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

154 BRAHMAGUPTA AND ARITHMETIC which is faultless.¹ This science of dhūlīkarma has not been imparted by great teachers for blasphemy. One who would be using it for this purpose would lose all good name.² Brahmagupta uses the term gaṇita only for those calculations which are of arithmetical in nature. The science of algebra, the foundations of which was laid by Āryabhaṭa I, was named as kuṭṭaka or kuṭṭākāra by Āryabhaṭa, and in the Brāhmasphuṭa- siddhānta also it is separately dealt with under Kuṭṭādhyāya or kuṭṭakādhāya (Chapter XVIII). Later on the term bījagaṇita was specifically given to the science of algebra. The Kuṭṭādhyāya of the Brāhmasphuṭasiddhānta deals with the (i) concept of kuṭṭaka (pulveriser), addition of positive and negative as well as zero quantities, equations in one unknown (eka-varṇa samīkaraṇa), equations in several unknowns (aneka- varṇa samīkaraṇa), equations involving products of unknowns (bhāvita) and quadratic equations (varga-prakṛtiḥ) (Chapter XVIII of the Brāhmasphuṭasiddhānta). Āryabhaṭa, Bhāskara and Brahmagupta use Place Value Notations. In Europe the first definite traces of the place-value nume- rals are found in the tenth and eleventh centuries, but the numerals came into general use in mathematical text books only in the seventeenth century. In India, however, Āryabhaṭa I (499), Bhāskara I (522), Lalla (c. 598) and Brahmagupta (628) all use the place value numerals. There is no trace of any other system in their works. Perhaps in this country we had the place value system as early as 200 B.C. if not earlier. The use of a symbol for zero is found in Piṅgala's Chandaḥ Sūtra (perhaps of 200 B.C.). In literature, we have an indication of the place value from about 100 B.C. and later in the Purāṇas from the second to the fourth century A.D. The Bakhāśalī Manuscript (perhaps of 200 A.D.) uses the place-value notations. The earliest use of the place value principle with the letter numerals

1. ग्रहयोगोत्र ग्रहयुतिरार्यांत्रिशतियुताष्टसप्तत्या । अध्याब्दैरभिभू लिकर्म वोच्चैर्विना भङ्गै ॥ —BrSpSi. X. 66.
2. गुरुणा न धूलिकर्म प्रतिकंचुककारिणे प्रदातव्यम् । दत्तं सुकृतप्रणाशं कुरुते प्रतिकंचुकं यस्य ॥ —BrSpSi. X. 67