MULTIPLICATION 19. Ekadasa-rasikam (the rule of eleven) 20. Bhanda-pratibhändam (barter and exchange) (B) Vyavahara or determinations 1. Miśrakah (mixture) 2. Średhi (progession or series) 3. Kşetram (plane figures) 4. Khatam (excavation) 5. Citih (stock) 6. Krakacikah (saw) 7. Rasih (mound) 8. Chaya (shadow) 157 Of the operations enlisted here, the first eight have been considered fundamental by later writers as Mahavira. 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 Aryabhta(499AD.) in his Aryabhatiya describing as a special section (Ganitapāda). Brahmagupta (628 A.D.) also followed Aryabhata in this respect and gave the science of calculation (ganita) a special place in his treatise on astronomy. The Siddhanta treatises, earlier than those of Aryabhata and Brahmagupta do not contain a chapter exclu AISA[ devoted to ganita (the Surya-Siddhanta and the Siddhantas of Vasiştha Pitamaha and Romaka are thus without ganita chapetrs). Later on Bhaskara I and Lalla also did not include ganita as a section or chapter in their treatises. It is said, howe- ver, that Lalla wrote a separate treatise on Patīganita. It may further be remarked here that Aryabhata 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 gunana', this term occurs in the Vedic literature also. The other terms for this logistics are hanana, vadha, kşaya etc., which all mean "killing' or 'destroying' The synonyms of 'hanana' (killing) for multiplication have been used by Aryabhata I (499). Brahma- gupta (628), Śrīdhara (c.750) and later writers, and these terms
