Ganita Kaumudi (Sanskrit:गणितकौमदी) is a treatise onmathematics written by Indian mathematicianNarayana Pandita in 1356. It was an arithmetical treatise alongside the other algebraic treatise called "Bijganita Vatamsa" byNarayana Pandit.

Gaṇita Kaumudī contains about 475 verses ofsūtra (rules) and 395 verses ofudāharaṇa (examples). It is divided into 14 chapters (vyavahāra):^[1]

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]

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]

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]

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

Excavations. 7 rules and 9 examples.^[1]

Stacks. 2 rules and 2 examples.^[1]

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

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

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

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

Contains factorization method,^[1] 11 rules and 7 examples.^[1]

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

Unit fractions were known inIndian mathematics in the Vedic period:^[3] theŚulba Sūtras give an approximation of√2 equivalent to${\displaystyle 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}}$. Systematic rules for expressing a fraction as thesum of unit fractions had previously been given in theGaṇita-sāra-saṅgraha ofMahāvīra (c. 850).[3] Nārāyaṇa'sGaṇita-kaumudi gave a few more rules: the sectionbhāgajāti in the twelfth chapter namedaṃśāvatāra-vyavahāra contains eight rules.^[3] The first few are:^[3]

• Rule 1. To express 1 as a sum ofn unit fractions:^[3]

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

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

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

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

Pick an arbitrary numberi such that${\displaystyle (q+i)/p}$ is an integerr, 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. Ifi is always chosen to be the smallest such integer, this is equivalent to thegreedy algorithm for Egyptian fractions, but the Gaṇita-Kaumudī's rule does not give a unique procedure, and instead statesevam 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]

${\displaystyle 1=\frac{(k_2-k_1)k_1}{k_2\cdot k_1}+\frac{(k_3-k_2)k_1}{k_3\cdot k_2}+\cdots +\frac{(k_n-k_{n-1})k_1}{k_n\cdot k_{n-1}}+\frac{1\cdot k_1}{k_n}}$

• 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

${\displaystyle 1=\frac{a_1}{1\cdot i_1}+\frac{a_2}{i_1\cdot i_2}+\frac{a_3}{i_2\cdot i_3}+\cdots +\frac{a_n}{i_{n-1}\cdot i_n}+\frac{1}{i_n}}$

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

Narayana Pandita noted the equivalence of thefigurate 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 ofn 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 bookThe Art of Computer Programming.^[5]

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

• "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 WalesSarasvati Bhavana Granthamala: Abhinava nibandhamālāPadmakara Dwivedi Jyautishacharya 1936)

Notes

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 ofDavid 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.

• Ganita Kaumudi Part 1 (1936)
• Ganita Kaumudi Part 2 (1942)
• Ganita Kaumudi and the Continued Fraction^{[permanent dead link‍]}

• Indian mathematics
• Social history of India
• 1356 works
• 14th century in science
• 1350s books

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