>Tom.Reding at 20:47, 17 December 2020

2020-12-17T20:47:53Z

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'''''Ganita Kaumudi''''' is a treatise on [[mathematics]] written by Indian mathematician [[Narayana Pandita (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 ''vyavahāra''s:<ref name=mii27>M. D. Srinivas, ''Mathematics In India'', Lecture 27.</ref>

=== 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.<ref name=mii25>M. S. Sriram, ''Mathematics in India'', Lecture 25.</ref> 63 rules and 82 examples<ref name=mii27/>

=== 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”<ref name=mii25/> 42 rules and 49 examples<ref name=mii27/>

=== 3. Śreḍhī-vyavahāra ===
Arithmetic and geometric progressions, sequences and series. The generalization here was crucial for finding the infinite series for sine and cosine.<ref name=mii25/> 28 rules and 19 examples.<ref name=mii27/>

=== 4. Kṣetra-vyavahāra ===
Geometry. 149 rules and 94 examples.<ref name=mii27/> Includes special material on cyclic quadratilerals, such as the “third diagonal”.<ref name=mii25/>

=== 5. Khāta-vyavahāra ===
Excavations. 7 rules and 9 examples.<ref name=mii27/>

=== 6. Citi-vyavahāra ===
Stacks. 2 rules and 2 examples.<ref name=mii27/>

=== 7. Rāśi-vyavahāra ===
Mounds of grain. 2 rules and 3 examples.<ref name=mii27/>

=== 8. Chāyā-vyavahāra ===
Shadow problems. 7 rules and 6 examples.<ref name=mii27/>

=== 9. Kuṭṭaka ===
Linear integer equations. 69 rules and 36 examples.<ref name=mii27/>

=== 10. Vargaprakṛti ===
Quadratic. 17 rules and 10 examples.<ref name=mii27/> Includes a variant of the [[Chakravala method]].<ref name=mii25/> Ganita Kaumudi contains many results from [[continued fraction]]s. In the text [[Narayana Pandita (mathematician)|Narayana Pandita]] used the knowledge of simple recurring continued fraction in the solutions of indeterminate equations of the type <math>nx^2+k^2=y^2</math>.

=== 11. Bhāgādāna ===
Factorization. Contains [[Fermat's factorization method]].<ref name=mii27/> 11 rules and 7 examples.<ref name=mii27/>

=== 12. Rūpādyaṃśāvatāra ===
Contains rules for writing a fraction as a sum of unit fractions. 22 rules and 14 examples.<ref name=mii27/>

Unit fractions were known in [[Indian mathematics]] in the Vedic period:<ref name=k497>{{Harvnb|Kusuba|2004|p=497}}</ref> the [[Shulba Sutras|Śulba Sūtras]] give an approximation of {{radic|2}} equivalent to <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math>. Systematic rules for expressing a fraction as the [[Egyptian fraction|sum of unit fractions]] had previously been given in the ''Gaṇita-sāra-saṅgraha'' of [[Mahāvīra (mathematician)|Mahāvīra]] ({{circa|850}}).<ref name=k497/> Nārāyaṇa's ''Gaṇita-kaumudi'' gave a few more rules: the section ''bhāgajāti'' in the twelfth chapter named ''aṃśāvatāra-vyavahāra'' contains eight rules.<ref name=k497/> The first few are:<ref name=k497/>

* '''Rule 1.''' To express 1 as a sum of ''n'' unit fractions:<ref name=k497/>
:: <math>1 = \frac1{1\cdot 2} + \frac1{2 \cdot 3} + \frac1{3 \cdot 4} + \dots + \frac1{(n-1)\cdot n} + \frac1n</math>

* '''Rule 2.''' To express 1 as a sum of ''n'' unit fractions:<ref name=k497/>
:: <math> 1 = \frac12 + \frac13 + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{2 \cdot 3^{n-2}}</math>

* '''Rule 3.''' To express a fraction <math>p/q</math> as a [[Egyptian fraction|sum of unit fractions]]:<ref name=k497/>
: Pick an arbitrary number ''i'' such that <math>(q+i)/p</math> is an integer ''r'', write
:: <math> \frac{p}{q} = \frac1r + \frac{i}{qr}</math>
: 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.")<ref name=k497/>

* '''Rule 4.''' Given <math>n</math> arbitrary numbers <math>k_1, k_2, \dots, k_n</math>,<ref name=k497/>
:: <math>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}</math>

* '''Rule 5.''' To express 1 as the sum of fractions with given numerators <math>a_1, a_2, \dots, a_n</math>:<ref name=k497/>
: Calculate <math>i_1, i_2, \dots, i_n</math> as <math>i_1 = a_1 + 1</math>, <math>i_2 = a_2 + i_1</math>, <math>i_3 = a_3 + i_2</math>, and so on, and write
:: <math> 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} </math>

=== 13. Aṅka-pāśa ===
Combinatorics. 97 rules and 45 examples.<ref name=mii27/> Generating permutations (including of a multiset), combinations, [[Partition (number theory)|partitions of a number]], binomial coefficients, generalized Fibonacci numbers.<ref name=mii25/>

[[Narayana Pandita (mathematician)|Narayana Pandita]] noted the equivalence of the [[figurate number]]s and the formulae for the number of combinations of different things taken so many at a time.<ref>{{cite book|last=Edwards|first=A. W. F.|title=Pascal's Arithmetical Triangle: The Story of a Mathematical Idea|publisher=JHU Press|page=16}}</ref>

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]]''.<ref>{{cite book|last=Knuth|first=Donald|title=[[The Art of Computer Programming]]|year=2006|publisher=[[Addison-Wesley]]|page=74}}</ref>

=== 14. Bhadragaṇita ===
Magic squares. 60 rules and 17 examples.<ref name=mii27/>

==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
{{Reflist}}
;Bibliography
* {{citation | last=Kusuba|first=Takanori | contribution=Indian Rules for the Decomposition of Fractions | year=2004 | title=Studies in the History of the Exact Sciences in Honour of [[David Pingree]] | publisher=[[Brill Publishers|Brill]] | isbn=9004132023 | issn=0169-8729 | editor1=Charles Burnett | editor2=Jan P. Hogendijk | editor3=Kim Plofker | editor3-link=Kim Plofker |display-editors = 3 | editor4=Michio Yano}}
* M. D. Srinivas, M. S. Sriram, K. Ramasubramanian, ''Mathematics in India - From Vedic Period to Modern Times''. [https://nptel.ac.in/courses/111101080/ Lectures] 25–27.

==External links==
*[https://archive.org/details/in.ernet.dli.2015.326561 Ganita Kaumudi Part 1 (1936)]
*[https://archive.org/details/ganithakoumudipa015625mbp Ganita Kaumudi Part 2 (1942)]
*[http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af9_1.pdf Ganita Kaumudi and the Continued Fraction]{{Dead link|date=December 2019 |bot=InternetArchiveBot |fix-attempted=yes }}

{{Indian mathematics}}

[[Category:Indian mathematics]]
[[Category:Social history of India]]
[[Category:1356 works]]
[[Category:14th century in science]]
[[Category:1350s books]]

Ganita Kaumudi - Revision history

>Tom.Reding at 20:47, 17 December 2020