
hails from Netherlands.

33. In praise of Brahmagupta
•
• Vrutta = circle.
• Vistara = area.
• Vrushabha = expert
• Vivrunvan = describing
• Vrutah = chosen
These five words starting with v create verbal
beauty in this verse.