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

202 BRAHMAGUPTA AS AN ALGEBRAIST Brahmagupta lays down the rules regarding evolution and involution as follows : The square of a positive or a negative number is positive .....The (sign of the) root is the same, as was that from which the square was derived¹. As regards the latter portion of this rule, Pṛthūdaka Svāmī has the following comment to make : "The square-root should be taken either negative or positive, as will be most suitable for subsequent operations to be carried on." It will be interesting to observe the following observation of Mahāvīra (850 A. D.) regarding square-root of a negative quantity "Since a negative number by its own nature is not a square, it has no square-root."² So says Śrīpati : "A negative number by itself is non-square, so its square-root is unreal; so the rule (for the square-root) should be applied in the case of a posi- tive number."³ Algebraic Operations Brahmagupta and other algebraists recognise six operations as fundamental in algebra : addition, subtraction, multiplication, division, squaring and the extraction of the square-root. Regarding addition and subtraction Brahmagupta says : Of the unknowns, their squares, cubes, fourth powers, fifth powers, sixth powers, etc., addition and subtraction are (performed) of the like; of the unlike (they mean simply their) statement severally.⁴ In place of "of the like", Bhāskara II uses the term "of those of the same species (jāti) amongst unknowns" : Addition and subtraction are performed of those of the same species (jāti) amongst unknowns; of different species they mean their separate statement.⁵

1. खोद्धृत मृणं धनं वा तच्छेदं खमृणधनविभक्तं वा । ऋणधनयोर्वर्गः स्वं खं खस्य पदं कृतैर्यत् तत् ॥ —BrSpSi, XVIII. 35
2. GSS. I, 52.
3. Śiśe, XIV, 5.
4. अव्यक्तवर्ग घनवर्ग वर्ग पंचगत षड्गतादीनाम् । तुल्यानां संकलित व्यवकलिते पृथगतुल्यानाम् ॥ —BrSpSi.XVIII. 41.
5. योगोऽन्तरं तेषु समान जात्योर्विभिन्न जात्योश्च पृथक् स्थितिश्च । — Bhāskara II, Bījagaṇita.