Ganita Kaumudi

Ganita Kaumudi is a treatise on mathematics written by Indian mathematician Narayana Pandita in 1356. It was an arithmetical treatise alongside the other algebraic treatise called "Bijganita Vatamsa" by Narayana Pandit. It was written as a commentary on the Līlāvatī by Bhāskara II.

Contents

Gaṇita Kaumudī contains about 475 verses of sūtra (rules), and 395 verses of udāharaṇa (examples). It is divided into 14 sections (chapters) known as vyavahāras:[1]

1. Prakīrṇaka-vyavahāra

Weights and measures, length, area, volume, etc. It describes addition, subtraction, multiplication, division, square, square root, cube and cube root. The problems of linear and quadratic equations described here are more complex than in earlier works.[2] 63 rules and 82 examples[1]

2. Miśraka-vyavahāra

Mathematics pertaining to daily life: “mixture of materials, interest on a principal, payment in instalments, mixing gold objects with different purities and other problems pertaining to linear indeterminate equations for many unknowns”[2] 42 rules and 49 examples[1]

3. Śreḍhī-vyavahāra

Arithmetic and geometric progressions, sequences and series. The generalization here was crucial for finding the infinite series for sine and cosine.[2] 28 rules and 19 examples.[1]

4. Kṣetra-vyavahāra

Geometry. 149 rules and 94 examples.[1] Includes special material on cyclic quadratilerals, such as the “third diagonal”.[2]

5. Khāta-vyavahāra

Excavations. 7 rules and 9 examples.[1]

6. Citi-vyavahāra

Stacks. 2 rules and 2 examples.[1]

7. Rāśi-vyavahāra

Mounds of grain. 2 rules and 3 examples.[1]

8. Chāyā-vyavahāra

Shadow problems. 7 rules and 6 examples.[1]

9. Kuṭṭaka

Linear integer equations. 69 rules and 36 examples.[1]

10. Vargaprakṛti

Quadratic. 17 rules and 10 examples.[1] Includes a variant of the Chakravala method.[2] Ganita Kaumudi contains many results from continued fractions. In the text Narayana Pandita used the knowledge of simple recurring continued fraction in the solutions of indeterminate equations of the type ${\displaystyle nx^{2}+k^{2}=y^{2}}$.

11. Bhāgādāna

Factorization. Contains Fermat's factorization method.[1] 11 rules and 7 examples.[1]

12. Rūpādyaṃśāvatāra

Contains rules for writing a fraction as a sum of unit fractions. 22 rules and 14 examples.[1]

Unit fractions were known in Indian mathematics in the Vedic period:[3] the Śulba Sūtras give an approximation of √2 equivalent to ${\displaystyle 1+{\tfrac {1}{3}}+{\tfrac {1}{3\cdot 4}}-{\tfrac {1}{3\cdot 4\cdot 34}}}$. Systematic rules for expressing a fraction as the sum of unit fractions had previously been given in the Gaṇita-sāra-saṅgraha of Mahāvīra (c. 850).[3] Nārāyaṇa's Gaṇita-kaumudi gave a few more rules: the section bhāgajāti in the twelfth chapter named aṃśāvatāra-vyavahāra contains eight rules.[3] The first few are:[3]

• Rule 1. To express 1 as a sum of n unit fractions:[3]

${\displaystyle 1={\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\dots +{\frac {1}{(n-1)\cdot n}}+{\frac {1}{n}}}$

• Rule 2. To express 1 as a sum of n unit fractions:[3]

${\displaystyle 1={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{3^{2}}}+\dots +{\frac {1}{3^{n-2}}}+{\frac {1}{2\cdot 3^{n-2}}}}$

• Rule 3. To express a fraction ${\displaystyle p/q}$ as a sum of unit fractions:[3]

Pick an arbitrary number i such that ${\displaystyle (q+i)/p}$ is an integer r, write

${\displaystyle {\frac {p}{q}}={\frac {1}{r}}+{\frac {i}{qr}}}$

and find successive denominators in the same way by operating on the new fraction. If i is always chosen to be the smallest such integer, this is equivalent to the greedy algorithm for Egyptian fractions, but the Gaṇita-Kaumudī's rule does not give a unique procedure, and instead states evam iṣṭavaśād bahudhā ("Thus there are many ways, according to one's choices.")[3]

• Rule 4. Given ${\displaystyle n}$ arbitrary numbers ${\displaystyle k_{1},k_{2},\dots ,k_{n}}$,[3]

${\displaystyle 1={\frac {(k_{2}-k_{1})k_{1}}{k_{2}\cdot k_{1}}}+{\frac {(k_{3}-k_{2})k_{1}}{k_{3}\cdot k_{2}}}+\dots +{\frac {(k_{n}-k_{n-1})k_{1}}{k_{n}\cdot k_{n-1}}}+{\frac {1\cdot k_{1}}{k_{n}}}}$

• Rule 5. To express 1 as the sum of fractions with given numerators ${\displaystyle a_{1},a_{2},\dots ,a_{n}}$:[3]

Calculate ${\displaystyle i_{1},i_{2},\dots ,i_{n}}$ as ${\displaystyle i_{1}=a_{1}+1}$, ${\displaystyle i_{2}=a_{2}+i_{1}}$, ${\displaystyle i_{3}=a_{3}+i_{2}}$, and so on, and write

${\displaystyle 1={\frac {a_{1}}{1\cdot i_{1}}}+{\frac {a_{2}}{i_{1}\cdot i_{2}}}+{\frac {a_{3}}{i_{2}\cdot i_{3}}}+\dots +{\frac {a_{n}}{i_{n-1}\cdot i_{n}}}+{\frac {1}{i_{n}}}}$

13. Aṅka-pāśa

Combinatorics. 97 rules and 45 examples.[1] Generating permutations (including of a multiset), combinations, partitions of a number, binomial coefficients, generalized Fibonacci numbers.[2]

Narayana Pandita noted the equivalence of the figurate numbers and the formulae for the number of combinations of different things taken so many at a time.[4]

The book contains a rule to determine the number of permutations of n objects and a classical algorithm for finding the next permutation in lexicographic ordering though computational methods have advanced well beyond that ancient algorithm. Donald Knuth describes many algorithms dedicated to efficient permutation generation and discuss their history in his book The Art of Computer Programming.[5]

14. Bhadragaṇita

Magic squares. 60 rules and 17 examples.[1]

Editions

• "Translation of Ganita Kaumudi with Rationale in modern mathematics and historical notes" by S L Singh, Principal, Science College, Gurukul Kangri Vishwavidyalaya, Haridwar
• Ganita Kaumudi, Volume 1–2, Nārāyana Pandita (Issue 57 of Princess of Wales Sarasvati Bhavana Granthamala: Abhinava nibandhamālā Padmakara Dwivedi Jyautishacharya 1936)

References

Notes

1. M. D. Srinivas, Mathematics In India, Lecture 27.
2. M. S. Sriram, Mathematics in India, Lecture 25.
3. Kusuba 2004, p. 497
4. Edwards, A. W. F. Pascal's Arithmetical Triangle: The Story of a Mathematical Idea. JHU Press. p. 16.
5. Knuth, Donald (2006). The Art of Computer Programming. Addison-Wesley. p. 74.

Bibliography

• Kusuba, Takanori (2004), "Indian Rules for the Decomposition of Fractions", in Charles Burnett; Jan P. Hogendijk; Kim Plofker; et al. (eds.), Studies in the History of the Exact Sciences in Honour of David Pingree, Brill, ISBN 9004132023, ISSN 0169-8729
• M. D. Srinivas, M. S. Sriram, K. Ramasubramanian, Mathematics in India - From Vedic Period to Modern Times. Lectures 25–27.

External links

• Ganita Kaumudi Part 1 (1936)
• Ganita Kaumudi Part 2 (1942)
• Ganita Kaumudi and the Continued Fraction