Appreciation of Mathematics. It is said that in ancient India no science did ever attain an independent existence and was cultivated in its own sake. Whatever of any science is found in Vedic India is supposed to have originated and grown as the handmaid of one or the other of the six "members of the Veda," and consequently with the primary object of helping the Vedic rituals. It is also supposed, sometimes, that any further culture of the science was somewhat discouraged by the Vedic Hindus in suspicion that it might prove a hindrance to their great quest of the knowledge of the Supreme by diverting the mind to other external channels. That is not, indeed, a correct view on the whole. It is perhaps true that in the earlier Vedic Age, sciences grew as help to religion. But it is generally found that the interest of people in a particular branch of knowledge, in all climes and times, has always been aroused and guided by specific reasons. Religion being the prime avocation of the earlier Hindus, it is not unnatural that the culture of other branches of knowledge grew as help to it and was kept subsidiary. but there is enough evidence to show that in course of time all the sciences outgrew their original purposes and were cultivated for their own sake. A new orientation had indeed set in in the latter part of the Vedic Age.
There is a story in the Chândogya Upaniṣad1 whose Page 4 value in support of our view cannot be over-estimated. It is said that once upon a time Nârada approached the sage Sanatkumâra and begged him the Brahmavidyâ or the supreme knowledge. Sanatkumâra asked Nârada to state what sciences and arts he had already studied to that he (Sanatkumâra) might judge what still remained to be learnt by him. Thereupon Nârada enumerated the various sciences and arts studied by him. This list included astronomy (nakṣatra-vidyâ) and arithmetic (râśi-vidyâ). Thus the culture of the science of mathematics or of any other branch of secular knowledge, was not considered to be a hindrance to spiritual knowledge. In fact, Aparâ-vidyâ ("secular knowledge") was then considered to be a helpful adjunct to Parâ-vidyâ ("spiritual knowledge").1
Importance to the culture of Gaṇita (mathematics) is also given by the Jainas. Their religious literature is generally classified into four branches, called anuyoga ("exposition of principles"). One of them is gaṇitânuyoga ("the exposition of the principles of mathematics"). The knowledge of Saṁkhyâna (literally, "the science of numbers," meaning arithmetic and astronomy) is stated to be one of the principal accomplishments of the Jaina priest.2 In Buddhist literature too, arithmetic (gaṇanâ, saṁkhyâna) is regarded as the first and the noblest of the arts.3 All these will give a fair idea of the importance and value set upon the culture of gaṇita in ancient India.
The following appreciation of mathematics, although belonging to a much later date, will be found to be interesting, especially, as it comes from the pen Page 5 of Mahâvȋra (850 A.D.), one of the best mathematicians of his time:
"In all transactions which relate to worldly, Vedic or other similar religious affairs calculation is of use. In the science of love, in the science of wealth, in music and in rama, in the art of cooking, in medicine, in architecture, in prosody, in poetics and poetry, in logic and grammar and such other things, and in relation to all that constitutes the peculiar value of the arts, the science of calculation (gaṇita) is held in high esteem. In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and conjunctions of planets, and in connection with the tripraśna (direction, position and time) and the course of the moon—indeed in all these it is utilised. The number, the diameter and the perimeter of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the world, of the interspace between the worlds, of the world of light, of the world of the gods and of the dwellers in hell, and other miscellaneous measurements of all sorts—all these are made out by the help of gaṇita. The configuration of living beings therein, the length of their lives, their eight attributes, and other similar things; their progress and other such things, their staying together, etc.—all these are dependent upon gaṇita (for their due comprehension). What is the good of saying much? Whatever there is in all the three worlds, which are possessed of moving and non-moving beings, cannot exist as apart from gaṇita (measurement and calculation).
"With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world, and of their disciples and disciples' disciples, who constitute the well-known series of preceptors, Page 6 I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold is from the stony rock and pearl from the oyster shell; and give out according to the power of my intelligence, the Sâra-saṁgraha, a small work on gaṇita, which is (however) not small in value."1
Mathematics in Hindu Education. The elementary stage in Hindu education lasted from the age of five till the age of twelve. This period slightly differed in the case of sons of kings and noblemen. The main subjects of study were lipi and lekhâ (alphabets, reading and writing), rûpa (drawing and geometry) and gaṇanâ (arithmetic). It is said in the Arthaśâstra of Kauṭilya (400 B.C.) that having undergone the ceremony of tonsure, the student shall learn the alphabets (lipi) and arithmetic (saṁkhyâna).2 We find in the Hâthȋgumphâ Inscription3 that king Khâravela (163 B.C.) of Kaliṅga spent nine years (from the age of sixteen to the age of 25) in learning lekhâ, rûpa and gaṇanâ. Prince Gautama began his education when he was eight years of age "firstly (with) writing and then arithmetic as the most important of the 72 sciences and arts."4 Mention of lekhâ, rûpa and gaṇanâ is also found in the Jaina canonical works.5
Page 3 1 Chândogya Upaniṣad, vii. 1, 2, 4.
Page 4 1 Muṇḍakopaniṣad, i. 1, 3–5.
2 Bhagavatȋ-sûtra, Sûtra 90; Uttarâdhyayana-sûtra, xxv. 7, 8, 38.
3 Vinaya Piṭaka, ed. Oldenberg, Vol. IV, p. 7; Majjhima Nikâya, Vol. I, p. 85; Cullaniddesa, p. 199.
Page 6 1 GSS, i. 9–19.
2 Arthaśâstra, ed. by R. Shamasastri, i. 5, 2; Eng. trans, p. 10.
3 Hathigumpha and three other inscriptions, ed. by Bhagavanlal Indraji, p. 22.
4 Antagaḍa-dasâo and Anuttarvavâiya-dasâo, Eng. trans. by L. D. Barnett, 1907, p. 30; cf. Kalpasûtra of Bradrabâhu, Sûtra 211.
5 E.g., Samavâyâṅga-sûtra, Sûtra 72.
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