Ganita Kaumudi

Ganita Kaumudi is a treatise on mathematics written by Indian mathematician Narayana Pandit 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 Lilāvati by Bhāskara II.

Contents

Ganita Kaumudi contains many results from combinatorics and continued fractions. In the text Narayana Pandit used the knowledge of simple recurring continued fraction in the solutions of indeterminate equations of the type $nx^2+k^2=y^2$. Narayana Pandit noted the equivalence of the figurate numbers and the formulae for the number of combinations of different things taken so many at a time.

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.

Rules for writing a fraction as a sum of unit fractions

Unit fractions were known in Indian mathematics in the Vedic period: the Śulba Sūtras give an approximation of √2 equivalent to $1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}$. 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). Nārāyaṇa's Gaṇita-kaumudi gave a few more rules: the section bhāgajāti in the twelfth chapter named aṃśāvatāra-vyavahāra contains eight rules. The first few are:

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

$1 = \frac1{1\cdot 2} + \frac1{2 \cdot 3} + \frac1{3 \cdot 4} + \dots + \frac1{(n-1)\cdot n} + \frac1n$

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

$1 = \frac12 + \frac13 + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{2 \cdot 3^{n-2}}$

• Rule 3. To express a fraction $p/q$ as a sum of unit fractions:

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

$\frac{p}{q} = \frac1r + \frac{i}{qr}$

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.")

• Rule 4. Given $n$ arbitrary numbers $k_1, k_2, \dots, k_n$,

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

• Rule 5. To express 1 as the sum of fractions with given numerators $a_1, a_2, \dots, a_n$:

Calculate $i_1, i_2, \dots, i_n$ as $i_1 = a_1 + 1$, $i_2 = a_2 + i_1$, $i_3 = a_3 + i_2$, and so on, and write

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

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)

External links

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