Pātīgaṇita is the term used in pre-modern Indian mathematical literature to denote the area of mathematics dealing with arithmetic and mensuration.
The term is a compound word formed by combining the words ''pātī'' and ''gaṇita''. The former is a non-Sanskrit word meaning a "board" and the latter is a Sanskrit word meaning "science of calculation". Thus the term ''pātīgaṇita'' literally means the science of calculations which requires a board (on which dust or sand is spread out) for performing the calculations, or "board-computation" in short. The usage of the term became popular among authors of Indian mathematical works about the beginning of the seventh century CE.
It may be noted that
Brahmagupta
Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
(c. 598 – c. 668 CE) has not used this term. Instead, he uses the term ''dhūlīkarma'' (''dhūlī'' is the Sanskrit term for dust). The terminology ''pātīgaṇita''may be contrasted with "bījagaṇita" which denotes the area of mathematics referred to as algebra.
The term ''Pātīgaṇita'' is also the title of a work composed by
Sridhara
Śrīdhara or Śrīdharācārya (8th–9th century) was an Indian mathematician, known for two extant treatises about arithmetic and practical mathematics, ''Pāṭīgaṇita'' and ''Pāṭīgaṇita-sāra'', and a now-lost treatise about algebra, ...
, an Indian mathematician who flourished during the 8th-9th century CE.
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Topics discussed in ''pātīgaṇita''
According to Brahmagupta there are 20 operations (''parikarma''-s) and 8 determinations (also called logistics) (''vyavahāra''-s) that come under ''pātīgaṇita''. He has stated as such in his ''Brahma-sphuṭa-siddhānta'' without specifying what these are. The commentators of ''Brahmasphuṭa-siddhānta'' have listed the following as the 20 operations and the 8 determinations.[
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''Parikarma'' (Operations)
''Vyavahāra''-s (determinations/logistics)
#''Miśrakah'' (mixture): Computations involving mixtures of several things.
#''Sreḍhi'' (progression or series): A ''sreḍhi''is that which has a beginning (first term) and an increase (common difference).
#''Kṣetram'' (plane figures): Calculations of the area of a figure having several angles.
#''Khātam'' (excavation): Finding the volumes of excavations.
#''Citih'' (stock): Computing the measure of a pile of bricks.
#''Krākacikah'' (saw): Finding the measure of the timber sawn.
#''Rāśih'' (mound): Calculations to find the amount of a heap of grain, etc.
#''Chāyā'' (shadow): Finding the time from the shadow of a gnomon, etc.
Works dealing with ''pāṭīgaṇita''
The earliest work dealing with the topics that come under ''pāṭīgaṇita'' that has survived to the present day is the Bakhshali manuscript some portions of which has been carbon dated as 224–383 CE. The following are the currently available texts which deal arithmetic and mensuration. They may contain more material than the 20 operations and the eight determinations that are listed as the topics that come under ''pāṭīgaṇita''.
#''Gaṇita-sāra-sañgraha'' of Mahavira (850 CE)
#''Pātīgaṇita'' and ''Pātīgaṇita-sāra'' (or ''Trisātikā'') of Śrīdharācarya
#''Gaṇita-tilaka'' of Srīpati (1039 CE) (incomplete)
#''Līlāvatī
''Līlāvatī'' is a treatise by Indian mathematician Bhāskara II on mathematics, written in 1150 AD. It is the first volume of his main work, the ''Siddhānta Shiromani'', alongside the ''Bijaganita'', the ''Grahaganita'' and the ''Golādhyāya ...
'' of Bhāskara II
Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
(1150 CE)
#'' Gaṇita-kaumudī'' of Nārāyaṇa
Narayana (, ) is one of the forms and epithets of Vishnu. In this form, the deity is depicted in yogic slumber under the celestial waters, symbolising the masculine principle and associated with his role of creation. He is also known as Pu ...
(1356 CE)
In these works one can see references to several older works, but none of them have survived to the present day. The lost works include ''Pātīgaṇita'' of Lalla (8th century CE) and ''Govindakṛti'' of Govindasvāmi
Govindasvāmi (or Govindasvāmin, Govindaswami) (c. 800 – c. 860) was an Indian mathematical astronomer most famous for his ''Bhashya'', a commentary on the ''Mahābhāskarīya'' of Bhāskara I, written around 830. The commentary contains many ...
(9th century CE).
The following astronomical treatises deal with arithmetic and mensuration in one of the chapters:
#''Brahma-sphuṭa-siddhānta'' of Brahmagupta (628 CE) (the twelfth chapter, entitled ''Gaṇitāddhyāya'')
#''Mahā-siddhānta'' of Āryabhaṭa II
Āryabhaṭa (c. 920 – c. 1000) was an Indian mathematician and astronomer, and the author of the ''Maha-Siddhanta''. The numeral II is given to him to distinguish him from the earlier and more influential Āryabhaṭa I. Scholars are ...
(c. 950 CE) (the fifteenth chapter, entitled ''Pātīgaṇita'')
#''Siddhānta-sekhara'' of Śrīpati
Śrīpati ( – 1066), also transliterated as Shri-pati, was an Indian astronomer, astrologer and mathematician. His major works include ''Dhīkotida-karana'' (1039), a work of twenty verses on solar eclipse, solar and lunar eclipses; ''Dhruva-mān ...
(1039 CE) (the thirteenth chapter, entitled ''Vyakta-gaṇitāddhyāya'')
Śrīdhara's ''Pāṭīgaṇita''
In Indian mathematical literature, Śrīdhara is the only author who has composed a work titled ''Pāṭīgaṇita''. He has composed another work titled ''Pāṭīgaṇita-sāra'' which is a short summary of his ''Pāṭīgaṇita''.
Full texts of Śrīdhara's works
*Full text of Śrīdhara's ''Pāṭīgaṇita'' is available in Internet Archive
The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
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*Full text of Śrīdhara's ''Pāṭīgaṇita-sāra'' is available in Internet Archive
The Internet Archive is an American 501(c)(3) organization, non-profit organization founded in 1996 by Brewster Kahle that runs a digital library website, archive.org. It provides free access to collections of digitized media including web ...
:
References
{{Indian mathematics
Indian mathematics
Arithmetic
Measurement