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

190 BRAHMAGUPTA AS AN ALGEBRAIST
A.D.). There can be no doubt regarding the age of these two
authors. Bhāskara II completed his great work on the Siddhānta-
śiromaṇi in 1072 Śaka, and Karaṇa-kutūhala a practical astro-
nomical treatise in 1105 Śaka; these dates are based on the
passages given by Bhāskara himself in his works. The Bīja-
gaṇita and the Līlāvatī form parts of the great treatise, the
Siddhānta-śiromaṇi. The genuineness of the text is established,
as Colebrooke says, with no less certainty by numerous commen-
tators in Sanskrit, besides a Persian version of it. Those com-
mentaries comprise a perpetual gloss, in which every passage of
the original is noticed and interpreted : and every word of it is
repeated and explained. From comparison and collation of
various texts, it appears then that the work of Bhāskara, exhibit-
ing the same uniform text which the modern transcripts of it
do, was in the hands of both Muhammedans and Hindus,
between two or three centuries ago : and numerous copies of it
having been diffused throughout India, at an earlier period, as of
a performance held in high estimation, It was the subject of
study and habitual reference in countries and places so remote
from each other as the north and west of India and the Southern
Peninsula.
This though not marking any extraordinary antiquity, nor
approaching to that of the author himself, was a material point
to be determined : as there will be in the sequel, so says Cole-
brooke, occasion to show, that modes of analysis, and in parti-
cular, general methods for the solution of indeterminate prob-
lems both of the first and second degrees, are taught in the
Bīja-gaṇita, and those for the first degrees repeated
in the Līlāvatī, which were unknown to the mathematicians of
the West, until invented anew in the last two centuries by
algebraists of France and England.¹ Bhāskara who himself
flourished more than six hundred and fifty years ago, was in
this respect a compiler and took those methods from Indian
authors as much more ancient than himself.
Regarding the age of the precursors of Bhāskara II, Cole-
brooke says : The age of his precursors cannot be determined
with equal precision. He then proceeds to examine the evidence
as follows :

1. Colebrooke, H. T., Miscellaneous Essays, p. 421.