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

QUADRATIC EQUATIONS 213
be performed all operations conformably to the state-
ment of the example and thus should be carefully
framed two or more sides and also equations. Equi-clear-
ance should be made first between two and two of
them and so on to the last : from one side one un-
known should be cleared, other unknowns reduced to
a common denominator and also the absolute numbers
should be cleared from the side opposite. The
residue of other unknowns being divided by the
residual coefficient of the first unknown will give the
value of the first unknown. If there be
obtained several such values, then with two and two
of them, equations should be formed after reduction to
common denominators. Proceeding in this way to the
end find out the value of one unknown. If that value
be (in terms of) another unknown then the coefficients
of those two will be reciprocally the values of the two
unknowns. If, however, there be present more un-
knows in that value, the method of the pulveriser
should be employed. Arbitrary values may then be
assumed for some of the unknowns.
Datta and Singh have said that the above rule of Brahma-
gupta, and also the one indicated in the commentary of Pṛthudaka
Svāmī, embraces the solution of indeterminate as well as the
determinate equations. In fact, all the examples given by
Brahmagupta in illustration of the rule are of indeterminate
character. So far as the determinate simultaneous equations are
concerned, Brahmagupta's method for solving them will be easily
recognised to be the same as our present one.
Quadratic Equations
The geometrical solution of a quadratic equation in this
country would take us to the Vedic Śulba period. The Bakha-
śālī Manuscript also contains certain problems which need the
solving of quadratic equations. I shall quote one out of the
numerous available :
A certain person travels s yojana on the first day and b
yojana more on each successive day. Another who
travels at the uniform rate of S yojana per day, has a
start of t days. When will the first man overtake the
second ?