160 BRAHMAGUPTA AND ARITHMETIC Thus for example: 96x237-(4x4x6)×237 = (4x237) x4x6=948×4×6 - (4x948)×6-3792×6 =22752 These methods of multiplication are found among the Arabs and the Italians, having obtained from people of India. They were known as the "Scapezzo" and "Repiego" methods respectively amongst Italians. (iii) Işta-gunana Method or the Algebraic Method. We have already quoted the relevant verse from the Brah- masphuta-siddhanta in this connection; (XII. 56): The multiplicand is multiplied by the sum or the difference of the multiplied and an assumed quantity and from the result the product of the assvmed quan- tity and the multiplicand is subtracted or added.¹ This method is of two kinds according as we (i) add or (b) subtract an assumed number. The assumed number is so chosen as to give two numbers with which multiplication will be easier than with the original multiplier. The two ways are illustrated below: (i) 93x13 (ii) 93x13 (93+7)x13-7x13-1300-91-1209. (90+3)13-90x13+3x13-1170+39 =1209 This method was in use among the Arabs and in Europe, obviously having gone out from this country. This process has been regarded as an inverse of multipli- Gation. The terms used for this operation are bhagahara, bhajana, harana, chedana, etc., all these terms more or less carrying the sense "to break into parts", "to divide" etc., excepting "harana" which denotes "to take away", This term shows the relation of division to the operation of subtraction. The dividend is termed as bhajya. harya etc., the divisor is known as 1. गुरथो राशि खकारराशिनेष्टाधिकोनकेन गुणः । गुरुदेष्टवषो न युतो गुणकेऽभ्यधिकोनके कार्यः || -BrSpSi. XII. 56
