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

224 BRAHMAGUPTA AS AN ALGEBRAIST
Hence ' by—ax=R₁—R₂
Putting c=R₁∽R₂
we get by-ax=±c
the upper or lower sign being taken according as R₁ is grea-
ter than or less than R₂.
Class II : To find a number (x) such that its product
with a given number (a) being increased or
decreased by another given number (8) and
then divided by a third given number (β)
will leave no remainder.
This means that in other words, we shall have to get the
solution of :
ax±r
———— = y
β
in positive integers.
Class III : Here we have to deal with an equation of the
form :
by ÷ ax= ± c
Kuṭṭaka, Kuṭṭākāra and Kuṭṭa : These are the three terms
which Brahmagupta has used in regards to the subject of indeter-
minate analysis of the first degree. Āryabhaṭa I has also descri-
bed this method in brief, but he does not use the word kuṭṭaka.
In the Mahābhāskarīya of Bhāskara I we have the terms kuṭṭā-
kāra and kuṭṭa (522 A.D.) MBh. I. 41,49). These words have
been translated into English as pulveriser or grinder. According
to Datta and Singh, the Hindu method of solving the equation
by-ax= ± c is essentially based on a process of deriving from it
successively other similar equations in which the values of the
coefficients (a,b) become smaller and smaller. Thus the process
is indeed the same as that of breaking a whole thing into smaller
pieces, and this accounts for its name kuṭṭaka or 'pulveriser'.
In the problems of the Class I, the quantities (a and b) are
called' divisors' bhāgahāra, bhajaka, cheda etc.) and R₁ and R₂
as 'remainders' (agra or seṣa etc.). while in a problem of the
Class II ; β is ordinarily called the 'divisor' (bhāgahāra or bhā-
jaka) and Y the 'interpolator' kṣepa, kṣepaka etc.) ; here a is called
the 'dividend' (bhāj