ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

MULTIPLICATION 157 19. Ekādaśa-rāśikam (the rule of eleven) 20. Bhāṇḍa-pratibhāṇḍam (barter and exchange) (B) Vyavahāra or determinations

1. Miśrakaḥ (mixture)
2. Śreḍhī (progession or series)
3. Kṣetram (plane figures)
4. Khātam (excavation)
5. Citiḥ (stock)
6. . Krākacikaḥ (saw)
7. Rāśiḥ (mound)
8. Chāyā (shadow) Of the operations enlisted here, the first eight have been considered fundamental by later writers as Mahāvīra. The opera- tions of duplation and mediation (doubling and halving) were considered fundamental by Arabs, Greeks and Egyptians; since they were not familiar with the place-value system. Mathematics in this country developed as an aid to astro- nomy, and therefore, for the first time we find Āryabhaṭa(499AD.) in his Āryabhaṭīya describing as a special section (Gaṇitapāda). Brahmagupta (628 A.D.) also followed Āryabhaṭa in this respect and gave the science of calculation (gaṇita) a special place in his treatise on astronomy. The Siddhānta treatises, earlier than those of Āryabhaṭa and Brahmagupta do not contain a chapter exclu- sively devoted to gaṇita (the Sūrya-Siddhānta and the Siddhāntas of Vasiṣṭha, Pitāmaha and Romaka are thus without gaṇita chapetrs). Later on Bhāskara I and Lalla also did not include gaṇita as a section or chapter in their treatises. It is said, howe- ver, that Lalla wrote a separate treatise on Pāṭīgaṇita. It may further be remarked here that Āryabhaṭa I gives the rules for finding the square and cube-roots only whilst Bra- hmagupta gives the cube-root rule only (BrSPSi. XII. 7). Multiplication Undoubtedly the common Indian name ‘multiplication’ is ‘guṇana’, this term occurs in the Vedic literature also. The other terms for this logistics are hanana, vadha, kṣaya etc., which all mean ‘killing’ or ‘destroying.’ The synonyms of ‘hanana’ (killing) for multiplication have been used by Āryabhaṭa I (499). Brahma- gupta (628), Śrīdhara (c.750) and later writers, and these terms