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